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<Document xmlns:gate="http://www.gate.ac.uk" name="A10_S12_Real-TimeWeighted_Pose-Space_Deformation_on_the_GPU_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1111/j.1467-6346.2008.01649.x
Volume 25 (2006), Number 3
EUROGRAPHICS 2006 / E. Gröller and L. Szirmay-Kalos (Guest Editors)
<Title>Real-Time Weighted Pose-Space Deformation on the GPU</Title>
Taehyun Rhee †1 J.P. Lewis 1 University of Southern California, U.S.A. 2 Stanford University, U.S.A.
<Abstract>WPSD (Weighted Pose Space Deformation) is an example based skinning method for articulated body animation. The per-vertex computation required in WPSD can be parallelized in a SIMD (Single Instruction Multiple Data) manner and implemented on a GPU. While such vertex-parallel computation is often done on the GPU vertex processors, further parallelism can potentially be obtained by using the fragment processors. In this paper, we develop a parallel deformation method using the GPU fragment processors. Joint weights for each vertex are automatically calculated from sample poses, thereby reducing manual effort and enhancing the quality of WPSD as well as SSD (Skeletal Subspace Deformation). We show sufficient speed-up of SSD, PSD (Pose Space Deformation) and WPSD to make them suitable for real-time applications.</Abstract>
Categories and Subject Descriptors (according to ACM CCS): I.3.1 [Computer Graphics]: Hardware ArchitectureParallel processing, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Curve, surface, solid and object modeling, I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Animation.
c The Eurographics Association
Blackwell Publishing 2006. Published by Blackwell Publishing
Garsington Road
Oxford OX4 2DQ
UK
Main Street
Malden
MA
USA.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
† [email protected] ‡ [email protected] § [email protected]
‡2 and Ulrich Neumann §1
<H1>1. Introduction</H1>
Skinning is an important part of realistic articulated body animation and is an important topic of computer graphics and animation. Generally, skinning can be categorized into algorithmic, physically-based, and example-based methods. Although widely used, simple algorithmic skinning schemes cannot capture the complexity and subtlety of real skin deformation, and revised approaches will be required to increase character animation realism. Physically-based skinning is based on the biomechanics of skin deformation arising from the motions of muscles and tendons. Although this approach can generate physically accurate simulations of each layer, it is not at present suitable for real time applications such as gaming due to the large computation required. Example-based methods capture some of the complexity of real skin deformation by interpolating scanned or sculpted examples of the desired skin shape in various poses. Al-
though this requires gathering a sufficient number of samples and some pre-calculation, example-based methods can potentially be used in real-time applications due to their relatively simple real-time computation. Weighted pose space deformation (WPSD) is an example based skinning method that generates high quality skinning with a limited number of sample poses
[KM04].
Although it can generate an accurate skinning, it requires more computation than the original pose space deformation (PSD)
[LCF00], since joint distances are computed independently for each vertex. As such, this method has not been suitable for real-time applications. Furthermore, both WPSD and SSD require joint weights for each vertex, and accurate joint weights are required to achieve good results. However, the weights are usually manually generated by artists, which requires effort and great skill in the case of a complex skeletal system such as the human hand. In this paper, we present a parallel WPSD algorithm (including automatic determination of joint weights) suitable for SIMD architectures such as current GPUs. The joint
weights for each vertex are automatically computed from the sample poses. This can enhance the skinning quality not only of SSD but also WPSD, since both methods require accurate joint weight values. The deformation required in WPSD and SSD is independent for each vertex and this per-vertex computation can be parallelized in a SIMD architecture. The GPU is a general SIMD architecture having one-sided (unidirectional) communication to texture memory. We demonstrate our parallel WPSD method using GPU fragment processors. In our experiments, we can speed up SSD, PSD, as well as WPSD to around 20 times faster than on the CPU (from 1.2FPS to 25FPS speed-up of WPSD on a detailed model having 22836 triangles with 11574 vertices) using a modern graphics card, thus making WPSD a feasible real-time skinning solution for various applications including games, virtual reality, and other real-time simulations.
<H1>2. Related work</H1>
Many commercial software packages generate skin deformation arising from joint movement using a method known as (linear blend) skinning, Skeletal Subspace Deformation (SSD), enveloping, etc., based in part on work published by Thalmann et al. [MTLT88]. SSD is based on the weighted blending of affine transformations of each joint and used in many real-time applications due to its simple and fast computation. However, it also exhibits some well known artifacts such as skin that collapses around the joints at increasing bend angles, and a variety of solutions for these problems have been published
[Web00, WP02, MTG03, KZ05]. Recently, example-based methods [LCF00, SRC01, ACP02, KJP02, KM04]
have permitted more complex skinning effects such as muscle bulges and major wrinkles, while also addressing the artifacts of simple algorithmic schemes. In these methods, a number of provided (scanned or sculpted) samples of the desired skin shape are simply interpolated based on the creature’s pose (and possibly additional abstract control “dimensions”). These example-based methods can also be considered as a non-parametric approach to skin deformation. In common with non-parametric sampling methods in texture synthesis (and more generally in statistical regression), the amount of memory for these methods grows with the number of training samples, but arbitrary distributions can be approximated. Some of the most impressive example-based results to date are those of Kurihara and Miyata’s hand model derived from medical images
[KM04]. Since acquiring 3D medical images is relatively expensive, they developed weighted pose space deformation (WPSD) to generate proper skinning from a limited number of pose samples. They modify the distance between poses using the joint weights of each vertex to provide a more appropriate distance measure for skinning. Although the joint weights for each vertex are important data for SSD and WPSD calculations, they have traditionally been manually generated by skilled artists. Least-squares based vertex weight estimation was shown in the skinning methods
[WP02, MTG03]. James et al. describe mesh based skinning including estimation of bone parameters and vertex weights for each bone
[JT05]. In their paper, the vertex weights of each joint are calculated by NNLS (non-negative least squares) and we derive a similar approach to calculate weights for SSD and WPSD. In recent years, since the performance of GPUs has been improving more rapidly than that of CPUs, and GPUs have many processing units serving as a SIMD parallel architecture, many algorithms have been accelerated by GPU programming [LHK ∗ 04, PF05, GPG]. Deformation and skinning algorithms can also be enhanced by GPUs and several papers have profited from this
[JP02, KJP02, BK05, JT05]. However, in previous research, since vertex information cannot be accessed in the fragment program, GPU-based vertex deformation is usually performed by vertex programs. In this paper, we develop a parallel WPSD method using the fragment processors to gain greater parallelism and performance. Person-specific data modeling and its deformation is also an interesting topic in realistic articulated body simulation. Rhee et al. described human hand modeling from surface anatomy of the person
[RNL06]. Anguelov et al. developed shape completion and animation of people, derived from the set of range scan data and example based deformation in pose and shape space [ASK ∗ 05]. Physically inspired skinning should be also recognized as another important area of articulated body animation. However, we entrust the review of the subject to the recent related papers [AHS03, CBC ∗ 05, PCLS05, SNF05].
<H1>3. Skin deformation</H1>
Example-based skinning problems can be described by the following general equation,
1
v(p a ) = S(v 0 + D(p a ))
where p a is an arbitrary pose, v(p a ) is a vertex of a deformed target surface of the arbitrary pose, v 0 is an undeformed (rest pose) vertex, S is the SSD function, and D(p a ) is a displacement as a function of the arbitrary pose. In skeletal subspace deformation the displacement D(p a ) is omitted and the target surface is calculated by SSD as a blend of affine transforms of v 0 [section 3.1]. Skinning methods related to PSD use the displacement of an arbitrary pose D(p a ), calculated by interpolation in pose space [section 3.2].
c The Eurographics Association and Blackwell Publishing 2006.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
(a)
(b)
(d)
(e)
Figure 1: Skinning result of each algorithm: (a) SSD, (b) PSD, (c) WPSD, (d) Difference between SSD and PSD (blue dotted area), (e) Difference between SSD and WPSD (blue dotted area), (f) Difference between PSD and WPSD (blue dotted area); areas around blue and red arrows represent unexpected results of SSD and PSD respectively.
<H2>3.1. Skeletal subspace deformation (SSD)</H2>
SSD [MTLT88] is based on the weighted blending of an affine transformation of each joint by equation 2.
2
n joint v a = S(v 0 ) = ( ∑ w j T j )v 0 j=1
where n joint is the number of joints, v a is a vertex in an arbitrary pose p a , v 0 is a vertex in the rest pose, and w j is a joint weight that defines the contribution of joint j’s transformations to the current vertex. The weight w j can be assigned by the artist to control deformation and usually ∑ n j=1 joint (w j ) = 1.0. This simple algorithm is used in many commercial graphics packages and real-time rendering applications but shows several limitations, because the deformation of this method is restricted to the subspace of the affine transformation of the joints
[LCF00].
<H2>3.2. Pose space deformation (PSD)</H2>
If we have a sufficient set of examples to describe the movement of an articulated object, we can interpolate displacement in “pose space” [LCF00]. Each sample pose consists of sample skin geometry and the related joint skeleton, and a vector containing the joint angles represents the pose. If we translate each skinning sample k to the rest coordinate frame using inverse SSD, the displacement between the
(c)
(f)
sample vertex v k and the rest pose vertex can be calculated: n joint
3
d k = ( ∑ w j T j ) −1 v k − v 0 j=1
where v k is a vertex in sample pose p k and d k is the displacement of this vertex relative to v 0 in the sample pose p k ; the other variables are defined as in equation 2. Note that the inverse here is of the weighted sum of affine transforms.
After defining the displacement of each pose, the displacement at an arbitrary pose can be calculated by RBF (Radial Basis Function)
[LCF00] or normalized radial basis function
[KM04] interpolation of the example poses’ displacements. The displacement d a of a vertex in an arbitrary pose p a can be calculated as n pose
4
d a = D(p a ) = ∑ r k (p a )d k k=1
where n pose is the number of sample poses, d a is a displacement of the vertex in an arbitrary pose p a , and the weight r k (p a ) defines the contribution of each sample pose. Normalized Radial Basis Functions can smoothly interpolate pose space using: n pose
5
f t (p a ) = ∑ λ t,k φ k (γ k ) k=1
where f t (p a ) is the radial basis weight function for example t evaluated at an arbitrary pose p a , n pose is the number of sample poses, λ t,k are real valued weights between pose t and k, φ k are the radial basis functions, and γ k is the distance between the pose p k and the arbitrary pose p a (defined as the Euclidian distance between the joint vectors of each pose). The weight r k (p a ) is calculated using normalized RBFs and is used in equation 4 to calculate the displacement d a of a vertex in an arbitrary pose p a :
c The Eurographics Association and Blackwell Publishing 2006.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
6
r k (p a ) = ∑ n pose f k (p f a (p ) )
t=1 Gaussian radial basis functions φ k (γ k ) = exp( −γ 2σ 2 k 2 ) are one possible choice of the basis and the constant σ can be specified experimentally [LCF00].
<H2>3.3. Weighted pose space deformation (WPSD)</H2>
WPSD is developed by Kurihara et al.
[KM04] to deform their example-based human hand model derived from medical images. In the original PSD, the distance between two poses p a and p k having n joint number of joints for each pose is defined as n joint
7
γ k (p a , p k ) = ∑ (p a, j − p k, j ) 2 j=1
In equation 7, since the γ k is the difference of n joint dimensional joint vectors of related poses, every vertex in the pose p k has same distance γ k resulting in the same weight r k (p a ) in every vertex of the pose p k . Furthermore, because each element of the joint vector equally contributes to the distance calculation, two vectors having a same value but different order generate same pose distance. For example, three different joint vectors p 1 = (θ, 0, 0), p 2 = (0, θ, 0), p 3 = (0, 0, θ) have same distance between them and it can cause unexpected results in PSD. In WPSD [KM04], Kurihara et al. modify the distance definition between poses using joint weight of each vertex i to give proper weight to each element of a joint vector,
8
n joint γ i,k (p a , p k ) = ∑ w i, j (p a, j − p k, j ) 2 j=1
where γ i,k (p a , p k ) is the distance between pose p a and p k of vertex i, and w i, j is weight of joint j of vertex i used in equation 2. From this definition, a more accurate pose distance is obtained and it generates better skinning in arbitrary poses, especially when the poses are far from the examples. Figure 1 shows result of three different skinning methods, but we entrust the detail comparison between quality of each algorithm to their original papers [MTLT88,LCF00,KM04].
<H1>4. Computing joint weights from samples</H1>
The joint weights of each vertex are important to generate accurate skinning in SSD (equation 2) as well as in WPSD (equation 8). In many applications, the weights are manually generated by skilled artists and it is hard to generate accurate values when a number of joints are involved in deforming a region. In this paper, we automatically calculate the joint weights of each vertex from the sample poses to enhance the accuracy of the weight value. This results in better skinning and reduces the elaborate manual work required to create weight maps. In each sample pose p k , we have following equation based on SSD: n joint
9
v ̃ k − e k = ( ∑ w j T j )v 0 j=1
where v ̃ k is a particular vertex from skin sample k, the right hand side is the SSD deformation of vertex v 0 from the rest pose, e k is a displacement between the SSD deformation and v ̃ k , and the other variables are as in equation 2. If we have sufficient examples involving the same set of n joint joints, we have n pose equations of the form: n joint
10
v ̃ k − e k = ( ∑ v j w j ) j=1
where v j is v 0 transformed by T j . Although the e k is unknown, we can solve for weights that minimize the e k in a least squares sense by stacking the equations 10 (with e k omitted) into a linear matrix system
11
2 v − Aw
where w is a n joint -dimensional weight vector, v is a 3n pose dimensional vector containing the vertex v ̃ i from every sample, and A is a 3n pose ×n joint matrix . >From equation 11, we can calculate w from the given value of v and A to reduce the error of this equation. We use the non-negative least square (NNLS) method to solve this problem and it determines positive weight values minimizing error in equation 10. The calculated weight vector w is normalized to satisfy ∑ n j=1 joint w j = 1.0. In order to avoid a singular matrix A, the number of poses should be greater or equal to the number of overall DOF (Degree Of Freedom) of the joint vector (each joint has 3 DOF), and the sample poses should be sufficiently different. James et al. used a similar approach to estimate vertex weights in each joint [JT05] and we demonstrate their efforts in our skinning method.
<H1>5. Parallel deformation on GPU</H1>
Skinning deformations vary across vertices. In SSD and WPSD, this per-vertex computation is independent for each vertex and can be parallelized by a SIMD parallel architecture. We developed a parallel skinning algorithm for SSD and WPSD that is suitable to GPUs having a SIMD architecture with one-side communication to texture memory.
c The Eurographics Association and Blackwell Publishing 2006.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
<H2>5.1. Parallel WPSD</H2>
The computation cost of the SSD skinning algorithm is O(n vertex × n joint ) from equations 1, 2, PSD is O(n vertex × n joint × n pose ) from equations 1, 2, 4, and WPSD is O(n vertex × n joint × n pose × n pose × n pose ) from equations 1, 2, 4, 5, 6. Where, computation cost of original PSD is defined by equation 1, 2, 4, since r i is same in all vertices and d i can be pre-calculated. The number of joints n joint and poses n pose can be reduced to the smaller numbers using the method developed by Kry et al.
[KJP02], as will be discussed in section 5.2.1 with efforts to reduce texture memory space. In previous research, the Eigenskin method based on PSD was developed using GPU vertex programming [KJP02]. The vertex program uses a relatively small number of slow processing units compared with the fragment processors, and the per-vertex computation cost of the original PSD is O(n joint × n pose ). Therefore WPSD, having higher pervertex computation cost O(n joint × n pose × n pose × n pose ), can clearly benefit from parallel computation on fragment processors.
<H2>5.2. Parallel WPSD on GPU</H2>
We developed parallel skinning using the GPU fragment processors and demonstrate our method using three rendering passes. In order to minimize real-time computation, we separate possible pre-calculation steps and save the results into texture memory using texture maps. Because the value in the texture memory is not changed in the successive deformation, it can be pre-computed and stored in the read-only texture memory. In the first and second pass, per-vertex deformation is calculated in the fragment program and the results are stored in texture maps using the FBO (Frame Buffer Object) extension [Gre05]. These texture maps are bound to the geometry of the rest pose with their texture coordinates. In the third pass, each vertex in the rest pose is changed by the deformed vertex stored in the output texture generated in the first and second passes using vertex texture fetch.
<H3>5.2.1. Packing data into textures</H3>
The fragment processors cannot access vertex information. Instead, we can use texture memory to send data to the fragment program. Information needed in the fragment program is packed into texture maps and stored into texture memory. Geometry information from the rest pose is stored into two RGB texture maps, a vertex texture T v and normal texture T n ; each has size n vertex × 3. These textures represent parameter v 0 in equation 2 and each 3D element (x, y, z) is stored into the (r, g, b) value of a texel [ Figure 2 ]. The joint weights calculated in section 4 are also stored
Figure 2: Packing data into textures: texture map T i can be
T v , T n , T w1 , T w2 , and T d j . V(vertex), N(normal), W(weight), J(joint index), and D(displacement) represent each texel (RGB(A)) value of the related texture. T d consist of eight T d j storing displacements of each pose j.
in the texture maps. In general, the distribution of skinning effects in an articulated body is local to several joints
[MMT97,KJP02], even in a region as complicated as a hand. For example, deformations arising from the PIP (Proximal Interphalangeal) joint of index finger do not propagate to the other fingers, and deformation on the middle phalanx of index finger is only affected by the movement of PIP and DIP(Distal phalanx) joints. From this observation, we can reduce joint weight storage from the actual number of joint n joint to a smaller number of “principal joints” n ̃ joint selected by sorting on the weight value. We threshold n ̃ joint at four in our tests with an additional four elements to hold the related joint index. As a result, we can save the joint weights of entire geometry in two RGBA textures T w1 , T w2 each with size n vertex × 4(rgba) and store the entire information required for SSD [equation 2] in four textures T v , T n , T w1 , and T w2 . The displacement values calculated by equation 3 can be stored in n pose displacement textures; n pose is the number of sample poses. In case of complex joint structures and a large DOF model, we need many sample poses to calculate accurate joint weights and PSD deformation. However, since the joint weights can be pre-calculated, we can reduce the number of sample poses needed in real-time PSD computation. PCA (Principal Component Analysis) of pose space can yield an orthogonal basis called “ Eigendisplacement ” [KJP02]. If we reduce the size of pose space from n pose to n ̃ pose “principal poses” ( n ̃ pose < n pose ), we can reduce the number of displacement textures. In our paper, we set n ̃ pose as eight in our experiment and save displacements of all poses into a RGB texture T d having size n vertex × 8( n ̃ pose ) × 3(rgb). Therefore, from the two important observations of “principal joints” and “principal poses”, the original computation cost for SSD, PSD, and WPSD discussed in section 5.1 can be reduced using n ̃ joint and n ̃ pose rather than n joint and n pose . In the original PSD, since the weight r i in equation 4 is the same at every vertex, we do not need to calculate this value in the GPU. Since the size of this value is just n ̃ pose , we can simply pass them to the GPU as parameters without generating a texture map. Therefore, we store all the information needed to calculate the original PSD at this point. In order to reduce real-time computation, we pre-calculate T j in equation 2 and λ in equation 5 and store them into another one channel texture T x having size n ̃ pose × ( n ̃ pose + n ̃ joint × 3(x, y, z)). As a result, we store all the variables required to calculate WPSD, PSD, and SSD in six texture maps: T v , T n , T w1 , T w2 , T d , and T x . The values in the texture maps are stored in the texture memory at setup time, since they are not changed during the deformation process. In current graphic card architectures, data transfer from CPU to GPU is slow compared with memory access within the GPU. Since the only data changed in each deformation and passed from CPU to GPU is a joint vector p a (size = n joint ) representing the current arbitrary pose, the memory access rate in our method is very efficient; In the original PSD method, an additional r k value (size = n ̃ pose ) is required.
c The Eurographics Association and Blackwell Publishing 2006.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
<H3>5.2.2. Configurations for fragment program</H3>
Variables: T out = output texture, T v = vertex texture 1 /* Set orthographic camera with same size of quad */; 2 gluOrtho2D(-1, 1, -1, 1); 3 bind(FBO); 4 /* Bind T out and set to FBO drawbuffer */; 5 bind(T out ), SetFBOdrawbuffer(T out ); 6 bind(T v ); 7 enable(fragment program); 8 /* Set viewport to the resolution of the texture */; 9 glViewport(0, 0, texWidth, texHeight); 10 /* Render a quad into T out using FBO */; 11 glBegin(GL_QUADS); 12 { glTexCoord2f(0, 0); glVertex3f(-1, -1, -0.5f); 13 glTexCoord2f(1, 0); glVertex3f( 1, -1, -0.5f); 14 glTexCoord2f(1, 1); glVertex3f( 1, 1, -0.5f); 15 glTexCoord2f(0, 1); glVertex3f(-1, 1, -0.5f); 16 }; 17 disable(fragment program); Algorithm 1: Configuration of fragment program for vertex refering and direct rendering into texture
Since vertex information cannot be accessed by the fragment program, vertex deformation on a GPU is usually performed by a vertex program
[KJP02, BK05]. Although, we cannot access vertex data in the fragment program, the efficiency of parallel computation on a fragment program is higher, since the fragment processor has more processing units and each of them has more computation power than a vertex processor. The fragment processing system is a general SIMD architecture using fragment streams as input data; each fragment is assigned to a fragment processor to calculate its final color value independently and in parallel. We developed a parallel WPSD algorithms using the fragment processors to enhance the extent of parallel computation. Geometry information like vertex positions and normals are stored in texture maps T v and T n as described in section 5.2.1 and the vertex information is referred in the fragment processors to calculate final color values. In order to assign each vertex value stored in a texture map to a fragment, we bind the geometry texture T v or T n to a quad and render it using an orthographic camera having the same width and height as the quad. Furthermore, since the viewport is set to the same resolution as the textures, each fragment is exactly matched with each texel holding the vertex information, and we can access each vertex using the texture coordinates of the fragment; vertex weights and displacements stored in the texture maps can also be accessed by similar methods. A similar idea was developed in [PBMH02] to calculate ray tracing in a fragment program and is used in GPGPU (General Purpose computation on GPUs) applications [GPG, LHK ∗ 04, PF05]. The FBO (Frame Buffer Object) extension
[Gre05]
supports rendering into an attached texture. This saves memory and time, since there is no copy operation from frame buffer to texture buffer. We implemented our WPSD algorithm using the fragment program with the FBO extension to store the result directly into texture maps accessed by vertex program in the next pass. A summary of this method is shown in Algorithm 1.
<H2>5.3. GPU implementation</H2>
We implemented GPU deformation using three rendering passes, and the basic architecture is described in figure 3 . In the first pass, we parallelize per-vertex deformation using GPU fragment processors. The data required to calculate this deformation is stored in the textures as described in section 5.2.1 and the deformation for each vertex is calculated in a fragment processor. In a given arbitrary pose defined by a joint vector, SSD is computed by equation 2 using texture maps T v , T w1 , T w2 and T x ; refer to the texture map notation in section 5.2.1. PSD is computed by equation 4 using T d , T x , after calculating r k (p a ) by equation 6. In the case of WPSD, the weighted distance is computed by equation 8 using T w1 , T w2 , and T x . In the first pass, the result of the deformation is rendered
c The Eurographics Association and Blackwell Publishing 2006.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
Figure 3: Overview of WPSD on GPU: Each T i is the texture
map storing the required data for the calculation (refer to section 5.2.1 for their values) , T i ′ s are the texture maps to store the output of the 1st and 2nd passes, P 0 is the geometry in the rest pose, and P a is a deformed model in an arbitrary pose.
into a texture map T v ′ , using the FBO, and passed to the third pass. In the second pass we calculate and store normal deformation with a similar method as in the first pass, and the results are stored in the texture map T n ′ . In the third pass, using a vertex program, each vertex of the rest pose is transformed to the final deformed position using the information from the texture maps computed in the previous two passes. In order to access related texture information in each vertex, we created texture coordinates of each texel in pre-processing and used them in the vertex program. Specifically, the two texture maps, T v ′ and T n ′ that are generated in the first and second passes are accessed in the vertex program using the texture coordinate of the current vertex. Alternatively, multiple render targets (MRTs) can combine the first and second pass, and vertex buffer objects (VBOs) could be used to render the deformed results back to the vertex array [OPE, GPG, LHK ∗ 04].
<H1>6. Results</H1>
We tested our methods using upper arm models consisting of four joints (collar, shoulder, elbow, and wrist). Each has three DOF and the wrist is the end joint having no DOF. Three different resolution meshes are used to test the performance of GPU parallel computation: the high-resolution model has 91460 triangles with 46036 vertices, the midresolution model has 22836 triangles with 11574 vertices, and the low-resolution model has 5762 triangles with 2972 vertices [ Figure 4 ]. Note that these models are considerably more detailed than those used in current games, so the reported frame rates would be much higher if typical gameresolution models were used. On the other hand, with the expected growth of GPU processing power, models such as these will be in wide use in a few years, and algorithms such as WPSD will be required to produce realistic deformations at this level of resolution. Eight sample poses were created by Poser [Cur] and the joints weights and displacements of each sample were derived from these models [ Figure 5 ]. Our parallel algorithm is based on three pass GPU computation. The fragment program for the 1st and 2nd pass, and the vertex program for the 3rd pass are implemented in the Cg language
[FK03]. For accuracy the GPU computation is performed by 32bit floating point operations with 32bit floating point texture maps. Table 2 shows the total memory space to store texture maps required by the fragment program. Note that the maximum required memory space for the highest resolution model is just 6.8 Mbytes; the size of the output texture T v ′ and T n ′ is the same as the size of T v and T n . The results of GPU-based deformation for SSD, PSD, and WPSD are shown in Figure 1 and 6, and the experiment is performed in a GeForce 6800 Ultra GPU and a 3.4Ghz Pentium 4 CPU. The timing results of each algorithm on the CPU and GPU are summarized in table 1 . On average, our GPU-based deformation shows around 20 times speed-up compared with CPU-based deformation. GPU-based WPSD has roughly the same speed as CPUbased SSD. Therefore, real-time applications using SSD can substitute WPSD running on the GPU without loosing their real-time performance. Since our algorithm shows speed-up for SSD and PSD as well as WPSD, applications can choose the most appropriate skinning method according to the required deformation and detail.
Figure 4: Mesh of test data: the top row is a low-resolution mesh, the second row is a mid-resolution mesh, and the bottom row is a high-resolution mesh
c The Eurographics Association and Blackwell Publishing 2006.
T. Rhee, J.P. Lewis, and U. Neumann / EG 2006
Method
Mesh
CPU(FPS)
GPU(FPS)
SSD
low
150
1425
middle
39
630
high
5
164
PSD
low
98
1230
middle
23
530
high
4.5
140
WPSD
low
5
85
middle
1.2
25
high
0.29
7
Method Mesh CPU(FPS) GPU(FPS) SSD low 150 1425 middle 39 630 high 5 164 PSD low 98 1230 middle 23 530 high 4.5 140 WPSD low 5 85 middle 1.2 25 high 0.29 7
Table 1: Timing results (in FPS) of each algorithm: the lowresolution mesh has 5762 triangles with 2972 vertices, the mid-resolution mesh has 22836 triangles with 11574 vertices, and the high-resolution mesh has 91460 triangles with 46036 vertices.
Vertices
T v &T n
T w1 &T w2
T d
T x
Tot
2972 (low)
35×2
46×2
278
1
441
11574 (mid)
135×2
180×2
1080
1
1711
46036 (high)
539×2
719×2
4315
1
6832
Vertices T v &T n T w1 &T w2 T d T x Tot 2972 (low) 35×2 46×2 278 1 441 11574 (mid) 135×2 180×2 1080 1 1711 46036 (high) 539×2 719×2 4315 1 6832
Table 2: Texture memory to store data required in fragment program (in Kbytes); refer to section 5.2.1 for texture notation .
<H1>7. Conclusions</H1>
In this paper, we present a parallel skinning algorithm suitable for SIMD architectures such as GPUs. The joint weights of each vertex are automatically computed by NNLS and used in the skinning computation for SSD and WPSD. Independent per-vertex deformation is parallelized on the GPU using three rendering passes. In the first and second passes, per-vertex deformation is calculated by the fragment processors and the results are stored in texture maps using FBO. In the third pass, using vertex processors, each vertex of the rest pose is changed by the deformed vertex stored in the textures generated by the first and second passes. Articulated body skinning using SSD, PSD, and WPSD are efficiently parallelized by our GPU-based method, and on a detailed model, we obtain around 20 times speed-up compared with CPU-based computation. Principal component compression of the examples and careful analysis of joint distributions can reduce the domain of computation
[KJP02] and other algorithms based on the SSD, PSD, and shape interpolation may be parallelized on GPU using our approach.
Figure 6: Arbitrary poses deformed by WPSD on GPU
<H1>Acknowledgments</H1>
This research has been funded by the Integrated Media System Center/USC, and Samsung Electronics. We wish to thank KyungKook Park, Changki Min, and Tim Foley for discussions about GPUs, and the anonymous reviewers for their sincere comments.
<H1>References</H1>
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612–619.
[AHS03] ALBRECHT I., HABER J., SEIDEL H. P.: Construction and animation of anatomically based human hand models. In Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA-03) (2003), pp. 98–109.
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[FK03] FERNANDO R., KILGARD M. J.: The Cg Tutorial; The Definitive Guide to Programmable Real-Time Graphics. Addison-Wesley, 2003. [GPG] GPGPU: General-purpose computation using graphics hardware. http://gpgpu.org .
[Gre05] GREEN S.: The OpenGL Framebuffer Object Extension. Game Developers Conference, 2005. http://developer.nvidia.com/object/gdc_2005_ presentations.html.
[JP02] JAMES D. L., PAI D. K.: Dyrt: dynamic response textures for real time deformation simulation with graphics hardware. In SIGGRAPH ’02: Proceedings of the 29 th annual conference on Computer graphics and interactive techniques (New York, NY, USA, 2002), ACMPress, pp. 582–585.
[JT05] JAMES D. L., TWIGG C. D.: Skinning mesh animations. ACM Trans. Graph. 24, 3 (2005), 399–407.
[KJP02] KRY P. G., JAMES D. L., PAI D. K.: Eigen Skin: Real time large deformation character skinning in hardware. In Proceedings of the 2002 ACM SIGGRAPH Symposium on Computer Animation (SCA-02) (2002), pp. 153–160.
[KM04] KURIHARA T., MIYATA N.: Modeling deformable human hands from medical images. In Proceedings of the 2004 ACM SIGGRAPH Symposium on Computer Animation (SCA-04) (2004), pp. 357–366.
[KZ05] KAVAN L., ZARA J.: Spherical blend skinning: A real-time deformation of articulated models. In 2005 ACMSIGGRAPH Symposium on Interactive 3D Graphics and Games (April 2005), ACM Press, pp. 9–16.
[LCF00] LEWIS J. P., CORDNER M., FONG N.: Pose space deformation: a unified approach to shape interpolation and skeleton-driven deformation. In SIGGRAPH ’00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques (New York, NY, USA, 2000), ACMPress/Addison-Wesley Publishing Co., pp. 165–172.
[LHK04] LUEBKE D., HARRIS M., KRUGER J., PURCELLT., GOVINDARAJU N., BUCK I., WOOLLEY C., LEFOHN A.: Gpgpu: general purpose computation ongraphics hardware. In GRAPH ’04: Proceedings of the conference on SIGGRAPH 2004 course
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[MTG03] MOHR A., TOKHEIM L., GLEICHER M.: Direct manipulation of interactive character skins. In SI3D ’03: Proceedings of the 2003 symposium on Interactive 3D graphics (New York, NY, USA, 2003), ACM Press, pp. 27–30.
[MTLT88] MAGNENAT-THALMANN N., LAPERRIÈRE R., THALMANN D.: Joint dependent local deformations for hand animation and object grasping. In Graphics In terface ’88 (June 1988), pp. 26–33.
[OPE] OPENGL: Opengl extension registry. http:// oss.sgi.com/projects/ogl sample/registry/.
[PBMH02] PURCELL T. J., BUCK I., MARK W. R., HANRAHAN P.: Ray tracing on programmable graphics hardware. ACM Transactions on Graphics 21, 3 (July 2002), 703–
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[PF05] PHARR M., FERNANDO R.: GPU Gems 2; Programming Techniques for HighPerformance Graphics and General-Purpose Computation. Addison-Wesley, 2005.
[RNL06] RHEE T., NEUMANN U., LEWIS J.: Human hand modeling from surface anatomy. In I3DG ’06: Proc. of ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games (2006).
[SNF05] SIFAKIS E., NEVEROV I., FEDKIW R.: Automatic determination of facial muscle activations from sparse motion capture marker data. ACM Trans. Graph. 24, 3 (2005), 417–425.
[SRC01] SLOAN P.-P. J., ROSE C. F., COHEN M. F.: Shape by example. In SI3D ’01: Proceedings of the 2001 symposium on Interactive 3D graphics (New York, NY, USA, 2001), ACM Press, pp. 135–143.
[Web00] WEBER J.: Run-time skin deformation. In Proceedings of Game Developers Conference (2000).
[WP02] WANG X. C., PHILLIPS C.: Multi-weight enveloping: least-squares approximation techniques for skin animation. In SCA ’02: Proceedings of the 2002 ACM
SIGGRAPH/Eurographics symposium on Computer animation (New York, NY, USA, 2002),
ACM Press, pp. 129–138.
Figure 5: Samples poses and displacements: the first row shows each sample poses, the second row shows displacement of each sample pose with the rest pose in the first column of the third row. Please enlarge to see details.
c The Eurographics Association and Blackwell Publishing 2006.
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"text": "['vertex information cannot be accessed in the fragment program']",
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"text": "['GPU-based vertex deformation is usually performed by vertex programs']",
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"text": "['in previous research']",
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"text": "['we develop a parallel WPSD method using the fragment processors to gain greater parallelism and performance']",
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"text": "['Person-specific data modeling and its deformation is also an interesting topic in realistic articulated body simulation']",
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"text": "['Rhee et al. described human hand modeling from surface anatomy of the person']",
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"text": "['RNL06']",
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"text": "['Anguelov et al. developed shape completion and animation of people, derived from the set of range scan data and example based deformation in pose and shape space']",
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"text": "['Physically inspired skinning should be also recognized as another important area of articulated body animation']",
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"text": "['AHS03']",
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],
"text": "['CBC ∗ 05']",
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"text": "['PCLS05']",
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],
"text": "['SNF05']",
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],
"text": "['Example-based skinning problems can be described by the following general equation']",
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"text": "['v(p a ) = S(v 0 + D(p a ))']",
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"text": "['In skeletal subspace deformation the displacement D(p a ) is omitted and the target surface is calculated by SSD as a blend of affine transforms of v 0']",
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"text": "['section 3.1']",
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"text": "['Skinning methods related to PSD use the displacement of an arbitrary pose D(p a ), calculated by interpolation in pose space']",
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"text": "['section 3.2']",
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"text": "['SSD']",
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],
"text": "['MTLT88']",
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"text": "['is based on the weighted blending of an affine transformation of each joint by equation']",
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"text": "['n joint v a = S(v 0 ) = ( ∑ w j T j )v 0 j=1']",
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"text": "['usually ∑ n j=1 joint (w j ) = 1.0']",
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"text": "['The weight w j can be assigned by the artist to control deformation']",
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"id": "T117"
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],
"text": "['This simple algorithm is used in many commercial graphics packages and real-time rendering applications']",
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"text": "['shows several limitations']",
"type": "background_claim",
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"text": "['the deformation of this method is restricted to the subspace of the affine transformation of the joints']",
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"text": "['LCF00']",
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"text": "['we have a sufficient set of examples to describe the movement of an articulated object']",
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"text": "['we can interpolate displacement in “pose space”']",
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"text": "['LCF00']",
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"text": "['we translate each skinning sample k to the rest coordinate frame using inverse SSD']",
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],
"text": "['the displacement between the']",
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"text": "['sample vertex v k and the rest pose vertex can be calculated: n joint']",
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"text": "['d k = ( ∑ w j T j ) −1 v k − v 0 j=1']",
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],
"text": "['After defining the displacement of each pose, the displacement at an arbitrary pose can be calculated by RBF']",
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],
"text": "['LCF00']",
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"text": "['or normalized radial basis function']",
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],
"text": "['KM04']",
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"text": "['interpolation of the example poses’ displacements']",
"type": "background_claim",
"id": "T133"
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"text": "['The displacement d a of a vertex in an arbitrary pose p a can be calculated as n pose']",
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"offsets": [
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"text": "['d a = D(p a ) = ∑ r k (p a )d k k=1']",
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"text": "['Normalized Radial Basis Functions can smoothly interpolate pose space using: n pose']",
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"id": "T136"
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{
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],
"text": "['f t (p a ) = ∑ λ t,k φ k (γ k ) k=1']",
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"text": "['The weight r k (p a ) is calculated using normalized RBFs and is used in equation 4 to calculate the displacement d a of a vertex in an arbitrary pose p a']",
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"text": "['r k (p a ) = ∑ n pose f k (p f a (p ) )']",
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"text": "['WPSD is developed by Kurihara et al.']",
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"text": "['KM04']",
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"text": "['to deform their example-based human hand model derived from medical images']",
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"text": "['the γ k is the difference of n joint dimensional joint vectors of related poses']",
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"text": "['every vertex in the pose p k has same distance γ k resulting in the same weight r k (p a ) in every vertex of the pose p k']",
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"id": "T144"
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],
"text": "['In the original PSD, the distance between two poses p a and p k having n joint number of joints for each pose is defined as n joint']",
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"id": "T145"
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"offsets": [
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"text": "['γ k (p a , p k ) = ∑ (p a, j − p k, j ) 2 j=1']",
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"text": "['each element of the joint vector equally contributes to the distance calculation']",
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"text": "['two vectors having a same value but different order generate same pose distance']",
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"text": "['three different joint vectors p 1 = (θ, 0, 0), p 2 = (0, θ, 0), p 3 = (0, 0, θ) have same distance between them and it can cause unexpected results in PSD']",
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"text": "['In WPSD']",
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"text": "['KM04']",
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"text": "['Kurihara et al. modify the distance definition between poses using joint weight of each vertex i to give proper weight to each element of a joint vector']",
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"id": "T152"
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],
"text": "['n joint γ i,k (p a , p k ) = ∑ w i, j (p a, j − p k, j ) 2 j=1']",
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"text": "['From this definition, a more accurate pose distance is obtained']",
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"text": "['it generates better skinning in arbitrary poses']",
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"id": "T155"
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"text": "['The joint weights of each vertex are important to generate accurate skinning in SSD']",
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"id": "T156"
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],
"text": "['equation 2']",
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"text": "['as well as in WPSD']",
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"text": "['equation 8']",
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"text": "['In many applications, the weights are manually generated by skilled artists']",
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"text": "['it is hard to generate accurate values']",
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"text": "['a number of joints are involved in deforming a region']",
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"text": "['we automatically calculate the joint weights of each vertex from the sample poses to enhance the accuracy of the weight value']",
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"text": "['This results in better skinning and reduces the elaborate manual work required to create weight maps']",
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"text": "['In each sample pose p k , we have following equation based on SSD: n joint']",
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"text": "['v ̃ k − e k = ( ∑ w j T j )v 0 j=1']",
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"text": "['we have sufficient examples involving the same set of n joint joints']",
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"text": "['v ̃ k − e k = ( ∑ v j w j ) j=1']",
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"text": "['the e k is unknown']",
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"text": "['we can solve for weights that minimize the e k in a least squares sense by stacking the equations 10 (with e k omitted) into a linear matrix system']",
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"text": "['2 v − Aw']",
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"text": "['From equation 11, we can calculate w from the given value of v and A to reduce the error of this equation']",
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"text": "['We use the non-negative least square (NNLS) method to solve this problem']",
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"text": "['it determines positive weight values minimizing error in equation 10']",
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"text": "['The calculated weight vector w is normalized to satisfy ∑ n j=1 joint w j = 1.0']",
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"text": "['In order to avoid a singular matrix A, the number of poses should be greater or equal to the number of overall DOF']",
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"text": "['of the joint vector']",
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"text": "['the sample poses should be sufficiently different']",
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"text": "['James et al. used a similar approach to estimate vertex weights in each joint']",
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"text": "['JT05']",
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"text": "['Skinning deformations vary across vertices']",
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18254,
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"text": "['In SSD and WPSD, this per-vertex computation is independent for each vertex and can be parallelized by a SIMD parallel architecture']",
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"offsets": [
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"text": "['We developed a parallel skinning algorithm for SSD and WPSD that is suitable to GPUs having a SIMD architecture with one-side communication to texture memory']",
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"id": "T184"
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{
"offsets": [
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"text": "['The computation cost of the SSD skinning algorithm is O(n vertex × n joint ) from equations 1, 2, PSD is O(n vertex × n joint × n pose ) from equations 1, 2, 4, and WPSD is O(n vertex × n joint × n pose × n pose × n pose ) from equations 1, 2, 4, 5, 6']",
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"id": "T185"
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19059,
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"text": "['r i is same in all vertices and d i can be pre-calculated']",
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"id": "T186"
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"text": "['computation cost of original PSD is defined by equation 1, 2, 4']",
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"offsets": [
[
19118,
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],
"text": "['The number of joints n joint and poses n pose can be reduced to the smaller numbers using the method developed by Kry et al.']",
"type": "own_claim",
"id": "T188"
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[
19245,
19250
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],
"text": "['KJP02']",
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19290
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],
"text": "['section 5.2.1']",
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],
"text": "['the Eigenskin method based on PSD was developed using GPU vertex programming']",
"type": "background_claim",
"id": "T191"
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{
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],
"text": "['KJP02']",
"type": "data",
"id": "T192"
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"offsets": [
[
19444,
19556
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],
"text": "['The vertex program uses a relatively small number of slow processing units compared with the fragment processors']",
"type": "background_claim",
"id": "T193"
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{
"offsets": [
[
19562,
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],
"text": "['the per-vertex computation cost of the original PSD is O(n joint × n pose )']",
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"id": "T194"
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{
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],
"text": "['WPSD']",
"type": "own_claim",
"id": "T195"
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19804
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],
"text": "['can clearly benefit from parallel computation on fragment processors']",
"type": "own_claim",
"id": "T196"
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"text": "['having higher pervertex computation cost O(n joint × n pose × n pose × n pose )']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
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],
"text": "['In order to minimize real-time computation, we separate possible pre-calculation steps and save the results into texture memory using texture maps']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
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],
"text": "['the value in the texture memory is not changed in the successive deformation']",
"type": "own_claim",
"id": "T199"
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{
"offsets": [
[
20227,
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"text": "['it can be pre-computed and stored in the read-only texture memory']",
"type": "own_claim",
"id": "T200"
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"offsets": [
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"text": "['The fragment processors cannot access vertex information']",
"type": "own_claim",
"id": "T201"
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"text": "['we can reduce joint weight storage from the actual number of joint n joint to a smaller number of “principal joints” n ̃ joint selected by sorting on the weight value']",
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"text": "['We threshold n ̃ joint at four in our tests with an additional four elements to hold the related joint index']",
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"text": "['we need many sample poses to calculate accurate joint weights and PSD deformation']",
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"text": "['a large DOF model']",
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"text": "['we set n ̃ pose as eight in our experiment and save displacements of all poses into a RGB texture T d having size n vertex × 8( n ̃ pose ) × 3(rgb)']",
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"text": "['from the two important observations of “principal joints” and “principal poses”, the original computation cost for SSD, PSD, and WPSD discussed in section 5.1 can be reduced using n ̃ joint and n ̃ pose rather than n joint and n pose']",
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23844,
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"text": "['In the original PSD']",
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"text": "['we do not need to calculate this value in the GPU']",
"type": "background_claim",
"id": "T233"
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"text": "['the size of this value is just n ̃ pose']",
"type": "own_claim",
"id": "T234"
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24029,
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"text": "['we can simply pass them to the GPU as parameters without generating a texture map']",
"type": "own_claim",
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24123,
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"text": "['we store all the information needed to calculate the original PSD at this point']",
"type": "own_claim",
"id": "T236"
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{
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"text": "['they are not changed during the deformation process']",
"type": "own_claim",
"id": "T237"
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{
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24561,
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"text": "['The values in the texture maps are stored in the texture memory at setup time']",
"type": "own_claim",
"id": "T238"
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{
"offsets": [
[
24204,
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],
"text": "['In order to reduce real-time computation, we pre-calculate T j in equation 2 and λ in equation 5 and store them into another one channel texture T x having size n ̃ pose × ( n ̃ pose + n ̃ joint × 3(x, y, z))']",
"type": "own_claim",
"id": "T239"
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{
"offsets": [
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24430,
24558
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"text": "['we store all the variables required to calculate WPSD, PSD, and SSD in six texture maps: T v , T n , T w1 , T w2 , T d , and T x']",
"type": "own_claim",
"id": "T240"
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{
"offsets": [
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24699,
24818
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"text": "['In current graphic card architectures, data transfer from CPU to GPU is slow compared with memory access within the GPU']",
"type": "background_claim",
"id": "T241"
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{
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24826,
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],
"text": "['the only data changed in each deformation and passed from CPU to GPU is a joint vector p a (size = n joint ) representing the current arbitrary pose']",
"type": "own_claim",
"id": "T242"
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"text": "['the memory access rate in our method is very efficient']",
"type": "own_claim",
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"text": "['In the original PSD method, an additional r k value (size = n ̃ pose ) is required']",
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"text": "['vertex information cannot be accessed by the fragment program']",
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26183,
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26261,
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"text": "['each fragment is exactly matched with each texel holding the vertex information']",
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"text": "['A similar idea was developed']",
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27803,
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"text": "['LHK ∗ 04']",
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27817
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"text": "['PF05']",
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27820,
27827
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"text": "['supports rendering into an attached texture']",
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"text": "['Gre05']",
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27941
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"text": "['This saves memory and time']",
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"text": "['there is no copy operation from frame buffer to texture buffer']",
"type": "background_claim",
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{
"offsets": [
[
28013,
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"text": "['We implemented our WPSD algorithm using the fragment program with the FBO extension to store the result directly into texture maps accessed by vertex program in the next pass']",
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{
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"text": "['Algorithm 1']",
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30485,
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[
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30588
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30595,
30603
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"offsets": [
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30656,
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"text": "['We tested our methods using upper arm models consisting of four joints']",
"type": "own_claim",
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{
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"text": "['Each has three DOF']",
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30788,
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"text": "['the wrist is the end joint having no DOF']",
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{
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"text": "['Three different resolution meshes are used to test the performance of GPU parallel computation']",
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{
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"text": "['the high-resolution model has 91460 triangles with 46036 vertices']",
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"text": "['the midresolution model has 22836 triangles with 11574 vertices']",
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31124
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"text": "['the low-resolution model has 5762 triangles with 2972 vertices']",
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31135
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"text": "['Figure 4']",
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"text": "['these models are considerably more detailed than those used in current games']",
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"text": "['the reported frame rates would be much higher']",
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"text": "['typical gameresolution models were used']",
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"text": "['models such as these will be in wide use in a few years']",
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"text": "['the expected growth of GPU processing power']",
"type": "own_claim",
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{
"offsets": [
[
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31552
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"text": "['algorithms such as WPSD will be required to produce realistic deformations at this level of resolution']",
"type": "own_claim",
"id": "T292"
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{
"offsets": [
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"text": "['Eight sample poses were created by Poser [Cur] and the joints weights and displacements of each sample were derived from these models']",
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"id": "T293"
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{
"offsets": [
[
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31698
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"text": "['Figure 5']",
"type": "data",
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"offsets": [
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31702,
31763
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"text": "['Our parallel algorithm is based on three pass GPU computation']",
"type": "own_claim",
"id": "T295"
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{
"offsets": [
[
31765,
31886
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"text": "['The fragment program for the 1st and 2nd pass, and the vertex program for the 3rd pass are implemented in the Cg language']",
"type": "own_claim",
"id": "T296"
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{
"offsets": [
[
31889,
31893
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"text": "['FK03']",
"type": "data",
"id": "T297"
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{
"offsets": [
[
32120,
32205
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],
"text": "['the maximum required memory space for the highest resolution model is just 6.8 Mbytes']",
"type": "own_claim",
"id": "T298"
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{
"offsets": [
[
31896,
32015
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"text": "['For accuracy the GPU computation is performed by 32bit floating point operations with 32bit floating point texture maps']",
"type": "own_claim",
"id": "T299"
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{
"offsets": [
[
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"text": "['the size of the output texture T v ′ and T n ′ is the same as the size of T v and T n']",
"type": "own_claim",
"id": "T300"
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{
"offsets": [
[
32556,
32660
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],
"text": "['On average, our GPU-based deformation shows around 20 times speed-up compared with CPU-based deformation']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
32546,
32553
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"text": "['table 1']",
"type": "data",
"id": "T302"
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{
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[
32662,
32719
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"text": "['GPU-based WPSD has roughly the same speed as CPUbased SSD']",
"type": "own_claim",
"id": "T303"
},
{
"offsets": [
[
32732,
32847
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"text": "['real-time applications using SSD can substitute WPSD running on the GPU without loosing their real-time performance']",
"type": "own_claim",
"id": "T304"
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{
"offsets": [
[
32855,
32915
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"text": "['our algorithm shows speed-up for SSD and PSD as well as WPSD']",
"type": "own_claim",
"id": "T305"
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{
"offsets": [
[
32917,
33026
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"text": "['applications can choose the most appropriate skinning method according to the required deformation and detail']",
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"id": "T306"
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{
"offsets": [
[
36388,
36473
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"text": "['we present a parallel skinning algorithm suitable for SIMD architectures such as GPUs']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
37012,
37115
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"text": "['Articulated body skinning using SSD, PSD, and WPSD are efficiently parallelized by our GPU-based method']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
37121,
37212
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],
"text": "['on a detailed model, we obtain around 20 times speed-up compared with CPU-based computation']",
"type": "own_claim",
"id": "T309"
},
{
"offsets": [
[
37214,
37342
]
],
"text": "['Principal component compression of the examples and careful analysis of joint distributions can reduce the domain of computation']",
"type": "background_claim",
"id": "T310"
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{
"offsets": [
[
37345,
37350
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],
"text": "['KJP02']",
"type": "data",
"id": "T311"
},
{
"offsets": [
[
37356,
37465
]
],
"text": "['other algorithms based on the SSD, PSD, and shape interpolation may be parallelized on GPU using our approach']",
"type": "own_claim",
"id": "T312"
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A01 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A01_S01_A_Powell_Optimization_Approach__for_Example-Based_Skinning_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1515/libr.2002.169
<Title>A Powell Optimization Approach for Example-Based Skinning in a Production Animation Environment</Title>
Xiao Xian ∗ John P. Lewis Nanyang Technological University Graphic Primitives Seah Hock Soon Nickson Fong Nanyang Technological University EggStoryCP Tian Feng Nanyang Technological University
<Abstract>We propose a layered framework for incorporating example-based skinning algorithms such as Pose Space Deformation or Shape-by-Example into an existing character animation system. The challenge in implementing example-based skinning in an existing system lies in the fact that it is generally believed that the interpolation of the examples is best performed before doing other skinning deformations (although there has been no analysis as to why this is the case), whereas the examples are specified by the user after the other deformations are performed. It is therefore necessary to invert the operation of these skinning and deformation operators. Existing systems typically allow layering of both basic skinning methods such as Skeleton Subspace Deformation (SSD) and other deformations such as lattices, etc., and commercial systems may further allow additional proprietary deformation algorithms as part of the character skinning. Unfortunately, understanding and accessing their various parameters can be laborious at best, and we do not have access to the algorithms in the case of commercial packages. The contributions of this paper are 1) a detailed analysis showing how inverting the skinning operations leads to better example interpolation, and 2) a demonstration</Abstract>
∗ [email protected]
that the black-box inverse can be accomplished in practice using Powell optimization, resulting in an improved example-based skinning capability for existing large animation systems.
<H2>Keywords: </H2>skinning, Powell optimization, computer animation
<H1>1 Introduction</H1>
With the help of modelling tools or capture devices, complicated 3D character models are widely used in fields of entertainment, virtual reality, medicine etc. The range of breathtaking realistic 3D models is only limited by the creativity of artists and resolution of devices. Driving 3D models in a natural and believable manner is not trivial, especially when the model is very detailed and playback of animation becomes quite heavy and time consuming. Each time when a frame goes wrong, a production cannot afford major revisions such as resculpting models or re-rigging skeletons. Therefore, providing a flexible and efficient solution to animation remains an open problem. Articulated character animation is a process of deforming the skin surface by manipulating influencing objects such as skeletons, IK, wire frames and Nurbs curves etc. Skeleton Subspace Deformation (SSD) is the predominant approach to character skinning at present. A nice review of SSD is given in [ 1 ]. SSD is widely used in games, virtual reality and other realtime applications due to its ease of implementation and low cost of computing. It provides the relation between characters and their underlying skeletons. Normally this relation is defined in the rest pose, and determines how characters move according to their skeletons thereafter. Sometimes, artists will edit the geometry of characters in the rest pose to fine-tune animations. This approach is not commonly applied, however, since editing in the rest pose will influence most other poses. On the other hand SSD is also notorious for artifacts at rotating elbows and extreme poses. For those applications that require visual fidelity, such as movies, SSD serves only as a basic framework, on which lots of more complicated deformation approaches are built as a compensation. Example based skinning methods such as Pose Space Deformation (PSD) are candidates for correcting SSD limitations. Example geometric models paired with underlying skeletons in different poses are provided by artists with carefully sculpting and posing. PSD smoothly interpolates these meshes in pose space and produces visually attractive animations. However, although PSD may be used as a compensation to the underlying SSD, and the animator specifies the PSD examples after the SSD has been performed, it is generally believed that the examples are best interpolated in the rest pose, before the SSD has been applied. Therefore the action of the SSD and any other deformations must be “inverted” in order to push the example compensation before these operations. Besides SSD, other skinning approaches such as rigid skinning, Free Form Deformation etc. can also be applied. Our goal is to incorporate examplebased skinning into a system having a variety of other skinning and deformation operations, and to be able to invert these operations regardless of their nature. Since SSD is the most representative in the family of basic skinning, we will discuss how it performs in the inverse operation of PSD scheme. For a simplified condition where only one joint rotation and two example poses are considered, we demonstrate this inverse strategy has a better performance than the same framework without it. The rest of this paper is organized as follows. After a review of related work, we will study details of SSD and PSD respectively in the third section. Then discussion of the inverse operation is presented in the fourth section including its implementation and reasons why this approach will improve the quality of deformation. In the fifth section, we propose a unified skinning framework by applying minimization theory with respect to other basic skinning schemes. Conclusion and some discussions of future work are presented in the last section.
1
<H1>2 Related Work</H1>
Besides the geometric solutions mentioned in the previous section, physical modelling and animation is another field providing realistic character simulations. Given physical principles, this category can generate more believable animation effects compared to its geometric counterpart. But they are seldom applied to interactive applications because of the high cost of computing and complicated algorithms. This paper is mainly dedicated to geometric solutions. Pose Space Deformation
[ 1 ] combines shape blending and Skeleton Subspace Deformation by formulating a scattered data interpolation problem over sculpted (or otherwise obtained) example poses. Character geometries in problematic poses will be re-sculpted by animators and then resulting displacement (referred as delta values in this paper) from the original geometries will be stored as “scattered data” for interpolation phase. The interpolation is performed in the pose space which consists of skeleton joints, or other potentially abstract controllers. Their values such as rotation degrees can be chosen as coordinates of the abstract pose space. After a model is posed and resculpted in different example poses, a multidimensional linear system is built by implementing an interpolation scheme using Radial Basis Functions (RBF), and the output of this system is the weights of each example pose. The final animation can be synthesized by linearly blending RBF functions with the solved weights. Related research efforts have improved the speed and power of example-based skinning. [ 2 ]
incorporate linear elements into RBF to produce constant changes between examples. [ 3 ] precompute principal components of the deformation influences for individual kinematic joints instead of storing displacements for key poses, thereby enabling realtime rendering large nonlinear finite element models of human hands. [ 4 ] introduce weighted pose space deformation for deforming realistic models of human hand. The latest work [ 5 ] identifies statistically relevant bones and approximates bone transforms from example mesh animations. Using established terminology from statistical modeling, these example-based approaches can be considered as non-parametric skin deformation methods. The data needed for these methods grows with the number of examples, but arbitrary deformations can be approximated as a result. Simpler parametric skinning approaches (of which SSD is the prototype) have a fixed number of parameters; these have also seen some development in recent years [ 6 ], [ 7 ]. Skinning using free form lattices [ 8 ], [ 9 ] or NURBS curves [ 10 ] instead of skeletons to drive character surface are also common practices in the entertainment production. Our framework implements existing PSD theory and the distinction is that we insert an optimization module into the PSD pipeline by applying a unified inverse approach assuming the knowledge of basic skinning is unavailable.
2
<H2>2.1 Our Contribution</H2>
We provide detailed reasons why and how the inverse operation can improve the results. For a simplified case, we show that the direction of deformed vertices from inverse skinning is a linear function of joint rotation, while in the forward approach, that direction is kept as a constant. This demonstration provides for the first time a clear theoretical reason why inverse operation is required. We formulate editing geometry in rest pose as an optimization problem and propose a unified framework which can be implemented on high-end commercial packages while allowing any proprietary skinning operators to be incorporated.
<H1>3 Skeleton Sub-Space Deformation</H1>
6 5 4 3 2 1 0
0 1 2 3 4 5 6 7 8 (a). (b).
Figure 1: (a). Skeleton Subspace Deformation; (b). Radial Basis Interpolation
Skeleton Subspace Deformation (SSD) is a basic algorithm that is used to define how the character surface deforms following movements of its underlying skeletons. The main idea is introduced by [ 11 ], and is also known as soft skinning, linear blending or Single Weight Enveloping (SWE). Due to its simplicity and efficiency, SSD is widely applied to interactive applications such as games and virtual reality, and it is implemented in most commercial animation packages. A skeleton should be rigged to a character surface beforehand, roughly based on the anatomy of the character and kinetic rules. The pose in which the skeleton is rigged normally is referred to as the rest pose. The basic relationship between surfaces and skeletons is defined at the rest pose, and all motions of the character will be influenced thereafter. If SSD is adopted to define this relation, each vertex or control point of the character surface is provided with a list of joints, that will influence it, along with the weight indicating the amount of influence. When the character is animated, the position of a vertex in the animated pose is the result of weighted linear blending of its transformation by each associated joint. We formulate SSD as:
1
M v p = SSD p (v r ) = ∑ ω k T pk v r k=1 r
For a vertex in rest pose v , its transformed position in pose p is v p . T pk means the kth joint’s transformation from rest pose to pose p. Readers can find details on how to compute T pk in
[ 1 ]. ω k is the corresponding weight. This weight is usually a function of distance between v r and its associated joints, and is defined when we apply SSD to the rigged character. Figure 1 (a). is a simple illustration of SSD with only two joint frames. Rectangles represent animated sections in each of two frames and the curve shows the blended result of both frames. Since vertex transformations can be easily implemented in the graphic card, SSD is very popular in circumstances that require animating a number of characters in real time. Some opportunities for control are provided to the animators. When a character goes wrong in some pose, animators can adjust joint influence weights. But the domain of adjusting one vertex in this way is strictly limited to the linear subspace formed by the vertex as transformed by joints influencing this vertex.
3
<H2>3.1 Pose Space Deformation</H2>
The famous SSD problem of “collapsed elbow” is recognized in
[ 1 ]
as being due to the fact that deforming is limited to a linear subspace. Because of this limitation SSD cannot synthesize many parts of a character skin involving complicated joint structures. Built on the SSD scheme, the Pose Space Deformation (PSD) is proposed by
[ 1 ] as a combination of SSD and shape blending providing nice solution to above mentioned problems. PSD can be performed in the following steps: • example pose setup: move the character to problematic poses, sculpt and store pose information (joint configuration x) including the amount of movement of each sculpted vertex (delta values d). • In the inverse PSD approach in this paper, we transform example models (d) to the rest pose (d r ): This step is trivial if the basic skinning, say SSD, is explicit. In our proposed framework, this step will be replaced by an optimization routine. • solving a linear system: we setup an interpolation scheme for delta Φ(x)ω = d and solve it to obtain weights ω for all example poses. • realtime synthesis: for an intermediate pose, we obtain the delta in the rest pose by interpolating example poses at runtime. We add this delta to the original character surface and then let SSD or any other skinning scheme finish the final transformation. For a vertex v, if sculpted in N example poses, then there are N delta d i , i = 0, . . . , N − 1 corresponding to each pose x i , i = 0, . . . , N − 1. These are converted to rest pose displacements using d i r = SSD −1 (d i ). We adopt Gaussian Radial Basis functions to interpolate d i r . First a N ∗ N matrix Φ is built with the (i, j)th element as φ ( x i − x j ), where x i − x j means the Euclidean distance between pose x i and pose x j , then we have a linear system:
2
W = Φ −1 D r
Here W and D r are column vectors with ith element ω i and d i r respectively. In the synthesis phase, for an intermediate pose x, we can obtain the delta d for this vertex by:
3
N−1 d = ∑ ω i φ ( x i − x ) i=0
For the Gaussian function φ (x) = e − σ x 2 2 , σ is used to control the “fall-off”. In Figure 1 (b)., we use Gaussian Radial Basis functions to interpolate 3 points. The blue and green curve represent σ = 1.0 and σ = 2.0 respectively. Other basis functions also can be candidates. Although PSD and improved example-based schemes have been discussed in many publications
[ 2 ], [ 3 ], [ 4 ]
, the reason why the inverse should be performed is still ambiguous. In the next section we will analyze this issue and demonstrate why inverting the SSD (and other deformations) in order to interpolate the examples in the rest pose is a the right choice.
<H1>4 Inverse Operation</H1>
This section will describe the implementation of our inverse algorithm and why it is an improvement. We still study SSD as the underlying skinning, since an explicit form of basic skinning can help to simplify our task of explanation. We call the PSD scheme without the inverse operation as “forward PSD”, and comparison to it will be used to demonstrate the superiority of the inverse method.
<H2>4.1 Implementation</H2>
In this sub-section, we will explain how inverse PSD works. For N examples, a vertex v is first transformed from rest pose by SSD to positions v i , i = 0, . . . , N − 1, then animators move it to example positions to obtain delta values d i , i = 0, . . . , N − 1. The final positions of v in example poses are v i + d i , i = 0, . . . , N − 1, and we call them target positions vt i . We summarize above operations as:
4
4
v i t = v i + d i = SSD i (v r ) + d i
where vr means the rest position of v and SSD i (∗) represents the equation 1. The “forward PSD” approach then concludes by interpolating d i as a function of pose. In the inverse approach we instead apply the inverse of SSD i (∗) to v i t to obtain a modified rest pose vertex v r i . The difference of vr i and vr produces new delta value d i r , which will be the input of linear system (equation 2) introduced in the previous section.
5
d i r = SSD −1 i (v t i ) − vr
In this step we need implement the inverse skinning operator SSD −1 . Since SSD is a 3D transformation, SSD −1 simply is the inverse transformation matrix generated by SSD. For the situation where other unknown skinning operations are adopted, we propose a unified framework which will be discussed in the following section. Next we build a new delta vector Dr with ith element as d i r , and replace D in equation 2 with d i r to get a new weight vector W r .
6
W r = Φ −1 Dr
In the synthesis phase, for an intermediate pose x we have:
7
N−1 d x r = ∑ ω i r φ ( x i − x )
i=0 then we add this d x r to v r and let SSD finish the rest of the job:
8
v x = SSD x (v r + d x r )
v x represents the final position of vertex v in pose x.
<H2>4.2 Deformation Direction Analysis</H2>
In this section we will analyze and compare the deforming effects of both forward PSD and inverse PSD, explaining why the inverse method is superior.. Given two examples as shown in Figure 2 (a) and (b) respectively, vertex v with the position v r in the rest pose ( 0 degrees ) is sculpted to a “target position” v ti in an example pose (90 degrees). The delta value in the first pose is zero. Then we apply forward and inverse PSD respectively to interpolate these two poses. For an intermediate pose x, we have two distinct deforming vertices resulting from two algorithms, as illustrated in Figure 3 , v ssd x , v x p , v x I p are the deformed positions from SSD, forward and inverse PSD in an intermediate pose x. We use two angles α p and α I p to analyze how directions of a deformed vertex change with the pose. In the forward case, α p is formed by the vector (v ssd x , v x p ) and the line y = Y v ssd x , where Y v x ssd is the y coordinate of v x ssd . For two examples shown in Figure 2 (a) and (b), we have delta values d 1 = [d 1x , d 1y ] and d 2 = [d 2x , d 2y ]. For equation 2: ω ω 2x 1x ω ω 2y 1y = φ φ 21 11 φ φ 12 22 −1 d d 2x 1x d d 1y 2y Because in the rest pose, we didn’t generate any movement for v , then d 1 = [d 1x , d 1y ] = [0, 0]. Taking the model in rest pose as an example is a common practice when applying shape interpolation, since interpolating effects from other examples should not change the original model in rest pose. Therefore, by solving above equation we have: ω 1x = φ 11 −1 d 1x + φ 12 −1 d 2x = φ 12 −1 d 2x ω 2x = φ 21 −1 d 1x + φ 22 −1 d 2x = φ 22 −1 d 2x = d 2x ω 1y = φ 11 −1 d 1y + φ 12 −1 d 2y = φ 12 −1 d 2y ω 2y = φ 21 −1 d 1y + φ 22 −1 d 2y = φ 22 −1 d 2y = d 2y where φ i −1 j is the (i, j)th element of Φ −1 , and if i = j, φ i −1 j = 1. Then in an intermediate pose x for α p , we have tan α p = d d x y . d y and d x are delta values in x, y coordinates computed from equation 3. With the above weight values, we have: tan α p = d d x y = ω ω 1x 1y φ φ (x (x − − x x 1 1 ) ) + + ω ω 2y 2x φ φ (x (x − − x x 2 2 ) ) = φ φ 12 12 −1 −1 d d 2x 2y φ φ (x (x − − x x 1 1 ) ) + + d d 2x 2y φ φ (x (x − − x x 2 2 ) ) = d 2y d 2x We can see this angle α p is a constant and depends only on the value of delta in the second pose d 2 = [d 2x , d 2y ]. Then we take a look at α I p in the inverse case. We transform two examples to rest pose to obtain delta values: d 1 r = [d 1x , d 1y ] = [0, 0] and d r 2 = [d 2x , d 2y ]. Since only the second joint is rotating, we simplify SSD as a rotation transformation ignoring other issues such as accumulating effects from the first joint: ⎛ ⎞ cos θ sin θ 0 SSD θ = ⎝ − sin θ cos θ 0 ⎠ . 0 0 1 where θ is the rotation angle of the second joint. Then for the vertex v r = [v 0x , v 0y ], SSD θ (v r ) transforms v from rest pose to [v SSD x , v SSD y ] = [v 0x cos θ − v 0y sin θ , v 0x sin θ + v 0y cos θ ]. In an intermediate pose x, we have its corresponding rest position as v r x = [v 0x + d x , v 0y + d y ], and here the [d x , d y ] are interpolated result computing from equation 3. We just apply the simplified SSD to v x r to obtain v I x p : v Inp x = (v 0x + d x ) cos θ − (v 0y + d y ) sin θ and v Inp y = (v 0x + d x ) sin θ + (v 0y + d y ) cos θ . Similarly, we compute the tangent of α I p : tan α I p = − v v Inp Inp x y − − v v SSD SSD x y = − d x sin θ + d y cos θ = − tan( β + θ ) d x cos θ − d y sin θ where tan β = d y = d 2y . Then we can see α I p = d x d 2x −( θ + β ), which is linearly proportional to the pose rotation θ . And now we take a look at a real cylinder model with one vertex sculpted in the second pose, shown in the Figure 4 . Forward PSD and the corresponding inverse PSD in the same poses (30, 45 and 60 degree of one rotated joint ) are illustrated respectively in Figure 5 . We can see that in forward case, the direction of deformed vertex always keeps the same with the example cylinder ( figure 4 ). For inverse PSD however, that direction is changed along with the rotation of the joint. The case described above is quite common in practice when animating shoulder, elbow, knee, hip-bone, neck, etc. All these parts would rotate from the rest pose with some angle to other poses. On the other hand, as a matter of experience, PSD is supposed to be a method as “local” correction, which means pose space should not be extended to a whole space that has to incorporate all influenced objects. Otherwise, large amount of unnecessary works of building examples will be required, and the distance between different poses is also meaningless. For example how to measure the distance between differing poses such as “lying down” and “pitching”?
Figure 2: (a). rest pose; (b). example pose with rotation of 90 degrees
Figure 3: deformation direction
5
Figure 4: A simple test case: two example poses with one vertex sculpted
<H1>5 A Unified Framework for Inverse Skinning Model</H1>
The above discussions assume that the basic skinning algorithm is SSD, but in many circumstances, other deformation schemes will be adopted
[ 9 ], [ 10 ], most of which have been implemented in most animation packages. Therefore we propose a unified framework in which no explicit inverse operation is necessitated. Given a basic skinning method supported by animation packages we can deform the original character model from rest pose to another specific pose. In the more general case, we need to replace SSD with SKINNING in the inverse skinning algorithm as such for equation 4: v i t = v i + d i = SKINNING i (v r ) + d i But this time we don’t implement the inverse of SKINNING as in equation 5. To find delta d i in the rest pose: v i = SKINNING i (v r ) + d i = SKINNING i (v r + d i ) we can setup a minimization problem to minimize the function:
6
Figure 5: Interpolated Poses: the rotated joint is at 30, 45 and 60 degree. left column: PSD; right column: inverse PSD.
9
f (d i ) = v i − SKINNING i (v r + d i ) 2
This function can be given to Powell’s method to find d i at the minimum of f (d ). For each example pose P i , we have a d i , then we can apply radial basis function to d i (i = 0 . . . n − 1) in pose space to obtain ω i (i = 0 . . . n − 1). In synthesis phase, a d x in an intermediate pose x can be computed by equation 3 based on its position x in pose space d x = ∑ n−1 i=0 ω i φ (||x − x i ||). Then we have the final synthesis result:
10
v x = SKINNING(v r + d x )
<H2>5.1 Direction Set (Powell’s) Methods for Minimization</H2>
For a minimization problem, there are many candidate algorithms according to the form of function, knowledge of the derivative, computing capacity, and requirements for the rate of convergence, etc. In our situation, the function form is not explicit, and the computing burden increases with the number of example poses increases. We will adopt Powell’s method as the solution to this minimization problem. One advantage of Powell’s classic method is that it does not need explicit computation of the function’s gradient
[ 12 ]. Because we are treating the skinning operations as a “black box”, their gradient is not available, so Powell’s method is suitable. Minimizing the function f (d ) in a particular direction is actually minimization problem of one variable, which is also called line minimization. Powell’s idea is trying to find each minimum of function f (d ) in different direction until f (d ) stops decreasing. How to choose the next direction is the main concern of Powell’s method, and it has been proved that after repeated cycles of M line minimizations on conjugate directions, the optimization will in due course converge to the minimum [ 12 ]. M is the dimensionality of the function f ().
<H2>5.2 System Overview</H2>
Figure 6: System Pipeline for Unified Inverse Skinning.
We implement this unified approach as a Maya plug-in. In Maya, “tweaking” is a procedure adding delta values to original surface vertices before any deformations. It is actu- ally Maya’s form of rest-pose editing for their built-in deformation operators. As presented in Figure 6 , the whole system is divided into two phases. The first phase is to find each delta in the rest pose corresponding to each example pose. Basic skinning provided by Maya is called in the loop of minimization scheme. The output of the first phase, the delta in the rest pose, is input to into the second phase that is a linear system performing RBF interpolation to obtain the PSD weights. In the synthesis process, for an intermediate pose x, a delta d x ( or d x r ) is synthesized by equation 7. The final deformed vertex is computed by Maya skinning as in equation 10.
7
<H2>5.3 Singular Poses</H2>
If the SSD transformation in equation 1 is singular, some types of inverse PSD deformation will not be possible, because any component of the desired deformation that lies in the null space of the SSD matrix will be ignored. Although singular cases are rare (one example is a joint with 180 o rotation and equal 2 1 , 2 1 weights on the two joint frames, which is an unrealistic case of selfintersection), it is possible to handle these cases with a small rearrangement of the inverse PSD approach. We reformulate the problem as f (y i ) = v i + w i − SKINNING i (v r + d i ) 2 + λ w i 2 where y i is a concatenated vector y i = [d i , w i ] and λ is an arbitrary small number. The final synthesis is then v x = SKINNING(v r + d x ) + w x where w x is interpolated after SKINNING by applying the same RBF scheme as used for d x (thus, only minimal code changed are required). The idea here is that, since w i is being minimized, it will be generally be zero, and will be non-zero only if it is not possible to obtain the desired deformation v i using SKINNING i (v r + d i ). In the case where the SSD transform is nearsingular, the solved d i can be much large than other d k , which can result in poorly posed interpolation. To address this case, we further modify the objective function as
f (y i ) = v i +w i −SKINNING i (v r +d i ) 2 + λ w i 2 + μ d i 2 where 0.0001 is a sufficient value for both λ and μ .
Figure 7: toad: a sequence of animated frames. left: PSD; right: inverse PSD.
Figure 8: toad: closeup of circled part from figure 9 . left: PSD; right: inverse PSD.
<H1>6 Conclusions and Discussions</H1>
Inverse skinning integrates SSD and shape interpolation more firmly than its forward rival. We demonstrate the direction of deformed vertex in inverse skinning is linearly proportional to joint rotations in a simplified example, while the forward PSD does not incorporate the direction information. Therefore the inverse approach presents better performance and more consistent interpolation ( Figure 7 to Figure 10 ). By formulating the inverse process as a minimization problem we propose a unified model not only for SSD but also for other skinning schemes, into which shape interpolation can be incorporated. But the minimizing process will introduce more cost. This cost depends on the size of deformed character, parameters of minimization methods (Powell) such as convergence precision, and the number of example poses. In addition the cost of the animation software must be considered (for example, the Maya API implements a run-time type interpretation system on all operations). The cost of the inverse operation is not critical, however, since it is a one time “setup” cost, and the compute time is insignificant compared to the human time required to sculpt the desired deformations. Once the linear system is solved, the synthesis is potentially realtime since no extra computing is involved in this process compared to the forward PSD. We implement this unified example-based approach as a Maya plugin. It interoperates with their closed-source “Smooth Skinning” deformation.
8
<H1>Acknowledgements</H1>
Authors would like to thank artists from EggStory Creative Production. We also thank several of the reviewers for comments which improved this work
<H1>References</H1>
[1] J. P. Lewis, Matt Cordner, and Nickson Fong. Pose space deformation: a unified approach to shape interpolation and skeleton-driven deformation. In SIGGRAPH 2000: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 165– 172, New York, NY, USA, 2000. ACM Press/Addison-Wesley Publishing Co.
[2] Peter-Pike J. Sloan, III Charles F. Rose, and Michael F. Cohen. Shape by example. In SI3D 2001: Proceedings of the 2001 symposium on Interactive 3D graphics, pages 135–143, New York, NY, USA, 2001. ACM Press.
[3] Paul G. Kry, Doug L. James, and Dinesh K. Pai. Eigenskin: real time large deformation character skinning in hardware. In SCA 2002: Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation, pages 153–159, New York, NY, USA, 2002. ACM Press.
[4] Tsuneya Kurihara and Natsuki Miyata. Modeling deformable human hands from medical images. In SCA 2004: Proceedings of the 2004 ACM SIGGRAPH/Eurographics symposium on Computer animation, pages 355–363, New York, NY, USA, 2004. ACM Press.
[5] Doug L. James and Christopher D. Twigg. Skinning mesh animations. ACM Trans. Graph., 24(3):399–407, 2005.
[6] Xiaohuan Corina Wang and Cary Phillips. Multi-weight enveloping: least-squares approximation techniques for skin animation. In SCA 2002: Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation, pages 129–138, New York, NY, USA, 2002. ACM Press.
[7] Alex Mohr and Michael Gleicher. Building efficient, accurate character skins from examples. ACM Trans. Graph., 22(3):562– 568, 2003.
[8] Karan Singh and Evangelos Kokkevis. Skinning characters using surface oriented free-form deformations. In Graphics Interface, pages 35–42, 2000.
[9] Ron MacCracken and Kenneth I. Joy. Free-form deformations with lattices of arbitrary topology. In SIGGRAPH 96: Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, pages 181–188, New York, NY, USA, 1996. ACM Press.
[10] Karan Singh and Eugene Fiume. Wires: a geometric deformation technique. In SIGGRAPH 98: Proceedings of the 25th annual conference on Computer graphics and interactive techniques, pages 405– 414, New York, NY, USA, 1998. ACM Press.
[11] N. Magnenat-Thalmann, R. Laperrire, and D. Thalmann. Joint-dependent local deformations for hand animation and object grasping. In Proceedings on Graphics interface ’88, pages 26–33, Toronto, Ont., Canada, Canada, 1988. Canadian Information Processing Society.
[12] William H. Press, William T. Vetterling, Saul A. Teukolsky, and Brian P. Flannery. Numerical Recipes in C++: the art of scientific computing. 2002.
9
Figure 9: PSD vs. Inverse PSD: we have two examples, one in rest pose and the other is sculpted when the thumb is rotated down 66.8 degrees. SSD shows an obvious collapse. The intermediate poses from SSD, PSD and inverse PSD are shown in the second (45 degrees) and third (60 degrees) row respectively.
examples
interpolated models computed by Powell optimization approach
Figure 10: human arm
10
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"text": "['this category can generate more believable animation effects compared to its geometric counterpart']",
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"text": "['Pose Space Deformation']",
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"text": "['combines shape blending and Skeleton Subspace Deformation by formulating a scattered data interpolation problem over sculpted (or otherwise obtained) example poses']",
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"text": "['[ 1 ]']",
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"text": "['Related research efforts have improved the speed and power of example-based skinning']",
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"text": "['incorporate linear elements into RBF to produce constant changes between examples']",
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"text": "['[ 2 ]']",
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"id": "T31"
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7988
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"text": "['precompute principal components of the deformation influences for individual kinematic joints instead of storing displacements for key poses']",
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"id": "T32"
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{
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"text": "['enabling realtime rendering large nonlinear finite element models of human hands']",
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"text": "['[ 3 ]']",
"type": "data",
"id": "T34"
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8086,
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"text": "['introduce weighted pose space deformation for deforming realistic models of human hand']",
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"text": "['[ 4 ]']",
"type": "data",
"id": "T36"
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"text": "['these example-based approaches can be considered as non-parametric skin deformation methods']",
"type": "background_claim",
"id": "T37"
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{
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],
"text": "['identifies statistically relevant bones and approximates bone transforms from example mesh animations']",
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"text": "['[ 5 ]']",
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"text": "['these have also seen some development in recent years']",
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"text": "['[ 6 ]']",
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],
"text": "['[ 7 ]']",
"type": "data",
"id": "T42"
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],
"text": "['are also common practices in the entertainment production']",
"type": "background_claim",
"id": "T43"
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"text": "['Skinning using free form lattices']",
"type": "background_claim",
"id": "T44"
},
{
"offsets": [
[
8802,
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],
"text": "['NURBS curves']",
"type": "background_claim",
"id": "T45"
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{
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[
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8821
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],
"text": "['[ 10 ]']",
"type": "data",
"id": "T46"
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],
"text": "['[ 8 ]']",
"type": "data",
"id": "T47"
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8793,
8798
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],
"text": "['[ 9 ]']",
"type": "data",
"id": "T48"
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"offsets": [
[
9263,
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],
"text": "['the inverse operation can improve the results']",
"type": "own_claim",
"id": "T49"
},
{
"offsets": [
[
9590,
9619
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],
"text": "['inverse operation is required']",
"type": "own_claim",
"id": "T50"
},
{
"offsets": [
[
10442,
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],
"text": "['SSD is widely applied to interactive applications such as games and virtual reality']",
"type": "background_claim",
"id": "T51"
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{
"offsets": [
[
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],
"text": "['it is implemented in most commercial animation packages']",
"type": "background_claim",
"id": "T52"
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{
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[
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10425
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],
"text": "['its simplicity']",
"type": "data",
"id": "T53"
},
{
"offsets": [
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10430,
10440
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],
"text": "['efficiency']",
"type": "data",
"id": "T54"
},
{
"offsets": [
[
12084,
12179
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],
"text": "['SSD is very popular in circumstances that require animating a number of characters in real time']",
"type": "background_claim",
"id": "T55"
},
{
"offsets": [
[
12014,
12082
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],
"text": "['vertex transformations can be easily implemented in the graphic card']",
"type": "data",
"id": "T56"
},
{
"offsets": [
[
12335,
12494
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],
"text": "['the domain of adjusting one vertex in this way is strictly limited to the linear subspace formed by the vertex as transformed by joints influencing this vertex']",
"type": "background_claim",
"id": "T57"
},
{
"offsets": [
[
12243,
12329
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],
"text": "['When a character goes wrong in some pose, animators can adjust joint influence weights']",
"type": "background_claim",
"id": "T58"
},
{
"offsets": [
[
12741,
12832
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],
"text": "['SSD cannot synthesize many parts of a character skin involving complicated joint structures']",
"type": "background_claim",
"id": "T59"
},
{
"offsets": [
[
12671,
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],
"text": "['deforming is limited to a linear subspace']",
"type": "background_claim",
"id": "T60"
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{
"offsets": [
[
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],
"text": "['[ 1 ]']",
"type": "data",
"id": "T61"
},
{
"offsets": [
[
12917,
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],
"text": "['a combination of SSD and shape blending providing nice solution to above mentioned problems']",
"type": "background_claim",
"id": "T62"
},
{
"offsets": [
[
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],
"text": "['[ 1 ]']",
"type": "data",
"id": "T63"
},
{
"offsets": [
[
14983,
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],
"text": "['PSD and improved example-based schemes have been discussed in many publications']",
"type": "background_claim",
"id": "T64"
},
{
"offsets": [
[
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],
"text": "['[ 2 ]']",
"type": "data",
"id": "T65"
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{
"offsets": [
[
15071,
15076
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],
"text": "['[ 3 ]']",
"type": "data",
"id": "T66"
},
{
"offsets": [
[
15078,
15083
]
],
"text": "['[ 4 ]']",
"type": "data",
"id": "T67"
},
{
"offsets": [
[
15086,
15151
]
],
"text": "['the reason why the inverse should be performed is still ambiguous']",
"type": "own_claim",
"id": "T68"
},
{
"offsets": [
[
15220,
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],
"text": "['inverting the SSD (and other deformations) in order to interpolate the examples in the rest pose is a the right choice']",
"type": "own_claim",
"id": "T69"
},
{
"offsets": [
[
15478,
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],
"text": "['it is an improvement']",
"type": "own_claim",
"id": "T70"
},
{
"offsets": [
[
15754,
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],
"text": "['the superiority of the inverse method']",
"type": "own_claim",
"id": "T71"
},
{
"offsets": [
[
16991,
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],
"text": "['SSD −1 simply is the inverse transformation matrix generated by SSD']",
"type": "own_claim",
"id": "T72"
},
{
"offsets": [
[
16963,
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],
"text": "['SSD is a 3D transformation']",
"type": "data",
"id": "T73"
},
{
"offsets": [
[
17950,
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]
],
"text": "['the inverse method is superior']",
"type": "own_claim",
"id": "T74"
},
{
"offsets": [
[
19098,
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],
"text": "['Taking the model in rest pose as an example is a common practice when applying shape interpolation']",
"type": "background_claim",
"id": "T75"
},
{
"offsets": [
[
19204,
19295
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],
"text": "['interpolating effects from other examples should not change the original model in rest pose']",
"type": "background_claim",
"id": "T76"
},
{
"offsets": [
[
19334,
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],
"text": "['we have: ω 1x = φ 11 −1 d 1x + φ 12 −1 d 2x = φ 12 −1 d 2x ω 2x = φ 21 −1 d 1x + φ 22 −1 d 2x = φ 22 −1 d 2x = d 2x ω 1y = φ 11 −1 d 1y + φ 12 −1 d 2y = φ 12 −1 d 2y ω 2y = φ 21 −1 d 1y + φ 22 −1 d 2y = φ 22 −1 d 2y = d 2y where φ i −1 j is the (i, j)th element of Φ −1']",
"type": "own_claim",
"id": "T77"
},
{
"offsets": [
[
20353,
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],
"text": "['we simplify SSD as a rotation transformation ignoring other issues such as accumulating effects from the first joint']",
"type": "own_claim",
"id": "T78"
},
{
"offsets": [
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],
"text": "['only the second joint is rotating']",
"type": "own_claim",
"id": "T79"
},
{
"offsets": [
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21667,
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],
"text": "['in forward case, the direction of deformed vertex always keeps the same with the example cylinder']",
"type": "own_claim",
"id": "T80"
},
{
"offsets": [
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],
"text": "['( figure 4 )']",
"type": "data",
"id": "T81"
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"offsets": [
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],
"text": "['For inverse PSD however, that direction is changed along with the rotation of the joint']",
"type": "own_claim",
"id": "T82"
},
{
"offsets": [
[
22106,
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],
"text": "['PSD is supposed to be a method as “local” correction']",
"type": "own_claim",
"id": "T83"
},
{
"offsets": [
[
22172,
22269
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],
"text": "['pose space should not be extended to a whole space that has to incorporate all influenced objects']",
"type": "own_claim",
"id": "T84"
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{
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],
"text": "['a matter of experience']",
"type": "data",
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],
"text": "['large amount of unnecessary works of building examples will be required']",
"type": "own_claim",
"id": "T86"
},
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"offsets": [
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22359,
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],
"text": "['the distance between different poses is also meaningless']",
"type": "own_claim",
"id": "T87"
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],
"text": "['most of which have been implemented in most animation packages']",
"type": "background_claim",
"id": "T88"
},
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],
"text": "['in many circumstances, other deformation schemes will be adopted']",
"type": "background_claim",
"id": "T89"
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{
"offsets": [
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],
"text": "['[ 10 ]']",
"type": "data",
"id": "T90"
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],
"text": "['[ 9 ]']",
"type": "data",
"id": "T91"
},
{
"offsets": [
[
23289,
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],
"text": "['we can deform the original character model from rest pose to another specific pose']",
"type": "own_claim",
"id": "T92"
},
{
"offsets": [
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25039,
25152
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],
"text": "['One advantage of Powell’s classic method is that it does not need explicit computation of the function’s gradient']",
"type": "background_claim",
"id": "T93"
},
{
"offsets": [
[
25154,
25160
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],
"text": "['[ 12 ]']",
"type": "data",
"id": "T94"
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"offsets": [
[
25264,
25291
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],
"text": "['Powell’s method is suitable']",
"type": "own_claim",
"id": "T95"
},
{
"offsets": [
[
25170,
25259
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],
"text": "['we are treating the skinning operations as a “black box”, their gradient is not available']",
"type": "data",
"id": "T96"
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"offsets": [
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25659,
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],
"text": "['after repeated cycles of M line minimizations on conjugate directions, the optimization will in due course converge to the minimum']",
"type": "background_claim",
"id": "T97"
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{
"offsets": [
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25790,
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],
"text": "['[ 12 ]']",
"type": "data",
"id": "T98"
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"offsets": [
[
26918,
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],
"text": "['If the SSD transformation in equation 1 is singular, some types of inverse PSD deformation will not be possible']",
"type": "own_claim",
"id": "T99"
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{
"offsets": [
[
27039,
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],
"text": "['any component of the desired deformation that lies in the null space of the SSD matrix will be ignored']",
"type": "data",
"id": "T100"
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"offsets": [
[
27152,
27175
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],
"text": "['singular cases are rare']",
"type": "own_claim",
"id": "T101"
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{
"offsets": [
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27324,
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],
"text": "['it is possible to handle these cases with a small rearrangement of the inverse PSD approach']",
"type": "own_claim",
"id": "T102"
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{
"offsets": [
[
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],
"text": "['one example is a joint with 180 o rotation and equal 2 1 , 2 1 weights on the two joint frames, which is an unrealistic case of selfintersection']",
"type": "data",
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"offsets": [
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],
"text": "['it will be generally be zero']",
"type": "own_claim",
"id": "T104"
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{
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],
"text": "['w i is being minimized']",
"type": "data",
"id": "T105"
},
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"offsets": [
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27881,
27992
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],
"text": "['will be non-zero only if it is not possible to obtain the desired deformation v i using SKINNING i (v r + d i )']",
"type": "own_claim",
"id": "T106"
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{
"offsets": [
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],
"text": "['Inverse skinning integrates SSD and shape interpolation more firmly than its forward rival.']",
"type": "own_claim",
"id": "T107"
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"offsets": [
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],
"text": "['the inverse approach presents better performance and more consistent interpolation']",
"type": "own_claim",
"id": "T108"
},
{
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],
"text": "['Figure 7 to Figure 10']",
"type": "data",
"id": "T109"
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{
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],
"text": "['the direction of deformed vertex in inverse skinning is linearly proportional to joint rotations in a simplified example, while the forward PSD does not incorporate the direction information']",
"type": "own_claim",
"id": "T110"
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],
"text": "['the minimizing process will introduce more cost']",
"type": "own_claim",
"id": "T111"
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],
"text": "['The cost of the inverse operation is not critical']",
"type": "own_claim",
"id": "T112"
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"offsets": [
[
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],
"text": "['it is a one time “setup” cost']",
"type": "own_claim",
"id": "T113"
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{
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[
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],
"text": "['the compute time is insignificant compared to the human time required to sculpt the desired deformations']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
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],
"text": "['a frame goes wrong']",
"type": "data",
"id": "T115"
},
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],
"text": "['physical principles']",
"type": "data",
"id": "T116"
},
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"offsets": [
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],
"text": "['the model is very detailed']",
"type": "data",
"id": "T117"
},
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"offsets": [
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2758,
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],
"text": "['playback of animation becomes quite heavy and time consuming']",
"type": "data",
"id": "T118"
},
{
"offsets": [
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],
"text": "['For those applications that require visual fidelity']",
"type": "background_claim",
"id": "T119"
},
{
"offsets": [
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],
"text": "['SSD serves only as a basic framework']",
"type": "background_claim",
"id": "T120"
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{
"offsets": [
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],
"text": "['PSD smoothly interpolates these meshes in pose space and produces visually attractive animations']",
"type": "background_claim",
"id": "T121"
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{
"offsets": [
[
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],
"text": "['Besides SSD, other skinning approaches']",
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"id": "T122"
},
{
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[
5041,
5060
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],
"text": "['can also be applied']",
"type": "background_claim",
"id": "T123"
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],
"text": "['rigid skinning']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
5014,
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],
"text": "['Free Form Deformation']",
"type": "data",
"id": "T125"
},
{
"offsets": [
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],
"text": "['physical modelling and animation is another field providing realistic character simulations']",
"type": "background_claim",
"id": "T126"
},
{
"offsets": [
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],
"text": "['Simpler parametric skinning approaches (of which SSD is the prototype) have a fixed number of parameters']",
"type": "background_claim",
"id": "T127"
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{
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],
"text": "['the direction of deformed vertices from inverse skinning is a linear function of joint rotation']",
"type": "own_claim",
"id": "T128"
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],
"text": "['in the forward approach, that direction is kept as a constant']",
"type": "own_claim",
"id": "T129"
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{
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],
"text": "['propose a unified framework which can be implemented on high-end commercial packages while allowing any proprietary skinning operators to be incorporated']",
"type": "own_claim",
"id": "T130"
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{
"offsets": [
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10276
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],
"text": "['Skeleton Subspace Deformation (SSD) is a basic algorithm that is used to define how the character surface deforms following movements of its underlying skeletons']",
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"id": "T131"
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{
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],
"text": "['[ 11 ]']",
"type": "data",
"id": "T132"
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],
"text": "['SSD is adopted to define this relation']",
"type": "data",
"id": "T133"
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[
10989,
11158
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],
"text": "['each vertex or control point of the character surface is provided with a list of joints, that will influence it, along with the weight indicating the amount of influence']",
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"id": "T134"
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{
"offsets": [
[
11192,
11326
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],
"text": "['the position of a vertex in the animated pose is the result of weighted linear blending of its transformation by each associated joint']",
"type": "background_claim",
"id": "T135"
},
{
"offsets": [
[
11165,
11190
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],
"text": "['the character is animated']",
"type": "data",
"id": "T136"
},
{
"offsets": [
[
12572,
12615
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],
"text": "['The famous SSD problem of “collapsed elbow”']",
"type": "background_claim",
"id": "T137"
},
{
"offsets": [
[
13915,
13942
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],
"text": "['sculpted in N example poses']",
"type": "data",
"id": "T138"
},
{
"offsets": [
[
13900,
13910
]
],
"text": "['a vertex v']",
"type": "data",
"id": "T139"
},
{
"offsets": [
[
13949,
14047
]
],
"text": "['there are N delta d i , i = 0, . . . , N − 1 corresponding to each pose x i , i = 0, . . . , N − 1']",
"type": "own_claim",
"id": "T140"
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A18 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A18_M10_On-line_Motion_Retargetting_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1002/1099-1778(200012)11:5<223::aid-vis236>3.3.co;2-x
<Title>On-line Motion Retargetting</Title>
Kwang-Jin Choi
Hyeong-Seok Ko SNU Human Animation Center School of Electrical Engineering Seoul National University E-mail: f g
kjchoi,ko @graphics.snu.ac.kr
<Abstract>This paper presents a method to retarget the motion of a character to another in real-time. The technique is based on inverse rate control, which computes the changes in joint angles corresponding to the changes in end-effector position. While tracking the multiple end-effector trajectories of the original subject or character, our on-line motion retargetting also minimizes the joint angle differences by exploiting the kinematic redundancies of the animated model. This method can generalize a captured motion for another anthropometry to perform slightly different motion, while preserving the original motion characteristics. Because the above is done in on-line, a real-time performance can be mapped to other characters. Moreover, if the method is used interactively during motion capture session, the feedback of retargetted motion on the screen provides more chances to get satisfactory results. As a by-product, our algorithm can be used to reduce measurement errors in restoring captured motion. The data enhancement improves the accuracy in both joint angles and end-effector positions. Experiments prove that our retargetting algorithm preserves the high frequency details of the original motion quite accurately.</Abstract>
<H1>1. Introduction</H1>
The dream of animating complex living creatures with pure computation (such as inverse kinematics, or dynamic control) proved impractical. Even though creatures are not free from physics, their motion is not a direct consequence of physics. Dynamic control can provide solutions based on simplified assumptions about human motion. However, the result tends to look quite mechanical. If a high quality character animation has to be produced during a short period of time, motion capture might be a most reasonable choice these days. The captured data itself is for a specific person in performing specific motion. Whenever the data needs to be reused, it has to be retar- getted to account for the differences in the anthropometry and motion. Therefore motion retargetting is emerging as an important technique in recent character animation. If the original motion characteristics are severely lost during motion retargetting, the technique loses its merit over the above pure computation approaches. The problem we try to solve in this paper can be summarized as: ( 1 ) finding in real-time the motion retargetted to a new character that has different anthropometric proportions, and ( 2 ) at the same time, preserving the features of the original motion during the retargetting. ( 3 ) As a by-product, it is possible to use the above retargetting algorithm for enhancing motion capture data so that the errors in joint angles and end-effector positions are reduced. On-line motion retargetting presented in this paper is based on inverse rate control [17] (or resolved motion rate control), which is a way to implement inverse kinematics based on Jacobian. It computes the changes in joint angles corresponding to the changes in end-effector position. While tracking the multiple end-effector trajectories of the original subject or character, our on-line motion retargetting imitates the joint motion of the original character by exploiting the kinematic redundancies of the animated model. Moreover, jerky motion is prevented since the next configuration is dependent on the previous configuration in inverse rate control. As will be shown in later experiments, the high frequency details of the original motion, which carries important characteristics of the motion, are also well preserved by our algorithm. Figure 1 shows the on-line retargetting process schematically. The input is a stream of joint angle vectors src of the measured subject in the source motion and another stream of the reference (or desired) end-effector positions x 1 of the animated character at discrete time ticks. The output is a stream of joint angle vectors des of the animated character during the destination motion at corresponding time ticks. The filter in the figure is causal. i.e., the output is calculated based on the current and immediately previous input values, but does not dependent on the future input. It explains why it is called on-line. If the retargetting can be done in on-line, real-time performance can be mapped to another character, or the feedback of the retargetted animation can facilitate motion capture session so that satisfactory results can be obtained with fewer trials. Since the memory required for on-line retargetting does not increase with time, our algorithm can handle an infinitely long sequence of motion. The primary goal of our on-line motion retargetting (OMR) is to track the given reference end-effector trajectory of joint x 1 angle t , and trajectory the secondary src t goal as much is to as imitate possible. the pattern Therefore src t carries the content to be retargetted, and x 1 t carries the variations needed during the retargetting. For example, when there is a bat-swing motion, we can obtain different swing motions aiming at different hit positions by specifying x 1 t appropriately. As a by-product, our OMR algorithm can be used to reduce measurement errors in restoring captured motion. For this data enhancement, we use captured position data for x 1 t even though it can be calculated from src t by forward kinematic positioning, to recover from possible measurement errors in joint angles. If the positioning of the pose is done from joint angles alone, the errors can accumulate as the forward kinematic positioning propagates toward the end-effector. The end-effector position data x 1 t can be utilized to limit the above error accumulation within a certain range. In Section 2 recent work related with motion retargetting is reviewed. Section 3 discusses inverse rate control and its implementation, and Section 4 presents the formulation of motion retagetting problem with inverse rate control. Section 5 discusses how our OMR can be used to reduce measurement errors in restoring captured motion. Section 6 shows the results obtained by our technique, and finally the conclusion follows.
src θ (t) Retarget Filter θ des (t) x (t) 1 original character new character
Figure 1. On-line Motion Retargetting Filter
<H1>2. Related Work</H1>
Several techniques have been proposed for reusing or altering existing motions. Witkin et al’s motion warping [19] and Bruderlin et al’s motion displacement mapping [4] discuss motion editing technique based on direct manipulation of data curves. Bruderlin et al [4] and Unuma et al [11] utilized signal processing techniques for motion editing. Wiley et al [18] proposed the interpolation synthesis algorithm that chooses and combines most relevant motions from the database to produce animation with a specific positional goals. Though some of the techniques above can be used for motion retargetting problem with user’s extra efforts, they don’t specifically address the motion retargetting problem. In [3], Boulic and Thalmann presented the combined direct and inverse kinematic control technique for motion editing. The concept called coach-trainee metaphor is very similar to the motion retargetting problem formulation. The fundamental idea is to consider the joint motion of coach as a reference input to trainee motion for the secondary task exploiting the null space of the Jacobian when solving inverse kinematics. The inverse kinematic constraint is given by half-space such as plane, cylinder, or sphere. Although their approach shares the technique of utilizing the redundancy in inverse kinematic control with ours, the problem they solved is not the motion retargetting but is rather a motion correction technique since the end-effector constraint specified by half-spaces is not general to solve the motion retargetting problem. A method which is devoted to the motion retargetting problem was proposed by Gleicher [6]. He used the spacetime constraint method that minimizes an objective function g x subject to the constrains of the form f x = c . The constraints can represent the ranges of parameters, or various kinds of spatial-temporal relationship among the body segments and the environment. The objective function is the time integral of the signal displacement between the source and destination motion. i.e.
( 1 )
Since the whole interval has to be integrated to find the optimal solution, the method is intrinsically an off-line process. The global method as above can correlate frames back and forth within the whole duration and thus generally produces more smooth results compared to the local method such as our OMR technique. But the look-ahead property of the global method is effective when the constraints are imposed only at sparse key frames. Our OMR takes continuous trajectories of constraints as input, so that it produces globally coherent motion in spite of local computation. The global coherence is also achieved from the effort to exploit the redundancy of the system in resembling the original motion. The local coherence of the motion comes from the fact that the adjacent frames are inter-related by the inverse rate control. Therefore, without significant degradation of quality, our algorithm provides much faster and interactive way of motion retargetting. Bindiganavale and Badler [2] presented a method to abstract and edit motion capture data. Their algorithm detects significant events and abstracts constraints from the motion, and imposes those constraints to other character. The constraints abstracted from the motion is solved by inverse kinematics at significant frames and then those frames are interpolated. Although the constraint abstraction is an improvement compared to the other techniques, the interpolation technique might fail to preserve the high frequency details if the key frames are sparsely spaced.
<H1>3. Inverse Rate Control</H1>
In an articulated figure, the joint configuration can be related to the position and orientation of the end effector by a kinematic mapping f : ! X , which maps the joint space to Cartesian space X . The mapping is usually a nonlinear equation given by
( 2 )
dimensional where x 1 is vector. an m -dimensional m =3 if we vector are interested and is only an n in position, or m =6 if we are interested in both position and orientation of the end-effector. m can be 12 or 18 if we want to impose multiple end-effector constraints. If we differentiate the above equation, we obtain
( 3 )
where x _ 1 and _ denote the end-effector positional velocity and joint angle velocity, respectively. J 1 is called Jacobian and is an m n matrix that linearly relates the end effector velocity and joint angle velocity at the moment. Given the end effector velocity, we can get joint angle velocity by inverting the Jacobian. However, most articulated figures have kinematic redundancy and thus the inverse of Jacobian is not unique. (more specifically, m n ) Therefore there are an infinite number of possible solutions that satisfy Equation 3. Some criteria can be specified to pick one that best fits for our purpose. One of popular criteria is called the minimal norm solution
( 4 )
where J 1 + = J 1 T J 1 J 1 T , 1 is the pseudo inverse 1 of J 1 . Equation 4 gives a particular solution, and can be generalized to include all possible solutions by adding a term from the null space of J 1 as in
( 5 ) 1 We , 1 actually , to get used consistent damped motion least squares near the solution, singularities [12, 1 16].
where projects y y is onto an arbitrary the null n space -dimensional of J 1 . This vector. null I space , J 1 + term J 1 corresponds to the redundant degrees of freedom, and can be utilized to perform secondary priority tasks [7, 10 , 20 , 21]. For example, consider the following task set.
primary task: thus secondary task: thus If the equation ( 6 )
is used for Equation 5, then the primary goal is accurately achieved in the case of continuous domain 2 and the secondary goal is also achieved in an optimal sense. Another way to utilize the redundancy of the system is to set y to the gradient , r g of a criterion function g in Equation 5. Then integration of Equation 5 tries to reduce the value of g while the end-effector is made to track the given trajectory [9].
<H2>3.1. Closed-loop Inverse Rate Control and Its Discrete Implementation</H2>
To control the articulated figure to follow given reference end-effector to give the value trajectory of t x 1 (Equation t , J 1 + x _ 1 4). t should But this be open-loop integrated fashion of integration can not eliminate the initial tracking error e 1 t 0 = x 1 t 0 , x des 1 t 0 , where x des 1 t is the resulting end-effector position at time t in the destination motion. Balestrino et al [1], Tsai and Orin [15], and Sciavicco and Siciliano [14, 13] proposed the closed-loop inverse kinematics (CLIK) scheme based on Jacobian pseudoinverse. CLIK leads to zero steady state error which means that the error is exponentially convergent to zero for a fixed target position. For CLIK, Equation 4 has to be modified to
( 7 )
where K 1 is a positive definite matrix we can provide. It can be easily shown that as the smallest eigen value of K 1 becomes large, the convergence rate increases accordingly since the error dynamics is governed by the relation e 1 + K 1 e 1 =0 . In a continuous time formulation such as Equation 7, a large value of K 1 is desirable. However, as will be shown below an arbitrarily large K 1 doesn’t guarantee convergence in implementing the discrete version of Equation 7.
2 In the discrete domain, since the Jacobian is a linear estimation of a non-linear function, the integration can produce errors, especially when the step size is large.
The difference equation corresponding to Equation 7 is given by
( 8 )
where and
( 9 )
Here u i is the value of the function u t at the discrete time t i . Equation 8 is implicit: to compute n we need to know in turn the requires value of the e 1 value n . But of computing x des 1 n , which e 1 n is by not Equation available 9 until n is known. Therefore, e 1 n should be estimated. Below we show that any estimation based on the old values (at n , 1 , n , 2 , : : : ) requires K 1 to be I for the best tracking performance. Suppose that we estimated e 1 n simply with e 1 n , 1 . Then Equation 8 becomes
( 10 ) To obtain the error equation, multiply at both sides of Equation 10 and we obtain ( 11 )
Assuming that the step size is small enough, we can rewrite the above equation as
( 12 ) With the relations = x 1 n , x des 1 n and e 1 n , 1 = , Equation 12 can be rewritten into ( 13 )
should Equation be within 13 reveals the interval that the , 1 eigen ; 1 to values prevent of the I , error K 1 from growing indefinitely. Even with K 1 = I the stability that is J 1 not n , guaranteed 1 n is due approximated to the nonlinearity as x des 1 of n f 1 in . Equa(Note tion 12.) But in practice, we found that instability rarely occurs at a usual sampling rate ( 30 60 Hz) in dealing with human motion. If we include the secondary task x 2 = f 2 , the openloop control law takes the form
( 14 )
To prevent from possible divergence due to errors, however, a closed-loop version needs to be considered again. The CLIK scheme including the secondary task based on Jacobian transpose is given by
( 15 )
where e 2 = x 2 , x des 2 . x des 2 is the actual result of the secondary task that tries to realize the given goal x 2 . It is proven that e 2 is ultimately bounded within a certain range and the tracking error for the primary task is not affected by the second term of Equation 15 [13]. But again, arbitrarily large K 2 is not allowed in discrete implementation. With the estimation of e 2 n based on old values, a K reasonable 2 The = I final is given choice CLIK by for scheme K 2 is in I . discrete domain with K 1 =
( 16 )
ẽ 1 n and ẽ 2 n in Equation 16 are the estimations of e 1 n and e 2 n , respectively. Although any estimation scheme based on the old values can not completely eliminate the era ror higher caused order by the estimation nonlinearity for e 1 of n f 1 than , it can e 1 n be , reduced 1 , and with consequently can give better tracking performance. We found that the estimation rule described below gives satisfactory results.
Step 1 : Step 2 : Step 3 :
The above procedures complete the discrete implementation of the CLIK algorithm with a secondary task.
<H2>3.2. Inverse Rate Control with Multiple Endeffector Trajectories</H2>
In this section, we discuss how to extend inverse rate control to track multiple end-effector trajectories. The serial chain is not suitable for modeling creatures since underlying articulated structures contain branches. An illustrative example is taken from human upper body, and is shown in Figure 2 . The model consists of spine and two arms. The waist is the root of the kinematic tree structure, and the two arms are branching at the top of the spine. If both hands have their own goals to reach, and if inverse kinematics is solved for these cases separately, then the spine angles will differ in the solutions. In [22], Zhao and Balder solved this problem by a weighted sum of independently obtained gradients, each of which directs its corresponding end-effector to a goal position. However, the effects of different weights are not easily predictable. Depending on the weight assignment, their algorithm can fail to find an inverse kinematic solution even if all the constraints can be actually met. Intrinsically, the problem of finding inverse kinematic solution of multiple constraints doesn’t require any weight or priority assignment: if all the end-effector constraints can be met, then it should be possible without considering weights or assigning priorities to each end-effector constraint. Compared with Zhao and Badler’s algorithm, Jacobian based inverse rate control gives a quite simple and intuitive solution to the problem. The only thing we have to do in order to incorporate multiple end-effector constraints is concatenating the end-effector vectors and composing the Jacobian appropriately. In the above example, the end effector vector J 1 becomes x 1 should 12 be 33 12 -dimensional matrix (two end-effector vector and the constraints Jacobian with six DOFs for each end-effector, and eleven joints with three DOFs for each joint). Of course, the Jacobian will have many zeroes where the joint angle and the end-effector have no relation such as left elbow joint and right hand. In inverse rate control, the above conflict of the spine angles is resolved during the computation of the pseudo inverse of the Jacobian.
joint9 joint6 joint4 joint5 joint10 joint7 joint3 joint2 joint11 joint8 joint1 left hand(end-effector1) right hand(end-effector2) waist(root)
Figure 2. Kinematic structure of human upper body
<H1>4. Motion Retargetting with Task Priority Strategy</H1>
In general, we can formulate the motion retargetting problem with the following task set, and can solve for des . primary task: x 1 = f 1 des ( 17 ) secondary task: x 2 = f 2 des ( 18 ) x 1 in the above is the desired end-effector trajectories which can be taken from the source character (and then be modified for necessary variations) or can be provided by the user. According to Equation 6, the open-loop control law for these tasks is given by
f ( . ) 2 − x 2 + + K 2 J 2 T I-J 1 + J 1 + θ . des θ des x . 1 + + Integrator x 1 + − + K 1 + + + J 1 + f ( . ) 1
Figure 3. Closed-loop control scheme with a secondary task
( 19 )
and the block diagram of its closed-loop version is shown in Figure 3 . Since joint angle trajectories contain important characteristics of a motion, and since the end-effector movements are already tracked by the primary task, an obvious and useful choice for the secondary task might be to imitate the joint motion of the source character. i.e.
secondary task: ( 20 )
which is simply the case when src and the identity function are used for x 2 and f 2 , respectively, in Equation 18. Then Equation 19 becomes
( 21 )
The block diagram of the closed-loop control scheme with the secondary task of joint motion imitation is shown in Figure 4 . Reasonable choices for K 1 and K 2 are I ’s in discrete implementation as stated before. But K 1 , K 2 can be adjusted based on the dexterity measure to get consistent motion near the kinematic singularities. A popular dexterity measure is min = max , where min and max are the minimum and maximum, respectively, among the singular values of the Jacobian. In this case, smaller K 1 and K 2 should be used if the dexterity measure turns out to be small. The adaptive scheme can be also useful if we apply the OMR algorithm to motion transition. Smaller gain (e.g. K 1 = K 2 = 0 : 1 I ) will produce sluggish tracking, but produces smooth motion. Therefore, if the animated model switches to another motion and there exists a large discrepancy at the motion boundary, smooth transition can be obtained by adjusting the gain matrix K 1 and K 2 appropriately. stability mated As briefly by since J 1 + x mentioned _ the 1 . The non-linear system before, function can K become 1 = _ I is does unstable linearly not guarantee when approxix _ 1 gets very large, or the sampling rate is very low. Therefore another provision for enforcing stability might be to clamp the value that goes into the box of J 1 + in Figure 4 whenever it is over a certain threshold. The provision might be effective when there is an excessively large acceleration, or when the model is fully stretched and almost no manipulative redundancy is left in the system. (In dealing with the human motion, however, the above provision was almost never needed.)
− x θ . 1 src + + K 2 I-J 1 + J 1 + + + θ . des Integrator θ des x 1 + − + K 1 + + + J 1 + f ( . ) 1
Figure 4. Closed-loop control scheme with the secondary task of joint motion imitation
<H1>5. Motion Capture Data Enhancement</H1>
When we capture a motion, we often measure the joint angles and use forward kinematics to reconstruct the motion. But the method can introduce large end-effector position errors since the joint angle error near the base is amplified when it comes to the end-effector, and joint angle errors are accumulated as the forward kinematic positioning propagates toward the end-effector. Choi et al’s interpolation/regression method [5], applies inverse kinematics at sparse keyframes and the resulting joint angles are interpolated with cubic spline curves. The interpolation is combined with least square fitting so that the characteristics of the original joint angle data is preserved in the resulting motion. The OMR algorithm described in the previous section can be used to reduce measurement errors in restoring the captured motion. The new method is an improvement over the above interpolation/regression method in three aspects: ( 1 ) inverse kinematics is done at every frame, which promises much closer end-effector tracking, ( 2 ) the joint angle imitation is done by exploiting redundant degrees of freedom rather than depending on the least square fit, and ( 3 ) the high frequency component of the original motion is preserved much better in the new method. For the enhancement, we measure both joint angle and end-effector trajectories during the motion capture session. The measured trajectories are supplied to our motion retargetting algorithm: the end-effector trajectories are supplied for x 1 , and the joint angle trajectories are supplied for src . Of course, the destination character has to be same with the source character, if pure data enhancement needs to be done. As the retargetting progresses, src will be adjusted to maintaining des so that the the joint end-effector angle pattern constraint of src . x 1 is met while Compared to the forward kinematic motion reconstruction our OMR algorithm reduces end-effector errors remarkably. In general, our algorithm also reduces the errors in joint angle measurements. While the joint angle errors can accumulate in forward kinematic reconstruction, once it is processed by our OMR, the total amount of accumulated error is limited by the amount of end-effector position error. Moreover, the joint angle error due to the end-effector position error is distributed among all the joints. Therefore unless the amount of end-effector position error is excessively larger than that of joint angle errors, our OMR produces more accurate result than the unprocessed data. Note that the above does not mean the retargetting and data enhancement should be done separately. If a different destination character is used, the two things are actually achieved at the same time. This is especially useful when a real-time performance is retargetted.
<H1>6. Experiments</H1>
This section describes the results of two experiments. In the first experiment, we show a retargetting example in which our OMR is applied to retarget a walking motion, to demonstrate that our OMR based on inverse rate control is not inferior in the quality to the retargetting based on spacetime constraints. Major error analysis of the algorithm is given in this example. In the second experiment, we show the retargetting of bat-swing motion. The motion clips mentioned below are available at http://graphics.snu.ac.kr/demo/omr/omr.mov .
<H2>6.1. Retargetting of Walking Motion</H2>
In this experiment, the source motion (refer to the video clip #1) is a curved path walking motion which was procedurally generated by Ko’s locomotion algorithm [8]. The walker took 13 steps and produced a total of 390 frames. The kinematic structure of the characters used for walking motion is shown in Figure 5 . Since the lower body motion is far more important than the upper body motion in walking example, we retargetted only the lower body motion. As shown in Figure 5 the lower body consists of pelvis, upper leg, lower leg, foot, and toes, and they are connected at the hip, knee, ankle, and ball joints. The total degree of freedom of the lower body is 8 3 + 6 = 30 . (All the joints were modeled by 3-DOF joints, and the base has extra 6 DOFs.) The destination character was about 60% scaled down from the source character with non-uniform proportions. They are shown in Figure 6 , and the lower body dimensions are compared in Table 1 . In the retargetting, the secondary task was set to src = des . To specify the primary task, we set the toe-tip of the stance leg as the base and the toe-tip of the swing leg as the end-effector. The source character’s toe-tip trajectory was used for x 1 without any modification. Therefore the destination character had to take relatively bigger steps considering his body size. At the boundaries of steps the base and end-effector were switched. It implies that there can be discontinuities at the boundary if the tracking error is large. The retargetted motion with the above task set is shown in the video clip #2. The tracking error of the swing foot was negligible and thus the produced motion was smooth at the step boundaries. But the pelvis motion showed non-uniform speed along the direction of progression (anterior-posterior), which wasn’t observable in the source motion. So we constrained the transverse plane motion of the pelvis. i.e. the pelvis was designated as another end-effector, and the x; z component of the source character’s pelvis movement was tracked in the destination motion. (Note that the pelvis motion along y -axis should be adapted to account for the height difference). After adding the constraint, we could obtain a satisfactory result as shown in the video clips #3 and #4. Even with the extra constraints, the end-effector trajectories of the source and the destination made an accurate match. The comparison is shown in Figure 7 . The dotted curves for the source motion are not visible because they overlap exactly with the solid curves, the end-effector trajectories of the destination motion. To show the tracking error microscopically, the area indicated with a small box near the 150th frame in Figure 7 was magnified in Figure 8 . The trajectories in the figure show that the tracking error is kept small where the velocity is nearly constant, but the error increases when the velocity makes sudden changes. The maximum error (1.0464 cm) occurred at the 128th frame where the y -coordinate (height) of the toe-tip reached its peak acceleration and this error was reduced to a negligible level at around the 135th frame as the acceleration decreased. The step boundary was taken from low-acceleration points so that the base to end-effector switch makes a smooth transition. The joint angle trajectories of the left leg during the original and retargetted motion are plotted in Figure 10 . Only the angles around the sideways direction (medial-lateral) axes are presented in the graphs. The comparison shows that the amplitude of the hip angle is increased in the destination motion to cover the given step length with the relatively smaller body. Other than that the original joint angle pattern was quite well preserved. 3 At the end of every step, the ball joint of the source character showed an abrupt change from a large negative value to zero. It corresponds to the toe-off moment when the toes take off the ground. After the retargetting, the sharp corner of the trajectory was well preserved. In general, our OMR preserves the high frequency content of the motion quite well, since inverse rate control is directed by Jacobian values. Big mountains or valleys are never missed. To recover tiny fluctuations as well, however, a high sampling rate is needed to avoid aliasing. If the sharp corners are undesirable, they can be prevented by adjusting the gain matrix K 2 or clamping some of the control input as stated in Section 4. The adjustment of K 2 does not affect the end-effector tracking performance.
head neck clavicle upper arm torso lower_arm hand pelvis upper leg lower leg foot toes
Figure 5. Kinematic model used for walking motion
src character
des character
ratio
pelvis (width)
30.0
30.0
1.00
upper leg
46.0
26.0
0.57
lower leg
46.0
34.0
0.74
foot
16.0
16.0
1.00
toes
8.0
8.0
1.00
src character des character ratio pelvis (width) 30.0 30.0 1.00 upper leg 46.0 26.0 0.57 lower leg 46.0 34.0 0.74 foot 16.0 16.0 1.00 toes 8.0 8.0 1.00
Table 1. Comparison of lower body dimensions
Figure 6. Characters used for walking motion (left: source character, right: destination character)
400 source pelvis X 300 destination left−toetip X 200 pelvis pelvis Z Z cm 100 left−toetip Z left−toetip left−toetip Y Y 0 −100 −200 0 50 100 150 200 250 300 350 frame
Figure 7. Comparison of end-effector trajectories
10 8 6 4 2 cm 0 −2 −4 −6 −8 −10 120 125 130 135 140 145 150 frame
Figure 8. Comparison of end-effector trajectories (a magnified view)
Figure 9. Three characters used for retargetting the bat swing motion
<H2>6.2. Retargetting of Bat Swing Motion</H2>
In this experiment, actual performance of a bat swing motion was processed by our OMR to produce the destination motion of three different characters shown in Figure 9 . The anthropometry of Character B is about the average. Character A has a longer torso but shorter limbs than average, and Character C has a shorter torso but longer limbs. Their kinematic structures are same as Figure 5 except that the torso is segmented to 5 parts and the feet are excluded. To set the primary task, the base and end-effectors should be specified as before. In this experiment, the pelvis was chosen as the base and two hands were chosen as the endeffectors. Three 6-DOF sensors were used to capture those positions and orientations. The end-effector motion was directly supplied for x 1 t without any modification. Therefore Character A, for example, had to make a relatively
3 Note that zero error in tracking the joint angle trajectories is unachievable due to the anthropometric difference.
Graph 1 : Hip angle source 0.5 destination destination radian 0 −0.5 0 50 100 150 200 Graph 2 : Knee angle 1.5 1 radian 0.5 0 0 50 100 150 200 Graph 3 : Ankle angle 1 0.5 radian 0 −0.5 0 50 100 150 200 Graph 4 : Ball angle 0.5 0 radian −0.5 −1 0 50 100 150
Figure 10. Comparison of joint angle trajectories
lower swing. Since all the torso segments can not be measured due to the limited number of sensors, we measured only the topmost segment. Therefore, the five joints from the waist to the top-most torso segment had to be generated from the orientation difference between the pelvis and the top-most torso segment. For this, the measured orientation of the topmost torso segment was added to the primary task, and zero angles for the unmeasured joints were added to the secondary task, expecting the primary task can be met with minimal joint angles along the torso. The other sensors were used to measure the joint angles. (Because we had only 13 sensors available, we had to give up capturing the foot motion.) In this experiment only the upper body motion was adapted by OMR. The lower body motion was reconstructed by applying the measured joint angles directly. The retargetting of the source motion to Characters A, B, C are shown in the video clips #5 ̃6, #7 ̃8, and #9 ̃10, respectively. The small green boxes in the video indicate the position of the end-effectors and base. In the video we can observe that end-effector positions are accurately tracked. Since the body dimensions of Character B and the real performer are similar, the retargetted motion does not contain any noticeable difference from the source motion. In the case of Character A, however, we can see the waist is bent to lower the hit position, and the torso is shifted forward to account for the shorter arms. In the case of Character C, the torso is bent backward and makes a bigger twist to account for the longer arms and shorter torso. Snap shots were taken during the retargetted motions to clearly demonstrate the above adaptation for the anthropometric differences and shown in Figure 11 .
250 300 350 250 300 350 250 300 350 200 250 300 350 frame
Figure 11. Snap shots taken from the retargetted motion. Observe different adaptations to compensate the anthropometric differences
walking
bat swing
number of frames
390
136
total DOFs
30
42
constraint DOFs
8
9
elapsed time (sec)
1.219
0.984
frame rate (Hz)
300.7
138.2
walking bat swing number of frames 390 136 total DOFs 30 42 constraint DOFs 8 9 elapsed time (sec) 1.219 0.984 frame rate (Hz) 300.7 138.2
Table 2. Computational time spent for retargetting ( The frame rate does not include the time for visualization )
<H2>6.3. Computational time</H2>
Since we had no privilege to install our program to the platform equipped with a motion capture system, we had to emulate the real-time motion capture. That is, the motion data captured at 30Hz was fed to the OMR system with the same sampling rate using a timer. At this sampling rate, not a single frame was lost even with the visualization included. We used an Intergraph GX1 system (dual P-III 550, wildcat 4000) for the experiments. Table 2 describes the computational time purely spent for the retargetting procedure in each motion. The code was not fully optimized and so the performance can be potentially improved further. The slower rate of the bat swing motion is due to the bigger size of the Jacobian matrix compared to the walking motion (8 30 vs. 9 42). As shown in the table, the OMR is fast enough to process motion capture data collected at a usual sampling rate (30 90Hz) in real-time for the models of reasonable complexity.
<H1>7. Discussion & Conclusion</H1>
This paper presented the on-line motion retargetting technique based on inverse rate control. The method is an improvement over the off-line retargetting based on spacetime constraints since real-time performances can be retargetted without degradation of retargetting quality. The OMR technique greatly helps to get more satisfactory results in motion capturing with fewer trials by giving the real-time feedback to the performer. Furthermore, the captured data are enhanced in both end-effector positions and joint angles by going through our OMR filter. One minor unsolved problem is that there is no easy way to guarantee full-proof stability of the system due to the non-linearity. We observed that at a very low sampling rate, and if the model goes near the kinematic singularity and thus very little manipulative redundancy is left, then the system can became unstable. However, experiments proved that the system never become unstable at 30Hz or higher sampling rate. If the source motion is available only at a low sampling rate, two remedies are recommended: ( 1 ) by interpolating the source motion curves, first produce more samples, and then use them as the input to the OMR filter, or ( 2 ) scale down the end-effector trajectory to avoid the singular configuration, or use both of ( 1 ) and ( 2 ). The above remedies are for an excessively bad situation. Our on-line motion retargetting produces satisfactory results in retargetting most human or creature motion. If the technique is well utilized, it can be very useful to people in character animation and game industries.
<H1>Acknowledgment</H1>
This work was supported by Creative Research Initiatives of the Korean Ministry of Science and Technology. This work was partially supported by ASRI (Automation and Systems Research Institution), Seoul National University.
<H1>References</H1>
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"text": "['later experiments']",
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"text": "['Figure 1']",
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"text": "['the retargetting can be done in on-line']",
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"text": "['the feedback of the retargetted animation can facilitate motion capture session']",
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"text": "['satisfactory results can be obtained with fewer trials']",
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"text": "['our algorithm can handle an infinitely long sequence of motion']",
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"text": "['the memory required for on-line retargetting does not increase with time']",
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"text": "['The primary goal of our on-line motion retargetting']",
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"text": "['is to track the given reference end-effector trajectory of joint x 1 angle t']",
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"text": "['src t carries the content to be retargetted']",
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"text": "['x 1 t carries the variations needed during the retargetting']",
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"text": "['there is a bat-swing motion']",
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"text": "['we can obtain different swing motions aiming at different hit positions by specifying x 1 t appropriately']",
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"text": "['this data enhancement']",
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"text": "['to recover from possible measurement errors in joint angles']",
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"text": "['the errors can accumulate as the forward kinematic positioning propagates toward the end-effector']",
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"text": "['the positioning of the pose is done from joint angles alone']",
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"text": "['The end-effector position data x 1 t can be utilized to limit the above error accumulation within a certain range']",
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"text": "['motion warping']",
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"text": "['and']",
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"text": "['discuss motion editing technique based on direct manipulation of data curves']",
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"text": "['19']",
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"text": "['4']",
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"text": "['utilized signal processing techniques for motion editing']",
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"text": "['4']",
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"text": "['11']",
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"text": "['the interpolation synthesis algorithm that chooses and combines most relevant motions from the database to produce animation with a specific positional goals']",
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"text": "['18']",
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"text": "['some of the techniques above can be used for motion retargetting problem with user’s extra efforts']",
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"text": "['3']",
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"text": "['The concept called coach-trainee metaphor is very similar to the motion retargetting problem formulation']",
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"text": "['The fundamental idea is to consider the joint motion of coach as a reference input to trainee motion for the secondary task exploiting the null space of the Jacobian']",
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"text": "['solving inverse kinematics']",
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"text": "['and trajectory the secondary src t goal as much is to as imitate possible. the pattern']",
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"text": "['plane']",
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"text": "['cylinder']",
"type": "data",
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"text": "['sphere']",
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"text": "['their approach shares the technique of utilizing the redundancy in inverse kinematic control with ours']",
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8352,
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"text": "['the problem they solved is not the motion retargetting']",
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"text": "['is rather a motion correction technique']",
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8457,
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"text": "['the end-effector constraint specified by half-spaces is not general to solve the motion retargetting problem']",
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"text": "['A method which is devoted to the motion retargetting problem was proposed']",
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"text": "['6']",
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"text": "['He used the spacetime constraint method that minimizes an objective function g x subject to the constrains of the form f x = c']",
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"text": "['the whole interval has to be integrated to find the optimal solution']",
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"text": "['The global method as above can correlate frames back and forth within the whole duration']",
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"text": "['the look-ahead property of the global method is effective when the constraints are imposed only at sparse key frames']",
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"text": "['The global coherence is also achieved from the effort to exploit the redundancy of the system in resembling the original motion']",
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"text": "['The local coherence of the motion comes from the fact that the adjacent frames are inter-related by the inverse rate control']",
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"text": "['Their algorithm detects significant events and abstracts constraints from the motion, and imposes those constraints to other character']",
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"text": "['2']",
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"text": "['The constraints abstracted from the motion is solved by inverse kinematics at significant frames and then those frames are interpolated']",
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"text": "['the interpolation technique might fail to preserve the high frequency details']",
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"text": "['the key frames are sparsely spaced']",
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"text": "['dimensional where x 1 is vector']",
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"text": "['are interested']",
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"text": "['in position']",
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"text": "['m can be 12 or 18']",
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"text": "['we differentiate the above equation']",
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"text": "['we obtain']",
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"text": "['3']",
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"text": "['where x _ 1 and _ denote the end-effector positional velocity and joint angle velocity, respectively']",
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"text": "['there are an infinite number of possible solutions that satisfy Equation 3']",
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"text": "['Given the end effector velocity']",
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"text": "['Some criteria can be specified to pick one that best fits for our purpose']",
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"text": "['One of popular criteria is called the minimal norm solution']",
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"text": "['4']",
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"text": "['where J 1 + = J 1 T J 1 J 1 T , 1 is the pseudo inverse 1 of J 1']",
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"text": "['Equation 4 gives a particular solution']",
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"text": "['the equation']",
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"text": "['is used for Equation 5']",
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"text": "['To control the articulated figure to follow given reference end-effector to give the value trajectory of t x 1']",
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"text": "['t should']",
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"text": "['this be open-loop integrated fashion of integration can not eliminate the initial tracking error e 1 t 0 = x 1 t 0 , x des 1 t 0']",
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"text": "['13']",
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"text": "['14']",
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"text": "['1']",
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"text": "['CLIK']",
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"text": "['the error is exponentially convergent to zero for a fixed target position']",
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"text": "['Equation 4 has to be modified to']",
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"text": "['the smallest eigen value of K 1 becomes large']",
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"text": "['the convergence rate increases accordingly']",
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"text": "['the error dynamics is governed by the relation e 1 + K 1 e 1 =0']",
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"text": "['In a continuous time formulation']",
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"text": "['a large value of K 1 is desirable']",
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"text": "['Equation 7']",
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"text": "['an arbitrarily large K 1 doesn’t guarantee convergence in implementing the discrete version of Equation 7']",
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"text": "['Equation 8 is implicit']",
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"text": "['to compute n we need to know in turn the requires value of the e 1 value n']",
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"text": "['e 1 n should be estimated']",
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"text": "['in practice, we found that instability rarely occurs at a usual sampling rate']",
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"text": "['in dealing with human motion']",
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"text": "['Even with K 1 = I the stability that is J 1 not n , guaranteed 1 n is']",
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"text": "['approximated to the nonlinearity as x des 1 of n f 1 in']",
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"text": "['any estimation scheme based on the old values can not completely eliminate the era ror higher caused order by the estimation nonlinearity for e 1 of n f 1 than']",
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"text": "['consequently can give better tracking performance']",
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"text": "['the estimation rule described below gives satisfactory results']",
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"text": "['The above procedures complete the discrete implementation of the CLIK algorithm with a secondary task']",
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"text": "['The serial chain is not suitable for modeling creatures']",
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"text": "['underlying articulated structures contain branches']",
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"text": "['the spine angles will differ in the solutions']",
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"text": "['inverse kinematics is solved for these cases separately']",
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"text": "['both hands have their own goals to reach']",
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"text": "['solved this problem by a weighted sum of independently obtained gradients, each of which directs its corresponding end-effector to a goal position']",
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"text": "['22']",
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"text": "['the effects of different weights are not easily predictable']",
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18510,
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"text": "['Depending on the weight assignment, their algorithm can fail to find an inverse kinematic solution even if all the constraints can be actually met']",
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"text": "['Intrinsically, the problem of finding inverse kinematic solution of multiple constraints doesn’t require any weight or priority assignment']",
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"text": "['it should be possible without considering weights or assigning priorities to each end-effector constraint']",
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"text": "['all the end-effector constraints can be met']",
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"text": "['Compared with Zhao and Badler’s algorithm, Jacobian based inverse rate control gives a quite simple and intuitive solution to the problem']",
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"text": "['In inverse rate control, the above conflict of the spine angles is resolved during the computation of the pseudo inverse of the Jacobian']",
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"text": "['an obvious and useful choice for the secondary task might be to imitate the joint motion of the source character']",
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"text": "['joint angle trajectories contain important characteristics of a motion']",
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"text": "['the end-effector movements are already tracked by the primary task']",
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"text": "['( 20 )']",
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"text": "['A popular dexterity measure is min = max']",
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"text": "['The adaptive scheme can be also useful']",
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"text": "['we apply the OMR algorithm to motion transition']",
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"text": "['will produce sluggish tracking']",
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"text": "['smooth transition can be obtained by adjusting the gain matrix K 1 and K 2 appropriately']",
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"text": "['the animated model switches to another motion']",
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"text": "['produces smooth motion']",
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"text": "['there exists a large discrepancy at the motion boundary']",
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"text": "['K 1 = K 2 = 0 : 1 I']",
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"text": "['K']",
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"text": "['1 = _ I']",
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"text": "['does']",
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"text": "['not guarantee']",
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"text": "['stability']",
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"text": "['The non-linear']",
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"text": "['function']",
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"text": "['linearly']",
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"text": "['approxi']",
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"text": "['mated']",
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"text": "['by']",
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"text": "['J 1 + x']",
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"text": "['system']",
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"text": "['x _ 1 gets very large, or the sampling rate is very low']",
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"text": "['it is over a certain threshold']",
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22712
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"text": "['Figure 4']",
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"text": "['the model is fully stretched and almost no manipulative redundancy is left in the system']",
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"text": "['the joint angle error near the base is amplified when it comes to the end-effector']",
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"text": "['joint angle errors are accumulated as the forward kinematic positioning propagates toward the end-effector']",
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"text": "['The new method is an improvement over the above interpolation/regression method in three aspects']",
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"text": "['the high frequency component of the original motion is preserved much better in the new method']",
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"text": "['the joint angle imitation is done by exploiting redundant degrees of freedom rather than depending on the least square fit']",
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"text": "['pure data enhancement needs to be done']",
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"text": "['Compared to the forward kinematic motion reconstruction our OMR algorithm reduces end-effector errors remarkably']",
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"text": "['our algorithm also reduces the errors in joint angle measurements']",
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"text": "['the joint angle errors can accumulate in forward kinematic reconstruction']",
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"text": "['it is processed by our OMR']",
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"text": "['the joint angle error due to the end-effector position error is distributed among all the joints']",
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"text": "['our OMR produces more accurate result than the unprocessed data']",
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"text": "['unless the amount of end-effector position error is excessively larger than that of joint angle errors']",
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"text": "['the above does not mean the retargetting and data enhancement should be done separately']",
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"text": "['the two things are actually achieved at the same time']",
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"text": "['This is especially useful']",
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"text": "['a different destination character is used']",
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"text": "['a real-time performance is retargetted']",
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"text": "['our OMR based on inverse rate control is not inferior in the quality to the retargetting based on spacetime constraints']",
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"text": "['we show a retargetting example in which our OMR is applied to retarget a walking motion']",
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"text": "['the first experiment']",
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"text": "['there can be discontinuities at the boundary']",
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"text": "['the tracking error is large']",
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"text": "['At the boundaries of steps the base and end-effector were switched']",
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"text": "['The tracking error of the swing foot was negligible']",
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"text": "['the produced motion was smooth at the step boundaries']",
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"text": "['After adding the constraint, we could obtain a satisfactory result']",
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"text": "['video clips #3']",
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"text": "['#4']",
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"text": "['Even with the extra constraints, the end-effector trajectories of the source and the destination made an accurate match']",
"type": "own_claim",
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"text": "['Figure 7']",
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"text": "['Other than that the original joint angle pattern was quite well preserved']",
"type": "own_claim",
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"text": "['After the retargetting, the sharp corner of the trajectory was well preserved']",
"type": "own_claim",
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"text": "['our OMR preserves the high frequency content of the motion quite well']",
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"text": "['inverse rate control is directed by Jacobian values']",
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"text": "['the tracking error is kept small']",
"type": "own_claim",
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"text": "['the error increases']",
"type": "own_claim",
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"text": "['the velocity is nearly constant']",
"type": "data",
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"text": "['the velocity makes sudden changes']",
"type": "data",
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"text": "['Figure 8']",
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"text": "['Big mountains or valleys are never missed']",
"type": "own_claim",
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"text": "['a high sampling rate is needed to avoid aliasing']",
"type": "own_claim",
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"text": "['To recover tiny fluctuations as well']",
"type": "own_claim",
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"text": "['they can be prevented by adjusting the gain matrix K 2 or clamping some of the control input']",
"type": "own_claim",
"id": "T283"
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"text": "['Section 4']",
"type": "data",
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"text": "['The adjustment of K 2 does not affect the end-effector tracking performance']",
"type": "own_claim",
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"text": "['the sharp corners are undesirable']",
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"text": "['the retargetted motion does not contain any noticeable difference from the source motion']",
"type": "own_claim",
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"text": "['the body dimensions of Character B and the real performer are similar']",
"type": "data",
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"text": "['the performance can be potentially improved further']",
"type": "own_claim",
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"text": "['The code was not fully optimized']",
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"text": "['the OMR is fast enough to process motion capture data collected at a usual sampling rate']",
"type": "own_claim",
"id": "T291"
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"text": "['in real-time for the models of reasonable complexity']",
"type": "own_claim",
"id": "T292"
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"text": "['At this sampling rate, not a single frame was lost even with the visualization included']",
"type": "own_claim",
"id": "T293"
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{
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38787,
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"text": "['The slower rate of the bat swing motion is due to the bigger size of the Jacobian matrix compared to the walking motion']",
"type": "own_claim",
"id": "T294"
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38945
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"text": "['table']",
"type": "data",
"id": "T295"
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"text": "['The method is an improvement over the off-line retargetting based on spacetime constraints']",
"type": "own_claim",
"id": "T296"
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"text": "['real-time performances can be retargetted without degradation of retargetting quality']",
"type": "data",
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"text": "['The OMR technique greatly helps to get more satisfactory results in motion capturing with fewer trials by giving the real-time feedback to the performer']",
"type": "own_claim",
"id": "T298"
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{
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39624,
39734
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"text": "['the captured data are enhanced in both end-effector positions and joint angles by going through our OMR filter']",
"type": "own_claim",
"id": "T299"
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"text": "['One minor unsolved problem is that there is no easy way to guarantee full-proof stability of the system due to the non-linearity']",
"type": "own_claim",
"id": "T300"
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"offsets": [
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"text": "['the model goes near the kinematic singularity']",
"type": "data",
"id": "T301"
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"text": "['very little manipulative redundancy is left']",
"type": "own_claim",
"id": "T302"
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39883,
39910
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"text": "['at a very low sampling rate']",
"type": "data",
"id": "T303"
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"offsets": [
[
40024,
40054
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"text": "['the system can became unstable']",
"type": "own_claim",
"id": "T304"
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{
"offsets": [
[
40065,
40153
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"text": "['experiments proved that the system never become unstable at 30Hz or higher sampling rate']",
"type": "own_claim",
"id": "T305"
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{
"offsets": [
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40246
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"text": "['two remedies are recommended']",
"type": "own_claim",
"id": "T306"
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40158,
40216
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"text": "['the source motion is available only at a low sampling rate']",
"type": "data",
"id": "T307"
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40254,
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"text": "['by interpolating the source motion curves, first produce more samples, and then use them as the input to the OMR filter']",
"type": "own_claim",
"id": "T308"
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40458
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"text": "['scale down the end-effector trajectory to avoid the singular configuration']",
"type": "own_claim",
"id": "T309"
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"text": "['use both of ( 1 ) and ( 2 )']",
"type": "own_claim",
"id": "T310"
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"text": "['The above remedies are for an excessively bad situation']",
"type": "own_claim",
"id": "T311"
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"text": "['it can be very useful to people in character animation and game industries']",
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"text": "['Our on-line motion retargetting produces satisfactory results in retargetting most human or creature motion']",
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"id": "T313"
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"text": "['the technique is well utilized']",
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Volume 31 (2012), Number 2
EUROGRAPHICS 2012 / P. Cignoni, T. Ertl (Guest Editors)
<Title>Data-Driven Estimation of Cloth Simulation Models</Title>
E. Miguel
1
2
D. Bradley 2 B. Thomaszewski 2 B. Bickel 2 W. Matusik
2
M. A. Otaduy 1 S. Marschner
1. URJC Madrid 2. Disney Research Zurich
Figure 1: We capture deformation behaviors of cloth materials with a dedicated setup (column 1 from left). The measurement images (2) are reconstructed into 3D geometry (3) yielding dense deformation fields. We use this data to fit parameters and investigate approximation qualities of three common cloth models: springs (4), soft constraints (5), and the StVK model (6).
<Abstract>Progress in cloth simulation for computer animation and apparel design has led to a multitude of deformation models, each with its own way of relating geometry, deformation, and forces. As simulators improve, differences between these models become more important, but it is difficult to choose a model and a set of parameters to match a given real material simply by looking at simulation results. This paper provides measurement and fitting methods that allow nonlinear models to be fit to the observed deformation of a particular cloth sample. Unlike standard textile testing, our system measures complex 3D deformations of a sheet of cloth, not just one-dimensional force–displacement curves, so it works under a wider range of deformation conditions. The fitted models are then evaluated by comparison to measured deformations with motions very different from those used for fitting.</Abstract>
<H1>1. Introduction</H1>
Today’s cloth simulators for animation, visual effects, games, and apparel design can mimic real cloth to a high degree of fidelity. But to fully exploit their capabilities, the constitutive models for cloth deformation must be tuned with great care. During this tuning process it is difficult to tell which models and which parameters are giving results more like the real material. This paper aims to solve this problem by introducing new techniques to measure complete cloth behavior under controlled conditions and to estimate cloth deformation models from these measurements. Most methods for testing cloth move the sample into a state of near-uniform strain, exercising one or at most two components of strain at once: pure stretching, pure shearing, or pure bending. One or two forces are measured to quantify the cloth’s resistance to deformation, and the resulting forcedisplacement curves are valuable in studying the differences between materials. However, this approach has certain limitations. The inevitable deviations from uniform strain create modeling error that cannot be quantified without knowing the actual strain variation; and force-displacement curves can be used directly to tune a cloth model, but do not provide any way to validate the resulting fit. The contributions of this paper are, first, a new, general system for observing cloth properties that measures more complete data than previous work in cloth capture or textile testing, and second, a new method for fitting parametric models to this type of data. Finally we show results that illustrate the performance of several widely used cloth models. Our measurement system applies forces to a sample of cloth using actuators and force sensors that let us know the complete applied force, in 3D. The resulting deformation is tracked by a stereo computer vision system that captures the complete deformation, also in 3D. Having deformation and force information makes our data well suited to model validation—the experiment measures the complete answer that should be predicted by a cloth simulator. Also, we do not need uniform strain, and in this paper we illustrate a range of tests, some that mimic traditional tests and some with more complex deformations. Our approach to model estimation is to numerically optimize nonlinear stress-strain curves to minimize errors in force and position compared to the measurement. We have designed a general fitting method, suited for the vast majority of existing cloth models, that leverages equilibrium conditions to guide the iteration. By estimating model parameters under a sequence of deformations of increasing complexity, we alleviate problems with convergence in the presence of abundant local minima. We have used our system to fit three membrane models and two bending models from the graphics literature, each based on a different strain measure, and to evaluate the resulting models against more complex motions.
3. MIT CSAIL 4. Cornell University
c 2012 The Author(s) Computer Graphics Forum c 2012 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
<H1>2. Prior Work</H1>
Cloth simulation has a comparatively long history in computer graphics.
Since the first physics-based approach by Terzopoulos et al. [ TPBF87 ] a multitude of different cloth models have emerged, ranging from simple mass-spring systems
[ Pro95 , CK02 ] over general particle systems
[ BHW94 , BW98 , EWS96 ] to elaborate models derived from continuum mechanics [ EKS03 , VMTF09 ] or even the discrete yarn structure
[ KJM08 ]. Considering the number of existing models, it is very hard to clearly identify or even quantify the advantages of individual approaches. Our goal is to define a platform for comparing cloth models to the observed behavior of real cloth. As a central component of any cloth model, material models describe the relation between deformation and resulting forces.
In the simplest case, this relationship is linear and thus completely described by a set of material constants, whose number depends on the cloth model: Massspring systems typically have four parameters (one per type of spring
[Pro95]), whereas continuum-based methods can have two (isotropic) to six (completely anisotropic) parameters for planar deformation
[EKS03] plus another one to three for bending [ GGWZ07 ]. Continuum-based approaches can accurately describe the directional variation of material properties, but regardless of the cloth model, a single set of material coefficients for the entire deformation range is not sufficient to faithfully capture the nonlinear response of typical fabrics. Bi-phasic models, typically implemented as strain limiting methods
[Pro95, BFA02, Mül08, TPS09, WOR10]
, improve on this by splitting the material behavior into an initial, weakly elastic range and a stiff, quasi-inextensible limit. At the extreme, the elastic range can be replaced altogether by inextensibil∗ ity constraints
[GHF 07, EB08]. A better approximation to the true material response can be obtained by making the material parameters functions of the deformation, rather than constants, and by fitting these functions to measured data. To this end, previous work [BHW94,EWS96, VMTF09 ] has mainly relied on the Kawabata Evaluation System (KES)
[Kaw80 ]
and corresponding machinery.
While the KES covers a comprehensive set of experiments, other devices have been used in more specific context such as the Picture Frame test
[ Cul79 ] for measuring shear properties and the Cantilever test [ CPGE90 ] for measuring bending properties (see also Pabst et al. [ PKST08 ]). These measurement-based approaches establish a valuable link between simulation and real-world behavior, but they rely on experiments that isolate individual deforma∗ tion modes. As an alternative, Bhat et al. [ BTH 03 ] (and recently Kunitomo et al.
[ KNM10 ]
) aim at avoiding the need for controlled conditions and try to extract parameters from casually captured videos of cloth. This approach appeals through a simple and inexpensive acquisition process, but it is not possible to accurately separate internal (i.e. material-specific) and external (e.g. friction, air drag) parameters. In a similar spirit, capture technology can be used to record time-varying geometry of complex cloth mo∗ ∗ tions [WCF07,BPS 08,SGdA 10]. But while capturing can provide accurate deformation data, parameter fitting remains very difficult without explicit control over boundary conditions, in particular loading forces. Closer to our work is the recent approach of Wang et al.
[WRO11]
, who propose a data-driven piecewise linear elastic cloth model comprising 39 material parameters. These parameters are fitted to experimentally acquired data obtained from planar and bending deformations. Their capture setup is appealingly simple, but ours is more general and powerful: it produces a 3D surface, rather than a 2D deformation, and it measures all forces applied to the cloth as they change during a range of different deformations.
<H1>3. Measurement system</H1>
The design goals of our measurement system are to create deformations in a sample of cloth that explore a substantial range of the material’s strain space, and to record complete information about the forces applied to the cloth and the deformation that it undergoes. Like other cloth testing systems, we focus primarily on tensile forces, because it is hard to repeatably produce and measure compression forces in a sheet that is inclined to buckle. Tests are performed on 100 mm square cloth samples using two kinds of plastic clips: small, rounded clips that grab a localized area, and long clips that grip one whole side of the sample. We measure the weights of all cloth samples as well as the clips (see Table 1 ) and use these values in the optimization process. Forces are applied to the clips by fine wire cords that are pulled to defined displacements by eight linear actuators, and the tension in the cords is monitored by miniature load cells located at the actuator ends (see Figure 2). Our actuators and load cells are capable of applying and measuring tensions up to 45 N, but in our experiments the maximum force is typically on the order of 10 N. The geometry of the cloth sample and the attached clips is monitored by a vision system composed of four highresolution cameras. The location and orientation of the cords attached to the clips (which reveal the direction of the applied force) are also tracked. Each output frame of a measurement session contains: • The configuration of the cloth sample, represented as a deformed mesh with 10K regularly sampled vertices. • The positions and orientations of all clips attached to the cloth, including a list of clamped cloth vertices. • The forces applied to all clips. The magnitudes are determined by the tension measurements, and the directions are determined by the observed directions of the cords. Note that the actuator positions themselves are not part of the output, since they are superseded by the displacements measured at the clips. This prevents stretching of the cord, or other factors affecting the distance between the clip and the actuator, from affecting displacement accuracy.
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
4
2
Pulleys & Wires 8 Force Sensors
8 Linear Actuators
Figure 2: Acquisition setup for the measurement system.
<H2>3.1. Reconstruction</H2>
Our vision system recovers the space-time geometry of the deforming cloth and attached rigid clips, as well as the directions of the forces applied to the clips. Initialization. The cloth sample starts flat on a table and we capture the rest pose without applied tensile forces. This initial frame serves to compute the geometry of the cloth without any occlusion from clips. We then attach the clips, and the measurement process continues automatically, following a defined script of actuations, and recording images and forces. We typically deform the cloth by moving the actuators at 0.5 mm/sec and capture a frame every 2 seconds. Cloth Geometry Reconstruction. The raw data for a single deformation consists of 20 to 200 individual measurement frames, with a set of camera images and simultaneous force sensor readings for each frame.
We compute the per-frame geometry using a state-ofthe-art stereo reconstruction technique
[ BBH08 ], which was specifically tailored for reconstructing cloth geome∗ try
[ BPS 08 ]. If the inherent texture of the cloth is not sufficiently random, it is printed with a wavelet noise pat∗ tern
[ AIH 08 ]
to provide texture that can be used for stereo reconstruction and tracking. The pattern is printed with a flatbed inkjet printer and does not have a noticeable effect on the material behavior. To represent inter-frame correspondence, we use optical flow to obtain a single triangle mesh that deforms over time, akin to the human face tracking method of Bradley et al. [BHPS10]. To start, the cloth vertices in the rest pose frame (frame 0) are projected onto the input images, where optical flow predicts the projection of each vertex at the next time step. Back-projecting onto the reconstructed geometry for the next frame gives new position estimates for the cloth vertices. The process is then repeated using the result from frame n to obtain frame n + 1. As with all sequential tracking methods, very small errors can accumulate over time and cause temporal drift in the reconstruction. To avoid drift, we subsequently match each frame independently back to the rest pose frame using the approach described in Bradley et al. [BHPS10]. The final solution is smoothed using Laplacian regularization to remove noise. Tracking Clips and Cords. In order to measure the complete answer that a simulator should predict, we need to determine the interaction between the rigid clips, the cloth, and the cords. The clips are produced, using rapid prototyping, with embedded codes
[Fia05] that allow us to determine their identity, position, and orientation automatically. The area of cloth occluded by the clips is used to automatically determine which cloth vertices are clamped by each clip and will therefore be constrained to it in the simulator. The vision system also finds the cords in the images and triangulates a 3D line for each cord. A few user scribbles on an input image indicate which cords are affecting each clip. Figure 3 illustrates the force measurements and clip locations for three different frames from one experiment. The forces are rendered as red vectors with lengths proportional to the force magnitudes.
Cloth Sample
Id
Mass (g)
cotton satin
#4
1.2
rayon/spandex knit
#12
3.1
cotton denim
#14
4.6
wool/cotton blend
#18
2.4
plastic clips (3 sizes)
1.9, 10.1, 13.3
Cloth Sample Id Mass (g) cotton satin #4 1.2 rayon/spandex knit #12 3.1 cotton denim #14 4.6 wool/cotton blend #18 2.4 plastic clips (3 sizes) 1.9, 10.1, 13.3
Table 1: Cloth and attachment clip masses.
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
Figure 3: Force measurements for selected frames of a corner pulling sequence. Forces are rendered as red vectors with magnitudes proportional to their values (in Newtons).
<H2>3.2. Measurements</H2>
The set of deformations to measure is motivated by the goals of the parameter fitting stage (Section 5): to fit model parameters for stretch, shear and bending that best describe the cloth, and to validate the parameter fits by comparing against other measurements. To reduce the risk of falling into local minima during parameter fits, we have designed deformation sequences that produce near-isolated strains, and allow estimating stretch, shear and bending properties in a separate and incremental manner. However, unlike standard textile evaluation practices [Kaw80]
, and thanks to our full 3D deformation capture solution, we relax the requirement of uniform strains. To isolate stretching we perform a uni-axial tension experiment, with forces applied to two long bar clips attached to either side of the cloth (see Figure 4 , 2nd column). The cloth is slowly stretched until a maximum force is reached and then slowly released back. The process is repeated three times, in both weft and warp directions separately. Shearing is captured using an approximate picture-frame experiment
[Cul79]
, where four long clips fix the cloth boundaries and shear stress is applied as the cords pull on opposite corners ( Figure 4 , 3rd column). To isolate bending deformation we slowly push the flat cloth sample off the edge of a table and measure its shape as it bends under its own weight ( Figure 4 , 4th column), for both weft and warp directions. Thus we have a total of five measurements per cloth sample that will be used for parameter fitting (two stretch, one shear, and two bending). We also capture two sequences with more complex deformation ( Figure 5 ) for validation after parameter fitting. In the first test, opposite edges of the cloth are pulled in opposite directions, causing shearing and buckling ( Figure 5 , top). The second is a four-corner pulling test, where opposite pairs of corners are pulled in alternation, resulting in diagonal wrinkles ( Figure 5 , bottom). Figures 4 and 5 show that our acquisition system is able to recover the 3D cloth geometry including temporal tracking (illustrated with an overlaid checkerboard), tracked 3D clip locations, and individual 3D force directions (shown as green lines). To our knowledge, our method presents the first system able to record such extensive information about the behavior of a cloth sample.
Figure 4: Selected frames from isolated measurements of stretching, shearing, and bending. The left column shows the cloth in its rest state. One input image is shown above each 3D reconstruction. The reconstruction includes parameterized cloth geometry, clip locations and the direction of the force vectors (shown as green lines).
<H2>3.3. Accuracy</H2>
In the vision system, the camera calibration accuracy is within 0.3 pixels, or about 0.075 millimeters at the distance of the cloth. The multi-view stereo algorithm of Bradley et al.
[BBH08] is among the most accurate available according to the Middlebury evaluation benchmark. It is difficult to quantify the accuracy of the temporal flow computation, but it can be visualized by compositing the reconstructed deformation on top of the input images (see accompanying video). The raw repeatability of our force sensors is about 3 millinewtons (RMS). The largest source of error in measuring the force indirectly through the cord is the internal friction in the cord as it bends around the pulleys, which introduces an artificial hysteresis of about 0.1 N.
<H1>4. Cloth Models</H1>
Our goal is to study the fidelity of constitutive models of cloth—models that predict the forces produced in the cloth in response to deformations. The input of such a model is the positions of the vertices x 1 , . . . , x n ∈ IR 3 that define the deformation state of the sheet (analogous to strain in continuum mechanics) and the output is the forces that act between those vertices in response (analogous to stress). Although some of the models we look at are discrete in nature, we will use the convenient terms stress and strain to describe them.
Figure 5: Selected frames from more elaborate cloth manipulation, demonstrating complex deformations. Input image (left) and 3D reconstruction shown from two viewpoints.
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
<H2>4.1. Strain Metrics</H2>
Most elastic cloth models separate membrane (i.e., stretch and shear) and bending deformation energies. In both cases, deformation energy density can be described by the product of strain (ε) and stress (σ), i.e., W = 2 1 σ · ε. Furthermore, most of these models define separable scalar stress components as linear functions of individual scalar strain metrics. In that case, the energy density of each deformation component i can be written as W i = 1 2 k i ε 2 i , where k i ε i = σ i and k i is the stiffness coefficient corresponding to the deformation component ε i . The force density due to each ε i follows as F i = − W i = −σ i ε i = −k i ε i ε i . We have evaluated three models for membrane deformation that fit this description (spring systems, the soft constraint model by Baraff and Witkin
[BW98]
and the diagonalized St.Venant-Kirchhoff (StVK) model by Volino et al. [VMTF09]), and two bending models (spring systems and the edge-based bending model in Discrete Shells [GHDS03]). Considering possible anisotropic behavior, we distinguish six different strain components on regularly triangulated cloth: weft-stretch (ε s,u ), warp-stretch (ε s,v ), shear (ε s,uv ), weft-bend (ε b,u ), warp-bend (ε b,v ), and diagonalbend (ε b,uv ). Next, we describe in detail the strain metrics for the individual deformation components in the selected models. Note that not all force models define the quantities below explicitly as strains, as they often rely on the resolution of the discretization, or they differ simply by scale factors that can be embedded in the stiffness k i . We use continuum strain definitions in all cases to fit them in a common formulation that allows us to easily compare the models. The details about the computation of strain gradients for the various cloth models, needed in the force computation, are given in their respective original papers. Spring Membrane. All deformation components are modeled based on springs, with weft and warp ring-1 springs for stretch, and diagonal ring-1 springs for shear. The springbased strain for each deformation component i can be defined as the relative change of edge length,
1
ε s,i = l / l 0 − 1,
with l the current length of the spring, and l 0 its rest length.
Diagonalized StVK. The membrane deformation is defined using the Green-Lagrange strain tensor, a formulation introduced to computer graphics by Terzopoulos et al. [TPBF87]. Given a per-triangle mapping function w from the undeformed 2D configuration (x a,0 , x b,0 , x c,0 ) to the deformed
3D configuration (x a , x b , x c ), the deformation gradient can be computed as −1 (w u w v ) = (x b − x a x c − x a ) x b,0 − x a,0 x c,0 − x a,0 .
Then, the components of the membrane Green-Lagrange strain are defined as:
3
w u 2 − 1 w v 2 − 1 T ε s,u = , ε s,v = , ε s,uv = w u w v . 2 2
Volino et al. [VMTF09] approximate the standard StVK model zeroing out off-diagonal terms in the matrix that relates strain and stress, σ = Eε. Then, in the diagonalized StVK, each membrane stress component depends only on its corresponding strain component, σ s,i (ε s,i ). Soft Constraints. Weftand warp-stretch are measured through a subtle modification of the Green-Lagrange strain tensor, defining terms that are quadratic in positions instead of quartic:
4
ε s,u = w u − 1, ε s,v = w v − 1, ε s,uv = w u T w v .
Spring Bending. The deformation is measured based on weft and warp ring-2 springs for weftand warp-bend, and diagonal ring-2 springs for diagonal-bend. Same as for membrane deformation, strain is measured as the relative change of edge length (1).
Discrete Shells Bending.
It is measured as the change of
dihedral angle at edges:
5
ε b,i = 1 / h 0 (θ − θ 0 ),
with θ the current angle, and θ 0 the undeformed angle. Grinspun et al. [GHDS03] and Bridson et al. [BMF03]
discovered concurrently the appropriate weighting of the angle change in order to model homogeneous bending on irregular triangle meshes with a homogeneous stiffness. Grinspun et al. define h 0 as a third of the average of the heights of the two triangles incident to the edge. This definition implies that bending energy density is integrated over edgecentered rectangles of size l 0 × h 0 . With our separation of weft-, warpand diagonal-bending to capture anisotropy, the bending models in Discrete Shells and by Baraff and Witkin
[BW98] are equivalent up to a stiffness scale factor.
<H2>4.2. Nonlinear Stress-Strain Curves</H2>
The generic force density model F = −σ ε defined above assumes a linear stress-strain curve σ = kε. However, stressstrain curves are potentially nonlinear functions. Then, for each deformation component, we model stress as a function σ i = k i (ε i )ε i , with a strain-dependent stiffness k i encoded using Hermite splines. We enforce non-negative constraints on the stiffness values at control points. The resulting nonlinear force density function, F i = −k i (ε i )ε i ε i yields a conservative force field, but note that the elastic energy density can no longer be defined simply as 1 2 kε 2 , and would now require the integration of the stiffness function. Although only Volino et al. [VMTF09] propose a general nonlinear stress-strain relationship (though many systems use some form of strain limiting instead), the same construction can easily be built on any of our selected models. Because linear models fit the data poorly, we used the nonlinear model in all cases, resulting in a consistent set of models, parameterized by the number of spline control points, which reduces to the widely used linear models when each spline has a single control point.
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
<H1>5. Fitting</H1>
The key question of how well a given model describes a particular piece of cloth is answered by fitting the model to the measurement data: adjusting its parameters to minimize the difference between the model’s predictions and the measured behavior, both in position and force. We do this by solving an optimization problem, leveraging that the cloth is at static equilibrium at the measured configurations. In principle all parameters of a cloth model can be fit to a sufficiently rich single deformation sequence, but this can result in a problem fraught with local minima. In order to achieve stable fits, we have designed an incremental optimization procedure that fits model parameters a few at a time using the isolated deformations described in Section 3.2.
<H2>5.1. Optimization Problem</H2>
For each different cloth sample, we have created a simulated replica with the same mass, uniformly distributed, and the same 100mm square geometry, discretized with a regular 25 × 25-node mesh, connected either with springs or with quadrilaterals split into triangles, depending on the model. In each measurement sequence, a different set of nodes is fixed to rigid bodies representing the clips. For the bending measurement sequences (see Figure 4 ), we fix all cloth nodes above the edge of the table. The measured pulling forces of the cords are applied as point forces on the rigid bodies at known locations, with known magnitudes and orientations. Given a set of captured static deformation frames, we wish to know the (nonlinear) stress-strain curves for the deformation components of a cloth model, such that a simulated cloth matches known positions and forces as well as possible. Specifically, we minimize the weighted error of cloth positions and clip forces over a sequence of measurement frames, subject to the constraint of static equilibrium on all frames. For the formulation of the objective function, we concatenate in vectors the positions, x n , and the net forces, F n , of free cloth nodes at all frames, as well as the forces, F c , applied by the cords on the clips. Due to equilibrium, the net force on the clips, produced by cord forces, gravity, and forces from fixed cloth nodes, must be zero. We indicate with x n and F ̃ c , respectively, the known cloth node positions and clip forces, measured as described in Section 3. We also concatenate in a vector k the (unknown) stiffness values at the control points of the nonlinear stress-strain curves for the deformation components of the cloth. Since the pieces of cloth are homogeneous, we use a single curve for each deformation component for all frames and all cloth elements. Then, the computation of model parameters based on the minimization of position and force errors subject to the static equilibrium condition can be formulated as the following nonlinear constrained least-squares problem: k = arg min μ x n (k) − x n 2 + λ F c (x n , k) − F ̃ c 2 ,
6
s.t. F n (x n , k) = 0.
In this optimization problem, we use the measured clip positions, x c , as known boundary conditions. For stretch tests, the objective function is based only on clip forces, i.e., μ = 0, λ = 1, while for bend tests it is based only on cloth positions (since there are no measured forces), i.e., μ = 1, λ = 0. For shear tests, the objective function is based only on clip forces parallel to the direction of the clips themselves. We observed that, in situations of near-homogeneous shear, the clip-parallel forces are dominated by shear, while clip-orthogonal forces are dominated by stretch. Then, by fitting only clip-parallel forces we reduce the sensitivity to potential errors in stretch stiffness. The optimization problem contains two unknowns: the parameter vector k and cloth node positions x n . We solve the optimization in an iterative manner, refining k and x n separately on two nested loops. In an outer loop, we refine k by local minimization of the error function and, in an inner loop, we recompute x n to satisfy the equilibrium constraint. Outer Loop. Assuming cloth positions that satisfy the (nonlinear) equilibrium constraints on all captured frames, we locally linearize those constraints w.r.t. both k and x n . As a result, we obtain a linear expression that relates node positions to parameter values:
7
−1 ∂F n ∂F n ∂F n ∂F n ∆x n + ∆k = 0 ⇒ ∆x n = − ∆k. ∂x n ∂k ∂x n ∂k
We also locally linearize clip forces,
8
∂F c ∂F c ∆F c = ∆x n + ∆k, ∂x n ∂k
and we turn (6) into a linear least squares problem, which we solve to refine the parameters k(i + 1) = k(i) + ∆k, with: −1 ∆k = arg min μ x n (i) − x n − ∂F n ∂F n ∆k 2 + ∂x n ∂k
9
−1 λ F c (i) − F ̃ c + ∂F c − ∂F c ∂F n ∂F n ∆k 2 . ∂k ∂x n ∂x n ∂k
We terminate the outer loop (and hence the overall optimization) when the residual is reduced by less than 1% between two consecutive iterations. To ensure convergence of the Newton-like iterations and to enforce non-negativity constraints on the components of k, we execute a line search from k(i) to the solution of (9) if the residual grows or if the solution violates some constraint. The solution to the linear least squares problem requires solving a system Ak = b, where the size of A is given by the number of unknown stiffness values, |k|. In our test examples, this number was always below 10, and we solved the linear systems using LDL factorization. The formulation of A, on the other hand, requires solving |k| linear systems of type ∂F n y = b, which ∂x n we did using the conjugate gradient method. Inner Loop. Once the parameter values k(i + 1) are refined, we bring the cloth to a static equilibrium position, x n (i + 1). We do this by solving quasi-static simulations until convergence on all captured frames, starting always from the measured configuration x n and using the measured clip positions x c as boundary conditions. We consider that a piece of cloth has converged to equilibrium when F n < 10μN. The quasi-static simulations involve linear-system solves with the cloth stiffness matrix ∂F n . We found that, during inter∂x n mediate iterations, the stiffness matrix may not always be well conditioned, therefore we have solved the quasi-static equilibrium problems using additive Levenberg-Marquardt, which effectively produces a modified stiffness matrix of the form ∂F n + μI. For improved conditioning, we also use this ∂x n modified stiffness matrix in the outer loop.
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
<H2>5.2. Incremental Parameter Fitting</H2>
The nonlinearity of cloth deformation, together with the complex interplay of various deformation components in the resulting forces and positions, make the optimization problem above extremely complex in the general case, prone to falling in local minima and sensitive to initialization values. However, we largely alleviate these issues with the design of the five isolated deformation measurements described in Section 3.2, which allow us to separately fit stiffness curves for the six deformation components described in Section 4.1, following an incremental parameter fitting procedure. First, we fit in parallel the weft-stretch stiffness curve, k s,u (ε s,u ), for the weft-stretch sequence, and the warp-stretch stiffness, k s,v (ε s,v ), for the warp-stretch sequence. We ignore shear and bend parameters for stretch fits, as we have observed that they have little effect. Second, using known stretch stiffness curves, we fit the shear stiffness k s,uv (ε s,uv ), for the shear sequence. Third, we fit in parallel the weftbending stiffness k b,u (ε b,u ), for the weft-bending measurement sequence, and the warp-bending stiffness k b,v (ε b,v ), for the warp-bending sequence. Finally, we fit the diagonalbending stiffness curve k b,uv (ε b,uv ), using both weftand warp-bending measurements. To better account for crossinfluence of shear and bending, we use their estimated values as initial guesses and run another fitting iteration. To fit each stiffness curve k i (ε i ), we iteratively subdivide the Hermite spline adding more control points until the residual error function (6) is reduced by less than 1% or a speci- fied maximum number of points, usually 4 or 5, is reached. First, we evaluate the strain histogram for the corresponding measurement sequence, and we determine maximum and minimum strains after removing outliers. We initialize the stiffness curve with one control point (i.e., constant stiffness), and subsequently we subdivide the strain range with equidistant control points.
10 cord 1 8 cord 2 (N) 6 cord 3 cord 4 tension 4 2 0 0 2 4 6 8 10 12 14 stretch (%)
Figure 6: Raw force measurements for X stretching in Sample #18. Cloth is held by two bar clips, with two cords attached to each clip.
<H1>6. Results</H1>
We tested our system on four fabric samples, including a knit and the three common weave patterns (plain weave, twill, and satin), and three fiber types (cotton, wool, and synthetic): cotton satin (#4), rayon/spandex knit (#12), cotton denim (#14), and wool/cotton blend (#18). Each fabric was tested with seven deformations (see Section 3.2): for fitting, stretch in X and Y, simple shear, and bending in X and Y; and for evaluation, complex shearing and corner pulling. Figure 6 shows the four individual tension measurements for a typical stretching test, plotted as a function of the overall extension of the cloth. The measurement shows the typical behavior of a woven fabric: a nonlinear curve with increasing stiffness for higher strain, and large hysteresis. The test repeats three times, retracing the same loop each time after the initial extension from rest. We worked with three cloth models built from the components described in Section 4. The Springs model uses the spring membrane model with the spring bending model; the Soft Constraints model uses Baraff and Witkin’s membrane model with the Discrete Shells bending model; and the St. VK model uses the diagonalized St. Venant-Kirchoff membrane model with the Discrete Shells bending model. We fit all the models in four variants: linear (constant stiffness for each deformation mode), isotropic (identical stiffness in warp and weft), linear and isotropic (the simplest variant), and nonlinear orthotropic (the most general variant). The results are too numerous to include in the paper; we refer the reader to the supplementary material, which illustrates the behavior of the nonlinear orthotropic variant of all three models for all four fabrics, and the behavior of the variants of the Soft Constraints model for denim, a largely nonlinear and anisotropic material. Observations. In Figure 7 and Table 2 we present results for fitting the Soft Constraints model to Sample #12. For each test we show a selected frame (near maximum distortion) with renderings illustrating the captured and fitted cloth geometry and forces. To illustrate the fitting residuals more quantitatively, we show a force-displacement plot comparing a summary of the measured forces to the predictions of the fitted model and a vector-field plot illustrating the position error over the geometry of the fitted mesh (see caption for details). Similar tables are shown for all cloth samples and several model variants in the supplementary document. The four selected fabrics span a large range of possible cloth behaviors. In a nutshell, #12 is isotropic and very compliant in stretch and bending; #4 is also isotropic, very stiff in stretch but compliant in bending; #14 is stiff and quite isotropic in stretch, but extremely anisotropic in bending (with 33/1 stiffness ratio in weft and warp); and #18 is anisotropic both in stretch (with 10/1 stiffness ratio) and in bending (with 13/1 stiffness ratio). The maximum stretch stiffness for #4 is 250 times higher than for #12, while #14 is 10 times stiffer in shear than any other fabric. All four fabrics show similar hysteresis behavior, with loading-to-unloading stretch stiffness ratios ranging from 1.4/1 to 1.8/1. Sample #12 is nearly linear in the test deformation range, while all other three fabrics exhibit nonlinearity. Interestingly, nonlinearity may arise in some deformation modes but not in others, with no clear pattern. For stretching, all three cloth models fit nicely to the average of the hysteresis bands, even in highly nonlinear cases. The fitting residual is larger for stiffer fabrics, and the nonlinear orthotropic model variants fit anisotropic fabrics best, as expected, while linear and/or isotropic variants reach a reasonable compromise but are not always able to remain inside the hysteresis band. For shearing, the fitting force residual is larger for #14, the stiffest fabric. Across models, the Soft Constraints and St. VK models fit to the average of the shearing hysteresis band, while the Springs model deviates at times. For bending, no forces are available, and we evaluate the position residual as well as profiles of sample curves orthogonal to the support plane. The fitting residual is similar for all fabrics, but distinctly higher for the Springs model. Often, the residual is dominated by a difference in curl near the edge of the sample, while the overall shape is well fit. The last two rows of Figure 7 show the evaluation tests; these data were not used in fitting the models. The behavior of sample #12, the most linear fabric, is predicted well in all cases, as seen in the force-displacement plots, the buckling behavior in corner pulling, and the (lower) effective shear stiffness of the sheet when allowed to buckle in the complex shear test. In the three other samples, however, the force-displacement plot of the corner pulling test reveals that the stiffness is underestimated for the Soft Constraints and
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
Scenario Stretch-X Stretch-Y Shear Bend-X Bend-Y Residual 0.53N 0.62N 0.22N 21.3mm 22.0mm
Table 2: RMS residuals of non-linear orthotropic fitting to Sample #12 with the Soft Constraints model.
St. VK models, and overestimated for the Springs model. Visually, the mismatch is more apparent in the complex shear test, where models with underestimated stiffness exhibit wider folds than the real fabrics. We have also evaluated the fitted models on new test samples of each fabric, to validate their generality. Specifically, we have tested stretching on new samples of rayon/spandex knit (#12.2) and cotton denim (#14.2), and shearing on new samples of cotton satin (#4.2) and wool/cotton blend (#18.2). The force-displacement plots of the real cloth samples, shown in the supplementary document, indicate very similar behavior between fitting and test samples for #12 and #14, and a larger disparity for #4 and #18. The evaluation plots for the simulation models behave similar for the test and fitting cases, but the matching quality depends on the actual disparity across cloth samples. Discussion. While overall force-displacement behavior is nicely matched, the actual folding shapes of simulated cloth may deviate largely from the captured cloth, because even a small change in material properties may lead to distant stable configurations in the L 2 sense. For this reason, the traditional L 2 metric is not appropriate for evaluating error in this case. The discontinuity of stable configurations is also the cause of flickering and twitches in some of our examples. The Springs model exhibits the worst fitting quality in shearing force-displacement curves, and the highest fitting residual for bending. This is probably due to the inherent coupling of stretch and bending deformation components in this model. Nevertheless, the overall deformations in complex shearing fit reasonably well. In contrast to continuum models, complex parameter tuning has often been regarded as a caveat of mass-spring models; but our results indicate that satisfactory parameter estimation is possible by incorporating anisotropy and nonlinearity into the model. The Soft Constraints and St. VK models produce results with very similar quality, which is expected as the models present only subtle differences as described in Section 4.1. At least three effects are missed by the tested models: hysteresis, Poisson effect (due to the diagonalization of the standard StVK model), and cross-modal stiffening (e.g., shear stiffening due to stretching). We indeed identified stretch stiffening in the shearing deformations, therefore we chose clip-parallel forces as objective function to minimize the effect of stretch errors on shear optimization. We conjecture that missing cross-modal stiffening may also be, to a large extent, the reason for stiffness underestimation in the corner pulling test for the Soft Constraints and St. VK models. An extension to the nonlinear model of Wang et al. [WRO11]
could help alleviate these problems.
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
Figure 7: Fitting results for Sample #12 with the Soft Constraints model. Top to bottom; Stretch-X, Simple shear, Bend-X, Corner pull, Complex shear. Left to right: captured geometry, equilibrium of fitted model, force comparison (thin line: measurement; thick line: model), position residual (vertex position minus corresponding measured position, magnified 5x). “Effective shear” is the shear angle of the best-fit transformation to the motion of the clamped cloth vertices. “Aggregated torque” is the torque about the center of the cloth applied by the clamps, with every other clamp counted negatively. “Stretching force difference” is the difference in the two corner-to-corner stretching forces.
<H1>7. Conclusion</H1>
This paper has demonstrated a novel system for observing cloth behavior, including complete information about deformation and forces, and a new method for fitting and evaluating cloth models using the measurements. Our system is different from standard textile testing systems because it captures detailed geometry information; it is different from previous cloth capture systems in that it captures complete force information and measures deformations of a 3D surface. The combination of very complete position and force information provides an unprecedented view into the complex behavior of cloth. Our measurement setup offers very accurate control over membrane deformations, but the bending tests require manual intervention and are thus less precise. Furthermore, the bending tests are most accurate for samples with straight edges, but some cloth materials (in particular knit) tend to curl up at free boundaries. In order to eliminate these problems, we would like to investigate alternative ways of controlling bending deformations in the future. The data from our experiments shows some of the limitations of current models. The most obvious of these is hysteresis—all widely used cloth models are elastic, but cloth is clearly far from elastic, resulting in quite large errors for any given point in the experiment. There are many paths for future work in measurement, including more complete exploration of strain space (including compression) and capture of dynamic properties, and in fitting, where new ways of evaluating fitting error are needed that can work when the cloth’s equilibrium state is unstable or non-deterministic. Acknowledgments. This work was funded in part by the Spanish Ministry of Science and Innovation, project TIN2009-07942, and by the European Research Council, project ERC-2011-StG-280135 Animetrics.
3 (N) force 2 stretching 1 0 0 20 40 stretch (%) 40 mm) 20 (N torque 0 −20 aggr. −40 −10 0 10 eff. shear (deg) (N) 1 diff. 0.5 force 0 stretching −0.5 −1 −20 0 20 eff. shear (deg) 40 mm) 20 (N torque 0 −20 aggr. −40 −10 0 10 eff. shear (deg)
c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
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],
"text": "['We have designed a general fitting method, suited for the vast majority of existing cloth models, that leverages equilibrium conditions to guide the iteration']",
"type": "own_claim",
"id": "T19"
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{
"offsets": [
[
4743,
4912
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],
"text": "['By estimating model parameters under a sequence of deformations of increasing complexity, we alleviate problems with convergence in the presence of abundant local minima']",
"type": "own_claim",
"id": "T20"
},
{
"offsets": [
[
5525,
5595
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],
"text": "['Cloth simulation has a comparatively long history in computer graphics']",
"type": "background_claim",
"id": "T21"
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{
"offsets": [
[
5599,
5640
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],
"text": "['Since the first physics-based approach by']",
"type": "background_claim",
"id": "T22"
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{
"offsets": [
[
5671,
5721
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],
"text": "['a multitude of different cloth models have emerged']",
"type": "background_claim",
"id": "T23"
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{
"offsets": [
[
5662,
5668
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],
"text": "['TPBF87']",
"type": "data",
"id": "T24"
},
{
"offsets": [
[
5736,
5762
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],
"text": "['simple mass-spring systems']",
"type": "background_claim",
"id": "T25"
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{
"offsets": [
[
5786,
5810
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],
"text": "['general particle systems']",
"type": "background_claim",
"id": "T26"
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{
"offsets": [
[
5840,
5889
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],
"text": "['elaborate models derived from continuum mechanics']",
"type": "background_claim",
"id": "T27"
},
{
"offsets": [
[
5917,
5944
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],
"text": "['the discrete yarn structure']",
"type": "background_claim",
"id": "T28"
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{
"offsets": [
[
5766,
5771
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],
"text": "['Pro95']",
"type": "data",
"id": "T29"
},
{
"offsets": [
[
5774,
5778
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],
"text": "['CK02']",
"type": "data",
"id": "T30"
},
{
"offsets": [
[
5814,
5819
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],
"text": "['BHW94']",
"type": "data",
"id": "T31"
},
{
"offsets": [
[
5822,
5826
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],
"text": "['BW98']",
"type": "data",
"id": "T32"
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{
"offsets": [
[
5829,
5834
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],
"text": "['EWS96']",
"type": "data",
"id": "T33"
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{
"offsets": [
[
5892,
5897
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],
"text": "['EKS03']",
"type": "data",
"id": "T34"
},
{
"offsets": [
[
5900,
5906
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],
"text": "['VMTF09']",
"type": "data",
"id": "T35"
},
{
"offsets": [
[
5948,
5953
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],
"text": "['KJM08']",
"type": "data",
"id": "T36"
},
{
"offsets": [
[
5957,
6092
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],
"text": "['Considering the number of existing models, it is very hard to clearly identify or even quantify the advantages of individual approaches']",
"type": "background_claim",
"id": "T37"
},
{
"offsets": [
[
6094,
6192
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],
"text": "['Our goal is to define a platform for comparing cloth models to the observed behavior of real cloth']",
"type": "own_claim",
"id": "T38"
},
{
"offsets": [
[
6194,
6315
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],
"text": "['As a central component of any cloth model, material models describe the relation between deformation and resulting forces']",
"type": "background_claim",
"id": "T39"
},
{
"offsets": [
[
6318,
6469
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],
"text": "['In the simplest case, this relationship is linear and thus completely described by a set of material constants, whose number depends on the cloth model']",
"type": "background_claim",
"id": "T40"
},
{
"offsets": [
[
6471,
6520
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],
"text": "['Massspring systems typically have four parameters']",
"type": "background_claim",
"id": "T41"
},
{
"offsets": [
[
6522,
6544
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],
"text": "['one per type of spring']",
"type": "data",
"id": "T42"
},
{
"offsets": [
[
6547,
6552
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],
"text": "['Pro95']",
"type": "data",
"id": "T43"
},
{
"offsets": [
[
6564,
6600
]
],
"text": "['continuum-based methods can have two']",
"type": "background_claim",
"id": "T44"
},
{
"offsets": [
[
6613,
6619
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],
"text": "['to six']",
"type": "background_claim",
"id": "T45"
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{
"offsets": [
[
6645,
6678
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],
"text": "['parameters for planar deformation']",
"type": "background_claim",
"id": "T46"
},
{
"offsets": [
[
6688,
6725
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],
"text": "['plus another one to three for bending']",
"type": "background_claim",
"id": "T47"
},
{
"offsets": [
[
6602,
6611
]
],
"text": "['isotropic']",
"type": "data",
"id": "T48"
},
{
"offsets": [
[
6621,
6643
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],
"text": "['completely anisotropic']",
"type": "data",
"id": "T49"
},
{
"offsets": [
[
6681,
6686
]
],
"text": "['EKS03']",
"type": "data",
"id": "T50"
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{
"offsets": [
[
6728,
6734
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],
"text": "['GGWZ07']",
"type": "data",
"id": "T51"
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"offsets": [
[
6738,
6837
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],
"text": "['Continuum-based approaches can accurately describe the directional variation of material properties']",
"type": "background_claim",
"id": "T52"
},
{
"offsets": [
[
6843,
7026
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],
"text": "['regardless of the cloth model, a single set of material coefficients for the entire deformation range is not sufficient to faithfully capture the nonlinear response of typical fabrics']",
"type": "background_claim",
"id": "T53"
},
{
"offsets": [
[
7028,
7094
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],
"text": "['Bi-phasic models, typically implemented as strain limiting methods']",
"type": "background_claim",
"id": "T54"
},
{
"offsets": [
[
7134,
7260
]
],
"text": "['improve on this by splitting the material behavior into an initial, weakly elastic range and a stiff, quasi-inextensible limit']",
"type": "background_claim",
"id": "T55"
},
{
"offsets": [
[
7097,
7102
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],
"text": "['Pro95']",
"type": "data",
"id": "T56"
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{
"offsets": [
[
7104,
7109
]
],
"text": "['BFA02']",
"type": "data",
"id": "T57"
},
{
"offsets": [
[
7111,
7116
]
],
"text": "['Mül08']",
"type": "data",
"id": "T58"
},
{
"offsets": [
[
7118,
7123
]
],
"text": "['TPS09']",
"type": "data",
"id": "T59"
},
{
"offsets": [
[
7125,
7130
]
],
"text": "['WOR10']",
"type": "data",
"id": "T60"
},
{
"offsets": [
[
7262,
7355
]
],
"text": "['At the extreme, the elastic range can be replaced altogether by inextensibil∗ ity constraints']",
"type": "background_claim",
"id": "T61"
},
{
"offsets": [
[
7358,
7364
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],
"text": "['GHF 07']",
"type": "data",
"id": "T62"
},
{
"offsets": [
[
7366,
7370
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],
"text": "['EB08']",
"type": "data",
"id": "T63"
},
{
"offsets": [
[
7373,
7576
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],
"text": "['A better approximation to the true material response can be obtained by making the material parameters functions of the deformation, rather than constants, and by fitting these functions to measured data']",
"type": "background_claim",
"id": "T64"
},
{
"offsets": [
[
7591,
7604
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],
"text": "['previous work']",
"type": "background_claim",
"id": "T65"
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{
"offsets": [
[
7628,
7679
]
],
"text": "['has mainly relied on the Kawabata Evaluation System']",
"type": "background_claim",
"id": "T66"
},
{
"offsets": [
[
7688,
7693
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],
"text": "['Kaw80']",
"type": "data",
"id": "T67"
},
{
"offsets": [
[
7606,
7611
]
],
"text": "['BHW94']",
"type": "data",
"id": "T68"
},
{
"offsets": [
[
7612,
7617
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],
"text": "['EWS96']",
"type": "data",
"id": "T69"
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{
"offsets": [
[
7619,
7625
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],
"text": "['VMTF09']",
"type": "data",
"id": "T70"
},
{
"offsets": [
[
7697,
7724
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],
"text": "['and corresponding machinery']",
"type": "background_claim",
"id": "T71"
},
{
"offsets": [
[
7733,
7782
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],
"text": "['the KES covers a comprehensive set of experiments']",
"type": "background_claim",
"id": "T72"
},
{
"offsets": [
[
7880,
7910
]
],
"text": "['for measuring shear properties']",
"type": "data",
"id": "T74"
},
{
"offsets": [
[
7784,
7837
]
],
"text": "['other devices have been used in more specific context']",
"type": "background_claim",
"id": "T73"
},
{
"offsets": [
[
7846,
7868
]
],
"text": "['the Picture Frame test']",
"type": "data",
"id": "T75"
},
{
"offsets": [
[
7872,
7877
]
],
"text": "['Cul79']",
"type": "data",
"id": "T76"
},
{
"offsets": [
[
7915,
7934
]
],
"text": "['the Cantilever test']",
"type": "data",
"id": "T77"
},
{
"offsets": [
[
7946,
7978
]
],
"text": "['for measuring bending properties']",
"type": "data",
"id": "T78"
},
{
"offsets": [
[
7937,
7943
]
],
"text": "['CPGE90']",
"type": "data",
"id": "T79"
},
{
"offsets": [
[
8004,
8010
]
],
"text": "['PKST08']",
"type": "data",
"id": "T80"
},
{
"offsets": [
[
8015,
8118
]
],
"text": "['These measurement-based approaches establish a valuable link between simulation and real-world behavior']",
"type": "background_claim",
"id": "T81"
},
{
"offsets": [
[
8124,
8192
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],
"text": "['they rely on experiments that isolate individual deforma∗ tion modes']",
"type": "background_claim",
"id": "T82"
},
{
"offsets": [
[
8194,
8211
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],
"text": "['As an alternative']",
"type": "background_claim",
"id": "T83"
},
{
"offsets": [
[
8279,
8398
]
],
"text": "['aim at avoiding the need for controlled conditions and try to extract parameters from casually captured videos of cloth']",
"type": "background_claim",
"id": "T84"
},
{
"offsets": [
[
8227,
8233
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],
"text": "['BTH 03']",
"type": "data",
"id": "T85"
},
{
"offsets": [
[
8269,
8274
]
],
"text": "['KNM10']",
"type": "data",
"id": "T86"
},
{
"offsets": [
[
8400,
8474
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],
"text": "['This approach appeals through a simple and inexpensive acquisition process']",
"type": "background_claim",
"id": "T87"
},
{
"offsets": [
[
8480,
8530
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],
"text": "['it is not possible to accurately separate internal']",
"type": "background_claim",
"id": "T88"
},
{
"offsets": [
[
8556,
8568
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],
"text": "['and external']",
"type": "background_claim",
"id": "T89"
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{
"offsets": [
[
8595,
8605
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],
"text": "['parameters']",
"type": "background_claim",
"id": "T90"
},
{
"offsets": [
[
8537,
8554
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],
"text": "['material-specific']",
"type": "data",
"id": "T91"
},
{
"offsets": [
[
8575,
8583
]
],
"text": "['friction']",
"type": "data",
"id": "T92"
},
{
"offsets": [
[
8585,
8593
]
],
"text": "['air drag']",
"type": "data",
"id": "T93"
},
{
"offsets": [
[
8628,
8719
]
],
"text": "['capture technology can be used to record time-varying geometry of complex cloth mo∗ ∗ tions']",
"type": "background_claim",
"id": "T94"
},
{
"offsets": [
[
8721,
8726
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],
"text": "['WCF07']",
"type": "data",
"id": "T95"
},
{
"offsets": [
[
8727,
8733
]
],
"text": "['BPS 08']",
"type": "data",
"id": "T96"
},
{
"offsets": [
[
8734,
8741
]
],
"text": "['SGdA 10']",
"type": "data",
"id": "T97"
},
{
"offsets": [
[
8754,
8801
]
],
"text": "['capturing can provide accurate deformation data']",
"type": "background_claim",
"id": "T98"
},
{
"offsets": [
[
8803,
8923
]
],
"text": "['parameter fitting remains very difficult without explicit control over boundary conditions, in particular loading forces']",
"type": "background_claim",
"id": "T99"
},
{
"offsets": [
[
8925,
8969
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],
"text": "['Closer to our work is the recent approach of']",
"type": "own_claim",
"id": "T100"
},
{
"offsets": [
[
8993,
9089
]
],
"text": "['who propose a data-driven piecewise linear elastic cloth model comprising 39 material parameters']",
"type": "background_claim",
"id": "T101"
},
{
"offsets": [
[
8984,
8989
]
],
"text": "['WRO11']",
"type": "data",
"id": "T102"
},
{
"offsets": [
[
9091,
9196
]
],
"text": "['These parameters are fitted to experimentally acquired data obtained from planar and bending deformations']",
"type": "background_claim",
"id": "T103"
},
{
"offsets": [
[
9198,
9239
]
],
"text": "['Their capture setup is appealingly simple']",
"type": "background_claim",
"id": "T104"
},
{
"offsets": [
[
9245,
9278
]
],
"text": "['ours is more general and powerful']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
9280,
9334
]
],
"text": "['it produces a 3D surface, rather than a 2D deformation']",
"type": "own_claim",
"id": "T106"
},
{
"offsets": [
[
9340,
9439
]
],
"text": "['it measures all forces applied to the cloth as they change during a range of different deformations']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
[
9502,
9656
]
],
"text": "['The design goals of our measurement system are to create deformations in a sample of cloth that explore a substantial range of the material’s strain space']",
"type": "own_claim",
"id": "T108"
},
{
"offsets": [
[
9662,
9768
]
],
"text": "['to record complete information about the forces applied to the cloth and the deformation that it undergoes']",
"type": "own_claim",
"id": "T109"
},
{
"offsets": [
[
9804,
9840
]
],
"text": "['we focus primarily on tensile forces']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
9850,
9951
]
],
"text": "['it is hard to repeatably produce and measure compression forces in a sheet that is inclined to buckle']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
10502,
10588
]
],
"text": "['Our actuators and load cells are capable of applying and measuring tensions up to 45 N']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
10594,
10664
]
],
"text": "['in our experiments the maximum force is typically on the order of 10 N']",
"type": "own_claim",
"id": "T113"
},
{
"offsets": [
[
11381,
11441
]
],
"text": "['the actuator positions themselves are not part of the output']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
11449,
11511
]
],
"text": "['they are superseded by the displacements measured at the clips']",
"type": "data",
"id": "T115"
},
{
"offsets": [
[
11513,
11662
]
],
"text": "['This prevents stretching of the cord, or other factors affecting the distance between the clip and the actuator, from affecting displacement accuracy']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
12096,
12256
]
],
"text": "['Our vision system recovers the space-time geometry of the deforming cloth and attached rigid clips, as well as the directions of the forces applied to the clips']",
"type": "own_claim",
"id": "T117"
},
{
"offsets": [
[
12762,
12934
]
],
"text": "['The raw data for a single deformation consists of 20 to 200 individual measurement frames, with a set of camera images and simultaneous force sensor readings for each frame']",
"type": "own_claim",
"id": "T118"
},
{
"offsets": [
[
12937,
13026
]
],
"text": "['We compute the per-frame geometry using a state-ofthe-art stereo reconstruction technique']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
13039,
13106
]
],
"text": "['which was specifically tailored for reconstructing cloth geome∗ try']",
"type": "background_claim",
"id": "T120"
},
{
"offsets": [
[
13030,
13035
]
],
"text": "['BBH08']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
13110,
13116
]
],
"text": "['BPS 08']",
"type": "data",
"id": "T122"
},
{
"offsets": [
[
13185,
13229
]
],
"text": "['it is printed with a wavelet noise pat∗ tern']",
"type": "background_claim",
"id": "T123"
},
{
"offsets": [
[
13123,
13183
]
],
"text": "['the inherent texture of the cloth is not sufficiently random']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
13243,
13317
]
],
"text": "['to provide texture that can be used for stereo reconstruction and tracking']",
"type": "background_claim",
"id": "T125"
},
{
"offsets": [
[
13233,
13239
]
],
"text": "['AIH 08']",
"type": "data",
"id": "T126"
},
{
"offsets": [
[
13319,
13434
]
],
"text": "['The pattern is printed with a flatbed inkjet printer and does not have a noticeable effect on the material behavior']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
13436,
13595
]
],
"text": "['To represent inter-frame correspondence, we use optical flow to obtain a single triangle mesh that deforms over time, akin to the human face tracking method of']",
"type": "own_claim",
"id": "T128"
},
{
"offsets": [
[
13612,
13618
]
],
"text": "['BHPS10']",
"type": "data",
"id": "T129"
},
{
"offsets": [
[
13801,
13919
]
],
"text": "['Back-projecting onto the reconstructed geometry for the next frame gives new position estimates for the cloth vertices']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
14044,
14133
]
],
"text": "['very small errors can accumulate over time and cause temporal drift in the reconstruction']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
14135,
14257
]
],
"text": "['To avoid drift, we subsequently match each frame independently back to the rest pose frame using the approach described in']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
[
14274,
14280
]
],
"text": "['BHPS10']",
"type": "data",
"id": "T133"
},
{
"offsets": [
[
14461,
14547
]
],
"text": "['we need to determine the interaction between the rigid clips, the cloth, and the cords']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
14397,
14459
]
],
"text": "['to measure the complete answer that a simulator should predict']",
"type": "data",
"id": "T135"
},
{
"offsets": [
[
14549,
14617
]
],
"text": "['The clips are produced, using rapid prototyping, with embedded codes']",
"type": "background_claim",
"id": "T136"
},
{
"offsets": [
[
14627,
14709
]
],
"text": "['that allow us to determine their identity, position, and orientation automatically']",
"type": "background_claim",
"id": "T137"
},
{
"offsets": [
[
14620,
14625
]
],
"text": "['Fia05']",
"type": "data",
"id": "T138"
},
{
"offsets": [
[
14711,
14831
]
],
"text": "['The area of cloth occluded by the clips is used to automatically determine which cloth vertices are clamped by each clip']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
14836,
14888
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],
"text": "['will therefore be constrained to it in the simulator']",
"type": "own_claim",
"id": "T140"
},
{
"offsets": [
[
14890,
14983
]
],
"text": "['The vision system also finds the cords in the images and triangulates a 3D line for each cord']",
"type": "own_claim",
"id": "T141"
},
{
"offsets": [
[
14985,
15068
]
],
"text": "['A few user scribbles on an input image indicate which cords are affecting each clip']",
"type": "own_claim",
"id": "T142"
},
{
"offsets": [
[
16762,
16853
]
],
"text": "['The set of deformations to measure is motivated by the goals of the parameter fitting stage']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
16855,
16864
]
],
"text": "['Section 5']",
"type": "data",
"id": "T144"
},
{
"offsets": [
[
16867,
16950
]
],
"text": "['to fit model parameters for stretch, shear and bending that best describe the cloth']",
"type": "data",
"id": "T145"
},
{
"offsets": [
[
16956,
17026
]
],
"text": "['to validate the parameter fits by comparing against other measurements']",
"type": "data",
"id": "T146"
},
{
"offsets": [
[
17028,
17269
]
],
"text": "['To reduce the risk of falling into local minima during parameter fits, we have designed deformation sequences that produce near-isolated strains, and allow estimating stretch, shear and bending properties in a separate and incremental manner']",
"type": "own_claim",
"id": "T147"
},
{
"offsets": [
[
17391,
17434
]
],
"text": "['we relax the requirement of uniform strains']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
17349,
17389
]
],
"text": "['our full 3D deformation capture solution']",
"type": "data",
"id": "T149"
},
{
"offsets": [
[
17436,
17579
]
],
"text": "['To isolate stretching we perform a uni-axial tension experiment, with forces applied to two long bar clips attached to either side of the cloth']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
17585,
17606
]
],
"text": "['Figure 4 , 2nd column']",
"type": "data",
"id": "T151"
},
{
"offsets": [
[
18776,
18867
]
],
"text": "['our acquisition system is able to recover the 3D cloth geometry including temporal tracking']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
18913,
18974
]
],
"text": "['tracked 3D clip locations, and individual 3D force directions']",
"type": "own_claim",
"id": "T153"
},
{
"offsets": [
[
19017,
19132
]
],
"text": "['our method presents the first system able to record such extensive information about the behavior of a cloth sample']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
18750,
18759
]
],
"text": "['Figures 4']",
"type": "data",
"id": "T155"
},
{
"offsets": [
[
18764,
18765
]
],
"text": "['5']",
"type": "data",
"id": "T156"
},
{
"offsets": [
[
18869,
18910
]
],
"text": "['illustrated with an overlaid checkerboard']",
"type": "data",
"id": "T157"
},
{
"offsets": [
[
18976,
18996
]
],
"text": "['shown as green lines']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
19692,
19726
]
],
"text": "['The multi-view stereo algorithm of']",
"type": "background_claim",
"id": "T159"
},
{
"offsets": [
[
19751,
19836
]
],
"text": "['is among the most accurate available according to the Middlebury evaluation benchmark']",
"type": "background_claim",
"id": "T160"
},
{
"offsets": [
[
19744,
19749
]
],
"text": "['BBH08']",
"type": "data",
"id": "T161"
},
{
"offsets": [
[
19838,
19911
]
],
"text": "['It is difficult to quantify the accuracy of the temporal flow computation']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
19917,
20009
]
],
"text": "['it can be visualized by compositing the reconstructed deformation on top of the input images']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
20015,
20033
]
],
"text": "['accompanying video']",
"type": "data",
"id": "T164"
},
{
"offsets": [
[
20110,
20256
]
],
"text": "['The largest source of error in measuring the force indirectly through the cord is the internal friction in the cord as it bends around the pulleys']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
20264,
20314
]
],
"text": "['introduces an artificial hysteresis of about 0.1 N']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
20036,
20102
]
],
"text": "['The raw repeatability of our force sensors is about 3 millinewtons']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
21360,
21403
]
],
"text": "['Most elastic cloth models separate membrane']",
"type": "background_claim",
"id": "T168"
},
{
"offsets": [
[
21430,
21462
]
],
"text": "['and bending deformation energies']",
"type": "background_claim",
"id": "T169"
},
{
"offsets": [
[
21411,
21418
]
],
"text": "['stretch']",
"type": "data",
"id": "T170"
},
{
"offsets": [
[
21423,
21428
]
],
"text": "['shear']",
"type": "data",
"id": "T171"
},
{
"offsets": [
[
22018,
22099
]
],
"text": "['We have evaluated three models for membrane deformation that fit this description']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
22101,
22115
]
],
"text": "['spring systems']",
"type": "data",
"id": "T173"
},
{
"offsets": [
[
22117,
22142
]
],
"text": "['the soft constraint model']",
"type": "data",
"id": "T174"
},
{
"offsets": [
[
22173,
22213
]
],
"text": "['and the diagonalized St.Venant-Kirchhoff']",
"type": "data",
"id": "T175"
},
{
"offsets": [
[
22221,
22226
]
],
"text": "['model']",
"type": "data",
"id": "T176"
},
{
"offsets": [
[
22166,
22170
]
],
"text": "['BW98']",
"type": "data",
"id": "T177"
},
{
"offsets": [
[
22245,
22251
]
],
"text": "['VMTF09']",
"type": "data",
"id": "T178"
},
{
"offsets": [
[
22255,
22277
]
],
"text": "['and two bending models']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
22279,
22293
]
],
"text": "['spring systems']",
"type": "data",
"id": "T180"
},
{
"offsets": [
[
22298,
22345
]
],
"text": "['the edge-based bending model in Discrete Shells']",
"type": "data",
"id": "T181"
},
{
"offsets": [
[
22347,
22353
]
],
"text": "['GHDS03']",
"type": "data",
"id": "T182"
},
{
"offsets": [
[
22734,
22804
]
],
"text": "['not all force models define the quantities below explicitly as strains']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
22809,
22864
]
],
"text": "['they often rely on the resolution of the discretization']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
22869,
22946
]
],
"text": "['they differ simply by scale factors that can be embedded in the stiffness k i']",
"type": "data",
"id": "T185"
},
{
"offsets": [
[
24324,
24435
]
],
"text": "['approximate the standard StVK model zeroing out off-diagonal terms in the matrix that relates strain and stress']",
"type": "background_claim",
"id": "T186"
},
{
"offsets": [
[
24316,
24322
]
],
"text": "['VMTF09']",
"type": "data",
"id": "T187"
},
{
"offsets": [
[
24451,
24558
]
],
"text": "['in the diagonalized StVK, each membrane stress component depends only on its corresponding strain component']",
"type": "background_claim",
"id": "T188"
},
{
"offsets": [
[
24437,
24443
]
],
"text": "['σ = Eε']",
"type": "data",
"id": "T189"
},
{
"offsets": [
[
24560,
24574
]
],
"text": "['σ s,i (ε s,i )']",
"type": "data",
"id": "T190"
},
{
"offsets": [
[
25418,
25583
]
],
"text": "['discovered concurrently the appropriate weighting of the angle change in order to model homogeneous bending on irregular triangle meshes with a homogeneous stiffness']",
"type": "background_claim",
"id": "T191"
},
{
"offsets": [
[
25410,
25415
]
],
"text": "['BMF03']",
"type": "data",
"id": "T192"
},
{
"offsets": [
[
25382,
25388
]
],
"text": "['GHDS03']",
"type": "data",
"id": "T193"
},
{
"offsets": [
[
25811,
25933
]
],
"text": "['With our separation of weft-, warpand diagonal-bending to capture anisotropy, the bending models in Discrete Shells and by']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
25960,
26005
]
],
"text": "['are equivalent up to a stiffness scale factor']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
25954,
25958
]
],
"text": "['BW98']",
"type": "data",
"id": "T196"
},
{
"offsets": [
[
26783,
26837
]
],
"text": "['propose a general nonlinear stress-strain relationship']",
"type": "background_claim",
"id": "T197"
},
{
"offsets": [
[
26902,
26973
]
],
"text": "['the same construction can easily be built on any of our selected models']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
26775,
26781
]
],
"text": "['VMTF09']",
"type": "data",
"id": "T199"
},
{
"offsets": [
[
27466,
27603
]
],
"text": "['The key question of how well a given model describes a particular piece of cloth is answered by fitting the model to the measurement data']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
27605,
27742
]
],
"text": "['adjusting its parameters to minimize the difference between the model’s predictions and the measured behavior, both in position and force']",
"type": "data",
"id": "T201"
},
{
"offsets": [
[
27874,
27980
]
],
"text": "['In principle all parameters of a cloth model can be fit to a sufficiently rich single deformation sequence']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
27986,
28040
]
],
"text": "['this can result in a problem fraught with local minima']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
28075,
28214
]
],
"text": "['we have designed an incremental optimization procedure that fits model parameters a few at a time using the isolated deformations described']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
28218,
28229
]
],
"text": "['Section 3.2']",
"type": "data",
"id": "T205"
},
{
"offsets": [
[
28054,
28073
]
],
"text": "['achieve stable fits']",
"type": "data",
"id": "T206"
},
{
"offsets": [
[
29000,
29019
]
],
"text": "['we wish to know the']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
29032,
29184
]
],
"text": "['stress-strain curves for the deformation components of a cloth model, such that a simulated cloth matches known positions and forces as well as possible']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
[
28949,
28998
]
],
"text": "['Given a set of captured static deformation frames']",
"type": "data",
"id": "T209"
},
{
"offsets": [
[
29607,
29716
]
],
"text": "['the net force on the clips, produced by cord forces, gravity, and forces from fixed cloth nodes, must be zero']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
29594,
29605
]
],
"text": "['equilibrium']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
30997,
31044
]
],
"text": "['the clip-parallel forces are dominated by shear']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
[
31052,
31099
]
],
"text": "['clip-orthogonal forces are dominated by stretch']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
30959,
30995
]
],
"text": "['situations of near-homogeneous shear']",
"type": "data",
"id": "T214"
},
{
"offsets": [
[
33715,
33807
]
],
"text": "['during inter∂x n mediate iterations, the stiffness matrix may not always be well conditioned']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
34013,
34078
]
],
"text": "['we also use this ∂x n modified stiffness matrix in the outer loop']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
33990,
34011
]
],
"text": "['improved conditioning']",
"type": "data",
"id": "T217"
},
{
"offsets": [
[
34323,
34617
]
],
"text": "['The nonlinearity of cloth deformation, together with the complex interplay of various deformation components in the resulting forces and positions, make the optimization problem above extremely complex in the general case, prone to falling in local minima and sensitive to initialization values']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
34628,
34723
]
],
"text": "['we largely alleviate these issues with the design of the five isolated deformation measurements']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
34750,
34844
]
],
"text": "['which allow us to separately fit stiffness curves for the six deformation components described']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
34861,
34913
]
],
"text": "['following an incremental parameter fitting procedure']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
34737,
34748
]
],
"text": "['Section 3.2']",
"type": "data",
"id": "T222"
},
{
"offsets": [
[
34848,
34859
]
],
"text": "['Section 4.1']",
"type": "data",
"id": "T223"
},
{
"offsets": [
[
37259,
37319
]
],
"text": "['The measurement shows the typical behavior of a woven fabric']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
37321,
37404
]
],
"text": "['a nonlinear curve with increasing stiffness for higher strain, and large hysteresis']",
"type": "data",
"id": "T225"
},
{
"offsets": [
[
38142,
38194
]
],
"text": "['The results are too numerous to include in the paper']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
38196,
38429
]
],
"text": "['we refer the reader to the supplementary material, which illustrates the behavior of the nonlinear orthotropic variant of all three models for all four fabrics, and the behavior of the variants of the Soft Constraints model for denim']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
39131,
39203
]
],
"text": "['The four selected fabrics span a large range of possible cloth behaviors']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
39220,
39278
]
],
"text": "['#12 is isotropic and very compliant in stretch and bending']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
39280,
39348
]
],
"text": "['#4 is also isotropic, very stiff in stretch but compliant in bending']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
39350,
39431
]
],
"text": "['#14 is stiff and quite isotropic in stretch, but extremely anisotropic in bending']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
39482,
39516
]
],
"text": "['#18 is anisotropic both in stretch']",
"type": "own_claim",
"id": "T232"
},
{
"offsets": [
[
39545,
39559
]
],
"text": "['and in bending']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
39438,
39475
]
],
"text": "['33/1 stiffness ratio in weft and warp']",
"type": "data",
"id": "T234"
},
{
"offsets": [
[
39523,
39543
]
],
"text": "['10/1 stiffness ratio']",
"type": "data",
"id": "T235"
},
{
"offsets": [
[
39566,
39586
]
],
"text": "['13/1 stiffness ratio']",
"type": "data",
"id": "T236"
},
{
"offsets": [
[
39722,
39851
]
],
"text": "['All four fabrics show similar hysteresis behavior, with loading-to-unloading stretch stiffness ratios ranging from 1.4/1 to 1.8/1']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
39589,
39658
]
],
"text": "['The maximum stretch stiffness for #4 is 250 times higher than for #12']",
"type": "own_claim",
"id": "T238"
},
{
"offsets": [
[
39666,
39720
]
],
"text": "['#14 is 10 times stiffer in shear than any other fabric']",
"type": "own_claim",
"id": "T239"
},
{
"offsets": [
[
39853,
39910
]
],
"text": "['Sample #12 is nearly linear in the test deformation range']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
39918,
39962
]
],
"text": "['all other three fabrics exhibit nonlinearity']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
39979,
40068
]
],
"text": "['nonlinearity may arise in some deformation modes but not in others, with no clear pattern']",
"type": "own_claim",
"id": "T242"
},
{
"offsets": [
[
40086,
40190
]
],
"text": "['all three cloth models fit nicely to the average of the hysteresis bands, even in highly nonlinear cases']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
40074,
40084
]
],
"text": "['stretching']",
"type": "data",
"id": "T244"
},
{
"offsets": [
[
40192,
40242
]
],
"text": "['The fitting residual is larger for stiffer fabrics']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
40248,
40317
]
],
"text": "['the nonlinear orthotropic model variants fit anisotropic fabrics best']",
"type": "own_claim",
"id": "T246"
},
{
"offsets": [
[
40338,
40461
]
],
"text": "['linear and/or isotropic variants reach a reasonable compromise but are not always able to remain inside the hysteresis band']",
"type": "own_claim",
"id": "T247"
},
{
"offsets": [
[
40477,
40542
]
],
"text": "['the fitting force residual is larger for #14, the stiffest fabric']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
[
40467,
40475
]
],
"text": "['shearing']",
"type": "data",
"id": "T249"
},
{
"offsets": [
[
40656,
40691
]
],
"text": "['the Springs model deviates at times']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
40544,
40586
]
],
"text": "['Across models, the Soft Constraints and St']",
"type": "own_claim",
"id": "T251"
},
{
"offsets": [
[
40588,
40648
]
],
"text": "['VK models fit to the average of the shearing hysteresis band']",
"type": "own_claim",
"id": "T252"
},
{
"offsets": [
[
40706,
40729
]
],
"text": "['no forces are available']",
"type": "own_claim",
"id": "T253"
},
{
"offsets": [
[
40697,
40704
]
],
"text": "['bending']",
"type": "data",
"id": "T254"
},
{
"offsets": [
[
40839,
40931
]
],
"text": "['The fitting residual is similar for all fabrics, but distinctly higher for the Springs model']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
40933,
41054
]
],
"text": "['Often, the residual is dominated by a difference in curl near the edge of the sample, while the overall shape is well fit']",
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41248,
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41286,
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"text": "['St.']",
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42009,
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41432,
41458
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"text": "['In the three other samples']",
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{
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41469,
41523
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"text": "['the force-displacement plot of the corner pulling test']",
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42061,
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42514,
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"text": "['even a small change in material properties may lead to distant stable configurations in the L 2 sense']",
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43191,
43270
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"text": "['the traditional L 2 metric is not appropriate for evaluating error in this case']",
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43272,
43383
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"text": "['The discontinuity of stable configurations is also the cause of flickering and twitches in some of our examples']",
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43385,
43521
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"text": "['The Springs model exhibits the worst fitting quality in shearing force-displacement curves, and the highest fitting residual for bending']",
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43547,
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"text": "['In contrast to continuum models, complex parameter tuning has often been regarded as a caveat of mass-spring models']",
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43962
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43993,
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"text": "['VK models produce results with very similar quality, which is expected as the models present only subtle differences']",
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44195,
44205
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44221
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44430,
44544
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"text": "['we chose clip-parallel forces as objective function to minimize the effect of stretch errors on shear optimization']",
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44418
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"text": "['We indeed identified stretch stiffening in the shearing deformations']",
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44565,
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44738
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44740,
44778
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"text": "['An extension to the nonlinear model of']",
"type": "own_claim",
"id": "T295"
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{
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44800,
44835
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"text": "['could help alleviate these problems']",
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44792,
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"text": "['WRO11']",
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"text": "['This paper has demonstrated a novel system for observing cloth behavior, including complete information about deformation and forces, and a new method for fitting and evaluating cloth models using the measurements']",
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46085,
46126
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"text": "['it captures detailed geometry information']",
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46128,
46179
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"text": "['it is different from previous cloth capture systems']",
"type": "own_claim",
"id": "T301"
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46188,
46226
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"text": "['it captures complete force information']",
"type": "data",
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46231,
46268
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"text": "['measures deformations of a 3D surface']",
"type": "data",
"id": "T303"
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46270,
46399
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"text": "['The combination of very complete position and force information provides an unprecedented view into the complex behavior of cloth']",
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46401,
46478
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"text": "['Our measurement setup offers very accurate control over membrane deformations']",
"type": "own_claim",
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"text": "['the bending tests require manual intervention and are thus less precise']",
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"text": "['some cloth materials']",
"type": "own_claim",
"id": "T307"
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46719
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"text": "['tend to curl up at free boundaries']",
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"text": "['the bending tests are most accurate for samples with straight edges']",
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"text": "['In order to eliminate these problems, we would like to investigate alternative ways of controlling bending deformations in the future']",
"type": "own_claim",
"id": "T310"
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46933
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"text": "['The data from our experiments shows some of the limitations of current models']",
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"text": "['resulting in quite large errors for any given point in the experiment']",
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"text": "['cloth is clearly far from elastic']",
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A27 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A27_F07_Highly_Adaptive_Liquid_Simulations__on_Tetrahedral_Meshes_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/2461912.2461982
<Title>Highly Adaptive Liquid Simulations on Tetrahedral Meshes</Title>
Ryoichi Ando ∗ Nils Thürey † Kyushu University ScanlineVFX
∗
∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
Figure 1: Our adaptive simulation framework allows us to efficiently simulate highly detailed splashes on large open surfaces. In this case, maximum BCC mesh resolutions from 8 to 1024 cells were used, leading to strong horizontal grading along the surface.
<Abstract>We introduce a new method for efficiently simulating liquid with extreme amounts of spatial adaptivity. Our method combines several key components to drastically speed up the simulation of largescale fluid phenomena: We leverage an alternative Eulerian tetrahedral mesh discretization to significantly reduce the complexity of the pressure solve while increasing the robustness with respect to element quality and removing the possibility of locking. Next, we enable subtle free-surface phenomena by deriving novel second-order boundary conditions consistent with our discretization. We couple this discretization with a spatially adaptive Fluid-Implicit Particle (FLIP) method, enabling efficient, robust, minimally-dissipative simulations that can undergo sharp changes in spatial resolution while minimizing artifacts. Along the way, we provide a new method for generating a smooth and detailed surface from a set of particles with variable sizes. Finally, we explore several new sizing functions for determining spatially adaptive simulation resolutions, and we show how to couple them to our simulator. We combine each of these elements to produce a simulation algorithm that is capable of creating animations at high maximum resolutions while avoiding common pitfalls like inaccurate boundary conditions and inefficient computation.</Abstract>
I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling Keywords: motion editing, motion signal-processing, spacetime constraints, motion capture.
Chris Wojtan ‡ IST Austria
Links:
DL PDF
<H1>1 Introduction</H1>
This paper aims to produce fluid simulations with a high degree of spatial adaptivity. We desire to enable a simulator to focus its computational resources on the visually interesting regions of a fluid flow, while remaining computationally efficient and avoiding common artifacts due to a spatially adaptive pressure solve. Previous approaches have made great strides towards this goal, but they often exhibit visual artifacts, a lack of computational robustness, or an unacceptably hefty computational expense. The groundbreaking work of
Losasso et al. [2004]
introduced an octree for spatial adaptivity, but it suffers from spurious flows at T-junctions. Finite volume methods
[Batty et al. 2010]
repair these spatial artifacts at the expense of solving a significantly larger system of equations and sacrificing computational stability near poorly-shaped elements. Furthermore, many existing methods still are not truly spatially adaptive in the sense that their computational complexity is still tied to a uniform grid or spatial parameter. We introduce a combination of techniques that successfully makes adaptive fluid simulation practical at large scales. We first reduce memory and computational costs by switching from a finite volume method to a discretization with a significantly smaller linear system for the pressure solve, which has the side effect of increasing the simulator’s robustness to poor-quality elements and effectively preventing locking artifacts. We next derive second-order Dirichlet boundary conditions consistent with our discretization to benefit from the subtle surface dynamics associated with an accurate pressure solve. We combine this robust and efficient tetrahedral meshbased fluid simulator with a spatially adaptive method for sampling particles for FLIP-based velocity advection, giving us a method free from any single spatial resolution. In addition to our adaptive FLIP simulator, we also introduce a new method for computing a surface from a distribution of particles with variable radii. We found that this method out-performs previous methods in cases of extreme spatial adaptivity by exhibiting smoother surfaces without sacrificing detail. Our fluid simulator works well with spatially adaptive tetrahedral meshes, but it is another question to decide exactly how these adaptive meshes should be generated. We investigate various methods for generating these adaptive meshes by experimenting with several sizing functions, allowing us to precisely dictate where simulation detail should occur. Some examples are a surface curvature-based metric that adds detail only where needed on the fluid surface, a turbulence metric that adds detail only where interesting fluid motion occurs, and a visibility metric that adds detail only in front of a virtual camera. Concretely, the contributions of our work are: • a novel tetrahedral discretization of the pressure projection step that is efficient to solve and robust to poor-quality elements; • an accurate treatment of second-order boundary conditions within the tetrahedral mesh; • a new technique for extracting a smooth surface from particles with varying radii; • and the inclusion of a flexible sizing function to focus computational resources on important areas of the flow with minimal overhead. These contributions work together to produce a practical fluid simulator that exhibits low computational and memory complexity, fewer visual artifacts, and a high effective simulation resolution.
<H1>2 Related Work</H1>
Our work is based on the Fluid-Implicit Particle (FLIP) method introduced to the computer graphics community by Zhu and Bridson [2005]
, which arguably represents the state-of the art for detailed and robust liquid simulations.
The algorithm still follows the general ideas of the Stable Fluid solver
[Stam 1999], and can be readily combined with second-order treatment of free surface boundary conditions
[Enright et al. 2003]. FLIP derives its success from the fact that it uses particles to compute an accurate, nondiffusive transport of flow quantities, in combination with a gridbased solve to accurately enforce constraints for mass conservation.
The FLIP algorithm is heavily used in the special effects industry, and recent advances have introduced accurate coupling with obstacles
[Batty et al. 2007], highly viscous materials
[Batty and Bridson 2008], and two-phase flows
[Boyd and Bridson 2012]. Traditionally, Cartesian grids are very popular for fluid simulations. The Marker-And-Cell (MAC) approach
[Harlow and Welch 1965]
, which stores velocity components at cell faces and pressure samples at cell centers, results in discretizations with good properties in terms of stability and accuracy. An inherent difficulty is that simulations on regular grids become prohibitively expensive for large resolutions. Thus, many works have proposed methods to focus the computations on regions that are of particular interest. One example are octrees, which were used by Losasso et al. [2004; 2005] to refine the computational grid in a controllable way. This approach, however, suffers from numerical diffusion and an inconsistent discretization near the tree’s T-junctions. Targeting a similar direction as our work,
Hong et al. [2009] and Ando et al. [2012]
have demonstrated methods to adapt the resolution of FLIP particles in a simulation. Both methods, in contrast to ours, focus on static computational grids and are restricted to smaller differences in particle size. Although Cartesian grids are widely used, they are limited in their flexibility to adapt to a simulation setup. Because of this, tetrahedral grids are popular for methods targeting adaptivity. In combination with a suitable method to discretize the problem at hand, they allow for very flexible computational grids. One example is the work of Klingner et al.
[Klingner et al. 2006]
which demonstrated the use of a Stable Fluids based solver for tetrahedral grids conforming to object boundaries. Another example is the non-linear fluid solver developed by
Mullen et al. [2009]
, which leads to an energy conserving solve. Unlike these methods, we make use of a non-conforming grid with Body-Centered Cubic (BCC) lattices. These meshes were also used by Chentanez et al. [2007] and by Batty et al. [2010]
for liquid simulations. We will denote this class of algorithms as Finite Volume Methods (FVM). These methods are primarily suitable for uniformly sampled particles, and we will demonstrate in Section 7 that their placement of pressure samples at tetrahedral circumcenters leads to numerical problems in combination with graded BCC meshes. Another direction of research performs fluid simulations based on arbitrary elements. Clausen et al. [2013] and Misztal et al. [2010]
have proposed a method to simulate liquids with a computational grid conforming to a triangulation of a liquid surface. Both methods lead to an increased computational cost in comparison to the more efficient tetrahedral BCC meshes. Sin et al. [2009]
proposed an alternative method for hybrid Lagrangian-Eulerian solvers which combines a Voronoi-based pressure solver and particles. Using this Vornoi-based approach for tetrahedral meshes would yield a pressure matrix similar to ours. Like our method,
Brochu et al. [2010]
used this discretization in combination with embedded second-order boundary conditions. Both of these approaches discretize velocities with per-face flux values, while we store velocity vectors at cell barycenters. Adaptive simulations have also been explored in the context of SPH simulations without Eulerian grids. The work of
[Adams et al. 2007]
shares similarities with our approach, as it is able to simulate a wider range of particle radii, and it proposes a surface reconstruction method in the adaptive setting. We will show in Section 5 that our surface creation method results in surfaces with fewer visual artifacts. Additionally, a robust and efficient method for adaptive SPH simulations was introduced by
Solenthaler et al. [2011]
, but this work primarily targets the coupling of two different particle resolutions. Several other methods have been proposed to reconstruct smooth surfaces around collections of particles without orientation. One approach that is commonly used is to compute a signed distance function with averaged particle radii and centroids
[Zhu and Bridson 2005]. A variant of this approach, taking into account information about the spatial variance of the particle’s neighborhood was proposed by
Yu et al. [2010]. Both methods primarily target particles with constant radius. More recently, a level-set based method was proposed that computes a constrained optimization with bihar- monic smoothing
[Bhattacharya et al. 2011]. However, such an optimization would be complicated to apply in our unstructured setting. In contrast to these methods, our approach for surface creation computes the union of convex hulls around triplets of particles, which leads to a smooth and closed surface around a collection of arbitrarily sized particles.
<H1>3 Fluid Solver</H1>
The aim of our method is to solve the Navier-Stokes equations, which for incompressible, Newtonian, inviscid flows can be written as ρDu/Dt = − p + f , with the additional constraint · u = 0 to enforce a divergence-free velocity field. Here, u, p and f denote velocity, pressure and external forces, respectively, while D/Dt denotes the material derivative. The density ρ is constant in our case. We solve these equations using operator splitting
[Stam 1999], and a level set φ(x) = 0 defines the position of the liquid-gas interface. Spatial Discretization An inherent strength of the FLIP algorithm is its hybrid nature. The motion of the fluid is computed in a Lagrangian manner using particles, while the pressure projection step is computed on an Eulerian grid. We will now describe how we compute the pressure projection using a tetrahedral discretization. This projection of the velocities into a divergence-free state can be formulated as the Poisson problem
1
∆t 2 p = · u ∗ , ρ
where u ∗ denotes an intermediate velocity after the advection. In the following, however, we prefer an alternate view that looks at this problem from an energy minimization perspective: we want to compute the minimal change in kinetic energy necessary to reach a divergence-free state of the flow similar to
[Batty et al. 2007]. This can be formulated as:
2
p = arg min 1 ||u ∗ − ∆t p|| 2 ρdV p Ω 2 ρ
Here, Ω represents the domain of the computational grid, and we choose to discretize this space using tetrahedral cells. This has the advantage of giving us a natural way to handle cells of different size, while yielding a consistent discretization of the differential operators involved. We store pressure samples at the nodes of the tetrahedral mesh, while velocities are stored at cell centers. This configuration is illustrated in Figure 2 . Note that by assuming a piece-wise constant velocity and a linear change of pressure within a cell, this setup results in a constant pressure gradient per tetrahedron, by construction. In the following, we denote the number of cells with m and the number of nodes with n, and we indicate discretized quantities with caret notation. Based on this representation we can discretize Eq. (2) with
3
m p = arg min 1 || u ˆ i ∗ − ∆t [ ] p|| 2 ρV i , p ˆ 2 ρ i
where we denote the volume of a cell with V i and the discretized gradient operator with [ ]. It consists of a m×n matrix, computing a per-tetrahedron gradient from nodal values. Consequently, we define the divergence operator to be the transpose of the discretized gradient. Before we go ahead to define [ ], we want to outline the rest of the steps for our pressure solve. We solve equation Eq. (3) with the commonly used least squares technique, yielding
4
∆t T T ∗ [ ] V [ ] p = [ ] V u ˆ , ρ
Figure 2:
Our discretization compared to previous approaches: the MAC grid stores velocity components normal to faces and pressure values at the center. The FVM discretizations follow along these lines and store velocities normal to faces and pressure values at the circumcenter of a cell. In our approach we store a 3component velocity vector at the barycenter of a cell and pressure values at each node.
where V denotes a matrix containing the V i as diagonal entries. The [ ] T [ ] matrix-matrix multiplication results in a square n × n matrix, which is symmetric and positive definite. In the following, we will denote the matrix on the left hand side of Eq. (4) as A. Given appropriate boundary conditions, we can use standard tools, such as a commonly used pre-conditioned conjugate gradient solver to compute a solution (we use the one suggested in
[Bridson 2008]). Now, all that is left to construct the left-hand side matrix and the right-hand side terms for Eq. (4) is to define [ ]. As we assume a linear change of the pressure for each cell, we can use simple barycentric interpolation to retrieve the pressure p at a position inside a cell. Given the nodal pressure values p 1..4 and barycentric weights σ 1..4 this means p = σ 1 p 1 + σ 2 p 2 + σ 3 p 3 + σ 4 p 4 . In line with finite element methods using linear elements, we define the gradient based on the partial derivatives of the barycentric interpolation. E.g., the first component of the gradient for a cell is computed with
5
∂ ∂σ 1 ∂σ 2 ∂σ 3 ∂σ 4 p = p 1 + p 2 + p 3 + p 4 . ∂x ∂x ∂x ∂x ∂x
To set up the final linear system of Eq. (4) for the pressure solve, we loop over all tetrahedra to compute the derivatives of the barycentric interpolation, adding their contributions to the global matrix. In contrast to previous work, our pressure solve is a linear system that has n degrees of freedom, n being the number of nodes in the tetrahedral mesh. For our BCC mesh, n is in practice smaller than the number of tetrahedra m (by a factor of 6 on average). A direct implication of this smaller linear system is that it is faster to solve. A second, less obvious implication of the smaller linear system is that it effectively prevents artifacts known as locking. These artifacts are commonly observed in finite element methods for problems in elasticity. Different methods have been proposed to circumvent these problems, e.g., using linear elements for pressure instead of piece-wise constant ones
[Irving et al. 2007]. Other works explicitly smooth the pressure field to reduce locking problems
[Misztal et al. 2010]. In general, locking can be observed if the pressure basis can represent more, and higher-frequency, functions than the basis for the velocity.
Thus, choosing a more restrictive basis for pressure, as in
[Irving et al. 2007], or explicitly removing high-frequency information from the pressure
[Misztal et al. 2010]
, reduces the chance of locking. Our method, by construction, has more degrees of freedom for representing velocity fields than pressure fields. Although we cannot prove that a local configuration over-constraining the velocities will never occur, the larger number of degrees of freedom for our velocities effectively prevents locking artifacts, and we have not encountered any in our tests. Boundary Conditions Second-order boundary conditions are a central component for accurate and visually appealing simulations with non-conforming grids. Achieving second-order accuracy for obstacle boundary conditions is straightforward with our discretization: we can rely on the formulation of previous work
[Batty et al. 2007], and set the volume of a cell V i in Eq. (4) to the volume that is filled with fluid. For the free surface, we have to ensure that the Dirichlet boundary condition p = 0 is satisfied at the interface position. Usually, this means computing a pressure value for nodes outside of the liquid so that a linear interpolation along an edge of a cell gives zero at the correct position
[Enright et al. 2005; Lew and Buscaglia 2008]. Considering two pressure samples along an edge, we’ll denote values inside the air with a G subscript, and values inside the liquid with an L subscript in the following. For a Cartesian MAC grid, the ghost pressure value p G is given by p G = p L φ G /φ L . In our case, however, this approach does not yield the desired result. The reason is that our velocity samples are not in line with the direct connections of the pressure samples – they are not locally orthogonal to each other. Instead, we have to ensure the boundary conditions result in the correct pressure value at the cell center. In the following we will show how to derive suitable free-surface boundary conditions to ensure second-order accuracy within our framework. In order to achieve accurate and smooth surface motions with our method, we compute the ghost pressure values p G with a linear combination of liquid pressure values as:
6
p G = w 1 p 1 + w 2 p 2 + w 3 p 3 ,
where w n and p n denote unknown coefficients and adjacent liquid pressures in the same tetrahedron. Note that for p i that are not inside of the liquid, we set w i = 0. In line with the traditional ghost fluid method, we define p G uniquely for each tetrahedron. To handle the most general case, let’s suppose that p 1..3 are all liquid pressure values. Once we have a value for p G , we can compute a pressure gradient for the tetrahedron and update the velocity at its center with:
7
u ˆ new = u ˆ ∗ − ∆t [ ] p G p 1 p 2 p 3 T ρ
In order to do this we need to compute the coefficients w n . p G can be rewritten in terms of a barycentric interpolation of the three values in the liquid as:
8
p G = p L φ G / φ ̃ L
with φ ̃ L = θ 1 φ 1 + θ 2 φ 2 + θ 3 φ 3 , and p L = θ 1 p 1 + θ 2 p 2 + θ 3 p 3 . Here the θ n are a set of barycentric coordinate coefficients such that θ 1 +θ 2 +θ 3 = 1, and a tilde superscript denotes a value interpolated with the barycentric weights. Substituting Eq.8 into Eq.6 yields
9
w n = θ n φ G / φ ̃ L .
That means the values w n are determined by those of the θ n coefficients, which we will compute in the following. Note that, theoretically, θ n could take any values as long as they add up to one. Before embedding the boundary conditions, the matrix entries of the pressure solve for a single tetrahedron are, according to Eq. (4), given by ∆t [ ] T V [ ]p = b. The computation of the ghost fluid ρ values is independent of the right-hand side b, so we will restrict the discussion to the left hand side. We denote the components of the local symmetric 4 × 4 matrix on the left hand side with: λ 1 α β γ
10
α λ 2 a b β a λ 3 c γ b c λ 4
Assuming, without loss of generality, that the first vertex is the one outside of the liquid volume, we embed the boundary condition into
M based on the w n coefficients. Then the first row of the system is changed to:
11
1 −w 1 −w 2 −w 3 p G 0 α λ 2 a b p 1 b 1 = . β a λ 3 c p 2 b 2 γ b c λ 4 p 3 b 3
We can extract two constraints for each θ n from this form, which, together with the barycentric coefficient constraint, give us a 3 × 3 matrix M that can be inverted analytically. A full derivation of these steps can be found in Appendix A. With this analytic expression we can compute the ghost pressure coefficients as:
12
φ G / φ ̃ L α w = α + β + γ β γ
This boundary condition ensures second-order accuracy while maintaining symmetry when it is assembled into the matrix of Eq. (4). If the quality of a tetrahedron is good, 0 < θ n < 1 is guaranteed. In this case, the resulting matrix is symmetric positivedefinite and can be easily inverted by the commonly used preconditioned conjugate gradient methods. However, positive off-diagonal terms of the matrix can result in values of θ outside of the range [0, 1], leading to an indefinite linear system. In these cases we consider the tetrahedron to have a poor quality. When using ghost fluid boundary conditions with a regular MAC grid, it is common practice to clamp small values in the denominator of Eq. (9) to prevent ill-conditioned pressure matrices. Effectively, this means reverting to first-order accuracy when second-order accuracy is intractable. We implement a similar step in our algorithm to overcome numerical problems resulting from badly shaped cells. We check whether the ghost fluid boundary conditions would violate diagonal dominance of an equation in our linear system. If we detect such a case, we smoothly transition to first order accuracy. Specifically, when we have computed M , we check if a resulting diagonal term M i,i is smaller than φλ 2..4 . Here, φ denotes a tolerance factor that we set to φ = 0.25. Whenever we detect such a case, we compute a coefficient k with
13
φ − 1 φ − 1 φ − 1 k = min λ 2 , λ 3 , λ 4 , w 2 α w 3 β w 4 γ
and multiply each w n with k when embedding. Note that this scaling does not break the symmetry of the resulting linear system. More specifically, for k = 1 this yields full second-order accuracy, while for badly shaped tetrahedra the resulting k = 0 means that we revert to the standard rounding strategy of a first order accurate method. With our BCC mesh, all regular BCC tetrahedra have very good quality and valid θ n values. The graded BCC tetrahedra, on the other hand, can be of lower quality and can require the use of Eq. (13). Luckily, in our tests these tetrahedra make up only a very small fraction of the mesh. Velocity Interpolation The FLIP advection step traces particles based on the velocities from the Eulerian grid. For this we need to construct a continuous velocity field based on the discrete values in our tetrahedral mesh. As we store velocities at the cell centers, the interpolation would ideally use the dual mesh consisting of the Voronoi cells of each node
[Brochu et al. 2010]. Unfortunately, performing interpolations within arbitrary Voronoi cells would be expensive and require a large amount of computation compared to the other steps of our simulator. Instead, we have found the following approach to yield high speed and good accuracy: we first interpolate the centered velocities to the nodes, similar to
[Chentanez et al. 2007]. Instead of interpolating these averaged values directly (which would result in smeared out motion), we temporarily subdivide the cells of our mesh by inserting a vertex at the center where we have an accurate velocity sample. We then perform barycentric interpolation based on these subdivided cells, ensuring a C 0 continuous velocity that retains the original velocities at cell centers. Note that these four smaller tetrahedra do not have to be stored explicitly. We construct them on the fly when a sample is requested from one of the original cells. Manipulating FLIP particles The spacing between FLIP particles may drift over time, and high-frequency errors may contribute to a bumpy surface. We combat these problems by directly manipulating particle positions. During each time step, we apply the position correction algorithm of
Ando et al. [2012]
; this algorithm essentially pushes each particle away from its neighbors to prevent clustering. We also introduce two special behaviors when the particles are close to the liquid surface (less than a distance of six times the particle radius). First, we impose the constraint that the position correction step may only move particles near the surface tangentially to the fluid interface. Secondly, particles near the surface may leave gaps when they spread out quickly. Our method naturally fills in these gaps by slightly pulling each particle towards the fluid interface. For particles near the interface, this pulling force acts in addition to the position correction. FLIP particles that partake in splashes and sprays can pose a significant burden on computational resources, especially in an adaptive framework like ours. This inefficiency stems from the fact that water droplets undergo extremely simple ballistic motion. Theoretically, we know that such a small region with purely free-surface boundary conditions will yield zero internal forces, so we simply detect individual FLIP particles that have no neighbors within six times their radius, remove them from the pressure solve, and accelerate them with gravity instead. When these particles eventually enter the neighborhood of other particles at some point in the future, we resume treating them like fluid by returning them to the pressure solve. This decision allows us to avoid aggressively refining the tetrahedral mesh in locations where the physical motion is uninteresting. Only a small percentage of the particles are simulated in this way, e.g., 1.7% on average for Figure 6 .
Figure 3: A simple geometric setup creating a big splash inside a container. This simulation, with a maximum resolution of 256 cells, took 1.3 minutes per frame on average.
<H1>4 Adaptivity</H1>
Our method achieves adaptivity by varying the mesh resolution over the computational domain. Our FLIP simulation performs computation on both a background volumetric mesh and on a set of particles. Given a sizing function that indicates the desired spatial level of detail, our method first creates a tetrahedral mesh with varying spatial resolution, and then it locally changes the particle density by splitting and merging operations. To compute the spatially-varying background grid, we start with the Delaunay tetrahedralization of a set of points distributed in a body-centered cubic lattice configuration.
In order to make the mesh resolution change over space, we use the octree-based grading method which was proposed by
Labelle and Shewchuk [2007] and later adopted in several adaptive simulation environments
[Chentanez et al. 2007; Wojtan and Turk 2008; Batty et al. 2010]. Similar to
[Batty et al. 2010]
, we generate a new tetrahedral mesh every ten time steps, instead of rebuilding the mesh on every consecutive step. Also, the tetrahedral mesh is only temporarily used for the pressure solver, so no information is transferred from one time step to the next by storing it on the grid. Thus, we do not worry about re-sampling data when computing a new tetrahedral mesh. We change the size and number of particles in our simulation with splitting and merging operations. For this, we modify the strategy of
Ando et al. [2012] to work within our framework: at each remeshing step, we loop through the particles and determine whether the resolution needs to be changed. If a particle is too small, then we merge it with its nearest neighboring particle, resulting in a particle whose radius is given by the combined volume of the two original particles. If a given particle is bigger than the desired size, then the particle is split in two. The two new particles are placed randomly within the original particle’s radius and redistributed with a heuristic that attempts to fill in nearby gaps: We first compute the 24 midpoints m i between this particle and its 24 nearest neighbors. Then we find the closest particle to each midpoint and store the squared distance as a weight ω i . The new particle’s position is equal to the weighted average of all nearby midpoints: x new = ω i m i / ω i . After a split or merge operation, the new particle’s velocity is computed using a volume-weighted average. Also, we must take care to ensure that particles close to the surface do not introduce interfacial bumps when they split or merge; whenever we create a new particle that is less than 1.25 times its radius away from the surface (through either a split or a merge event), we move it in the surface normal direction such that its sphere lies exactly tangent to the liquid interface. Numerical viscosity in fluid simulations is tightly coupled to the spatial resolution resolving the flow. We compensate for spatiallyvarying numerical viscosity caused by particles of various sizes in our simulation by adjusting the PIC/FLIP blending parameter in our FLIP simulation
[Bridson 2008]. Given quantities Q i,PIC and Q i,FLIP computed at particle i from PIC and FLIP simulations, respectively, the new quantity Q i is computed as a weighted blend between the two:
Figure 4: For the perfectly planar particle placement shown in (a), the methods [Zhu and Bridson 2005] (b), [Yu and Turk 2010] (c), and [Adams et al. 2007] (d) do not result in a flat surface, while our method produces the desired result (e).
14
ν∆t ν∆t Q i = r i 2 Q i,PIC + (1 − r i 2 )Q i,FLIP
where r i is the radius of particle i and ν is the viscosity of the flow. We found that this strategy adequately eliminates any artifacts due to spatially varying numerical viscosity. Sizing Functions We define the level of detail in our simulations with a spatially varying sizing function S(x). We have experimented with several different sizing functions depending on factors such as distance to a camera, distance to the liquid surface, curvature of the liquid surface, measures of fluid turbulence, and arbitrary analytical number fields. Our simulator is versatile enough to cope with any of these sizing functions, resulting in efficient simulations with highly variable levels of detail. E.g., Figure 1 showcases a simulation where smallest and largest cells differ by a factor of 128. In all of the examples in this paper, the sizing function is defined as a combination of five different metrics:
15
S(x) = max ( d(x), V (x, min(κ liquid (x), κ solid (x), e(x)) )
where x is the position of a point in space, and d(x) encodes the depth of the liquid by returning the absolute level set value of the liquid surface. This has the effect that motion near the surface has higher priority than motions far inside the bulk volume of the liquid. V (x, y) is a view-dependent function that returns the value y if x is within the camera’s visible region and returns the maximum particle radius r max (representing the minimum surface resolution) otherwise. The next two metrics are designed to prioritize geometric detail of the liquid surface and of obstacles by computing a desired resolution based on cuvature. κ liquid (x) returns 0.8 divided by the extrapolated curvature of the liquid interface. Similarly, κ solid (x) returns 1.6 W smooth (d solid , r max ) divided by the extrapolated curvature of the solid interface, where W smooth (x, h) is a smooth kernel function (1 − ||x|| 2 /h 2 ) 3 and d solid is the closest distance to the solid boundary. As a last component of our sizing function we found it beneficial to invest computational resources into keeping interesting motion of the flow field alive. Inspired by turbulence models (used e.g. in
[Pfaff et al. 2010]), we have found that the strain tensor of the flow field reliably indicates detailed motions, and we compute e(x) as 30 divided by the Frobenius norm of the fluid strain tensor computed from the velocity field.
<H1>5 Surface Representation</H1>
We also introduce a new method for computing an implicit surface from a set of particles. Given our set of FLIP particles with variable radii, we aim to implicitly represent the fluid surface by computing its signed distance function. Several useful methods for computing a surface from a collection of particles have been proposed in the past
[Zhu and Bridson 2005; Adams et al. 2007; Yu and Turk 2010]
, but they tend to produce undesirably bumpy surfaces when considering particles of highly variable radii ( Figure 4 ). In this section, we introduce a new strategy for computing an implicit surface from a set of particles of various sizes. The main idea is to approximate the fluid surface with the union of the convex hulls of each triplet of nearby particles close to the surface. For each set of three FLIP particles near the surface, the convex hull forms a thickened triangle shape with rounded edges ( Figure 5 ). We only consider particles that are less than a given distance apart, with the maximum distance equal to a constant scale factor l times the sum of the two particle radii. A small l shows more surface details, while a larger l tends to fill in small concavities. We used l = 2 for most of the simulations in this paper. We ultimately represent our surface as the union of all such local convex hull shapes, and the minimum signed distance from these shapes to a point in space defines the outer part of our level set function. In practice, we compute the local convex hull by finding the two outermost planes tangent to a set of three spheres. We efficiently compute the distance to these planes by analytically solving the polynomial system: ax 1 +by 1 +cz 1 +d = r 1 , ax 2 +by 2 +cz 2 +d = r 2 , ax 3 + by 3 + cz 3 + d = r 3 , a 2 + b 2 + c 2 = 1 . where r i and x i , y i , z i are the radius and x, y, z coordinates of particle i, respectively. a, b, c, d are the variables defining our plane with the signed distance function ax + by + cz + d = 0. Intuitively, the first three equations ensure that the plane is the right distance away from each particle with the normal facing away from them, and the final equation ensures that the plane equation is normalized to a distance function. These four equations represent the intersection of three hyperplanes and a hypercylinder in 4D {a, b, c, d} space. We solve this system analytically by first finding the line of common intersection of the first three equations, and then intersecting this line with the cylinder represented by the final equation. The system has two solutions, representing the top and bottom planes of our convex hull shape. The above calculation describes how to find the planar regions of the convex hull of a set of three spheres. By computing the conic and spherical convex hull facets ( Figure 5 , bottom right) in a similar fashion, we can easily compute the signed distance between this convex hull and a point in space. To evaluate our final level set value, we compute the minimum signed distance from a query point to all nearby convex hulls. We evaluate the level set on each of the vertices of our adaptive BCC mesh, and we extract a triangle mesh using a marching tetrahedra algorithm. We then perform a light mesh smoothing to increase the reliability of any curvature computations. The algorithm as described works perfectly for computing the level set outside of our particle surface, but it may lead to small gaps inside. To avoid the creation of holes, we temporarily reassign each particle’s radius: r i temp = max(r i , −kφ i ) where φ i is the particle’s stored level set value from the previous time step, and k is a constant set to 0.75 in our simulations. Using this temporary radius to compute the signed distance as described above will remove erroneous gaps inside the liquid. We need to compute liquid surfaces both for final visualization as well as for several calculations during the progress of our simulation. For the final visualization, we compute an especially highresolution BCC mesh from all of our particles and proceed with the algorithm above. The final surface creation is trivially parallelized, and takes around five minutes average per frame for all of our simulations. We attempt to speed up the surface creation routine used for simulation computations by computing on the moderate-resolution BCC mesh used for simulation and ignoring ballistic particles (Section 3). We compare our surface creation routine with a few existing methods in Figure 4 . Most previous algorithms perform poorly in this comparison because they were not designed for particles with varying radii.
Figure 5: The 2D version of our surface creation algorithm takes a pair of particles (top left) and computes their convex hull (bottom left). In 3D, we convert three nearby particles (top right) into a convex hull (bottom right) consisting of spherical (orange), conic (dark blue), and planar (light blue) segments.
<H1>6 Implementation</H1>
At this point we have described all of the components of our simulator. The resulting algorithm can be seen in Algorithm 1. In the beginning of each step (line 2), we typically compute the level-set for the current particle configuration as described in Section 5. We require the distance to the surface in several steps of our algorithm, so we store the level set values for each particle (line 3). When enough time has passed to trigger an update of the mesh, it becomes necessary to evaluate the sizing function. At this point, additional user-defined sizing functions could be computed as well. Having the information from the sizing functions ready, we create a new BCC mesh and perform particle merging and splitting. For mapping the particle velocities onto the grid (line 9) we use an SPH-like kernel function, which is weighted by the particle volume: max(v i (4r i 2 /d 2 − 1), 0), where v i is the particle volume and d is
Algorithm 1: One step of our simulation algorithm. 1 begin 2 Compute simulation surface S 3 Pre-compute φ for all particles 4 Correct particle positions x i 5 if Mesh update necessary then 6 Evaluate sizing function S(x) at x i 7 Build octree and BCC mesh 8 Merge and split particles 9 Compute mesh velocity u from particle velocities 10 Extrapolate u outside the liquid 11 Solve pressure p on the tetrahedral mesh, update u 12 Update particle velocities with gradient of p 13 Advect particles with u
the distance to the particle center normalized with its radius. The particle velocity update of line 12 uses barycentric interpolations as explained in Section 3. Likewise, the grid-based velocity extrapolation of line 10 uses the nodal velocities of Section 3. A time step is completed by performing the pressure projection and advecting the particles in the resulting divergence-free velocity field.
<H1>7 Results and Discussion</H1>
To evaluate the performance and robustness of our method in comparison to previous work we have performed an extensive series of tests. A selection of these can be found in the accompanying comparison video. One comparison that is particularly interesting is the one comparing our method to an FVM based simulation. Using a graded BCC mesh leads to problems with the latter, as the position of the circumcenter lies exactly on a face for the graded tetrahedrons. In the graded region, this can result in two pressure samples from adjacent tetrahedra being placed at the exact same position. To alleviate this problem,
[Batty et al. 2010]
propose to slightly offset the pressure samples from the faces. However, our implementation of their method exhibited slow convergence and the velocity artifacts despite this fix. The influence of the different components of our sizing function on the evolution of a simulation is difficult to depict with static images, so we refer to the accompanying video for a comparison. To evaluate the basis of our adaptive model without any influence of the camera dependent sizing function, we have simulated the simple geometric configuration shown in Figure 3 . For this setup, resolutions from 8 to 256 were used, resulting in 6 levels of adaptivity. Timing information for the main steps of our algorithm over the course of this simulation can be found in Figure 8 and Figure 9 . For this simulation, the initial configuration consisted of 168, 161 particles, and momentarily peaked up to 1, 048, 776 during the maximal extent of the splash (settling down again to around 250 thousand in the end). Note that a full sampling of the initial configuration with a regular grid would have required approximately 6 million particles. Our measurements show that the run-time of our method has a strong linear relationship to the number of particles, and thus the visual complexity of the simulation. The per-frame time is low at the beginning and end of the simulation, but strongly peaks during the complex splash in its middle. Our visibility sizing function is highlighted by the setup shown in Figure 6 . Here the computational resources are focused on the visible region of a rotating camera, as the liquid splashes around a U-shaped corridor. Our solver efficiently resolves the complex motion near the camera, while effectively reducing the computational cost for parts that are not visible. The simulation of Figure 7 shows
Figure 6: Our visibility sizing function: as the liquid flows along the U-shaped corridor, the visible volume is simulated with a high resolution (surface shown in top row), while regions outside of the view frustrum are coarsened (particle view in bottom row).
Setup Min. ∆x −1 Min. ∆x s −1 Max. ∆x −1 Duration frames
Figure 3
8
16
256
5h:09m
240
Figure 6
16
32
256
4h:26m
330
Figure 7
16
64
512
5h:55m
480
Figure 1
8
32
1024
12h:8m
160
Setup Min. ∆x −1 Min. ∆x −1 s Max. ∆x −1 Duration frames Figure 3 8 16 256 5h:09m 240 Figure 6 16 32 256 4h:26m 330 Figure 7 16 64 512 5h:55m 480 Figure 1 8 32 1024 12h:8m 160
Table 1: Resolutions and running times for our simulations (not including final surface creation). Here, ∆x −1 and ∆x −1 s denote the
number of BCC cells along one spatial axis for the simulation and for the surface generation, respectively. The simulations were run on a workstation with an Intel Core i7-3960X CPU with 3.30GHz running under Linux.
a liquid interacting with a highly detailed obstacle. The κ solid component of our sizing function ensures that geometrically complex regions near the obstacle are simulated with higher accuracy. In this way, we can resolve the detailed flow of liquid through the holes in the obstacle. Figure 1 shows a situation that would be challenging to simulate with a regular solver. Without adaptivity, the large open liquid surface with complex splashes in a localized region would require huge amounts of computational resources. Our method can simulate this setup very efficiently, and in a fully coupled manner with an effective high resolution. The large open region is successfully coarsened by our sizing function, resulting in subtle wave motions around the splashes. In this case, the whole simulation with 8 different octree levels and a maximum resolution of up to 1024 cells took on average only 4.6 minutes per frame to compute. Just to illustrate the amount of detail in this setup – our adaptive version initially used 1.7 million particles, while a regular sampling at the finest resolution would have required roughly 400 million. Further runtime and resolution details for our simulations can be found in Table 1 . Discussion We found our discretization (Section 3) beneficial in a number of ways. We store the pressure variables on tetrahedral vertices, and there are far fewer vertices than tetrahedra in a given mesh. Consequently, the pressure solve has fewer variables and is faster to solve. Also, by counting degrees of freedom and constraints, we can see that our discretization prevents the locking ar- tifacts which are common in other methods. However, the lower number of pressure constraints also implies that the highest frequencies of the velocity field may be unconstrained. In our case a regularization via PIC interpolation acts to diminish any highfrequency artifacts. Our method uses a FLIP scheme instead of a purely Eulerian method. We store all physical variables on the FLIP particles, so information is carried from one time step to the next in a Lagrangian manner. As a result, we are allowed to aggressively remesh the tetrahedral background grid without worrying about excessive damping or re-sampling artifacts. On the other hand, FLIP simulations have a well-known problem of creating noisy particle distributions, because there are typically several times more particles than velocity variables on the background grid. We utilize particle repositioning to improve the distribution quality, at the expense of slight inaccuracies due to displacing physical variables. We noticed that our new surface creation routine is essential for maintaining detailed simulations in the presence of accurate freesurface boundary conditions. One major benefit of our method is that it can easily create perfectly flat surfaces from a mixture of differently-sized particles. These flat surfaces represent the equilibrium state of a fluid simulation, so our animations are able to smoothly settle down as time progresses. Without a method for accurately reproducing flat surfaces, second-order boundary conditions will introduce additional forces in the locations of surface bumps, which artificially prevent a simulation from settling down. While we believe that our surface creation routine is indispensable, it is quite expensive to compute. In the future we would like to optimize the surface computation. Our simulations perform quite well for large differences in resolutions, but we have only been able to push them to a certain point in our current implementation. We found that using too sharp of a grading in our sizing function can place coarse and fine simulation elements too close together and potentially result in artifacts. For example, when small particles land in very coarse cells after violent splashes, these particles can get stuck in mid air. Occasionally, this can also lead to an overly strong weight for such particles during the velocity mapping, resulting in momentum artifacts. Our adaptive numerical viscosity in Eq. (14) can also exhibit dangerously small damping values for very fine resolutions, so we clamped the blending coefficient to a minimum value of 0.1 in Figure 1 . In Section 4 we introduced a novel collection of sizing functions for adaptively selecting details from a fluid simulation. While we presented specific parameters for the sake of reproducibility, these values were not meticulously tuned and are certainly not optimal. The task of choosing an ideal sizing function is still an open problem that we are interested in pursuing in the future. In particular, we are interested in taking more temporal information into account. This could lead to more gradual changes in resolution, at the expense of a slightly higher particle count.
Figure 8: This graph shows durations for the different parts of our algorithm over the course of the simulation from Figure 3 . Note that we only include the computationally more expensive steps here, and not the re-meshing (which is done in intervals).
Figure 9: The time required for our re-meshing and particle merging & splitting computations over time for Figure 3 . Note that we perform them only every ten simulation steps.
Figure 7: A simulation of detailed flow through a complex obstacle. The liquid correctly flows through the orifices at the center of the filigree.
<H1>8 Conclusions and Outlook</H1>
We have presented a novel framework for highly adaptive liquid simulation. In our method, a novel, robust discretization works together with accurate embedded boundary conditions and a flexible sizing function to allow for aggressive adaptivity and high computational performance. In this way, we can efficiently compute tough simulation setups, such as large surfaces with very localized details. We have additionally presented a novel surface creation method that yields smooth surfaces in the presence of strongly varying particle radii, which turned out to be an important building block for our framework. We chose a BCC mesh generation because it is, to the best of our knowledge, the fastest way to generate high-quality meshes. However, despite its efficiency, mesh generation is still a bottleneck for our simulation. This is partly due to the fact that it is a mostly serial operation that is difficult to parallelize (most other steps of our algorithm parallelize easily). So, instead of computing the mesh from scratch each time, we are interested in exploring techniques for continuous re-meshing. Also, our choice of piece-wise constant basis functions for velocity indicates that our discretization could lead to difficulties when it is used for diffusion or viscosity solves. It will be interesting to see how these could be incorporated into our framework. Finally, we are highly interested in applying our method to other types of phenomena, such as smoke and fire simulations, or visco-elastic materials. It will be very interesting to leverage the benefits of our framework for extreme adaptivity in these situations.
<H1>Acknowledgements</H1>
We would like to thank the anonymous reviewers for their helpful feedback. We also wish to thank Reiji Tsuruno for providing us with computational resources, and Pascal Clausen as well as Ramprasad Sampath for constructive discussions. This work was supported by the Japan Society for the Promotion of Science (JSPS). Finally, we would like to express our gratitude to the authors of the ANN library, which we use for kd-tree look-ups, and to the Mitsuba renderer, which we have used to render all images in this paper.
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A Ghost Fluid Coefficients
Here we describe how to compute the ghost fluid coefficients θ n given the 4 × 4 pressure matrix entries of a single tetrahedron. Reorganizing Eq. (11) gives:
16
λ 2 + αw 1 a + αw 2 b + αw 3 p 1 b 1 a + βw 1 λ 3 + βw 2 c + βw 3 p 2 = b 2 . b + γw 1 c + γw 2 λ 4 + γw 3 p 3 b 3
Note that each of the θ n has two degree of freedom, and thus each w n also has two degrees of freedom. As we know that the resulting matrix needs to be symmetric, which gives us the following constraints: a + αw 2 = a + βw 1 , b + γw 1 = b + αw 3 , and c + βw 3 = c + γw 2 . As the w n linearly depend on θ n , that means: αθ 2 = βθ 1 , γθ 1 = αθ 3 , and βθ 3 = γθ 2 . When we re-write these constraints in matrix form, and include the barycentric coordinate constraint θ 1 + θ 2 + θ 3 = 1 we get the following linear system:
17
−β α 0 0 γ 0 −α θ 1 0 0 1 −γ 1 β 1 θ θ 3 2 = 1 0
As the rank of top three rows of the matrix is 2, we can drop one of them. Removing the first row from the system gives us the following full-rank, 3 × 3 matrix:
18
γ 0 −α θ 1 0 0 −γ β θ 2 = 0 1 1 1 θ 3 1
The analytical solution of this system is:
19
θ 1 1 α θ 2 = β . θ 3 α + β + γ γ
This means that for our discretization, the ghost pressure coefficients θ n are given by the tetrahedron’s matrix entries from Eq.10. More specifically, by those for the vertex that is located outside of the liquid, i.e., α, β and γ. Substituting this equation into Eq.9 and Eq.6 yields the final equation for the ghost pressure Eq. (12).
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"offsets": [
[
13928,
14016
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],
"text": "['by assuming a piece-wise constant velocity and a linear change of pressure within a cell']",
"type": "data",
"id": "T110"
},
{
"offsets": [
[
14250,
14278
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],
"text": "['Based on this representation']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
14341,
14399
]
],
"text": "['m p = arg min 1 || u ˆ i ∗ − ∆t [ ] p|| 2 ρV i , p ˆ 2 ρ i']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
15031,
15172
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],
"text": "['Our discretization compared to previous approaches: the MAC grid stores velocity components normal to faces and pressure values at the center']",
"type": "own_claim",
"id": "T113"
},
{
"offsets": [
[
15554,
15617
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],
"text": "['a square n × n matrix, which is symmetric and positive definite']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
15834,
15855
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],
"text": "['to compute a solution']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
15741,
15766
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],
"text": "['we can use standard tools']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
15702,
15739
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],
"text": "['Given appropriate boundary conditions']",
"type": "data",
"id": "T117"
},
{
"offsets": [
[
15776,
15833
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],
"text": "['a commonly used pre-conditioned conjugate gradient solver']",
"type": "data",
"id": "T118"
},
{
"offsets": [
[
15887,
15899
]
],
"text": "['Bridson 2008']",
"type": "data",
"id": "T119"
},
{
"offsets": [
[
15857,
15884
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],
"text": "['we use the one suggested in']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
16878,
16998
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],
"text": "['our pressure solve is a linear system that has n degrees of freedom, n being the number of nodes in the tetrahedral mesh']",
"type": "own_claim",
"id": "T121"
},
{
"offsets": [
[
16848,
16876
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],
"text": "['In contrast to previous work']",
"type": "own_claim",
"id": "T122"
},
{
"offsets": [
[
17018,
17074
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],
"text": "['n is in practice smaller than the number of tetrahedra m']",
"type": "own_claim",
"id": "T123"
},
{
"offsets": [
[
17000,
17016
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],
"text": "['For our BCC mesh']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
17106,
17186
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],
"text": "['A direct implication of this smaller linear system is that it is faster to solve']",
"type": "own_claim",
"id": "T125"
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{
"offsets": [
[
17188,
17310
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],
"text": "['A second, less obvious implication of the smaller linear system is that it effectively prevents artifacts known as locking']",
"type": "own_claim",
"id": "T126"
},
{
"offsets": [
[
17312,
17402
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],
"text": "['These artifacts are commonly observed in finite element methods for problems in elasticity']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
17404,
17469
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],
"text": "['Different methods have been proposed to circumvent these problems']",
"type": "background_claim",
"id": "T128"
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{
"offsets": [
[
17477,
17547
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],
"text": "['using linear elements for pressure instead of piece-wise constant ones']",
"type": "data",
"id": "T129"
},
{
"offsets": [
[
17550,
17568
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],
"text": "['Irving et al. 2007']",
"type": "data",
"id": "T130"
},
{
"offsets": [
[
17571,
17646
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],
"text": "['Other works explicitly smooth the pressure field to reduce locking problems']",
"type": "background_claim",
"id": "T131"
},
{
"offsets": [
[
17649,
17668
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],
"text": "['Misztal et al. 2010']",
"type": "data",
"id": "T132"
},
{
"offsets": [
[
17683,
17812
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],
"text": "['locking can be observed if the pressure basis can represent more, and higher-frequency, functions than the basis for the velocity']",
"type": "background_claim",
"id": "T133"
},
{
"offsets": [
[
17821,
17867
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],
"text": "['choosing a more restrictive basis for pressure']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
17901,
17965
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],
"text": "['explicitly removing high-frequency information from the pressure']",
"type": "own_claim",
"id": "T135"
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{
"offsets": [
[
17991,
18020
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],
"text": "['reduces the chance of locking']",
"type": "own_claim",
"id": "T136"
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{
"offsets": [
[
17877,
17895
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],
"text": "['Irving et al. 2007']",
"type": "data",
"id": "T137"
},
{
"offsets": [
[
17968,
17987
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],
"text": "['Misztal et al. 2010']",
"type": "data",
"id": "T138"
},
{
"offsets": [
[
18022,
18032
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],
"text": "['Our method']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
18051,
18132
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],
"text": "['has more degrees of freedom for representing velocity fields than pressure fields']",
"type": "own_claim",
"id": "T140"
},
{
"offsets": [
[
18143,
18235
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],
"text": "['we cannot prove that a local configuration over-constraining the velocities will never occur']",
"type": "own_claim",
"id": "T141"
},
{
"offsets": [
[
18237,
18334
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],
"text": "['the larger number of degrees of freedom for our velocities effectively prevents locking artifacts']",
"type": "own_claim",
"id": "T142"
},
{
"offsets": [
[
18382,
18532
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],
"text": "['Boundary Conditions Second-order boundary conditions are a central component for accurate and visually appealing simulations with non-conforming grids']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
18534,
18641
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],
"text": "['Achieving second-order accuracy for obstacle boundary conditions is straightforward with our discretization']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
18693,
18710
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],
"text": "['Batty et al. 2007']",
"type": "data",
"id": "T145"
},
{
"offsets": [
[
18847,
18920
]
],
"text": "['Dirichlet boundary condition p = 0 is satisfied at the interface position']",
"type": "own_claim",
"id": "T146"
},
{
"offsets": [
[
18931,
19090
]
],
"text": "['this means computing a pressure value for nodes outside of the liquid so that a linear interpolation along an edge of a cell gives zero at the correct position']",
"type": "background_claim",
"id": "T147"
},
{
"offsets": [
[
19093,
19112
]
],
"text": "['Enright et al. 2005']",
"type": "data",
"id": "T148"
},
{
"offsets": [
[
19114,
19136
]
],
"text": "['Lew and Buscaglia 2008']",
"type": "data",
"id": "T149"
},
{
"offsets": [
[
19309,
19333
]
],
"text": "['For a Cartesian MAC grid']",
"type": "data",
"id": "T150"
},
{
"offsets": [
[
18798,
18818
]
],
"text": "['For the free surface']",
"type": "data",
"id": "T151"
},
{
"offsets": [
[
19335,
19394
]
],
"text": "['the ghost pressure value p G is given by p G = p L φ G /φ L']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
19419,
19466
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],
"text": "['this approach does not yield the desired result']",
"type": "own_claim",
"id": "T153"
},
{
"offsets": [
[
19397,
19408
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],
"text": "['In our case']",
"type": "data",
"id": "T154"
},
{
"offsets": [
[
19468,
19575
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],
"text": "['The reason is that our velocity samples are not in line with the direct connections of the pressure samples']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
19578,
19623
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],
"text": "['they are not locally orthogonal to each other']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
19634,
19731
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],
"text": "['we have to ensure the boundary conditions result in the correct pressure value at the cell center']",
"type": "own_claim",
"id": "T157"
},
{
"offsets": [
[
20767,
20861
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],
"text": "['p G can be rewritten in terms of a barycentric interpolation of the three values in the liquid']",
"type": "own_claim",
"id": "T158"
},
{
"offsets": [
[
21026,
21082
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],
"text": "['the θ n are a set of barycentric coordinate coefficients']",
"type": "own_claim",
"id": "T159"
},
{
"offsets": [
[
21093,
21110
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],
"text": "['θ 1 +θ 2 +θ 3 = 1']",
"type": "data",
"id": "T160"
},
{
"offsets": [
[
21314,
21376
]
],
"text": "['the values w n are determined by those of the θ n coefficients']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
21444,
21499
]
],
"text": "['θ n could take any values as long as they add up to one']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
21666,
21749
]
],
"text": "['The computation of the ghost fluid ρ values is independent of the right-hand side b']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
22042,
22098
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],
"text": "['the first vertex is the one outside of the liquid volume']",
"type": "background_claim",
"id": "T164"
},
{
"offsets": [
[
23023,
23163
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],
"text": "['the resulting matrix is symmetric positivedefinite and can be easily inverted by the commonly used preconditioned conjugate gradient methods']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
23012,
23021
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],
"text": "['this case']",
"type": "data",
"id": "T166"
},
{
"offsets": [
[
23174,
23309
]
],
"text": "['positive off-diagonal terms of the matrix can result in values of θ outside of the range [0, 1], leading to an indefinite linear system']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
23446,
23514
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],
"text": "['it is common practice to clamp small values in the denominator of Eq']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
23520,
23564
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],
"text": "['to prevent ill-conditioned pressure matrices']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
23590,
23665
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],
"text": "['reverting to first-order accuracy when second-order accuracy is intractable']",
"type": "own_claim",
"id": "T170"
},
{
"offsets": [
[
23927,
23973
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],
"text": "['we smoothly transition to first order accuracy']",
"type": "own_claim",
"id": "T171"
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{
"offsets": [
[
23904,
23925
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],
"text": "['we detect such a case']",
"type": "data",
"id": "T172"
},
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"offsets": [
[
24177,
24208
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],
"text": "['we compute a coefficient k with']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
24241,
24302
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],
"text": "['φ − 1 φ − 1 φ − 1 k = min λ 2 , λ 3 , λ 4 , w 2 α w 3 β w 4 γ']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
24154,
24175
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],
"text": "['we detect such a case']",
"type": "data",
"id": "T175"
},
{
"offsets": [
[
24467,
24476
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],
"text": "['for k = 1']",
"type": "data",
"id": "T176"
},
{
"offsets": [
[
24477,
24515
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],
"text": "['this yields full second-order accuracy']",
"type": "own_claim",
"id": "T177"
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{
"offsets": [
[
24523,
24550
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],
"text": "['for badly shaped tetrahedra']",
"type": "data",
"id": "T178"
},
{
"offsets": [
[
24551,
24658
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],
"text": "['the resulting k = 0 means that we revert to the standard rounding strategy of a first order accurate method']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
24751,
24776
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],
"text": "['The graded BCC tetrahedra']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
24797,
24850
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],
"text": "['can be of lower quality and can require the use of Eq']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
24852,
24856
]
],
"text": "['(13)']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
25213,
25307
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],
"text": "['the interpolation would ideally use the dual mesh consisting of the Voronoi cells of each node']",
"type": "own_claim",
"id": "T183"
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{
"offsets": [
[
25310,
25328
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],
"text": "['Brochu et al. 2010']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
25595,
25664
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],
"text": "['we first interpolate the centered velocities to the nodes, similar to']",
"type": "own_claim",
"id": "T185"
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{
"offsets": [
[
25667,
25688
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],
"text": "['Chentanez et al. 2007']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
26091,
26156
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],
"text": "['these four smaller tetrahedra do not have to be stored explicitly']",
"type": "own_claim",
"id": "T187"
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{
"offsets": [
[
26246,
26328
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],
"text": "['Manipulating FLIP particles The spacing between FLIP particles may drift over time']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
26334,
26389
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],
"text": "['high-frequency errors may contribute to a bumpy surface']",
"type": "own_claim",
"id": "T189"
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{
"offsets": [
[
26484,
26529
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],
"text": "['we apply the position correction algorithm of']",
"type": "own_claim",
"id": "T190"
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{
"offsets": [
[
26531,
26549
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],
"text": "['Ando et al. [2012]']",
"type": "data",
"id": "T191"
},
{
"offsets": [
[
27223,
27330
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],
"text": "['FLIP particles that partake in splashes and sprays can pose a significant burden on computational resources']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
27343,
27377
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],
"text": "['in an adaptive framework like ours']",
"type": "data",
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{
"offsets": [
[
27379,
27478
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],
"text": "['This inefficiency stems from the fact that water droplets undergo extremely simple ballistic motion']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
27964,
28095
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],
"text": "['This decision allows us to avoid aggressively refining the tetrahedral mesh in locations where the physical motion is uninteresting']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
28097,
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],
"text": "['Only a small percentage of the particles are simulated in this way']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
28171,
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],
"text": "['1.7% on average']",
"type": "data",
"id": "T197"
},
{
"offsets": [
[
28191,
28199
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],
"text": "['Figure 6']",
"type": "data",
"id": "T198"
},
{
"offsets": [
[
28466,
28557
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],
"text": "['Our method achieves adaptivity by varying the mesh resolution over the computational domain']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
[
29135,
29195
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],
"text": "['we use the octree-based grading method which was proposed by']",
"type": "own_claim",
"id": "T200"
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{
"offsets": [
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29225,
29286
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],
"text": "['and later adopted in several adaptive simulation environments']",
"type": "own_claim",
"id": "T201"
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{
"offsets": [
[
29197,
29224
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],
"text": "['Labelle and Shewchuk [2007]']",
"type": "data",
"id": "T202"
},
{
"offsets": [
[
29289,
29310
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],
"text": "['Chentanez et al. 2007']",
"type": "data",
"id": "T203"
},
{
"offsets": [
[
29312,
29332
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],
"text": "['Wojtan and Turk 2008']",
"type": "data",
"id": "T204"
},
{
"offsets": [
[
29334,
29351
]
],
"text": "['Batty et al. 2010']",
"type": "data",
"id": "T205"
},
{
"offsets": [
[
29367,
29384
]
],
"text": "['Batty et al. 2010']",
"type": "data",
"id": "T206"
},
{
"offsets": [
[
29354,
29364
]
],
"text": "['Similar to']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
29388,
29501
]
],
"text": "['we generate a new tetrahedral mesh every ten time steps, instead of rebuilding the mesh on every consecutive step']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
[
29509,
29578
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],
"text": "['the tetrahedral mesh is only temporarily used for the pressure solver']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
29583,
29669
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],
"text": "['no information is transferred from one time step to the next by storing it on the grid']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
29865,
29890
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],
"text": "['we modify the strategy of']",
"type": "own_claim",
"id": "T211"
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{
"offsets": [
[
29892,
29910
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],
"text": "['Ando et al. [2012]']",
"type": "data",
"id": "T212"
},
{
"offsets": [
[
29855,
29863
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],
"text": "['For this']",
"type": "data",
"id": "T213"
},
{
"offsets": [
[
29911,
29939
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],
"text": "['to work within our framework']",
"type": "own_claim",
"id": "T214"
},
{
"offsets": [
[
30056,
30079
]
],
"text": "['a particle is too small']",
"type": "data",
"id": "T215"
},
{
"offsets": [
[
30086,
30135
]
],
"text": "['we merge it with its nearest neighboring particle']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
30240,
30288
]
],
"text": "['a given particle is bigger than the desired size']",
"type": "data",
"id": "T217"
},
{
"offsets": [
[
30295,
30323
]
],
"text": "['the particle is split in two']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
30668,
30752
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],
"text": "['The new particle’s position is equal to the weighted average of all nearby midpoints']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
30754,
30775
]
],
"text": "['x new = ω i m i / ω i']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
31265,
31369
]
],
"text": "['Numerical viscosity in fluid simulations is tightly coupled to the spatial resolution resolving the flow']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
31371,
31548
]
],
"text": "['We compensate for spatiallyvarying numerical viscosity caused by particles of various sizes in our simulation by adjusting the PIC/FLIP blending parameter in our FLIP simulation']",
"type": "own_claim",
"id": "T222"
},
{
"offsets": [
[
31551,
31563
]
],
"text": "['Bridson 2008']",
"type": "data",
"id": "T223"
},
{
"offsets": [
[
31672,
31740
]
],
"text": "['the new quantity Q i is computed as a weighted blend between the two']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
31566,
31656
]
],
"text": "['Given quantities Q i,PIC and Q i,FLIP computed at particle i from PIC and FLIP simulations']",
"type": "data",
"id": "T225"
},
{
"offsets": [
[
32212,
32306
]
],
"text": "['this strategy adequately eliminates any artifacts due to spatially varying numerical viscosity']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
32668,
32744
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"text": "['Our simulator is versatile enough to cope with any of these sizing functions']",
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"text": "['the sizing function is defined as a combination of five different metrics']",
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"text": "['S(x) = max ( d(x), V (x, min(κ liquid (x), κ solid (x), e(x)) )']",
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"text": "['In all of the examples in this paper']",
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"text": "['This has the effect that motion near the surface has higher priority than motions far inside the bulk volume of the liquid']",
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"text": "['V (x, y) is a view-dependent function that returns the value y']",
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"text": "['x is within the camera’s visible region']",
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33529,
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"text": "['returns the maximum particle radius r max']",
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"offsets": [
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34181,
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"text": "['it beneficial to invest computational resources into keeping interesting motion of the flow field alive']",
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"text": "['the strain tensor of the flow field reliably indicates detailed motions']",
"type": "own_claim",
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"text": "['Pfaff et al. 2010']",
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"offsets": [
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34286,
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"text": "['Inspired by turbulence models']",
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34769,
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"text": "['we aim to implicitly represent the fluid surface by computing its signed distance function']",
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34716,
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"text": "['Given our set of FLIP particles with variable radii']",
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"text": "['Several useful methods for computing a surface from a collection of particles have been proposed in the past']",
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"text": "['Zhu and Bridson 2005']",
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[
34994,
35011
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],
"text": "['Adams et al. 2007']",
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[
35013,
35029
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],
"text": "['Yu and Turk 2010']",
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35037,
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],
"text": "['they tend to produce undesirably bumpy surfaces']",
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35090,
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"text": "['considering particles of highly variable radii']",
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"offsets": [
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35139,
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"text": "['Figure 4']",
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"offsets": [
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"text": "['The main idea is to approximate the fluid surface with the union of the convex hulls of each triplet of nearby particles close to the surface']",
"type": "own_claim",
"id": "T248"
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"offsets": [
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"text": "['a larger l tends to fill in small concavities']",
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"text": "['A small l shows more surface details']",
"type": "own_claim",
"id": "T250"
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{
"offsets": [
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35963,
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"text": "['the minimum signed distance from these shapes to a point in space defines the outer part of our level set function']",
"type": "own_claim",
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{
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"text": "['We efficiently compute the distance to these planes by analytically solving the polynomial system']",
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"text": "['the plane is the right distance away from each particle with the normal facing away from them']",
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"id": "T253"
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"text": "['the first three equations']",
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36788,
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"text": "['the plane equation is normalized to a distance function']",
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36756,
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"text": "['the final equation']",
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36845,
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"text": "['These four equations represent the intersection of three hyperplanes and a hypercylinder in 4D {a, b, c, d} space']",
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{
"offsets": [
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37467,
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"text": "['we can easily compute the signed distance between this convex hull and a point in space']",
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"id": "T258"
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"offsets": [
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37362,
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"text": "['By computing the conic and spherical convex hull facets']",
"type": "data",
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{
"offsets": [
[
37445,
37465
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"text": "['in a similar fashion']",
"type": "data",
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{
"offsets": [
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37420,
37428
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],
"text": "['Figure 5']",
"type": "data",
"id": "T261"
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{
"offsets": [
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37925,
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],
"text": "['The algorithm as described works perfectly for computing the level set outside of our particle surface']",
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"offsets": [
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38033,
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"text": "['it may lead to small gaps inside']",
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"text": "['The final surface creation is trivially parallelized']",
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39125,
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"text": "['Most previous algorithms perform poorly in this comparison']",
"type": "own_claim",
"id": "T265"
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"offsets": [
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39192,
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],
"text": "['they were not designed for particles with varying radii']",
"type": "data",
"id": "T266"
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"offsets": [
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40191,
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"text": "['additional user-defined sizing functions could be computed as well']",
"type": "own_claim",
"id": "T267"
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"offsets": [
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40176,
40189
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"text": "['At this point']",
"type": "data",
"id": "T268"
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"offsets": [
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40122,
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"text": "['it becomes necessary to evaluate the sizing function']",
"type": "own_claim",
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"offsets": [
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40443,
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"text": "['we use an SPH-like kernel function']",
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"id": "T270"
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40384,
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"text": "['For mapping the particle velocities onto the grid']",
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40479,
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"text": "['which is weighted by the particle volume']",
"type": "own_claim",
"id": "T272"
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40521,
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"text": "['max(v i (4r i 2 /d 2 − 1), 0)']",
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41175,
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"text": "['The particle velocity update of line 12 uses barycentric interpolations']",
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"text": "['Section 3']",
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"text": "['the grid-based velocity extrapolation of line 10 uses the nodal velocities']",
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"text": "['Section 3']",
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"offsets": [
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41373,
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"text": "['A time step is completed by performing the pressure projection and advecting the particles in the resulting divergence-free velocity field']",
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"id": "T278"
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{
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"text": "['One comparison']",
"type": "own_claim",
"id": "T279"
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{
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[
41800,
41832
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],
"text": "['that is particularly interesting']",
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41833,
41891
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"text": "['is the one comparing our method to an FVM based simulation']",
"type": "own_claim",
"id": "T281"
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{
"offsets": [
[
41893,
41950
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],
"text": "['Using a graded BCC mesh leads to problems with the latter']",
"type": "own_claim",
"id": "T282"
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{
"offsets": [
[
41955,
42038
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],
"text": "['the position of the circumcenter lies exactly on a face for the graded tetrahedrons']",
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"id": "T283"
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{
"offsets": [
[
42062,
42166
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"text": "['this can result in two pressure samples from adjacent tetrahedra being placed at the exact same position']",
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"id": "T284"
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{
"offsets": [
[
42040,
42060
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"text": "['In the graded region']",
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"offsets": [
[
42168,
42193
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"text": "['To alleviate this problem']",
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"id": "T286"
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{
"offsets": [
[
42197,
42214
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"text": "['Batty et al. 2010']",
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"offsets": [
[
42217,
42279
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"text": "['propose to slightly offset the pressure samples from the faces']",
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"id": "T288"
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{
"offsets": [
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"text": "['The influence of the different components of our sizing function on the evolution of a simulation is difficult to depict with static images']",
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{
"offsets": [
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"text": "['we refer to the accompanying video for a comparison']",
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"id": "T290"
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"offsets": [
[
43222,
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],
"text": "['a full sampling of the initial configuration with a regular grid would have required approximately 6 million particles']",
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"id": "T291"
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{
"offsets": [
[
43369,
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"text": "['the run-time of our method has a strong linear relationship to the number of particles']",
"type": "own_claim",
"id": "T292"
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{
"offsets": [
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"text": "['the visual complexity of the simulation']",
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{
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"text": "['The per-frame time is low at the beginning and end of the simulation']",
"type": "own_claim",
"id": "T294"
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"offsets": [
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43581,
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"text": "['strongly peaks during the complex splash in its middle']",
"type": "own_claim",
"id": "T295"
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43721,
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"text": "['the computational resources are focused on the visible region of a rotating camera']",
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"id": "T296"
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{
"offsets": [
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43808,
43854
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"text": "['the liquid splashes around a U-shaped corridor']",
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43856,
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"text": "['Our solver efficiently resolves the complex motion near the camera']",
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43930,
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"text": "['effectively reducing the computational cost for parts that are not visible']",
"type": "own_claim",
"id": "T299"
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{
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45906,
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"text": "['geometrically complex regions near the obstacle are simulated with higher accuracy.']",
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45848,
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"text": "['The κ solid component of our sizing function']",
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46003,
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"text": "['we can resolve the detailed flow of liquid through the holes in the obstacle']",
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"id": "T302"
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45990,
46001
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"text": "['In this way']",
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"offsets": [
[
46189,
46316
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],
"text": "['the large open liquid surface with complex splashes in a localized region would require huge amounts of computational resources']",
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"id": "T304"
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{
"offsets": [
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46169,
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"text": "['Without adaptivity']",
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"offsets": [
[
46318,
46369
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"text": "['Our method can simulate this setup very efficiently']",
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"id": "T306"
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46375,
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"text": "['in a fully coupled manner with an effective high resolution']",
"type": "own_claim",
"id": "T307"
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{
"offsets": [
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46436,
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"text": "['The large open region is successfully coarsened by our sizing function']",
"type": "own_claim",
"id": "T308"
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46508,
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"text": "['resulting in subtle wave motions around the splashes']",
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"text": "['the whole simulation with 8 different octree levels and a maximum resolution of up to 1024 cells took on average only 4.6 minutes per frame to compute']",
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"id": "T310"
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{
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46562,
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"text": "['In this case']",
"type": "data",
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46784,
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"text": "['our adaptive version initially used 1.7 million particles']",
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46849,
46932
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"text": "['a regular sampling at the finest resolution would have required roughly 400 million']",
"type": "own_claim",
"id": "T313"
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{
"offsets": [
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47030,
47057
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],
"text": "['We found our discretization']",
"type": "own_claim",
"id": "T314"
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{
"offsets": [
[
47070,
47100
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"text": "['beneficial in a number of ways']",
"type": "own_claim",
"id": "T315"
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{
"offsets": [
[
47163,
47223
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],
"text": "['there are far fewer vertices than tetrahedra in a given mesh']",
"type": "own_claim",
"id": "T316"
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"offsets": [
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47239,
47277
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"text": "['the pressure solve has fewer variables']",
"type": "own_claim",
"id": "T317"
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47282,
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"text": "['is faster to solve']",
"type": "own_claim",
"id": "T318"
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{
"offsets": [
[
47372,
47457
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],
"text": "['our discretization prevents the locking ar- tifacts which are common in other methods']",
"type": "own_claim",
"id": "T319"
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{
"offsets": [
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47527,
47593
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],
"text": "['the highest frequencies of the velocity field may be unconstrained']",
"type": "own_claim",
"id": "T320"
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"offsets": [
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47468,
47508
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"text": "['the lower number of pressure constraints']",
"type": "data",
"id": "T321"
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{
"offsets": [
[
47908,
48043
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],
"text": "['we are allowed to aggressively remesh the tetrahedral background grid without worrying about excessive damping or re-sampling artifacts']",
"type": "own_claim",
"id": "T322"
},
{
"offsets": [
[
47817,
47893
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],
"text": "['information is carried from one time step to the next in a Lagrangian manner']",
"type": "data",
"id": "T323"
},
{
"offsets": [
[
48064,
48148
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],
"text": "['FLIP simulations have a well-known problem of creating noisy particle distributions,']",
"type": "background_claim",
"id": "T324"
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{
"offsets": [
[
48157,
48252
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],
"text": "['there are typically several times more particles than velocity variables on the background grid']",
"type": "background_claim",
"id": "T325"
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{
"offsets": [
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48417,
48559
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],
"text": "['our new surface creation routine is essential for maintaining detailed simulations in the presence of accurate freesurface boundary conditions']",
"type": "own_claim",
"id": "T326"
},
{
"offsets": [
[
48561,
48691
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],
"text": "['One major benefit of our method is that it can easily create perfectly flat surfaces from a mixture of differently-sized particles']",
"type": "own_claim",
"id": "T327"
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{
"offsets": [
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48771,
48837
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],
"text": "['our animations are able to smoothly settle down as time progresses']",
"type": "own_claim",
"id": "T328"
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{
"offsets": [
[
48693,
48766
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],
"text": "['These flat surfaces represent the equilibrium state of a fluid simulation']",
"type": "data",
"id": "T329"
},
{
"offsets": [
[
48999,
49057
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],
"text": "['which artificially prevent a simulation from settling down']",
"type": "own_claim",
"id": "T330"
},
{
"offsets": [
[
48898,
48997
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],
"text": "['second-order boundary conditions will introduce additional forces in the locations of surface bumps']",
"type": "data",
"id": "T331"
},
{
"offsets": [
[
49081,
49126
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],
"text": "['our surface creation routine is indispensable']",
"type": "own_claim",
"id": "T332"
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{
"offsets": [
[
49128,
49160
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],
"text": "['it is quite expensive to compute']",
"type": "own_claim",
"id": "T333"
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{
"offsets": [
[
49227,
49298
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],
"text": "['Our simulations perform quite well for large differences in resolutions']",
"type": "own_claim",
"id": "T334"
},
{
"offsets": [
[
49304,
49388
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],
"text": "['we have only been able to push them to a certain point in our current implementation']",
"type": "own_claim",
"id": "T335"
},
{
"offsets": [
[
49404,
49520
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],
"text": "['using too sharp of a grading in our sizing function can place coarse and fine simulation elements too close together']",
"type": "own_claim",
"id": "T336"
},
{
"offsets": [
[
49537,
49557
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],
"text": "['result in artifacts.']",
"type": "own_claim",
"id": "T337"
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{
"offsets": [
[
49642,
49682
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],
"text": "['these particles can get stuck in mid air']",
"type": "own_claim",
"id": "T338"
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{
"offsets": [
[
49576,
49640
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],
"text": "['small particles land in very coarse cells after violent splashes']",
"type": "data",
"id": "T339"
},
{
"offsets": [
[
49698,
49790
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],
"text": "['this can also lead to an overly strong weight for such particles during the velocity mapping']",
"type": "own_claim",
"id": "T340"
},
{
"offsets": [
[
49792,
49823
]
],
"text": "['resulting in momentum artifacts']",
"type": "own_claim",
"id": "T341"
},
{
"offsets": [
[
49825,
49857
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],
"text": "['Our adaptive numerical viscosity']",
"type": "own_claim",
"id": "T342"
},
{
"offsets": [
[
49870,
49945
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],
"text": "['can also exhibit dangerously small damping values for very fine resolutions']",
"type": "own_claim",
"id": "T343"
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{
"offsets": [
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49950,
50011
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http://dx.doi.org/10.1145/545261.545286
<Title>EigenSkin: Real Time Large Deformation Character Skinning in Hardware</Title>
Paul G. Kry
Doug L. James
Dinesh K. Pai University of British Columbia
{pgkry|djames|pai}@cs.ubc.ca
a) FEM simulated pose b) SSD only
Figure 1: Comparison of EigenSkin and Skeletal-Subspace Deformation for an extreme pose not in the training data. Note significant differences in the thumb between a) the new pose computed from our finite element hand model, b) skeletal-subspace deformation only, and c) EigenSkin with one eigendisplacements and one normal correction per support. Figure d) shows our EigenSkin hand example being animated using a CyberGlove. The hand model shown here consists of 55,904 triangles and is drawn using display lists with a GeForce3 vertex program.
<Abstract>We present a technique which allows subtle nonlinear quasi-static deformations of articulated characters to be compactly approximated by data-dependent eigenbases which are optimized for real time rendering on commodity graphics hardware. The method extends the common Skeletal-Subspace Deformation (SSD) technique to provide efficient approximations of the complex deformation behaviours exhibited in simulated, measured, and artist-drawn characters. Instead of storing displacements for key poses (which may be numerous), we precompute principal components of the deformation influences for individual kinematic joints, and so construct error-optimal eigenbases describing each joint’s deformation subspace. Pose-dependent deformations are then expressed in terms of these reduced eigenbases, allowing precomputed coefficients of the eigenbasis to be interpolated at run time. Vertex program hardware can then efficiently render nonlinear skin deformations using a small number of eigendisplacements stored in graphics hardware. We refer to the final resulting character skinning construct as the model’s EigenSkin. Animation results are presented for a very large nonlinear finite element model of a human hand rendered in real time at minimal cost to the main CPU.</Abstract>
CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically-based modeling I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism— Animation, Virtual reality Keywords: skeletal-subspace deformation, pose-space deformation, principal component analysis, hardware rendering
c) EigenSkin d) Real time hardware simulation
<H1>1 Introduction</H1>
<H2>1.1 Previous Work</H2>
Significant work has occurred in graphics for deforming articulated characters using geometric methods
[Magnenat-Thalmann et al. 1988; Singh and Kokkevis 2000; Lewis et al. 2000] and physicallybased methods
[Wilhelms and van Gelder 1997; Scheepers et al. 1997; Gourret et al. 1989]. Despite this, most character animation in interactive applications, such as video games, is based on a geometric skeletal deformation technique commonly referred to as linear blending, or matrix palette skinning, or Skeletal-Subspace Deformation (SSD), in which vertex locations are weighted averages of points in several coordinate frames (see
[Magnenat-Thalmann et al. 1988; Magnenat-Thalmann and Thalmann 1991]). One alternative is to store a large database of character poses, and interpolate between them
[Maestri 1999]. While these approaches give animators great control over character deformation, they have the disadvantage of requiring a potentially very large number of poses for animation, and also lack an underlying kinematic model. Nevertheless, such approaches are common, especially for facial animation
[Parke et al. 1996]. A hybrid approach which effectively combines SSD and morphing, is the work of Lewis et al. who introduced “Pose Space Deformations” (PSD) [Lewis et al. 2000]
to overcome the limitations of linear transform blending while retaining a kinematic approach. Starting with a (simple) SSD model, they then store vertex displacement offsets between the SSD surface and various character poses. At run time, the character may be simulated by mapping interpolated displacements onto the underlying SSD character model, thereby providing a kinematic deformation model which also has artist-drawn poses. While this is a big improvement over character morphing, and sufficiently interactive for animators, storing surface displacements for each pose in a large pose space is a memory inefficient approach for hardware applications. Similar to PSD, Sloan et al. show a more efficient method of interpolating an articulated figure using example shapes scattered in an abstract space [Sloan et al. 2001]. The abstract space consists of dimensions describing global properties of the shape, such as age and gender, but also includes dimensions used to describe configuration, such as the amount of bend at an elbow. Like our method, interpolation occurs in the rest pose before SSD is applied, however, the interpolation involves blending over all of the example shapes for every vertex. This becomes inefficient and difficult to map to hardware with the large number of examples required for a highly articulated figure since the independence of abstract space dimensions is not taken into account (e.g., bend in left elbow and bend in right elbow). In addition to character poses created by 3D artists, we also wish to efficiently render deformation behaviour computed using physically-based and reality-based deformable models. Such models have been widely used
[Terzopoulos and Fleischer 1988; Terzopoulos and Witkin 1988; Metaxas and Terzopoulos 1992; CaniGascuel 1998; O’Brien and Hodgins 1999; Pai et al. 2001; Allen et al. 2002]
, although most approaches are not intended for real time (hardware) rendering. Recently, approaches for fast simulation of physical dynamic volumetric deformations have appeared [Zhuang and Canny 1999; Debunne et al. 2001; Picinbono et al. 2001] for interactive applications, such as surgical simulation. Our interest is more closely related to quasi-static deformation, for which fast deformation techniques also exist [Cotin et al. 1999; James and Pai 1999]
but are unfortunately restricted to small deformations unlike those associated with articulated characters (although see [James and Pai 2002b]).
More closely related to character animation is anatomically based modeling of physical deformable models
[Wilhelms and van Gelder 1997]; examples include musculature
[Chen and Zeltzer 1992; Scheepers et al. 1997] and faces
[Lee et al. 1995]. We note that a large class of pose-dependent quasi-static deformations can be described using the EigenSkin approach, largely independent of their origin, whether artist-drawn, measured, or anatomically based physical models. For example, pose-space parameterization of nonhysteretic cloth on articulated characters has recently been considered
[Herman 2001]
, and could be optimized for hardware rendering using the techniques presented herein. Finally, the use of reduced eigenbasis representations for highdimensional models has a long history in science, with foundations on Principal Component Analysis and Karhunen-Loeve theory
[Jolliffe 1986; Hyvarinen et al. 2001].
Related deformation topics include a morphable model for face synthesis
[Blanz and Vetter 1999], modal analysis for dynamic vibrations
[Pentland and Williams July 1989 ; James and Pai 2002a], decomposition of static deformations
[Bookstein 1989], and recognition applications in computer vision, e.g., face recognition
[Turk and Pentland 1991].
<H2>1.2 Our Contribution</H2>
We introduce a method for extending SSD that enhances its range of modeling capabilities at very little cost, and in a manner optimized for real time graphics hardware. EigenSkin constitutes an error-optimal set of eigenbases for approximating the original deformation model, for a given amount of per-vertex displacement memory. We illustrate our method by rendering a very large finite element model (which took several hundred hours to compute) at interactive rates on a PC with negligible cost to the main CPU. Using commodity graphics hardware, EigenSkin enables the simulation of subtle nonlinear surface deformations of geometrically complex models at little more than the cost of rendering.
Rendering of complex physical deformation models for character animation remains a significant hurdle for interactive applications, but one that has been largely overcome for off-line animation. Currently, most real time character animation, e.g., for video games, is done using a very common linear transform blending technique called (among other things) Skeletal-Subspace Deformation (SSD)
[Magnenat-Thalmann et al. 1988]. It is extremely popular for its simplicity and plausibility, and is also widely supported by graphics hardware accelerators. Despite this, it is also widely known to suffer from several key problems:
• buckling of skin near joints, e.g., elbows, in extreme poses; • poor behaviour near more complicated joints, such as shoulders and thumbs; • restrictions on the range of deformations that can be easily modeled and displayed for various character poses.
While methods have been proposed to address this and have been effectively employed by the motion picture industry [Lewis et al. 2000], due to memory and graphics hardware constraints nearly all video game character animation is still done using traditional SSD. In this paper, we present a practical technique which overcomes all aforementioned SSD problems, and can be achieved using a memory-efficient linear correction to the traditional SSD method. The resulting EigenSkin construct allows subtle character deformations for skin and clothing, such as those derived from highly realistic artist-drawn poses, measurements from the real world, or laboriously computed anatomically and physically-based models. The deformations can be compactly represented in an efficient datadependent basis and rendered in real time using vertex shaders in commodity graphics hardware, e.g., see
[Lindholm et al. 2001]. Our approach is to start with an artist’s SSD approximation of the character in question, as well as with geometry corresponding to particular key poses not well approximated by SSD. Vertex displacements between a given pose and the SSD model are mapped back to the neutral character pose, providing a displacement field pose correction. Instead of storing these displacement fields for each key pose and then interpolating between them at runtime, as in Pose Space Deformation (PSD) [Lewis et al. 2000], we use Principal Component Analysis (PCA) to construct an error-optimal eigendisplacement basis for representing this potentially large set of pose corrections. However, we do not simply use PCA on the displacement field defined over the entire surface, since this would lead to a large number of important basis functions and be inefficient for hardware rendering. Instead, we decompose the model into locally supported domains learned from the influence of individual joints on the displacement fields (described in detail in Section 2.2). The resulting memory sensitive set of locally supported eigendisplacement basis functions constitutes the EigenSkin approximation, and is well suited to rendering in graphics hardware. Please see Figure 1 for an example of EigenSkin results.
<H1>2 Method</H1>
In this section we describe the process of augmenting an existing SSD model with EigenSkin. Although the process is shown for displacements, it applies similarly to the construction of linear normal corrections, allowing EigenSkin to correct SSD for both shape and shading.
<H2>2.1 Notation: SSD and Bone Weights</H2>
Let B be the set of all bone indices, and denote the bones affecting vertex i by the subset of indices B i ⊂ B. For a given skeletal configuration, with bone transforms {T b } b∈B , the position of the i th vertex after SSD is
1
v ̃ i = ∑ b∈B i w ib T b v i
where v i is the position of vertex i in the neutral pose, and w ib give the affine combination of bone transforms for this vertex. In the character’s neutral pose we assume that T b = I, ∀b ∈ B. Starting with a reasonable set of bone weights is important because the added displacements only correct the SSD predicted mesh shape near observed configurations. We compute our SSD bone weights as a function of vertex bone distances in the neutral pose. This yields reasonable bone weights which change smoothly over the mesh. Filtering may be required to force each bone’s weights to zero at the edges of its influence to prevent discontinuities. In principle, the weights can be computed to optimize the quality of the EigenSkin correction, and this is a topic of future research.
<H2>2.2 Locally Supported Joint Displacements</H2>
Let P be the set of indices of observed poses with 0 ∈ P representing the rest pose and let the observed vertex positions and bone transforms for pose p ∈ P be denoted as v p and T p , respectively. The differences between the SSD vertex positions and the observed pose positions mapped back into the rest pose yield displacements (see Figure 2 ), The observed mesh shapes result when these displacements are added to the rest pose before applying the bone weighted transformation. If the deformations vary smoothly over pose space, then interpolated displacements provide a good approximation of deformations at configurations between observations. To make our hardware implementation possible, we exploit the observation that localized changes to the configuration of an articulated character often result in local deformations. This independence occurs in most articulated characters, and certainly exists in realistic human hands. Bending a single joint in one finger, though difficult without bending any other joints, does not cause noticeable deformations in the other fingers. Likewise, bending one finger of our finite element hand model does not cause noticeable deformations in the others (see Figure 4 ). Although the finite element model deformations resulting from a change to a single joint are global, the displacement magnitudes are imperceptible at vertices that are far from the joint. We refer to the set of vertices significantly affected by a joint motion as the joint support. Note that the joint supports depend on the SSD weights and in general they do not correspond to the sets of vertices influenced by bone transforms. To find the support of a joint we compute the deformations that result from moving the joint to different positions in its full range of motion while keeping all other joints fixed to the rest pose position. The set of vertices having a displacement larger than a given threshold in any of these computed poses then becomes the support of this joint. For example, in our case we used four percent of the maximum observed displacement (we will see that memory constraints also play a large part). Several joint supports of our finite element hand model are shown in Figure 3 . Note that we consider only single joint perturbations due to the high dimensionality of our hand model’s configuration space. Nevertheless, we can still approximate linear coupling effects since we let the joint supports overlap.
−1 u i p = ∑ b∈B i w ib T b p v i p − v i 0 .
v p ~ v ( Σ wT) −1 u v 0
Figure 2: The displacement for vertex i and pose p, denoted u p , i is the difference between its observed position, v p , and its position i under SSD, v ̃ p , mapped back into the rest pose. i
thumb CM thumb MP thumb IP index MP
Figure 3: Joint supports for thumb carpal-metacarpal, metacarpophalangeal, inter-phalangeal joints and index metacarpo-phalangeal joint.
For notational convenience, suppose the articulated figure has a tree structure, i.e., does not have loops, such as for humanoids, and joints are denoted by the index of the adjacent bone furthest from the root of the hierarchy. Denoting 0 ∈ B as the root, joints have nonzero index. Let P j ⊂ P be the set of pose indices used to compute the support for joint j and let S j be the set of vertex indices in the joint support. Furthermore, let J i be the set of joints whose supports contain vertex i. That is, J i = { j|i ∈ S j } ⊂ B\{0}.
<H2>2.3 Eigendisplacements</H2>
Although the pose displacements computed for independently perturbed joints may be used as a basis for describing displacements of new configurations, significant redundancy exists in the pose displacements, e.g., skin bulging in similar directions. Principal Component Analysis (PCA) of joint support displacements yields an orthogonal displacement basis, which we term eigendisplacements. As guaranteed by PCA, adding successive corrections with the eigendisplacement basis provides approximations which are better in a formal, least squares, sense
[Golub and van Loan 1996]. Computing principal components with the Euclidean norm is equivalent to computing the singular value decomposition (in the case of a square symmetric matrix it is equivalent to eigenanalysis). For each joint j we construct a rectangular matrix, A j , of size 3|S j | × |P j |, whose columns consist of the x,y, and z components of the vertex displacements on the joint support. In the singular value decomposition, A j = U j D j V T j , the matrix U j has the same size as A j and consists of columns of eigendisplacements for support j in the same block column format that was used to build A j . The singular values, in the diagonal matrix D j , identify the importance that each eigendisplacement has in reproducing the observed poses (they relate to the proportion of variation explained by each principal component). Note that the matrix V j and the singular values combine to gives the coordinates of our observed displacements in the eigendisplacement basis. We denote u ˆ jk the eigendisplacement i of vertex i in the basis of support j with importance k where k goes from 1 (the principal component) up to |P j |. Figure 5 shows the first four eigendisplacements of the thumb carpal-metacarpal joint support in our hand example. At this point we can truncate each eigendisplacement basis expansion knowing that the error will be minimized in the least squares sense. The hardware limits the size of each truncated basis set as there is a limited amount of per vertex data memory in which we can send the eigendisplacements to the EigenSkin vertex program (see Section 2.5). Letting n j < |P j | be the size of the truncated basis set of joint support j, this constraint can be written as max n j |J i | ≤ maximum possible displacements. i Instead of choosing each n j individually, we take an equal number of eigendisplacements from each support. The equation for computing the deformed mesh shape for an arbitrary configuration with bone transforms {T b } b∈B can then be written as
Figure 4: A subset of the training data showing some of the thumb and index finger poses.
2
n j v ̃ i = ∑ w ib T b v 0 i + ∑ ∑ α ̃ jk u ˆ i jk b∈B i j∈J i k=1
where α ̃ gives the coordinates of the displacement correction in jk terms of the reduced eigendisplacement basis. These coordinates are computed to interpolate between observed displacements, as shown below. Note that Equation 2 provides a powerful model for shape deformation (see, in particular,
[James and Pai 2002a]).
<H2>2.4 Interpolating Eigendisplacement Coordinates</H2>
As an articulated character moves between observed configurations, its shape should interpolate the observed poses. To do this we interpolate the eigendisplacement coordinates of the observed configurations. For the truncated set of eigendisplacements at each support, we need the coordinates in the truncated basis which give displacements closest to the observed displacements. That is, we want to solve for α p in n j u i p = ∑ ∑ α jk p u ˆ i jk . j∈J i k=1 This is an over constrained linear system which we can solve using least squares to get the best fit to our observed displacements. Conveniently, the least squares solution for any number of eigendisplacements, n j , is available from the singular value decomposition computed in Section 2.3. For joint support j, column p of D j V T j contains α jk p for k = 1..|P j |. This leads us to the problem of computing the eigendisplacement coordinates for arbitrary configurations. Radial basis functions [Powell 1987] (RBF) are a common choice for interpolating scattered data, and have been used by Lewis et al. [Lewis et al. 2000] for pose space deformation and by Sloan et al. [Sloan et al. 2001] for shape interpolation with articulated figures. Our interpolation is one dimensional since all our observations involved perturbations of individual joints. Although we could use a simpler interpolant, we also choose RBFs because they extend easily to the higher dimensional domains needed to let EigenSkin capture nonlinear multi-joint coupling effects (a subject of future work). We use Gaussian interpolation shape functions, φ (r) = exp(−r/r 0 ). In our one dimensional case, the α jk only depend on the distance of joint j from its settings in poses P j . For revolute joints, we can easily compute the distance, r, by comparing the joint angles directly. For joints with more than one rotational degree of freedom, we compute distance as the angle in the axis-angle representation of the joint’s rotation matrix. Ideally, with a large number of observed joint perturbations per support we would interpolate using fewer interpolation basis functions ( φ ) than observations. In the case of our hand model, however, we only have approximately half a dozen pose perturbations for each joint degree of freedom (for a total of approximately 120 poses). This justifies our use of interpolation basis functions since the total cost of constructing and evaluating the RBF interpolant for half a dozen poses is negligible. The interpolated eigendisplacement coordinates for a new pose are computed as α ̃ jk = ∑ λ q jk φ (r jq ) q∈P j where r jq is the distance of joint j in the new pose from its setting in pose q, and the λ q jk for q ∈ P j are given by the solution to the linear system, α jk p = ∑ λ q jk φ (r p jq ), for p ∈ P j . q∈P j Here r p is the distance between joint j’s position in pose p and jq its position in pose q (and thus r p = 0). The system of equations jp
#0 ( σ 0 ≈ 0.093) #1 ( σ 1 ≈
Figure 5: Eigendisplacements and singular values, σ , for thumb carpal-metacarpal joint in left-right order of importance. XYZ components of displacement are represented using an RGB colour correspondence.
is square, and invertible provided P j does not contain two observations with identical joint settings.
<H2>2.5 EigenSkin Vertex Programming</H2>
Modern vertex programming hardware (e.g., [Lindholm et al. 2001]
) is ideally suited to performing the per-vertex weighted linear superposition of eigendisplacements (contained in the large brackets of Equation 2) performed prior to the SSD weighted transformation. Depending on the number of eigendisplacements used, the weighted eigendisplacement vector accumulations are about as costly as the weighted transform matrix-vector multiplyaccumulate operations. Current vertex programs limit per vertex data to 16 4-tuples of floats. In our implementation we impose a limit of 10 eigendisplacements per vertex (or 5 eigendisplacements and 5 normal corrections), which still leaves room for texture coordinates after specifying the vertex position, normal, colour, and bone weights. Notice that this limit is not hard since careful choices and packing of per vertex data permit more than 10 of the 16 available tuples to be allocated for EigenSkin data. If a vertex is in many supports then the number of eigendisplacements renderable by current hardware may be too severely restricted. In this case it is useful to smoothly mask the support groups to smaller regions, otherwise fewer eigendisplacements must be used.
<H1>3 Results</H1>
To illustrate our EigenSkin method, we have constructed a finite element model of the human hand (see Figure 6 ) which exhibits subtle nonlinear skin deformations. The surface skin model and matching skeleton are based on Loop subdivision [Loop 1987] of a hand mesh exported from Curious Labs Poser [Curious Labs Inc.]. A finite element mesh containing 11,171 high-order 10-node tetrahedral elements was generated using NETGEN
[Schoberl 1997] (and subsequent simplification). The hand was moved into various poses by applying position constraints to vertices adjacent to the rigid bones, and computing the resulting tissue deformation using geometrically nonlinear static finite element analyses
[Zienkiewicz 1977] with (a modified version of) the CalculiX program [Dhondt and Wittig]. Approximately half a dozen poses were computed for each joint degree of freedom to estimate the locally supported joint eigendisplacements, and 25 additional poses were computed for validation. Finite element analyses were performed on a cluster of modern workstations and consumed several hundred CPU hours.
0.065) #2 ( σ 2 ≈ 0.023) #3 ( σ 3 ≈ 0.018)
The model was not intended to reproduce detailed skin wrinkling effects, and lacks anatomical details such as tendons, blood vessels, and skin layers. Despite these limitations, the model reasonably describes bulk tissue deformations and was sufficient to illustrate our method. As shown in Figure 7 , the eigendisplacement approximations of the hand model produce a clear improvement over the traditional SSD algorithm. Even with only five leading eigendisplacements, the EigenSkin approximation is essentially indistinguishable from the original FEM model. Our interactive simulation uses a CyberGlove [Immersion Corporation] input device to interactively drive our EigenSkin hand model, while graphical feedback is rendered using OpenGL and a GeForce3 graphics card. Radial basis function interpolation of the pose-space data is performed on the main CPU, with eigendisplacement amplitudes and bone transforms set as input parameters to the EigenSkin vertex programs which are compiled as static display lists. Currently, our unoptimized implementation renders the EigenSkinned hand model only slightly slower than the traditional SSD model. A large 55,904 triangle hand model renders at 47 frames per second (FPS), while a coarser 13,976 triangle model achieves 181 FPS. Please see our accompanying video for a demonstration of the real time simulation.
Figure 6: Skeleton used to drive finite element hand model.
<H1>4 Conclusions and Discussion</H1>
Our results confirm that the EigenSkin method is an effective tool for character skinning when compressed hardware renderable approximations are required for an articulated character’s nonlinear quasi-static deformations. EigenSkin works best when SSD corrections are localized, providing independence between different parts of the mesh, and are stable (i.e., corrections vary slowly over posespace), allowing accurate and efficient interpolation. Under these
Skeletal-Subspace Deformation Only relErr=1.000
3 eigendisplacements relErr=0.100
Figure 7: EigenSkin approximation for 0–5 eigendisplacements of the thumb’s carpal-metacarpal joint shown with the 13,976 triangle hand model. The pose geometry (skin colour) is approximated by the EigenSkin model (green). The l 2 relative displacement error (relErr) is also printed below each image. Remarkably, the SSD model is substantially improved after the addition of only one eigendisplacement.
conditions, very practical results can be obtained in which only one or two eigendisplacements per joint produce a visually dramatic improvement over commonplace Skeletal-Subspace Deformation. Limitations and Future Work: Despite the advantages of EigenSkin, as presented there are several limitations to the method which are topics of current research. We assume that an initial SSD model is provided and then show how the EigenSkin corrections are beneficial. However, an alternate approach involves optimizing bone weights to allow better EigenSkin approximations of the displacements and normals. While good eigendisplacement bases can often be constructed using displacements resulting from single joint motions, in practice it is desirable to allow general pose sets and to recover nonlinear joint-joint coupling phenomena.
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1 eigendisplacement 2 eigendisplacements relErr=0.161 relErr=0.160
4 eigendisplacements 5 eigendisplacements relErr=0.098 relErr=0.065
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"text": "['Singh and Kokkevis 2000']",
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"text": "['Lewis et al. 2000']",
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"text": "['and physicallybased methods']",
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"text": "['Wilhelms and van Gelder 1997']",
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"text": "['Scheepers et al. 1997']",
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"text": "['Gourret et al. 1989']",
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"text": "['most character animation in interactive applications']",
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"text": "['is based on a geometric skeletal deformation technique']",
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"text": "['in which vertex locations are weighted averages of points in several coordinate frames']",
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"text": "['linear blending']",
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"text": "['matrix palette skinning']",
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"text": "['Skeletal-Subspace Deformation']",
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"text": "['Magnenat-Thalmann et al. 1988']",
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"text": "['Magnenat-Thalmann and Thalmann 1991']",
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"text": "['One alternative is to store a large database of character poses, and interpolate between them']",
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"text": "['Maestri 1999']",
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"text": "['these approaches give animators great control over character deformation']",
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"text": "['they have the disadvantage of requiring a potentially very large number of poses for animation, and also lack an underlying kinematic model']",
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"text": "['such approaches are common']",
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"text": "['Parke et al. 1996']",
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"text": "['A hybrid approach which effectively combines SSD and morphing, is the work of Lewis et al. who introduced “Pose Space Deformations”']",
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"text": "['Lewis et al. 2000']",
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"text": "['to overcome the limitations of linear transform blending while retaining a kinematic approach']",
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"text": "['this is a big improvement over character morphing, and sufficiently interactive for animators']",
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"text": "['storing surface displacements for each pose in a large pose space is a memory inefficient approach for hardware applications']",
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"text": "['providing a kinematic deformation model which also has artist-drawn poses']",
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"text": "['At run time, the character may be simulated by mapping interpolated displacements onto the underlying SSD character model']",
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"text": "['Sloan et al. show a more efficient method of interpolating an articulated figure using example shapes scattered in an abstract space']",
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"text": "['Sloan et al. 2001']",
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"text": "['interpolation occurs in the rest pose before SSD is applied']",
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"text": "['the interpolation involves blending over all of the example shapes for every vertex']",
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"text": "['the independence of abstract space dimensions is not taken into account']",
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"text": "['This becomes inefficient and difficult to map to hardware with the large number of examples required for a highly articulated figure']",
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"text": "['we also wish to efficiently render deformation behaviour computed using physically-based and reality-based deformable models']",
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"text": "['Such models have been widely used']",
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"text": "['Terzopoulos and Fleischer 1988']",
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"text": "['Terzopoulos and Witkin 1988']",
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"text": "['Metaxas and Terzopoulos 1992']",
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"text": "['CaniGascuel 1998']",
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"text": "['O’Brien and Hodgins 1999']",
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"text": "['Pai et al. 2001']",
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"text": "['Allen et al. 2002']",
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"text": "['most approaches are not intended for real time (hardware) rendering']",
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"text": "['Recently, approaches for fast simulation of physical dynamic volumetric deformations have appeared']",
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"text": "['Zhuang and Canny 1999']",
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"text": "['Debunne et al. 2001']",
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"text": "['Picinbono et al. 2001']",
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"text": "['for interactive applications']",
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"text": "['quasi-static deformation, for which fast deformation techniques also exist']",
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"text": "['Cotin et al. 1999']",
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"text": "['James and Pai 1999']",
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"text": "['are unfortunately restricted to small deformations']",
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"text": "['James and Pai 2002b']",
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"text": "['More closely related to character animation is anatomically based modeling of physical deformable models']",
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"text": "['Wilhelms and van Gelder 1997']",
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"text": "['examples include musculature']",
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"text": "['Chen and Zeltzer 1992']",
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"text": "['and faces']",
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"text": "['Lee et al. 1995']",
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"text": "['pose-space parameterization of nonhysteretic cloth on articulated characters has recently been considered']",
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"text": "['a large class of pose-dependent quasi-static deformations can be described using the EigenSkin approach']",
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"text": "['and could be optimized for hardware rendering using the techniques presented herein']",
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"text": "['the use of reduced eigenbasis representations for highdimensional models has a long history in science, with foundations on Principal Component Analysis and Karhunen-Loeve theory']",
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"text": "['decomposition of static deformations']",
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"text": "['and recognition applications in computer vision']",
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"text": "['Turk and Pentland 1991']",
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"text": "['We introduce a method for extending SSD that enhances its range of modeling capabilities at very little cost, and in a manner optimized for real time graphics hardware']",
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"text": "['EigenSkin constitutes an error-optimal set of eigenbases for approximating the original deformation model, for a given amount of per-vertex displacement memory']",
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"text": "['at interactive rates on a PC with negligible cost to the main CPU']",
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"text": "['We illustrate our method by rendering a very large finite element model']",
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"text": "['Using commodity graphics hardware, EigenSkin enables the simulation of subtle nonlinear surface deformations of geometrically complex models at little more than the cost of rendering']",
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"text": "['Rendering of complex physical deformation models for character animation remains a significant hurdle for interactive applications']",
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"text": "['Skeletal-Subspace Deformation']",
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"text": "['buckling of skin near joints']",
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"text": "['in extreme poses']",
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"text": "['restrictions on the range of deformations that can be easily modeled and displayed for various character poses']",
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"text": "['methods have been proposed to address this and have been effectively employed by the motion picture industry']",
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"text": "['Lewis et al. 2000']",
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"text": "['nearly all video game character animation is still done using traditional SSD']",
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"text": "['we present a practical technique which overcomes all aforementioned SSD problems, and can be achieved using a memory-efficient linear correction to the traditional SSD method']",
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"text": "['The resulting EigenSkin construct allows subtle character deformations for skin and clothing']",
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],
"text": "['The deformations can be compactly represented in an efficient datadependent basis and rendered in real time using vertex shaders in commodity graphics hardware']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
10703,
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],
"text": "['Lindholm et al. 2001']",
"type": "data",
"id": "T106"
},
{
"offsets": [
[
11401,
11483
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],
"text": "['we do not simply use PCA on the displacement field defined over the entire surface']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
[
11491,
11595
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],
"text": "['this would lead to a large number of important basis functions and be inefficient for hardware rendering']",
"type": "own_claim",
"id": "T108"
},
{
"offsets": [
[
11231,
11390
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],
"text": "['we use Principal Component Analysis (PCA) to construct an error-optimal eigendisplacement basis for representing this potentially large set of pose corrections']",
"type": "own_claim",
"id": "T109"
},
{
"offsets": [
[
11606,
11734
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],
"text": "['we decompose the model into locally supported domains learned from the influence of individual joints on the displacement fields']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
11759,
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],
"text": "['Section 2.2']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
11773,
11956
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],
"text": "['The resulting memory sensitive set of locally supported eigendisplacement basis functions constitutes the EigenSkin approximation, and is well suited to rendering in graphics hardware']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
12157,
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],
"text": "['the process is shown for displacements']",
"type": "own_claim",
"id": "T113"
},
{
"offsets": [
[
12197,
12266
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],
"text": "['it applies similarly to the construction of linear normal corrections']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
12268,
12328
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],
"text": "['allowing EigenSkin to correct SSD for both shape and shading']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
12585,
12629
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],
"text": "['the position of the i th vertex after SSD is']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
12667,
12696
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],
"text": "['v ̃ i = ∑ b∈B i w ib T b v i']",
"type": "own_claim",
"id": "T117"
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{
"offsets": [
[
12556,
12582
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],
"text": "['bone transforms {T b } b∈B']",
"type": "data",
"id": "T118"
},
{
"offsets": [
[
12515,
12549
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],
"text": "['For a given skeletal configuration']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
12914,
12973
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],
"text": "['Starting with a reasonable set of bone weights is important']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
12982,
13076
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],
"text": "['the added displacements only correct the SSD predicted mesh shape near observed configurations']",
"type": "own_claim",
"id": "T121"
},
{
"offsets": [
[
13170,
13241
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],
"text": "['This yields reasonable bone weights which change smoothly over the mesh']",
"type": "own_claim",
"id": "T122"
},
{
"offsets": [
[
13078,
13168
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],
"text": "['We compute our SSD bone weights as a function of vertex bone distances in the neutral pose']",
"type": "own_claim",
"id": "T123"
},
{
"offsets": [
[
13243,
13362
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],
"text": "['Filtering may be required to force each bone’s weights to zero at the edges of its influence to prevent discontinuities']",
"type": "own_claim",
"id": "T124"
},
{
"offsets": [
[
13378,
13457
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],
"text": "['the weights can be computed to optimize the quality of the EigenSkin correction']",
"type": "own_claim",
"id": "T125"
},
{
"offsets": [
[
13787,
13918
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],
"text": "['The differences between the SSD vertex positions and the observed pose positions mapped back into the rest pose yield displacements']",
"type": "own_claim",
"id": "T126"
},
{
"offsets": [
[
13924,
13932
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],
"text": "['Figure 2']",
"type": "data",
"id": "T127"
},
{
"offsets": [
[
13936,
14068
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],
"text": "['The observed mesh shapes result when these displacements are added to the rest pose before applying the bone weighted transformation']",
"type": "own_claim",
"id": "T128"
},
{
"offsets": [
[
14073,
14119
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],
"text": "['the deformations vary smoothly over pose space']",
"type": "data",
"id": "T129"
},
{
"offsets": [
[
14121,
14236
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],
"text": "['then interpolated displacements provide a good approximation of deformations at configurations between observations']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
14284,
14417
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],
"text": "['we exploit the observation that localized changes to the configuration of an articulated character often result in local deformations']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
14419,
14521
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],
"text": "['This independence occurs in most articulated characters, and certainly exists in realistic human hands']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
[
14523,
14559
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],
"text": "['Bending a single joint in one finger']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
14612,
14671
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],
"text": "['does not cause noticeable deformations in the other fingers']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
14683,
14787
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],
"text": "['bending one finger of our finite element hand model does not cause noticeable deformations in the others']",
"type": "own_claim",
"id": "T135"
},
{
"offsets": [
[
14793,
14801
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],
"text": "['Figure 4']",
"type": "data",
"id": "T136"
},
{
"offsets": [
[
14814,
14904
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],
"text": "['the finite element model deformations resulting from a change to a single joint are global']",
"type": "own_claim",
"id": "T137"
},
{
"offsets": [
[
14906,
14991
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],
"text": "['the displacement magnitudes are imperceptible at vertices that are far from the joint']",
"type": "own_claim",
"id": "T138"
},
{
"offsets": [
[
15098,
15234
]
],
"text": "['the joint supports depend on the SSD weights and in general they do not correspond to the sets of vertices influenced by bone transforms']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
15688,
15729
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],
"text": "['memory constraints also play a large part']",
"type": "own_claim",
"id": "T140"
},
{
"offsets": [
[
15822,
15865
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],
"text": "['we consider only single joint perturbations']",
"type": "own_claim",
"id": "T141"
},
{
"offsets": [
[
15873,
15936
]
],
"text": "['the high dimensionality of our hand model’s configuration space']",
"type": "own_claim",
"id": "T142"
},
{
"offsets": [
[
15952,
16000
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],
"text": "['we can still approximate linear coupling effects']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
16007,
16040
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],
"text": "['we let the joint supports overlap']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
17282,
17422
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],
"text": "['the pose displacements computed for independently perturbed joints may be used as a basis for describing displacements of new configurations']",
"type": "own_claim",
"id": "T145"
},
{
"offsets": [
[
17424,
17479
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],
"text": "['significant redundancy exists in the pose displacements']",
"type": "own_claim",
"id": "T146"
},
{
"offsets": [
[
17487,
17521
]
],
"text": "['skin bulging in similar directions']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
17523,
17628
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],
"text": "['Principal Component Analysis (PCA) of joint support displacements yields an orthogonal displacement basis']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
17686,
17823
]
],
"text": "['adding successive corrections with the eigendisplacement basis provides approximations which are better in a formal, least squares, sense']",
"type": "own_claim",
"id": "T149"
},
{
"offsets": [
[
17826,
17849
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],
"text": "['Golub and van Loan 1996']",
"type": "data",
"id": "T150"
},
{
"offsets": [
[
17852,
17966
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],
"text": "['Computing principal components with the Euclidean norm is equivalent to computing the singular value decomposition']",
"type": "own_claim",
"id": "T151"
},
{
"offsets": [
[
18009,
18042
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],
"text": "['it is equivalent to eigenanalysis']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
17983,
18008
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],
"text": "['a square symmetric matrix']",
"type": "data",
"id": "T153"
},
{
"offsets": [
[
18684,
18816
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],
"text": "['the matrix V j and the singular values combine to gives the coordinates of our observed displacements in the eigendisplacement basis']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
19090,
19226
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],
"text": "['At this point we can truncate each eigendisplacement basis expansion knowing that the error will be minimized in the least squares sense']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
19228,
19284
]
],
"text": "['The hardware limits the size of each truncated basis set']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
19288,
19415
]
],
"text": "['there is a limited amount of per vertex data memory in which we can send the eigendisplacements to the EigenSkin vertex program']",
"type": "own_claim",
"id": "T157"
},
{
"offsets": [
[
19421,
19432
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],
"text": "['Section 2.5']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
20358,
20416
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],
"text": "['Equation 2 provides a powerful model for shape deformation']",
"type": "own_claim",
"id": "T159"
},
{
"offsets": [
[
20440,
20459
]
],
"text": "['James and Pai 2002a']",
"type": "data",
"id": "T160"
},
{
"offsets": [
[
20625,
20672
]
],
"text": "['its shape should interpolate the observed poses']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
20561,
20623
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],
"text": "['an articulated character moves between observed configurations']",
"type": "data",
"id": "T162"
},
{
"offsets": [
[
20766,
20936
]
],
"text": "['For the truncated set of eigendisplacements at each support, we need the coordinates in the truncated basis which give displacements closest to the observed displacements']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
20947,
21018
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],
"text": "['we want to solve for α p in n j u i p = ∑ ∑ α jk p u ˆ i jk . j∈J i k=1']",
"type": "own_claim",
"id": "T164"
},
{
"offsets": [
[
21019,
21149
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],
"text": "['This is an over constrained linear system which we can solve using least squares to get the best fit to our observed displacements']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
21165,
21310
]
],
"text": "['the least squares solution for any number of eigendisplacements, n j , is available from the singular value decomposition computed in Section 2.3']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
21333,
21388
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],
"text": "['column p of D j V T j contains α jk p for k = 1..|P j |']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
21316,
21331
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],
"text": "['joint support j']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
21496,
21518
]
],
"text": "['Radial basis functions']",
"type": "background_claim",
"id": "T169"
},
{
"offsets": [
[
21539,
21627
]
],
"text": "['are a common choice for interpolating scattered data, and have been used by Lewis et al.']",
"type": "background_claim",
"id": "T170"
},
{
"offsets": [
[
21520,
21531
]
],
"text": "['Powell 1987']",
"type": "data",
"id": "T171"
},
{
"offsets": [
[
21629,
21646
]
],
"text": "['Lewis et al. 2000']",
"type": "data",
"id": "T172"
},
{
"offsets": [
[
21648,
21694
]
],
"text": "['for pose space deformation and by Sloan et al.']",
"type": "background_claim",
"id": "T173"
},
{
"offsets": [
[
21696,
21713
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],
"text": "['Sloan et al. 2001']",
"type": "data",
"id": "T174"
},
{
"offsets": [
[
21715,
21763
]
],
"text": "['for shape interpolation with articulated figures']",
"type": "background_claim",
"id": "T175"
},
{
"offsets": [
[
21765,
21801
]
],
"text": "['Our interpolation is one dimensional']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
21808,
21872
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],
"text": "['all our observations involved perturbations of individual joints']",
"type": "own_claim",
"id": "T177"
},
{
"offsets": [
[
21883,
21917
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],
"text": "['we could use a simpler interpolant']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
21919,
21938
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],
"text": "['we also choose RBFs']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
21947,
22070
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],
"text": "['they extend easily to the higher dimensional domains needed to let EigenSkin capture nonlinear multi-joint coupling effects']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
22197,
22275
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],
"text": "['the α jk only depend on the distance of joint j from its settings in poses P j']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
22171,
22195
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],
"text": "['our one dimensional case']",
"type": "data",
"id": "T182"
},
{
"offsets": [
[
22299,
22376
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],
"text": "['we can easily compute the distance, r, by comparing the joint angles directly']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
22282,
22297
]
],
"text": "['revolute joints']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
22382,
22436
]
],
"text": "['joints with more than one rotational degree of freedom']",
"type": "data",
"id": "T185"
},
{
"offsets": [
[
22438,
22534
]
],
"text": "['we compute distance as the angle in the axis-angle representation of the joint’s rotation matrix']",
"type": "own_claim",
"id": "T186"
},
{
"offsets": [
[
22609,
22695
]
],
"text": "['we would interpolate using fewer interpolation basis functions ( φ ) than observations']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
22550,
22608
]
],
"text": "['a large number of observed joint perturbations per support']",
"type": "data",
"id": "T188"
},
{
"offsets": [
[
22737,
22828
]
],
"text": "['we only have approximately half a dozen pose perturbations for each joint degree of freedom']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
22712,
22726
]
],
"text": "['our hand model']",
"type": "data",
"id": "T190"
},
{
"offsets": [
[
22871,
22926
]
],
"text": "['This justifies our use of interpolation basis functions']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
22933,
23035
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],
"text": "['the total cost of constructing and evaluating the RBF interpolant for half a dozen poses is negligible']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
23917,
23984
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],
"text": "['P j does not contain two observations with identical joint settings']",
"type": "data",
"id": "T193"
},
{
"offsets": [
[
23470,
23493
]
],
"text": "['The system of equations']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
23882,
23907
]
],
"text": "['is square, and invertible']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
24066,
24100
]
],
"text": "['Modern vertex programming hardware']",
"type": "background_claim",
"id": "T196"
},
{
"offsets": [
[
24109,
24129
]
],
"text": "['Lindholm et al. 2001']",
"type": "data",
"id": "T197"
},
{
"offsets": [
[
24133,
24231
]
],
"text": "['is ideally suited to performing the per-vertex weighted linear superposition of eigendisplacements']",
"type": "background_claim",
"id": "T198"
},
{
"offsets": [
[
24280,
24330
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],
"text": "['performed prior to the SSD weighted transformation']",
"type": "background_claim",
"id": "T199"
},
{
"offsets": [
[
24332,
24525
]
],
"text": "['Depending on the number of eigendisplacements used, the weighted eigendisplacement vector accumulations are about as costly as the weighted transform matrix-vector multiplyaccumulate operations']",
"type": "background_claim",
"id": "T200"
},
{
"offsets": [
[
24527,
24597
]
],
"text": "['Current vertex programs limit per vertex data to 16 4-tuples of floats']",
"type": "background_claim",
"id": "T201"
},
{
"offsets": [
[
24621,
24674
]
],
"text": "['we impose a limit of 10 eigendisplacements per vertex']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
24727,
24845
]
],
"text": "['which still leaves room for texture coordinates after specifying the vertex position, normal, colour, and bone weights']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
24859,
24881
]
],
"text": "['this limit is not hard']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
24888,
25016
]
],
"text": "['careful choices and packing of per vertex data permit more than 10 of the 16 available tuples to be allocated for EigenSkin data']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
25021,
25049
]
],
"text": "['a vertex is in many supports']",
"type": "data",
"id": "T206"
},
{
"offsets": [
[
25055,
25149
]
],
"text": "['the number of eigendisplacements renderable by current hardware may be too severely restricted']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
25164,
25231
]
],
"text": "['it is useful to smoothly mask the support groups to smaller regions']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
[
25233,
25280
]
],
"text": "['otherwise fewer eigendisplacements must be used']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
25376,
25436
]
],
"text": "['we have constructed a finite element model of the human hand']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
25442,
25450
]
],
"text": "['Figure 6']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
25453,
25502
]
],
"text": "['which exhibits subtle nonlinear skin deformations']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
[
25504,
25578
]
],
"text": "['The surface skin model and matching skeleton are based on Loop subdivision']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
25580,
25589
]
],
"text": "['Loop 1987']",
"type": "data",
"id": "T214"
},
{
"offsets": [
[
25591,
25638
]
],
"text": "['of a hand mesh exported from Curious Labs Poser']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
25660,
25766
]
],
"text": "['A finite element mesh containing 11,171 high-order 10-node tetrahedral elements was generated using NETGEN']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
25769,
25782
]
],
"text": "['Schoberl 1997']",
"type": "data",
"id": "T217"
},
{
"offsets": [
[
25817,
26036
]
],
"text": "['The hand was moved into various poses by applying position constraints to vertices adjacent to the rigid bones, and computing the resulting tissue deformation using geometrically nonlinear static finite element analyses']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
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A14 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A14_M05_Example-Based_Control_of_Human_Motion_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1028523.1028534
Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2004) R. Boulic, D. K. Pai (Editors)
<Title>Example-Based Control of Human Motion</Title>
Eugene Hsu 1 Sommer Gentry
1 Computer Science and Artificial Intelligence Laboratory 2 Laboratory for Information and Decision Systems Massachusetts Institute of Technology
<Abstract>In human motion control applications, the mapping between a control specification and an appropriate target motion often defies an explicit encoding. We present a method that allows such a mapping to be defined by example, given that the control specification is recorded motion. Our method begins by building a database of semantically meaningful instances of the mapping, each of which is represented by synchronized segments of control and target motion. A dynamic programming algorithm can then be used to interpret an input control specification in terms of mapping instances. This interpretation induces a sequence of target segments from the database, which is concatenated to create the appropriate target motion. We evaluate our method on two examples of indirect control. In the first, we synthesize a walking human character that follows a sampled trajectory. In the second, we generate a synthetic partner for a dancer whose motion is acquired through motion capture.</Abstract>
Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Animation
Jovan Popović
<H1>1. Introduction</H1>
Authoring human motion is difficult for computer animators, as humans are exceptionally sensitive to the slightest of errors. This process involves an animator providing a control specification which is mapped to a target motion by some means. In traditional keyframe animation, for instance, the keyframes are the control specification, and the target motion is achieved through spline interpolation. Due to advances in data acquisition technology and computational power, techniques have been developed that allow desired target motion to be specified using a human performance. This is natural for traditional keyframe animators, who often use recorded or live human motion for reference. Motion capture is the most direct method to map performances to animated humans, as it is essentially an identity mapping. However, a generalization of this approach to allow for more indirect mappings creates an array of fantastic possibilities, such as mapping voice signals to facial motion [Bra99] or gestural actions to animated reactions [JP99]. Indirect mappings, however, must still be encoded in some way. Manually, this can be an exceptionally challenging task requiring detailed, domain-specific knowledge. Consider a partner dance scenario in which an animator wishes to con-
trol a follower using the captured motion of a leader. The mapping from leader to follower motion must minimally encode a significant amount of knowledge about the structure of the dance; this knowledge, unfortunately, would be out of reach to an animator who is not a skilled dancer. Indeed, it would still be difficult for a skilled dancer to state the precise mapping. Human dancers learn their skills by observation and practice; our objective is to emulate this process on a computer for situations, such as partner dance, when the control specification takes the form of one dancer’s motion. To learn indirect mappings, we adopt a memory-based approach which implicitly encodes the desired mapping using a database of semantically meaningful example instances. These instances store segments of synchronized control and target motion, which provide examples of how the mapping should be applied to input control motions. In partner dance, an instance might contain an example control motion of a leader pushing his or her partner forward. The corresponding example target motion would be that of the follower, taking a step backward in response. A new input control motion can be interpreted as a sequence of rigidly transformed and temporally stretched control segments from the mapping database. Through the mapping instances, a given interpretation also corresponds to a sequence of target segments that can be assembled to form a target motion. We use dynamic programming to select a sequence that balances the quality of interpretation with the continuity of the induced target motion. Various postprocessing techniques can be then be applied to smooth and adjust the desired target motion. Our approach is evaluated on two applications. In the first, we demonstrate its ability to map low-dimensional input to high-dimensional motion by controlling walk motion from mouse trajectories. In the second, we highlight our method’s capability to handle complex, stylized mappings by controlling a dance follower with the motion of a dance leader.
c The Eurographics Association 2004.
Hsu, Gentry, and Popović / Example-Based Control of Human Motion
<H1>2. Background</H1>
Performance-driven animation, or computer puppetry, derives its broad appeal from its ability to map human performances automatically to animated characters
[Stu98]. While these mappings can be as simple as a direct copy of joint angles, the ability to discover more complex mappings gives the approach a tremendous amount of power and flexibility. In online techniques [JP99]
, computational speed and instantaneous results are of paramount importance; offline techniques [Bra99] allow quality and global optimality to take precedence. Our method falls into the latter category. Complex mappings often defy purely physical or mathematical encodings. As a result, many methods assume that mappings are described by parametric probabilistic models
[Bra99, DB01, DYP03, JP99]. An advantage of these techniques is their ability to generalize to a variety of inputs. However, this comes at a price: statistical learning often necessitates large volumes of training data or severe restrictions on model complexity. For certain applications, this is a worthwhile tradeoff, but for others, it can result in impractically long training times or loss of important detail. A memory-based approach like ours does not suffer from these disadvantages. An important benefit of this design choice is the ability to use segments, rather than frames, as the primitive unit of motion. This allows for explicit preservation of higherlevel motion semantics. Kim et al. demonstrate that a semantically guided segmentation of rhythmic motion allows for highly realistic motion synthesis, even using simple transition models
[KPS03]. Although this work, like ours, uses partner dance for evaluation, it does not address the problem of generating a follower given the motion of a leader. In the segment modeling domain, we consider our method most similar to that of Pullen and Bregler
[PB02]. While Pullen and Bregler’s method was shown to be an effective solution for the chosen application of texturing keyframed motion, its applicability to our problem is limited by several factors. First, their method assumes no spatial dependencies between the control (keyframed curves) and the target (textured motion). Second, there is no enforcement of motion continuity, other than a heuristic for consecutively observed segments. Our approach generates target motion segments that are amenable to simple blending. Finally, their method assumes that the input motion can be presegmented analogously to the examples, which is achieved in their work by observing sign changes in velocity. One could extend this approach for rhythmic motions using the automated approach of Kim et al.
[ KPS03 ]. In the general case, however, a control motion may not admit any intuitive presegmentation. One may wish, for instance, to generate walk motion from a constant-velocity trajectory. Our method requires no presegmentation; moreover, it produces a semantically guided segmentation as part of the optimization. In this context, our algorithm could be viewed as an extension of speech recognition methods that use connected word models
[ RJ93 ]. Arikan et al. describe an example-based approach to synthesizing human motion that satisfies sparse temporal annotation and pose constraints
[ AFO03 ]. Although their work differs from ours in intent, they also employ a dynamic programming algorithm that optimizes a weighted combination of interpretation and motion continuity. Our formulation differs in two subtle but important ways. First, our notion of continuity is dependent on the interpretation; that is, the continuity between two motion segments is undefined until a candidate interpretation specifies a coordinate frame for comparison. Second, their objective function is defined over frames instead of segments. As a result, they must use coarse-to-fine iterations of their dynamic programming algorithm to gain the temporal consistency that is intrinsic to our segment-based approach.
Other related methods based on motion capture clip rearrangement include work by Kovar et al.
[ KGP02 ], Lee et al. [ LCR ∗ 02 ], and Arikan and Forsyth
[ AF02 ]. Although these do not aim to discover control by example, they have nevertheless provided inspiration for our work. An additional distinction is that these methods do not use continuous control from human performance and focus on sparser specifications such as keyframes and nontemporal paths. Our method is not designed to handle such control specifications and therefore should be viewed as an alternative to these approaches, rather than a replacement. Many motion rearrangement techniques are derived from previous work in texture synthesis. Here, we consider our work most similar in intent to image analogies [ HJO ∗ 01 ]. This method, given an unfiltered and filtered version of the same image, applies an analogous filter to a novel image. Our method, given a set of synchronized control and target motions, applies an analogous mapping to a new input control motion. Image analogies was shown to be an elegant method with applications such as texture transfer, textureby-numbers, and super-resolution. It is our hope that our method will have the same versatility for motion. Our dance evaluation suggests an alternative view of our method as one of interaction modeling. In this domain, tech- niques have been developed that specify the mappings between character motions with explicit models of character interaction. Adaptive autonomous characters have used rules to exhibit complex flocking, herding, and locomotory behaviors
[Rey87, TT94].
Approaches to explicit interaction modeling have included layered architectures
[BG95], procedural descriptions
[PG96], and even cognitive models
[FTT99]. In this context, our work might be viewed as a competency module that enhances the skills of characters to enable their participation in complex interactive performances.
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Control Target
Figure 1: Segmentation of Lindy Hop motion into two-beat rhythm units.
<H1>3. Database Construction</H1>
We begin by acquiring examples of synchronized control motions A and target motions B. Each frame of motion is encoded by a point cloud. For human motion, we use skeletal joint positions, since this representation provides a more intuitive space than joint angle representations for comparing poses
[KGP02]. Furthermore, point cloud representations allow for generalization to control motions without skeletal representations, such as mouse input. The examples are divided into control segments a 1 , . . . , a N and target segments b 1 , . . . , b N , where a i and b i are synchronized motions that together represent a primitive semantic instance of the mapping. Our dance motions are segmented into two-beat rhythm units, since they are a basic unit of interaction for the specific type of dance (Lindy Hop), as shown in Figure 1 . Our walk motions, on the other hand, are segmented according to gait cycles. In both cases, we use manual transcription, since each example motion must only be segmented once. Methods exist to automate this process if desired. Dance motion could be segmented using motion beat analysis
[KPS03]. More general motions could be segmented using annotation
[AFO03] or curve clustering [CGMS03].
<H1>4. Algorithm Description</H1>
Given a control motion x with T frames, our goal is to generate an appropriate target motion. This is achieved by selecting a sequence of appropriate target segments from the database. To make the database motions more flexible, we allow each selected target segment to be spatially transformed and uniformly stretched in time. The proper selection of segments can be achieved using an efficient dynamic programming algorithm.
Figure 2: An example instance from the database is stretched and transformed to align the control segment with the input motion. The same stretch and transform can then be applied to the target segment.
<H2>4.1. Single Segment</H2>
Before developing our general algorithm, we address the simpler problem of interpreting the input as a single control segment from the database. We quantify the similarity of the input motion x and a control segment a s with a distance function:
1
D(x, T s ) ≡ − s T )a s T 2 .
a x M(x, a
Here, a T s represents the control segment a s , uniformly stretched in time to T frames, and M(x, a s T ) is a rigid transformation that optimally aligns x and a s T :
2
2
2
M(x, a s T ) ≡ arg min M x − Ma s T .
This optimization is the solution to the Procrustes problem, which has several efficient numerical solutions [ELF97]. Since our example dance and walk motions only differ by ground translation and vertical rotation, our implementation uses a closed form solution
[KGP02]. To compute the optimal interpretation, we determine the segment a s ∗ that is most similar to the input motion:
3
s ∗ = arg min s D(x, a T s ).
The index s ∗ also identifies, by construction of the database, an appropriate target b s ∗ for both the control segment a s ∗ and the input motion x. The stretch T completes the specification of the optimal interpretation, M(x, a T s ∗ )a T s ∗ , and the optimal target, M(x, a T s ∗ )b T s ∗ . This is illustrated in Figure 2 . The optimal target may not precisely satisfy desired physical or kinematic constraints. However, given a descriptive database, it can provide a good approximation which can be adjusted appropriately during postprocessing. In practice, we limit the allowed amount of uniform time stretch by a constant factor since the distance metric does not distinguish between motions of varying speed. A dancer that pushes his partner slowly, for instance, will elicit quite a different response if he pushes quickly. Limiting the amount of stretch also has the practical benefit of reducing the search space of our general algorithm, which we will now describe.
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1 2 3 4 5 1 3 4 1 5 4
Figure 3: A good interpretation may not account for the continuity of the target (middle). Our scoring function strikes a balance between the two (bottom).
<H2>4.2. Multiple Segments</H2>
In general, we must handle the case where the optimal control and target consist of a sequence of segments. We can specify this sequence analogously to the single segment case by the number of segments L ∗ , the segment indices s ∗ 1 , . . . , s ∗ L , and the segment durations d 1 ∗ , . . . , d L ∗ . As in the single segment case, the distance metric D evaluates the interpretation quality of each segment in the sequence. However, the quality of the interpretation alone does not account for the continuity of the target motion, as shown in Figure 3 . To offset this problem, we introduce a function which measures the continuity between segments v and w:
4
C(v, w) = ω(v) − α(w) 2 .
Here, α and ω represent the head and tail functions, which respectively extract the positions of the first and last frame of a segment. One could also use more frames to measure higher-order continuity if desired. Given a sequence specification L, s 1 , . . . , s L , and d 1 , . . . , d L , we define a scoring function that accounts for both the quality of interpretation and the continuity of the target:
5
L L−1 ∑ D(x i , a s d i i ) + k ∑ C M i b s d i i , M i+1 b s d i+1 i+1 . i=1 i=1
Here, x i is the subinterval of the input that is implied by the segment durations d 1 , . . . , d i . These in turn induce the transformations M i ≡ M(x i , a d s i i ). The user-specified constant k defines the balance of interpretation and continuity. The optimal substructure property of the score function, as defined by the following recurrence, can be used to find a globally optimal solution using dynamic programming:
6
Q s,d [t] = min r,c Q r,c [t − d] + D(x d,t , a s d )
6
a + kC(M r,c,t−d b r c , M s,d,t b d s )
7
Q s,d [d] = D(x d,d , a s d ).
Here, x d,t represents the subsequence of input frames starting at frame t − d and ending at frame t, which in turn induces the alignment matrix M s,d,t ≡ M(x d,t , a d s ). Q s,d [t] is defined as the score of the optimization on the subsequence x t,t , given that the last segment is indexed by s and stretched to duration d. By minimizing Q s,d [T ] over all s and d, we can compute the score of the optimal sequence specification and recover it by backtracking. In the following section, we describe this process in more detail.
<H2>4.3. Implementation</H2>
To solve the recurrence efficiently, values of Q are stored in a two-dimensional array. Cells in this array are indexed by the time t on one axis and by all legal combinations of s and d on the other (recall from Section 4.1 that the amount of allowed stretch is limited). First, all legal values of Q s,d [d] are initialized according to the base case given in Equation 7, and all other array cells are set to infinity. The algorithm proceeds by iterating forward through time. At each time t, all non-infinite cells are located and scores are conditionally propagated forward in time according to Equation 6. More specifically, suppose that we are currently processing the array cell Q r,c [t]. For each legal combination of s and d, the candidate value z is computed:
8
z = Q r,c [t] + D(x d,t+d , a s d ) + kC(M r,c,t b r c , M s,d,t+d b d s ).
If the value in the array cell Q s,d [t + d] is greater than z, we set it to z and store a backpointer to cell Q r,c [t]. By continuing this process, the entire array is filled. Since the indexing of each cell encodes a segment identifier and duration, the optimal sequence specification can be recovered by following backpointers from the best score at time T .
<H2>4.4. Efficiency</H2>
At each time t, O(P) noninfinite cells are processed, where P is the number of legal combinations of s and d. Since processing an individual cell is an O(P) operation, the total asymptotic time complexity of the algorithm is O(P 2 T ). To increase its efficiency, we apply several heuristic optimizations. Beam search. Rather than process all O(P) noninfinite cells at each time t, we only process cells with scores less than min s,d Q s,d [t] + w, where w is a user-specified constant. This technique is known as beam search, and w is known as the beam width. This is motivated by the fact that cells with worse scores are unlikely to be on the optimal backtracking path, and thus can be pruned from the search. Clustering. In Section 3, we described the construction of a motion database by storing all instances derived from the examples. Since the time complexity of the algorithm scales quadratically with the database size, this leads to inefficiency when the number of instances is large. To resolve this issue, redundant instances are eliminated using complete-linkage clustering
[DHS00]. For this, the distances between instances is defined by Equation 1. The advantage of complete-linkage clustering over other methods (such as k-means) is that it explicitly limits the distance of any two instances in a cluster by a user-defined threshold. After clusters are formed, a representative instance is chosen at random from each cluster to remain in the database, and all other instances are discarded. An additional benefit of this process is that it helps beam search; since clustering reduces ambiguity in interpretation, a larger proportion of search paths can be pruned. Downsampling. High sampling rates are common for systems such as motion capture, but they are generally unnecessary for interpreting the input control motion. By downsampling motions by a user-chosen constant, we can effectively reduce the length of the input sequence. However, the resulting optimal sequence specification will also be at the lower frame rate, and it is generally desirable to have it at the frame rate of the original input. Simple upsampling often introduces slight but undesirable temporal errors. To remedy this, we run a highly constrained version of our dynamic programming algorithm that only adjusts the durations appropriately. Constraints can be easily encoded by making appropriate cells in the Q array illegal. For instance, we can force the result to contain a certain target segment b s at some time t by disallowing any processing on cells Q r,c [u], where r = s and u − c ≤ t ≤ u.
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<H1>5. Postprocessing</H1>
As described in Section 4, the output of our optimization is a specification of an appropriate target motion in terms of target segments in a database. Specifically, it provides a sequence of target segment indices s ∗ 1 , . . . , s L and durations d 1 ∗ , . . . , d L ∗ . The corresponding target segments can be copied from the database, stretched, transformed by the induced matrices M ∗ 1 , . . . , M ∗ L , and concatenated. The result is a moving point cloud that approximates the desired result. Of course, the same selections, stretches, and transformations can just as easily be applied to the source motions that generated the point cloud. From the perspective of motion synthesis, the main problem with our approach is that the raw result will generally contain some kinematic errors. In our dance example, footplant and handhold constraints are never explicitly enforced. For such constraints, existing methods can be applied to postprocess the data
[KSG02], but such methods often require some amount of manual constraint annotation. Like similar motion clip rearrangement techniques, we can propagate constraints by example. In other words, each example instance can be annotated with constraints that can be transferred to the target motion. This is demonstrated by our propagation of handhold constraints, shown in Figure 4 . We do not aim to introduce novel solutions for motion blending or constraint satisfaction. Instead, our goal is to provide motion that is amenable to postprocessing with these approaches. To demonstrate our method’s capabilities in this regard, we show that it can generate realistic and compelling motion, even with extremely simple postprocessing. Our results, shown in the following section and in our accompanying video, are filtered with a basic smoothing operation that linearly adjusts motion curves to match across segment boundaries.
Figure 4: A handhold constraint, indicated by the line connecting the characters, is propagated from annotated examples to this generated motion. In this two-beat sequence, the leader begins in an open crosshand stance and pulls the follower in (1,2). The follower releases handhold and performs an inside turn toward the leader (3,4). Nearing completion of the turn, the follower prepares to catch the leader’s hand and enter embrace (5), and handhold is reestablished in closed stance (6).
<H1>6. Results and Evaluation</H1>
We evaluate our technique with two examples. In the first, we animate a realistic walking human from time-sampled mouse movement. Walk motions, however, do not show the full ability of our technique to discover complex mappings. To better demonstrate this aspect, we apply our method to a partner dance called Lindy Hop. Specifically, we use the complex motion of the dance leader to drive the motion of the follower. In the following sections, all human motions were acquired in a motion capture studio and standard commercial tools were used to estimate joint positions
[Vic03]. For the point cloud representation of body motion, we used only the positions of the hands and feet, as we found that these endeffectors were sufficient to evaluate interpretation and continuity in both evaluations. To generate the motion, we applied the resulting sequence specification to the source motion and used basic smoothing. All timings were performed on a workstation with dual 2.4 Ghz Intel Xeon processors. Where applicable, we state the clock times for the dynamic programming algorithm (Section 4.3), upsampling (Section 4.4), and postprocessing (Section 5). The continuity constant, defined in Section 4.2, and the stretch limit were chosen experimentally.
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<H2>6.1. Walk</H2>
We acquired 2 minutes of motion captured walk footage at 30 Hz. The subject was directed to walk within the capture area with random changes in direction and speed. We artificially constructed a synchronized example control motion by projecting the positions of the hip joints onto the floor and normalizing their distance. As stated previously, the target motions were represented by end-effector positions. The walk footage was transcribed manually according to the gait cycle. More specifically, a segmentation point was manually placed at each footplant. From this process, we created 200 segments, which we reduced to 70 using clustering. In our tests, we downsampled these motions to 10 Hz and allowed each segment to be stretched ±0.2 seconds. Our first evaluation involved creating control motions from new walk motions that were not in the database. As before, we projected the hip joints onto the ground and normalized their distance. We ran our algorithm on these control motions and compared our results to the original source motions. Experimentally, we found that larger values of the continuity constant were more effective. For short walks, the generated motion was highly realistic. The frequency of the generated gait cycle nearly matched the frequency of the source, but phase differed. In more concrete terms, the generated motion might choose to start on the left foot, whereas the original source motion might start on the right. This was expected, as the control signals did not encode any phase information. For longer walks, however, we were surprised to discover that the generated motions often kept in nearly perfect phase with the source. The reason for this was that the subject preferred to make sharp turns with the same footwork pattern. These served as synchronizing signals which were propagated throughout the generated gait cycle due to the global optimization. In our timing tests, we used a 57 second control motion. We first ran the algorithm without the beam search optimiza- tion. The dynamic programming algorithm took 12.5 seconds, upsampling from 10 Hz to 30 Hz took 0.4 seconds, and postprocessing took 1.1 seconds. With the beam search optimization on, we were able to reduce the clock time of the algorithm to 1.2 seconds (47 seconds of input processed per second of clock time) while retaining visually perfect results. The upsampling and postprocessing times remained the same. We ran the algorithm on shorter and longer inputs and experimentally confirmed the asymptotic linear dependency of running time on input length, described in Section 4.4. In our second evaluation, we built an interface that allowed users to draw paths using mouse input, as shown in Figure 5 . The position of the mouse pointer was sampled at 30 Hz, and Frenet frames were used to generate a control motion. For a wide variety of user inputs, our method was capable of generating highly realistic walking motion. Since the timing of the path was important, we found that users required minor training to understand the concept of performing a path instead of drawing it. It was often tempting, for instance, to rapidly move the mouse to draw a straight line. This would correspond to a impossibly fast run, well beyond the capabilities of a human. To resolve these issues, our interface allows a user to overlay the playback of an existing motion on the drawing canvas to get a sense of speed. Furthermore, it provides options to smooth the trajectory spatially and temporally. The speed of the algorithm allows for rapid feedback.
Figure 5: A synthetic character walks along a trajectory from mouse input. The spacing of the points indicates the speed.
<H2>6.2. Dance</H2>
Our choice of partner dance as a demonstration was primarily motivated by the complexity of its style and mappings. From a small segmented set of example instances, we generate a follower’s motion to accompany a leader’s motion. Generating partner dance motion would be a difficult trial for both physical methods, which would yield underdetermined systems, and statistical methods, which would typically require a very large database in place of our small segmented one. Swing dance also allows for a more principled evaluation of our results than most types of motion, since the performance of the algorithm at generating valid mappings can be evaluated independently of style considerations or subjective judgments of motion quality. Lindy Hop is a subgenre of swing dance that, at a basic level, can be described as a state machine. A dance couple moves between four basic stances: open (◦), closed (•), open crosshand (◦), and closed crosshand (•). Open and closed refer to whether the couple is apart or in embrace, respectively. Crosshand refers to the case when the leader and follower hold right hands (we could also refer to it as a handshake). Basic Lindy Hop motions switch between these four stances by means of transitions: an inside turn ( ), when the follower spins towards the leader, an outside turn ( ), when the follower spins away from the leader, and a simple step (→). At the end of each transition, the dancers may also change their handhold to instantly transition between crosshand states (◦, •) and non-crosshand states (◦, •). Figure 4 shows a couple transitioning from open crosshand stance to closed stance using an outside turn: ◦ •. Each of these transitions occurs over four beats of music, which are assembled from two-beat segments; this was our motivation for performing two-beat segmentation, as described in Section 3. Figure 4 shows only the last two beats of a four-beat transition that starts with a two-beat rocking motion. Skilled Lindy Hop dancers use a greater variety of moves, ranging from more complex transitions such as double outside turns to complex aerial maneuvers. We did not include the entire range of motions. Instead, we constructed a smaller database with seven basic 8-beat dance patterns that every Lindy Hop dancer knows (shown in the first column of Table 1 ). We constructed the motion database from a set of 12 short dances, each containing the seven basic 8-beat patterns, giving a total of 5 minutes of motion. These dances were segmented into 364 two-beat mapping instances, with lengths varying from approximately 0.6 seconds to 1 second due to different music. For our evaluations, we captured three longer test dances (approximately 2-3 minutes each) in which the dancers were instructed to improvise with the transitions and stances included in the database. Their improvisations led to dances which included thirteen new 8-beat patterns not found in the database (shown in the last column of Table 1 ) as well as some repeats of patterns in the database. These test dances spanned a tempo range from about 120 beats per minute to about 190 beats per minute. We used the motion of the leader to control a synthetic follower, which was then compared with the actual follower. Visually, the results exhibited the fluidity, grace, and style of the original dancer. Some footskate and handhold violations are visible because we wanted to show the output in its almost raw form, with smoothing applied only for visual coherence. In a direct comparison with the actual follower motions, we found that the synthetic follower matched very well in closed stances. In open stances, the follower was much freer to include stylistic variations, so the generated motions often differed visually from the actual motions. Additionally, the synthesized dancers almost always kept in perfect rhythm with the leader. Our algorithm ably recreated the semantics of the leader to follower mapping, even for novel patterns. When the algorithm encountered a pattern that was not in the database (one of 14 such patterns shown in Table 1 ), it was able to correctly reconstruct the novel sequence by rearranging the two-beat segments. Of the 91 patterns (21 unique) in our three test dances, the synthetic dancer matched the pattern of the actual dancer in all but 5 cases, one of which is shown in Figure 6. When the algorithm did differ from the real dancer in the composition of the pattern, the leader and follower still executed a valid Lindy Hop pattern. In these misinterpreted instances, the leader’s motion is quite similar across two different follower patterns. To disambiguate these, we might add information to the control signal, such as forceplate readings, or we might accept these rare mismatches because they are in fact valid mappings. Furthermore, all 5 mismatched patterns differed by a single two-beat segment, so, of 91 × 4 = 364 two-beat segments in the test dances, the algorithm misinterpreted the signal in 5 cases for an error rate of less than 2%. For all our evaluations and timing tests, we reduced the size of the database from 364 to 168 with clustering, downsampled to 7.5 Hz, and allowed a segment stretch of ±0.15 seconds. We cite our efficiency figures for generating, from leader motion only, a particular 150 second dance motion. Without beam search, the dynamic programming algorithm ran for 78 seconds, 2 seconds were spent on upsampling, and 26 seconds were spent on postprocessing. With beam search enabled with modest parameters, we were able to drive the runtime of the dynamic programming to 10 seconds while maintaining excellent visual and semantic results. As with our walk motion evaluation, we found that clock times scaled linearly with the length of the input.
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Database Patterns
Test Patterns
1 ◦→•→◦
1 ◦→• ◦
8 ◦ •→•
2 ◦→• ◦
2 •→• ◦
9 ◦ • ◦
3 ◦→•→•
3 •→•→•
10 ◦ •→◦
4 ◦→• ◦
4 •→• ◦
11 ◦→• ◦
5 ◦→• ◦
5 •→• ◦
12 ◦ • ◦
6 •→•→◦
6 •→•→◦
13 ◦ • ◦
7 ◦ •→◦
7 •→• ◦
14 ◦→•→◦
Database Patterns Test Patterns 1 ◦→•→◦ 1 ◦→• ◦ 8 ◦ •→• 2 ◦→• ◦ 2 •→• ◦ 9 ◦ • ◦ 3 ◦→•→• 3 •→•→• 10 ◦ •→◦ 4 ◦→• ◦ 4 •→• ◦ 11 ◦→• ◦ 5 ◦→• ◦ 5 •→• ◦ 12 ◦ • ◦ 6 •→•→◦ 6 •→•→◦ 13 ◦ • ◦ 7 ◦ •→◦ 7 •→• ◦ 14 ◦→•→◦
Table 1: A notational description of the dance patterns stored in the database and the novel test patterns performed in our three test dances. Our technique adapts by rearranging the segments in the database to recreate the patterns it has not seen before.
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Figure 6: On the top, a clip of an actual dance is displayed. Here, the leader performs a regular handhold change during a step transition. This transition never occurs in our motion database. In response to the same motion cue, our algorithm generates a leaping outside turn, as show on the bottom. This is one of five two-beat segments (out of 380 two-beat segments in our three test dances), where the algorithm differs in its selection of response from an experienced dance follower. In other instances of this regular handhold change during a step transition in the test data, the algorithm correctly sequences motions to discover this novel vocabulary element.
<H1>7. Conclusion</H1>
We have presented a method for example-based performance control of human motion. Our dynamic programming algorithm uses segments of motion along with an objective function that accounts for both the quality of control interpretation and the continuity of the target motion to generate visually and semantically correct motions. The semantic accuracy of the generated motion was evaluated in the setting of partner dance, where the follower’s motion is generated from the leader’s motion. The algorithm generated semantically correct partner motion even from test sequences of leader motions that did not appear in the training set. Our dynamic programming algorithm performs a global optimization, which precludes the local decisions that are required for online applications. However, we demonstrate in our evaluations that it can compute results significantly faster than input motion can be recorded, thus making it suitable for rapid-feedback motion authoring applications. We believe that segmental approaches like ours hold great promise for real-time performance-driven animation, and consider it a promising area of future research. To preserve spatial dependencies in mappings, we apply rigid transformations to optimally align control segments with input control motions. Target segments inherit these transformations. This approach is effective for our applications or whenever the control signal indicates appropriate spatial and temporal cues. It is also possible to select other transformations for applications outside the domain of human motion control. For instance, allowing arbitrary homogeneous transformations in two dimensions might form an alternative segmental solution to the curve analogies prob- lem [HOCS02]. Eliminating transformations entirely might also be appropriate for applications such as synthesis of facial motion from speech signals
[Bra99]. We have shown that our segment similarity metric is effective for our experiments. However, we acknowledge the fact that other metrics may be more appropriate for different types of motion and believe that it is a promising direction for future research. In the process of generating target motion, our dynamic programming algorithm performs a semantically guided segmentation of the input control motion. The entire process, however, relies on the availability of semantically segmented examples. For our evaluations, we were able to perform this segmentation manually by tapping a key in response to the rhythm of music or the gait pattern of a walk cycle. While specific methods exist to automate this segmentation for the cases of dance and walk, a more general method is desirable. For this, we could begin with a few manually segmented examples and grow the set of example instances by iterative application of our algorithm. This approach would be similar in spirit to the semiautomatic SVM-based annotation approach of Arikan et al.
[AFO03]. The annotation propagation we describe above suggests that our method could be used for interpretation rather than control. Paralleling our automatic annotation of handholds, it is possible to annotate any new control motion given a set of labeled example instances. This could be used to transcribe the motion into a symbolic representation, such as the one used in this paper, or even Laban notation [Hut73]. Such a representation could then be analyzed or summarized using natural language processing techniques.
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<H1>Acknowledgments</H1>
We would like to thank Jonathan Chu, Jim Glass, Kevin Murphy, the members of the MIT CSAIL graphics group, and the anonymous reviewers for their advice and assistance. Our dance evaluations would not have been possible without the generous help of Marilee Annereau, Bethany Certa, Rebecca Drzewiczewski, Steve Drzewiczewski, Amanda Gruhl, Shawn Hershey, Reuben Pharms, Paolo Piselli, Dorry Segev, Peter Strom, and Gary Ulaner. Funding for this work was provided by the MIT Oxygen Project. Eugene Hsu was partially supported by an MIT Presidential Fellowship. Sommer Gentry was supported by a Department of Energy Computational Science Graduate Fellowship.
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[Hut73] H UTCHINSON A.: Labanotation: The System of Analyzing and Recording Movement, third ed. Routledge, New York, 1973. 8
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[KGP02] K OVAR L., G LEICHER M., P IGHIN F.: Motion graphs. ACM Transactions on Graphics 21, 3 (July 2002), 473– 482. 2, 3
[KPS03] K IM T., P ARK S., S HIN S.: Rhythmic-motion synthesis based on motion-beat analysis. ACM Transactions on Graphics 22, 3 (July 2003), 392–401. 2, 3
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[LCR ∗ 02] L EE J., C HAI J., R EITSMA P. S. A., H ODGINS J. K., P OLLARD N. S.: Interactive control of avatars animated with human motion data. ACM Transactions on Graphics 21, 3 (July 2002), 491–500. 2
[PB02] P ULLEN K., B REGLER C.: Motion capture assisted animation: Texturing and synthesis. ACM Transactions on Graphics 21, 3 (July 2002), 501–508. 2
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c The Eurographics Association 2004.
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"text": "['we consider our work most similar in intent to image analogies']",
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"text": "['HJO ∗ 01']",
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"text": "['This method, given an unfiltered and filtered version of the same image, applies an analogous filter to a novel image']",
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"text": "['Our method, given a set of synchronized control and target motions, applies an analogous mapping to a new input control motion']",
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"text": "['Image analogies was shown to be an elegant method with applications such as texture transfer, textureby-numbers, and super-resolution']",
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"text": "['Our dance evaluation suggests an alternative view of our method as one of interaction modeling']",
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"text": "['In this domain, tech- niques have been developed that specify the mappings between character motions with explicit models of character interaction']",
"type": "background_claim",
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"text": "['Adaptive autonomous characters have used rules to exhibit complex flocking, herding, and locomotory behaviors']",
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"text": "['Rey87']",
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"text": "['TT94']",
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"text": "['Approaches to explicit interaction modeling have included layered architectures']",
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"text": "['BG95']",
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"text": "['procedural descriptions']",
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"text": "['PG96']",
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"text": "['and even cognitive models']",
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"text": "['In this context, our work might be viewed as a competency module that enhances the skills of characters to enable their participation in complex interactive performances']",
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"text": "['Each frame of motion is encoded by a point cloud']",
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"text": "['human motion']",
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"text": "['we use skeletal joint positions']",
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"text": "['this representation provides a more intuitive space than joint angle representations for comparing poses']",
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"text": "['KGP02']",
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"text": "['point cloud representations allow for generalization to control motions without skeletal representations']",
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"text": "['they are a basic unit of interaction for the specific type of dance']",
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"text": "['Our dance motions are segmented into two-beat rhythm units']",
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"text": "['Figure 1']",
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"text": "['In both cases, we use manual transcription']",
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"text": "['each example motion must only be segmented once']",
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"text": "['Our walk motions']",
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"text": "['are segmented according to gait cycles']",
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"text": "['Methods exist to automate this process']",
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"text": "['Dance motion could be segmented using motion beat analysis']",
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"text": "['KPS03']",
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"text": "['More general motions could be segmented using annotation']",
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"text": "['AFO03']",
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"text": "['or curve clustering']",
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"text": "['This is achieved by selecting a sequence of appropriate target segments from the database']",
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"text": "['To make the database motions more flexible, we allow each selected target segment to be spatially transformed and uniformly stretched in time']",
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"text": "['The proper selection of segments can be achieved using an efficient dynamic programming algorithm']",
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"text": "['M(x, a s T ) is a rigid transformation that optimally aligns x and a s T']",
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"text": "['We quantify the similarity of the input motion x and a control segment a s with a distance function']",
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"text": "['D(x, T s ) ≡ − s T )a s T 2']",
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"text": "['a x M(x, a']",
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"text": "['2']",
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"text": "['M(x, a s T ) ≡ arg min M x − Ma s T']",
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"text": "['This optimization is the solution to the Procrustes problem, which has several efficient numerical solutions']",
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"text": "['ELF97']",
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"text": "['our example dance and walk motions only differ by ground translation and vertical rotation']",
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"text": "['our implementation uses a closed form solution']",
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"text": "['To compute the optimal interpretation, we determine the segment a s ∗ that is most similar to the input motion']",
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"text": "['s ∗ = arg min s D(x, a T s )']",
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"text": "['The index s ∗ also identifies, by construction of the database, an appropriate target b s ∗ for both the control segment a s ∗ and the input motion x']",
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"text": "['The stretch T completes the specification of the optimal interpretation, M(x, a T s ∗ )a T s ∗ , and the optimal target, M(x, a T s ∗ )b T s ∗']",
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"text": "['Figure 2']",
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"text": "['The optimal target may not precisely satisfy desired physical or kinematic constraints']",
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"text": "['given a descriptive database, it can provide a good approximation which can be adjusted appropriately during postprocessing']",
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"text": "['In practice, we limit the allowed amount of uniform time stretch by a constant factor']",
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"text": "['we must handle the case where the optimal control and target consist of a sequence of segments']",
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"text": "['We can specify this sequence analogously to the single segment case by the number of segments L ∗ , the segment indices s ∗ 1 , . . . , s ∗ L , and the segment durations d 1 ∗ , . . . , d L ∗']",
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"text": "['Figure 3']",
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"text": "['L L−1 ∑ D(x i , a s d i i ) + k ∑ C M i b s d i i , M i+1 b s d i+1 i+1']",
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"text": "['Q s,d [t] = min r,c Q r,c [t − d] + D(x d,t , a s d )']",
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"text": "['the optimal sequence specification can be recovered by following backpointers from the best score at time T']",
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"text": "['To increase its efficiency, we apply several heuristic optimizations']",
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"text": "['can be pruned from the search']",
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"text": "['the time complexity of the algorithm scales quadratically with the database size']",
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"text": "['the number of instances is large']",
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"text": "['this leads to inefficiency']",
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"text": "['To resolve this issue, redundant instances are eliminated using complete-linkage clustering']",
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"text": "['DHS00']",
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"text": "['The advantage of complete-linkage clustering over other methods']",
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"text": "['is that it explicitly limits the distance of any two instances in a cluster by a user-defined threshold']",
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"text": "['An additional benefit of this process is that it helps beam search']",
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"text": "['clustering reduces ambiguity in interpretation']",
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"text": "['a larger proportion of search paths can be pruned']",
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"text": "['High sampling rates are common for systems such as motion capture']",
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"id": "T209"
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21751,
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"text": "['they are generally unnecessary for interpreting the input control motion']",
"type": "own_claim",
"id": "T210"
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21934
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"text": "['By downsampling motions by a user-chosen constant, we can effectively reduce the length of the input sequence']",
"type": "own_claim",
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22026
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"text": "['the resulting optimal sequence specification will also be at the lower frame rate']",
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22032,
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"text": "['it is generally desirable to have it at the frame rate of the original input']",
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22110,
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"text": "['Simple upsampling often introduces slight but undesirable temporal errors']",
"type": "background_claim",
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22185,
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],
"text": "['To remedy this, we run a highly constrained version of our dynamic programming algorithm that only adjusts the durations appropriately']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
22321,
22405
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],
"text": "['Constraints can be easily encoded by making appropriate cells in the Q array illegal']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
22421,
22548
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],
"text": "['we can force the result to contain a certain target segment b s at some time t by disallowing any processing on cells Q r,c [u]']",
"type": "own_claim",
"id": "T217"
},
{
"offsets": [
[
22783,
22906
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],
"text": "['the output of our optimization is a specification of an appropriate target motion in terms of target segments in a database']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
22922,
23026
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],
"text": "['it provides a sequence of target segment indices s ∗ 1 , . . . , s L and durations d 1 ∗ , . . . , d L ∗']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
23029,
23183
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],
"text": "['The corresponding target segments can be copied from the database, stretched, transformed by the induced matrices M ∗ 1 , . . . , M ∗ L , and concatenated']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
23185,
23256
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],
"text": "['The result is a moving point cloud that approximates the desired result']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
23269,
23403
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],
"text": "['the same selections, stretches, and transformations can just as easily be applied to the source motions that generated the point cloud']",
"type": "own_claim",
"id": "T222"
},
{
"offsets": [
[
23405,
23549
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],
"text": "['From the perspective of motion synthesis, the main problem with our approach is that the raw result will generally contain some kinematic errors']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
23551,
23637
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],
"text": "['In our dance example, footplant and handhold constraints are never explicitly enforced']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
23639,
23716
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],
"text": "['For such constraints, existing methods can be applied to postprocess the data']",
"type": "own_claim",
"id": "T225"
},
{
"offsets": [
[
23719,
23724
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],
"text": "['KSG02']",
"type": "data",
"id": "T226"
},
{
"offsets": [
[
23731,
23801
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],
"text": "['such methods often require some amount of manual constraint annotation']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
23854,
23893
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],
"text": "['we can propagate constraints by example']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
23911,
24011
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],
"text": "['each example instance can be annotated with constraints that can be transferred to the target motion']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
24087,
24095
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],
"text": "['Figure 4']",
"type": "data",
"id": "T230"
},
{
"offsets": [
[
24198,
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],
"text": "['our goal is to provide motion that is amenable to postprocessing with these approaches']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
24356,
24446
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],
"text": "['it can generate realistic and compelling motion, even with extremely simple postprocessing']",
"type": "own_claim",
"id": "T232"
},
{
"offsets": [
[
25390,
25464
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],
"text": "['do not show the full ability of our technique to discover complex mappings']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
25367,
25379
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],
"text": "['Walk motions']",
"type": "own_claim",
"id": "T234"
},
{
"offsets": [
[
25501,
25556
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],
"text": "['we apply our method to a partner dance called Lindy Hop']",
"type": "own_claim",
"id": "T235"
},
{
"offsets": [
[
25296,
25365
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],
"text": "['we animate a realistic walking human from time-sampled mouse movement']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
25745,
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],
"text": "['standard commercial tools were used to estimate joint positions']",
"type": "own_claim",
"id": "T238"
},
{
"offsets": [
[
25811,
25816
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],
"text": "['Vic03']",
"type": "data",
"id": "T239"
},
{
"offsets": [
[
25937,
26033
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],
"text": "['these endeffectors were sufficient to evaluate interpretation and continuity in both evaluations']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
25819,
25918
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],
"text": "['For the point cloud representation of body motion, we used only the positions of the hands and feet']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
26659,
26721
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],
"text": "['We acquired 2 minutes of motion captured walk footage at 30 Hz']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
26824,
26981
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],
"text": "['We artificially constructed a synchronized example control motion by projecting the positions of the hip joints onto the floor and normalizing their distance']",
"type": "own_claim",
"id": "T242"
},
{
"offsets": [
[
27737,
27797
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],
"text": "['larger values of the continuity constant were more effective']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
27803,
27814
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],
"text": "['short walks']",
"type": "data",
"id": "T244"
},
{
"offsets": [
[
27816,
27857
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],
"text": "['the generated motion was highly realistic']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
27859,
27943
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],
"text": "['The frequency of the generated gait cycle nearly matched the frequency of the source']",
"type": "own_claim",
"id": "T246"
},
{
"offsets": [
[
27949,
27963
]
],
"text": "['phase differed']",
"type": "own_claim",
"id": "T247"
},
{
"offsets": [
[
27989,
28048
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],
"text": "['the generated motion might choose to start on the left foot']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
[
28058,
28109
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],
"text": "['the original source motion might start on the right']",
"type": "own_claim",
"id": "T249"
},
{
"offsets": [
[
28111,
28128
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],
"text": "['This was expected']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
28133,
28189
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],
"text": "['the control signals did not encode any phase information']",
"type": "own_claim",
"id": "T251"
},
{
"offsets": [
[
28195,
28207
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],
"text": "['longer walks']",
"type": "data",
"id": "T252"
},
{
"offsets": [
[
28253,
28325
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],
"text": "['the generated motions often kept in nearly perfect phase with the source']",
"type": "own_claim",
"id": "T253"
},
{
"offsets": [
[
28356,
28428
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],
"text": "['the subject preferred to make sharp turns with the same footwork pattern']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
28430,
28556
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],
"text": "['These served as synchronizing signals which were propagated throughout the generated gait cycle due to the global optimization']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
28821,
28928
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],
"text": "['With the beam search optimization on, we were able to reduce the clock time of the algorithm to 1.2 seconds']",
"type": "own_claim",
"id": "T256"
},
{
"offsets": [
[
28986,
29026
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],
"text": "['while retaining visually perfect results']",
"type": "own_claim",
"id": "T257"
},
{
"offsets": [
[
29028,
29085
]
],
"text": "['The upsampling and postprocessing times remained the same']",
"type": "own_claim",
"id": "T258"
},
{
"offsets": [
[
29087,
29230
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],
"text": "['We ran the algorithm on shorter and longer inputs and experimentally confirmed the asymptotic linear dependency of running time on input length']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
[
29245,
29256
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],
"text": "['Section 4.4']",
"type": "data",
"id": "T260"
},
{
"offsets": [
[
29258,
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],
"text": "['In our second evaluation, we built an interface that allowed users to draw paths using mouse input']",
"type": "own_claim",
"id": "T261"
},
{
"offsets": [
[
29370,
29378
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],
"text": "['Figure 5']",
"type": "data",
"id": "T262"
},
{
"offsets": [
[
29495,
29598
]
],
"text": "['For a wide variety of user inputs, our method was capable of generating highly realistic walking motion']",
"type": "own_claim",
"id": "T263"
},
{
"offsets": [
[
29606,
29642
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],
"text": "['the timing of the path was important']",
"type": "own_claim",
"id": "T264"
},
{
"offsets": [
[
29658,
29756
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],
"text": "['users required minor training to understand the concept of performing a path instead of drawing it']",
"type": "own_claim",
"id": "T265"
},
{
"offsets": [
[
29758,
29779
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],
"text": "['It was often tempting']",
"type": "own_claim",
"id": "T266"
},
{
"offsets": [
[
29795,
29844
]
],
"text": "['to rapidly move the mouse to draw a straight line']",
"type": "own_claim",
"id": "T267"
},
{
"offsets": [
[
29935,
30079
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],
"text": "['To resolve these issues, our interface allows a user to overlay the playback of an existing motion on the drawing canvas to get a sense of speed']",
"type": "own_claim",
"id": "T268"
},
{
"offsets": [
[
30094,
30163
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],
"text": "['it provides options to smooth the trajectory spatially and temporally']",
"type": "own_claim",
"id": "T269"
},
{
"offsets": [
[
30165,
30217
]
],
"text": "['The speed of the algorithm allows for rapid feedback']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
30430,
30544
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],
"text": "['Our choice of partner dance as a demonstration was primarily motivated by the complexity of its style and mappings']",
"type": "own_claim",
"id": "T271"
},
{
"offsets": [
[
30659,
30743
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],
"text": "['Generating partner dance motion would be a difficult trial for both physical methods']",
"type": "own_claim",
"id": "T272"
},
{
"offsets": [
[
30788,
30811
]
],
"text": "['and statistical methods']",
"type": "own_claim",
"id": "T273"
},
{
"offsets": [
[
30751,
30786
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],
"text": "['would yield underdetermined systems']",
"type": "own_claim",
"id": "T274"
},
{
"offsets": [
[
30819,
30900
]
],
"text": "['would typically require a very large database in place of our small segmented one']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
30902,
30999
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],
"text": "['Swing dance also allows for a more principled evaluation of our results than most types of motion']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
31007,
31165
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],
"text": "['the performance of the algorithm at generating valid mappings can be evaluated independently of style considerations or subjective judgments of motion quality']",
"type": "own_claim",
"id": "T277"
},
{
"offsets": [
[
31585,
31666
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],
"text": "['Basic Lindy Hop motions switch between these four stances by means of transitions']",
"type": "background_claim",
"id": "T278"
},
{
"offsets": [
[
31267,
31314
]
],
"text": "['A dance couple moves between four basic stances']",
"type": "own_claim",
"id": "T279"
},
{
"offsets": [
[
31316,
31324
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],
"text": "['open (◦)']",
"type": "data",
"id": "T280"
},
{
"offsets": [
[
31326,
31336
]
],
"text": "['closed (•)']",
"type": "data",
"id": "T281"
},
{
"offsets": [
[
31338,
31356
]
],
"text": "['open crosshand (◦)']",
"type": "data",
"id": "T282"
},
{
"offsets": [
[
31362,
31382
]
],
"text": "['closed crosshand (•)']",
"type": "data",
"id": "T283"
},
{
"offsets": [
[
31668,
31686
]
],
"text": "['an inside turn ( )']",
"type": "data",
"id": "T284"
},
{
"offsets": [
[
31732,
31751
]
],
"text": "['an outside turn ( )']",
"type": "data",
"id": "T285"
},
{
"offsets": [
[
31803,
31820
]
],
"text": "['a simple step (→)']",
"type": "data",
"id": "T286"
},
{
"offsets": [
[
31822,
31983
]
],
"text": "['At the end of each transition, the dancers may also change their handhold to instantly transition between crosshand states (◦, •) and non-crosshand states (◦, •)']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
32095,
32196
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],
"text": "['Each of these transitions occurs over four beats of music, which are assembled from two-beat segments']",
"type": "own_claim",
"id": "T288"
},
{
"offsets": [
[
32396,
32452
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],
"text": "['Skilled Lindy Hop dancers use a greater variety of moves']",
"type": "background_claim",
"id": "T289"
},
{
"offsets": [
[
32607,
32713
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],
"text": "['we constructed a smaller database with seven basic 8-beat dance patterns that every Lindy Hop dancer knows']",
"type": "own_claim",
"id": "T290"
},
{
"offsets": [
[
32550,
32596
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],
"text": "['We did not include the entire range of motions']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
32728,
32751
]
],
"text": "['first column of Table 1']",
"type": "data",
"id": "T292"
},
{
"offsets": [
[
33262,
33366
]
],
"text": "['Their improvisations led to dances which included thirteen new 8-beat patterns not found in the database']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
33381,
33403
]
],
"text": "['last column of Table 1']",
"type": "data",
"id": "T294"
},
{
"offsets": [
[
33406,
33457
]
],
"text": "['as well as some repeats of patterns in the database']",
"type": "own_claim",
"id": "T295"
},
{
"offsets": [
[
33678,
33763
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],
"text": "['Visually, the results exhibited the fluidity, grace, and style of the original dancer']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
33765,
33815
]
],
"text": "['Some footskate and handhold violations are visible']",
"type": "own_claim",
"id": "T297"
},
{
"offsets": [
[
33824,
33875
]
],
"text": "['we wanted to show the output in its almost raw form']",
"type": "own_claim",
"id": "T298"
},
{
"offsets": [
[
33927,
34056
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],
"text": "['In a direct comparison with the actual follower motions, we found that the synthetic follower matched very well in closed stances']",
"type": "own_claim",
"id": "T299"
},
{
"offsets": [
[
34058,
34134
]
],
"text": "['In open stances, the follower was much freer to include stylistic variations']",
"type": "own_claim",
"id": "T300"
},
{
"offsets": [
[
34139,
34208
]
],
"text": "['the generated motions often differed visually from the actual motions']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
34224,
34300
]
],
"text": "['the synthesized dancers almost always kept in perfect rhythm with the leader']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
34302,
34378
]
],
"text": "['Our algorithm ably recreated the semantics of the leader to follower mapping']",
"type": "own_claim",
"id": "T303"
},
{
"offsets": [
[
34410,
34474
]
],
"text": "['the algorithm encountered a pattern that was not in the database']",
"type": "data",
"id": "T304"
},
{
"offsets": [
[
34520,
34612
]
],
"text": "['it was able to correctly reconstruct the novel sequence by rearranging the two-beat segments']",
"type": "own_claim",
"id": "T305"
},
{
"offsets": [
[
34645,
34751
]
],
"text": "['in our three test dances, the synthetic dancer matched the pattern of the actual dancer in all but 5 cases']",
"type": "own_claim",
"id": "T306"
},
{
"offsets": [
[
34614,
34632
]
],
"text": "['Of the 91 patterns']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
34778,
34786
]
],
"text": "['Figure 6']",
"type": "data",
"id": "T308"
},
{
"offsets": [
[
34793,
34872
]
],
"text": "['the algorithm did differ from the real dancer in the composition of the pattern']",
"type": "data",
"id": "T309"
},
{
"offsets": [
[
34874,
34938
]
],
"text": "['the leader and follower still executed a valid Lindy Hop pattern']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
34940,
35050
]
],
"text": "['In these misinterpreted instances, the leader’s motion is quite similar across two different follower patterns']",
"type": "own_claim",
"id": "T311"
},
{
"offsets": [
[
35052,
35121
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],
"text": "['To disambiguate these, we might add information to the control signal']",
"type": "own_claim",
"id": "T312"
},
{
"offsets": [
[
35152,
35192
]
],
"text": "['or we might accept these rare mismatches']",
"type": "own_claim",
"id": "T313"
},
{
"offsets": [
[
35131,
35150
]
],
"text": "['forceplate readings']",
"type": "data",
"id": "T314"
},
{
"offsets": [
[
35201,
35209
]
],
"text": "['they are']",
"type": "own_claim",
"id": "T315"
},
{
"offsets": [
[
35218,
35232
]
],
"text": "['valid mappings']",
"type": "own_claim",
"id": "T316"
},
{
"offsets": [
[
35247,
35310
]
],
"text": "['all 5 mismatched patterns differed by a single two-beat segment']",
"type": "own_claim",
"id": "T317"
},
{
"offsets": [
[
35316,
35454
]
],
"text": "['of 91 × 4 = 364 two-beat segments in the test dances, the algorithm misinterpreted the signal in 5 cases for an error rate of less than 2%']",
"type": "own_claim",
"id": "T318"
},
{
"offsets": [
[
35953,
36083
]
],
"text": "['we were able to drive the runtime of the dynamic programming to 10 seconds while maintaining excellent visual and semantic results']",
"type": "own_claim",
"id": "T319"
},
{
"offsets": [
[
35909,
35951
]
],
"text": "['beam search enabled with modest parameters']",
"type": "data",
"id": "T320"
},
{
"offsets": [
[
36135,
36191
]
],
"text": "['clock times scaled linearly with the length of the input']",
"type": "own_claim",
"id": "T321"
},
{
"offsets": [
[
38729,
38809
]
],
"text": "['We have presented a method for example-based performance control of human motion']",
"type": "own_claim",
"id": "T322"
},
{
"offsets": [
[
38811,
39056
]
],
"text": "['Our dynamic programming algorithm uses segments of motion along with an objective function that accounts for both the quality of control interpretation and the continuity of the target motion to generate visually and semantically correct motions']",
"type": "own_claim",
"id": "T323"
},
{
"offsets": [
[
39218,
39360
]
],
"text": "['The algorithm generated semantically correct partner motion even from test sequences of leader motions that did not appear in the training set']",
"type": "own_claim",
"id": "T324"
},
{
"offsets": [
[
39362,
39505
]
],
"text": "['Our dynamic programming algorithm performs a global optimization, which precludes the local decisions that are required for online applications']",
"type": "own_claim",
"id": "T325"
},
{
"offsets": [
[
39555,
39632
]
],
"text": "['it can compute results significantly faster than input motion can be recorded']",
"type": "own_claim",
"id": "T326"
},
{
"offsets": [
[
39639,
39706
]
],
"text": "['making it suitable for rapid-feedback motion authoring applications']",
"type": "own_claim",
"id": "T327"
},
{
"offsets": [
[
39724,
39816
]
],
"text": "['segmental approaches like ours hold great promise for real-time performance-driven animation']",
"type": "own_claim",
"id": "T328"
},
{
"offsets": [
[
39871,
40010
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<Document xmlns:gate="http://www.gate.ac.uk" name="A32_C02_Animating_Wrinkles_on_Clothes_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1109/visual.1999.809885
<Title>Animating Wrinkles on Clothes</Title>
Sunil Hadap Endre Bangerter Pascal Volino Nadia Magnenat-Thalmann MIRALab
CUI University of Geneva
Switzerland email: {sunil
endre
pascal
thalmann}@cui.unige.ch
<Abstract>This paper describes a method to simulate realistic wrinkles on clothes without fine mesh and large computational overheads. Cloth has very little in-plane deformations, as most of the deformations come from buckling. This can be looked at as area conservation property of cloth. The area conservation formulation of the method modulates the user defined wrinkle pattern, based on deformation of individual triangle. The methodology facilitates use of small in-plane deformation stiffnesses and a coarse mesh for the numerical simulation, this makes cloth simulation fast and robust. Moreover, the ability to design wrinkles (even on generalized deformable models) makes this method versatile for synthetic image generation. The method inspired from cloth wrinkling problem, being geometric in nature, can be extended to other wrinkling phenomena.
</Abstract>
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation; Keywords: cloth modeling, wrinkle modeling, deformable models.
<H1>1 INTRODUCTION</H1>
Wrinkles add life to garments in fashion. Figure 1 illustrates how wrinkles are important for visual realism. In this section we outline the problem and the motivation behind the work. In order to capture realistic wrinkles on a real-life garment, from a mere geometric point of view, the number of triangles required can be easily upto a hundred thousand. Such a large number of triangles put cloth simulation off from interactive speeds, even with adaptive time steps, introduced recently
[ 1 ]. Apart from simulation time, the large triangle count increases the rendering time and the cost significantly. In order to avoid these, one can increase fineness of triangles only in the potential regions where wrinkling might occur. This is very well possible due to advances in the triangulation and interactive systems developed
[ 2 , 7 , 8 , 13 ]. Even then, a significant problem remains: how to estimate the regions and the orientations of wrinkles. Cloth has very large in-plane deformation stiffnesses compared to its ability to bend and shear. This gives rise to very stiff equations of motion. The problem of solving stiff equations is successfully dealt with by the use of an implicit method for numerical integration by Baraff et al
[ 1 ]. Here, though the problem of stiff equations has been tackled, it has been the strong motivation for the authors behind developing the methodology specifically for wrinkles. Even if one wishes to have a fine triangular mesh, using robust and fast numerical solvers and having patience for long computations, it is not guaranteed that the wrinkles will be satisfactory.
Accurate and fast collision detection methods
[ 12 ], constraint methods
[ 5 , 6 ] and good deformable models
[ 6 , 9 , 10 ] have proved to give quality cloth animation. However, real-life wrinkling is a complex phenomenon. It is characterized by frictional forces (especially between body and cloth) which are difficult to model. Collision response methods and friction models developed so far have been rather simple for such a complex problem and robust numerics too. We take a geometric and texture based approach to wrinkling. As it is difficult to physically simulate real life wrinkles, the user designs them interactively as a bump map on a coarse mesh cloth/garment. It is then animated by modulating it as per cloth deformation. The key theme is conservation of cloth area. The work is continuation of earlier work
[ 11 ].
Other attempts to model wrinkles include those by Gotoda et al
[ 3 , 4 ] and Wu et al
[ 14 ]. The following section dwells on issues of animating cloth using a coarse mesh and describes how a geometric wrinkling method can be developed based on area conservation. Section 3 introduces the algorithm, leaving mathematical details of the formulation to Appendix A. Section 4 describes implementation issues of the algorithm and extends it for practical use. Section 5 gives a case study of a typical animation using the algorithm. Finally a conclusion and future work are given.
Figure 1: Wrinkles in fashion
<H1>2 ANIMATING CLOTH USING COARSE MESH</H1>
We would like to animate the cloth using coarse triangular mesh (typically a few thousand triangles per garment), for the reasons mentioned in the Introduction. However, Figure 2 depicts how the geometry of coarse mesh would be unable to capture fine wrinkles on cloth. Real cloth has very little in-plane deformation as most of the deformations come from buckling. For the coarse mesh, setting high metric (in-plane deformation) stiffnesses will not work properly. For the vivid picture of the situation, refer to the triangles undergoing deformations in Figure 3A . Real cloth would wrinkle to this deformation (see typical wrinkles in Figure 3A ). Consider an edge of a triangle, as shown in Figure 3B . In reality, the compression forces will buckle the edge as shown by dotted line. As the bending stiffness of the cloth is small, the buckled edge exerts small forces on the vertices. However, in the coarse mesh situation, the buckled edge is approximated by a straight line between the vertices. Consequently, the real life buckling is attributed to the compression of the edge. If we assume a high metric stiffness associated to this compression, the corresponding forces on the vertices will be high. This is in contrast with real cloth situation. Thus, to animate the cloth realistically with a coarse mesh, we need to set small metric stiffnesses. This allows otherwise possible wrinkling/buckling which is embedded in the deformation of triangle. Very little in-plane deformations can be looked at as area conservation property of cloth. Note the change in area of the triangles due to the deformations in Figure 3 in case of large triangle deformation. We propose to capture gross cloth movements and deformations using a coarse mesh and the fine deformations (wrinkles) using a bump map (or a displacement map). Let us assume the wrinkle pattern is given by the user. We will try to modulate the amplitude of the wrinkle pattern such that, though there is a change in the area of a triangle (with the displacement map), it is invariant after applying the modulated displacement map. Compare the edge shown in Figure 3 to that in Figure 4 after the application of the proposed algorithm. In the next section, we derive the area conser- vation formulation. The method is inspired by the area conservation property, even though Section 3.3 points out that the empiricism introduced later does not actually conserve the area.
Figure 2: Wrinkles and coarse mesh
Figure 3: Large triangle deformations due to buckling
displacement mapped triangle edge modulation original triangle edge
Figure 4: Edge with modulated wrinkle pattern
<H1>3 GEOMETRIC WRINKLES</H1>
<H2>3.1 Overview</H2>
In this subsection, we would like to give a heuristic overview of our geometric wrinkling algorithm and the context in which it is embedded. We also introduce the key terms that will be used throughout the paper, whereas their accurate definitions will be given in section 3.2. Please refer to Figure 5 , which illustrates this overview. First, let us state what serves as an input to the algorithm. We start with a user defined wrinkle pattern, which is given in the form of a texture and an initial undeformed triangular mesh in 3D space. This mesh may represent a garment or another deformable model. The wrinkle pattern is bump or displacement mapped onto the initial mesh by the user. Thus, we obtain a static wrinkled garment. Note that the texture mapping coordinates do not change throughout the computations described below. By introducing a fixed scale for the displacement or bump map on a mesh triangle, we obtain a function which, we call the wrinkle function. Using these inputs in step 1 (refer to Figure 5 ), the algorithm computes a set of four parameters termed wrinkling coefficients for each triangle of the mesh. The initial mesh serves as an input to a simulation engine, which in the context of cloth simulation would be the physical model with a numerical solver. The mesh that is the output by the simulation engine will be the deformed mesh. This deformed mesh is then further processed by the proposed algorithm. For each triangle, we compute the deformation transformation that relates the corresponding triangle of the initial and the deformed mesh. Using this deformation transformation and the already computed wrinkling coefficients, we compute the modulation factors. These modulation factors are used to compute a modulation map which modulates the wrinkle pattern. The modulated wrinkle pattern, which reflects the response of the wrinkled surface to the deformation of the underlying coarse triangular mesh, is used for the rendering. In the course of animation, as the simulation engine recalculates the deformed mesh, the procedure described above is iterated. However, note that the wrinkling coefficients need not to be recalculated during the animation. They are constant with respect to the animation process. They depend only on the initial wrinkle pattern, the initial mesh and the mapping coordinates.
<H2>3.2 Derivation of the Wrinkling Algorithm</H2>
This subsection introduces the mathematical formulation of our geometric wrinkling algorithm. Refer to Figure 5 and consider the grayed triangle undergoing a deformation ending up as the deformed triangle . The deformation of the triangle can be described by a general 4D homogeneous coordinate transformation. However, the rotational and translational parts of the transformation are irrelevant to the derivation of the algorithm. We introduce a local rectangular right handed two dimensional coordinate system, which is defined by choosing any edge of the triangle as the x axis ( refer to Figure 5 ). Hence, x, y denote the local coordinates of the initial triangle and x , y that of the deformed triangle . They are related by the deformation transformation
y x user defined wrinkle pattern initial mesh 1 wrinkling coefficient for each triangle C 1 , C 2 , C 3 , C 4 deformable models such as cloth simulation y' 2 ' x' modulation factor for each triangle (h) deformed mesh modulated wrinkle pattern bump mapping final mesh with animated wrinkles
Figure 5: Wrinkles and coarse mesh
1
x a b x T : = y 0 d y T 1 (x, y) ≡ T 2 (x, y)
The matrix elements a and d represent scaling in the x and y direction respectively, whereas b describes a shear. Let us also introduce the inverse deformation transformation and denote it by
2
T −1 : x = a b x y 0 d y ! ≡ T T 1 2 −1 −1 (x (x , , y y ) )
We define the wrinkle function f (x, y) as the function in the coordinate system xy that results from mapping the wrinkle pattern onto the initial triangular mesh. Formally, we write
3
2 f (x, y) : ∈ R → R
Further we require f (x, y) to be continuous and that its first partial derivatives exist and also be continuous. The wrinkle function f (x , y ) on the triangle is related to f (x, y) by
2 f (x , y ) : ∈ R → R = f T 1 −1 (x , y ), T 2 −1 (x , y ) ¡ where = T ( )
Now let us introduce the area A( , f (x, y)) of the wrinkle function f (x, y) over the triangle and in complete analogy the area of the deformed wrinkle function A ( , f (x , y )) over the deformed triangle . Generally, these two areas will be unequal and therefore the overall wrinkled surface area over the whole mesh will not be conserved during the animation However as pointed out in Section 2, real life examples such as in cloth wrinkling, the surface area is conserved. Note that this is a purely geometric requirement. One might think of several different approaches to meet this requirement. Our approach realizes overall area conservation by achieving area conservation on a per triangle basis of the mesh. We parameterize the area of the deformed triangle by h, that scales f (x , y ) on each triangle of the mesh. Formally, this parameterized surface area is given by Using this notation the requirement of area conservation on a per triangle basis is expressed by the following equation
A( , f (x, y)) = A ( , f (x , y ))
A (h, , f (x , y )) ≡ A ( , hf (x , y ))
4
A( , f (x, y)) = A (h, , f (x , y ))
This is an equation for h, which we call modulation factor. In Appendix A, we present an approximate solution to this equation that yields
5
1 h = 1− (C 1 (a −1) + C 2 b + C 3 (d −1)) C 4
The constants C 1 , C 2 , C 3 , C 4 are here after referred as wrinkling coefficients. The following section gives significance of the formulation.
<H2>3.3 Explanations</H2>
Following rather abstract and mathematical derivation in the previous subsection, we would like to point out and discuss some properties of the algorithm we have presented so far:
• The algorithm is computationally inexpensive during animation, as the only quantities that have to be determined for each triangle are the elements a , b , d of the inverse deformation matrix and the resulting modulation factor by using the formula given by equation 5. The wrinkling coefficients on the other hand, are computationally expensive. However, as one can see from equation 11 in Appendix A, they depend only on quantities that are known prior to entering the animation loop and can therefore be calculated once at the beginning. • As we have pointed out in the discussion so far, we have derived an algorithm that is based on the area conservation property. Here we explain the role played by the area conservation property in our work. From a mathematical point of view, it is clear that we have presented a solution within the approximation of small deformations. There are several possibilities to deal with this restriction. One could decide to restrict the simulations to small deformations where the approximation is valid and/or take into account the higher order terms in the series expansion to extend the range of validity of the approximation. Instead, we propose a pragmatic approach. We declare that equation 5 describes the overall wrinkling behavior only. This frees us conceptually from the “burden of mathematical correctness”. This is because, we are more concerned with the visual results of the animation, rather than precise area conservation. Moreover, the deformations during cloth simulation are moderate in general. Hence, higher order terms in the expansion may become significant but not predominant. The modulation factor h is a function of the deformation of triangle and has value around one. If the triangle is net compressed, h will be greater than one. For the elongation, it will be less than one. One can scale, translate and clip it to introduce a finer control required for the animation. Note that this transformation of the modulation factor no longer satisfies the area conservation property. Also, Section 4 introduces a modulation map, which further violates area conservation. • Another very important property of our algorithm is that it is local. By local, we mean that wrinkling effects caused by deformations are confined to the deformed areas. This is crucial to obtain realistic wrinkling. For example, a garment wrinkles around the shoulder of an animated character as she lifts her arm, while it is stretched on the corresponding side. Locality is introduced in our algorithm by working on a per triangle basis. The size of the mesh triangles actually governs the extension of local wrinkling effects. • Wrinkling coefficients are sensitive to the wrinkle function and therefore to the wrinkle patterns. Wrinkling coefficients for two different patterns on the same triangle will generally differ. Therefore, the same deformation applied to a triangle will yield two different modulation factors (one for each pattern). Each pattern, for instance, features a “principal wrinkling direction”. Assume that the wrinkling patterns are orthogonal to each other. Then, a deformation in the orthogonal direction of one pattern will result in a smaller modulation factor as compared to a modulation factor for the other pattern. In other words, the direction of the deformation “favors one pattern over the other”. This property can be used for developing multi-fold wrinkling techniques. Section 4.1 describes a multi-fold wrinkling implementation.
<H1>4 IMPLEMENTATION OF GEOMETRIC WRINKLES</H1>
The numerical computation of the formulation is trivial. For the numerical integrals of the wrinkling coefficients, we use adaptive sampling in the triangular domain to give a fixed user defined error. The following issues are worth mentioning about the implementation.
1. The wrinkle function and the wrinkle pattern, though referred to as the same entity, they differ in implementation. The wrinkle pattern is gray scale texture image defining the bump map. The user defines an overall normalization factor for the texture to map wrinkle pattern to wrinkle function. The normalization factor is important as the formulation assumes real distances for the bump map (or more precisely the displacement map). The factor should be some fraction of the overall dimensions of the average triangle of the mesh. The wrinkle function is a continuous real valued function, which is a spline approximation of the normalized texture as described in next item. 2. The wrinkling coefficient computation involves partial derivatives of wrinkle function f (x, y) with respect to (x, y). For the reasonable numerical accuracy and stability, the wrinkling pattern needs to be smooth. We fit a spline function to the pattern to smooth out any discontinuities in the input. In addition to this, the user is advised to blur the pattern. 3. As stated in the formulation (Appendix A), solution to equation 4 exists if the input pattern is not constant. As the pattern is user defined, one needs to watch for the invalidity of the solution (constant C 4 in equation 5 turn out to be zero) and therefore eliminates it. In this case, we define the modulation factor to be one. 4. The modulation factor varies significantly across triangles. If we treat a constant modulation factor for a triangle (see Figure 6), wrinkles appear patchy and one can distinctly see the triangular granules. To avoid this, the modulation factors are linearly interpolated across triangles to give smooth Modulation Map ( Figure 6 ).
Figure 6: Smoothing modulation factors gives modulation map
5. The user is given additional control for the animation by transforming the modulation map by a scale factor, clip, and bias. The final bump/displacement map is the product of the modulation map and the wrinkling pattern.
The implementation of multi-fold wrinkling is discussed in detail in the next section.
<H2>4.1 Multi-fold Wrinkling</H2>
Animating a single wrinkle pattern is not satisfactory (particularly for cloth). In real-life, the wrinkles are not mere modulations of a fixed wrinkle pattern. Rather, the pattern changes according to the deformation. Hence, we would like to apply the technique using multiple wrinkle patterns. As stated in Section 3.3, two different wrinkle patterns give different wrinkling coefficients for the same triangle geometry. Hence, for the same deformation of the triangle, corresponding to each pattern, the modulation factors will be different. It all depends on how the wrinkle pattern is oriented with respect to the deformation direction. If a pattern is orthogonal to the deformation direction (as compared to the other), corresponding modulation factor will be small. In other words, the direction of the deformation favors one pattern over the other. To illustrate this, let us consider simple cloth animation as shown in the Figure 7 . In Stage 1 cloth is undeformed. It is then stretched to the bottom left corner (Stage 2). Comes back to the neutral position (Stage 3) and finally in Stage 4, stretches to the bottom right corner. We would like to apply multiple wrinkle patterns for this animation. For simplicity of the discussion, we consider only two wrinkle patterns, though the methodology is developed for multiple patterns. The wrinkle patterns chosen are orthogonal to each other as shown in Figure 8 . As the marked triangle undergoes a series of deformations (Figure 7, Stages 1-4), it may compute different values for the modulation factor for each of the wrinkle patterns ( Figure 8 ). These two modulation factors are then plotted against each other in Figure 9 . For Stages 1 & 3 both the modulation factors are 1 as cloth is undeformed. However, for Stages 2 & 4 the modulation factors differ significantly, depending upon the direction of the deformation. The relatively small modulation factor (say M F 1 is smaller for Stage 2) indicates that the corresponding wrinkle pattern is well oriented towards the direction of the deformation. We choose this pattern for wrinkling for the deformation. This selective application of the wrinkle pattern (along with its modulation) will give a change of one pattern to the other as the deformation direction changes. However, in the animation a sudden switch of the pattern is not temporally coherent and is visually quite disturbing. Figure 10 shows typical the movement of the wrinkle functions in the situation (the movement does not refer to the animation in Figure 7 ). The dotted line (when two modulation factors are equal) indicates the boundary between the patterns. As the modulation factors cross the dotted line, there is a sudden switch in the pattern. To avoid this sudden switch of pattern, we introduce a user definable variance around the mean value of the wrinkling coefficients, which defines a transition zone. This is indicated by a gray area around dotted line in Figure 10 . There will be a smooth transition between wrinkling patterns in this zone. We employ a wrinkling pattern weight function as shown in Figure 11 to achieve the smooth transition. This is in fact a simple power function with an appropriate scaling and clipping. If M F 1 is much smaller than M F 2 (stage 1 in Figure 7 ), M F 1 will be smaller than (1 − variance)(M F 1 + M F 2 )/2 and M F 2 will be bigger than (1 + variance)(M F 1 + M F 2 )/2. This gives maximum weight (W 1 = 1,W 2 = 0) to pattern 1. In the transition zone, when M F 1 and M F 2 are comparable, the two patterns will be blended smoothly. The user definable power n is representative of the tightness of the transition and n = ∞ is a sudden switch of pattern. Figure 7 shows the two patterns animated according to deformation. Note that for lower left triangles in Stage 2 of the animation, both wrinkle patterns get blended. On the other hand, for lower right triangle in Stage 2, the deformation direction favors one pattern clearly.
Figure 7: Simple two-fold wrinkling
Figure 8: Orthogonal patterns for two-fold wrinkling
> 2) compression 2 (pattern 2 4 MF 3 1 (1,1) compression > MF 1 (pattern 1)
Figure 9: Movement of wrinkling coefficients
> MF 1 + MF 2 2) compression stage 1 va ria nc 2 e (pattern transition zone 2 MF stage 2 (1,1) compression > MF 1 (pattern 1)
Figure 10: Transition of the patterns
variance 1 W 1 n=4 n=2 W 2 n=1 0 compression MF 1 MF 2 MF 1 + MF 2 2
Figure 11: Smooth switch between two coefficients
Figure 12: Animated garment without wrinkles
Figure 13: Visualizing strains in the garment
pattern modulated = (M F 1 W 1 pattern 1 +M F 2 W 2 pattern 2 )
<H1>5 ANIMATING GARMENT WRINKLES</H1>
Until now, the Geometric Wrinkle formulation is developed keeping in mind a general deformable model. In this section, we apply the method specifically for cloth animation. Figure 12 shows representative frames of a typical garment animation without Geometric Wrinkle. The garment is animated using a coarse mesh and low metric stiffnesses for the reasons explained in section 2. Though the user can design the wrinkles according to her wish, it is worthwhile to study the strain patterns in the garment. This is because, inappropriately placed wrinkles in the region where there is no deformation will not animate satisfactorily. In addition, the pattern should be orthogonal to the deformations in general, as explained in section 4.1. Figure 13 is a strain pattern for the second frame in the Figure 12 . Dark triangles are triangles with compression and depict the regions where wrinkles might appear. Based on such strain patterns (corresponding to two distinct frames of the garment animation in Figure 12 ), two wrinkling patterns are designed as shown in the Figure 14 . The patterns have distinct wrinkles and additional irregularities to smooth out the sharp appearance of wrinkles. Each pattern represents a distinct direction of deformation. Note that they are considerably orthogonal. Figure 17 (color plate) shows the result after applying the Geometric Wrinkles. It is interesting to see the smooth switch of the wrinkling patterns in the animation because of multi-fold wrinkling. Figure 15 and Figure 16 (color plates) use a single wrinkle pattern (different from Figure 14 ). The frames on the left side correspond to the animation without Geometric Wrinkles. Note that, there are very few wrinkles in the second figure as there is very little deformation of the mesh. The first figure shows the modulation of the wrinkles as per the deformation. As the calculations of the wrinkling coefficients are done on a per triangle basis, the computational time is linear with respect to number of triangles. Typically, it takes 5 minutes per thousand triangles on a MIPS R10000 200 MHz processor. Once the wrinkling coefficients are computed, the time spent on modulating wrinkle pattern is negligible compared to rendering time. In fact, for small meshes (upto a hundred polygons) the modulation of wrinkle pattern can be real time (20 fps).
Figure 14: Wrinkling patterns for the garment
<H1>6 CONCLUSION AND FUTURE WORK</H1>
We have developed a fast and versatile method for animating realistic wrinkles, which is geometric in nature. Hence, it can be applied to general deformable models such as cloth. We would like to extend the method by automatically creating wrinkle patterns from the strain pattern, which is currently a time consuming task.
<H1>7 ACKNOWLEDGMENT</H1>
This work is supported by the Swiss National Research Foundation. The reviewers comments were very helpful and encouraging. Many thanks to Chris Joslin for proof reading this document.
<H1>References</H1>
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[2] Martin Courshesnes, Pascal Volino, and Nadia Magnenat Thalmann. Versatile and Efficient Techniques for Simulating Cloth and other Deformable Objects. In Robert Cook, editor, SIGGRAPH 95 Conference Proceedings, Annual Conference Series, pages 137–144. ACM SIGGRAPH, Addison Wesley, August 1995. held in Los Angeles, California, 06-11 August 1995.
[3] Tosiyasu L. Kunii and Hironobu Gotoda. Modeling and Animation of Garment Wrinkle Formation Processes. In N. Magnenat-Thalmann and D. Thalmann, editors, Computer Animation ’90 (Second workshop on Computer Animation), pages 131–147. Springer-Verlag, April 1990.
[4] Tosiyasu L. Kunii and Hironobu Gotoda. Singularity Theoretical Modeling and Animation of Garment Wrinkle Formation Processes. The Visual Computer, 6(6):326–336, December 1990.
[5] Jen-Duo Liu, Ming-Tat Ko, and Ruei-Chuan Chang. Collision Avoidance in Cloth Animation. The Visual Computer, 12(5):234–243, 1996. ISSN 0178-2789.
[6] John C. Platt and Alan H. Barr. Constraint Methods for Flexible Models. In John Dill, editor, Computer Graphics (SIGGRAPH ’88 Proceedings), volume 22, pages 279–288, August 1988.
[7] Jim Ruppert. A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation. Journal of Algorithms, 1995.
[8] Jonathan Richard Shewchuk. Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In Dinesh Manocha Ming C. Lin, editor, Applied Computational Geometry: Towards Geometric Engineering, volume 1148, pages 203–222. The First ACM Workshop on Applied Computational Geometry, Springer-Verlag, May 1996.
[9] Demetri Terzopoulos and Kurt Fleischer. Modeling Inelastic Deformation: Viscoelasticity, Plasticity, Fracture. Computer Graphics (SIGGRAPH ’88 Proceedings), 22(4):269–278, August 1988. Held in Atlanta, Georgia.
[10] Demetri Terzopoulos and Kurt Fleischer. Deformable Models. The Visual Computer, 4(6):306–331, December 1988.
[11] Pascal Volino and Nadia Magnenat-Thalmann. Fast Geometric Wrinkles on Animated Surfaces. In WSCG’99, Plzen, Feb 1999.
[12] Pascal Volino and Nadia Magnenat Thalmann. Efficient SelfCollision Detection on Smoothly Discretized Surface Animations using Geometrical Shape Regularity. Computer Graphics Forum, 13(3):155–166, 1994. Eurographics ’94 Conference issue.
[13] Pascal Volino, Nadia Magnenat Thalmann, Shen Jianhua, and Daniel Thalmann. An Evolving System for Simulating Clothes on Virtual Actors. IEEE Computer Graphics and Applications, 16(5):42–51, September 1996 1996. ISSN 02721716.
[14] Yin Wu, Prem Kalra, Laurent Moccozet, and Nadia Magnenat-Thalmann. Simulating Wrinkles and Skin Aging. The Visual Computer, 15(4):183–198, 1999. ISSN 01782789.
A FORMULATION DETAILS
In order to present a solution to equation 4 we first give the explicit expressions for A( , f (x, y)) and A (h, , f (x , y )) respectively
6
2 2 ∂f ∂f A( , f (x, y)) = 1+ + dx dy ∂x ∂y
and 2 2 ∂f ∂f A (h, , f (x , y )) = 1+h 2 +h 2 dx dy ∂x ∂y (7)
To evaluate the integral in equation 7, we perform a change of integration variables as follows
8
x = T −1 x y y
By applying this change of variables to equation 7 (refer to equation 2 for a , b , d ), we obtain
A (h, , f (x , y )) 2 2 2 ∂ = 1+ h a f ∂x
9
1/2 2 2 ∂ ∂ 1 +h b +d f dx dy ∂x ∂y a d
≡ I(h, a , b , d ) dx dy
Now let us perform a series expansion of equation 9 in the transformation parameters and the modulation factor. As far as the transformation parameters are concerned, we choose as expansion point the identity transformation and for the modulation factor we expand around h = 1
A (h, , f (x , y )) = (I(h = 1, a = 1, b = 0, d = 1) ∂I ∂I + (a −1) + b ∂a ∂b ∂I ∂I + (d −1) + (h − 1) ∂d ∂h + . . . ) dx dy (10)
In the following, we discard higher order terms in the expansion above. For small deformations around the identity transformation and h = 1, a first order expansion represents a good approximation for the value of the surface area over a deformed triangle. Retaining terms up to first order in the series expansion and by plugging expression equation 10 into equation 4 we obtain
A( , f (x, y)) = C 0 + C 1 (a −1) + C 2 b + C 3 (d −1) + C 4 (h−1) where C 0 = I(1, 1, 0, 1) dx dy ∂I C 1 = dx dy ∂a ∂I C 2 = dx dy ∂b ∂I C 3 = dx dy ∂d
11
∂I C 4 = dx dy ∂h
We call these expansion coefficients wrinkling coefficients. C 1 , C 2 , C 3 , C 4 relate changes in the parameters a , b , d , h to changes of the area of the wrinkle function on the triangle. By observing that C 0 = A( , f (x, y)) and solving equation 11 for the modulation factor we obtain
12
1 h = 1− (C 1 (a −1) + C 2 b + C 3 (d −1)) C 4
Equation 12 represents the solution to equation 4 in the first order approximation to A (h, , f (x , y )), if f (x , y ) is not constant.
Figure 15: Garment animation without (left) and with (right) Geometric Wrinkles, frame 1, blouse 686 triangles, skirt 1274 triangles
Figure 16: Garment animation without (left) and with (right) Geometric Wrinkles, frame 128. Observe very few Geometric Wrinkles because of low mesh deformations.
Figure 17: Multi-fold Wrinkling, frames [53, 67, 81], 25fps. Observe the pattern switch with blending.
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"text": "['This is very well possible']",
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"text": "['advances in the triangulation and interactive systems developed']",
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"text": "['2']",
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"text": "['a significant problem remains']",
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"text": "['Cloth has very large in-plane deformation stiffnesses compared to its ability to bend and shear']",
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"text": "['The problem of solving stiff equations is successfully dealt with by the use of an implicit method for numerical integration by Baraff et al']",
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"text": "['constraint methods']",
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"text": "['have proved to give quality cloth animation']",
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"text": "['which are difficult to model']",
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"text": "['between body and cloth']",
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"text": "['have been rather simple for such a complex problem and robust numerics too']",
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"text": "['The wrinkling coefficients on the other hand, are computationally expensive']",
"type": "own_claim",
"id": "T97"
},
{
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[
14589,
14671
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],
"text": "['they depend only on quantities that are known prior to entering the animation loop']",
"type": "own_claim",
"id": "T98"
},
{
"offsets": [
[
14676,
14725
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],
"text": "['can therefore be calculated once at the beginning']",
"type": "own_claim",
"id": "T99"
},
{
"offsets": [
[
14970,
15062
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],
"text": "['it is clear that we have presented a solution within the approximation of small deformations']",
"type": "own_claim",
"id": "T100"
},
{
"offsets": [
[
14935,
14968
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],
"text": "['From a mathematical point of view']",
"type": "data",
"id": "T101"
},
{
"offsets": [
[
15064,
15125
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],
"text": "['There are several possibilities to deal with this restriction']",
"type": "own_claim",
"id": "T102"
},
{
"offsets": [
[
15362,
15393
]
],
"text": "['we propose a pragmatic approach']",
"type": "own_claim",
"id": "T103"
},
{
"offsets": [
[
15127,
15351
]
],
"text": "['One could decide to restrict the simulations to small deformations where the approximation is valid and/or take into account the higher order terms in the series expansion to extend the range of validity of the approximation']",
"type": "own_claim",
"id": "T104"
},
{
"offsets": [
[
15411,
15467
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],
"text": "['equation 5 describes the overall wrinkling behavior only']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
15469,
15541
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],
"text": "['This frees us conceptually from the “burden of mathematical correctness”']",
"type": "own_claim",
"id": "T106"
},
{
"offsets": [
[
15673,
15737
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],
"text": "['the deformations during cloth simulation are moderate in general']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
[
15746,
15804
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],
"text": "['higher order terms in the expansion may become significant']",
"type": "own_claim",
"id": "T108"
},
{
"offsets": [
[
15809,
15824
]
],
"text": "['not predominant']",
"type": "own_claim",
"id": "T109"
},
{
"offsets": [
[
15826,
15894
]
],
"text": "['The modulation factor h is a function of the deformation of triangle']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
15899,
15919
]
],
"text": "['has value around one']",
"type": "own_claim",
"id": "T111"
},
{
"offsets": [
[
15956,
15982
]
],
"text": "['h will be greater than one']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
15924,
15954
]
],
"text": "['the triangle is net compressed']",
"type": "data",
"id": "T113"
},
{
"offsets": [
[
16004,
16028
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],
"text": "['it will be less than one']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
15984,
16002
]
],
"text": "['For the elongation']",
"type": "data",
"id": "T115"
},
{
"offsets": [
[
16030,
16122
]
],
"text": "['One can scale, translate and clip it to introduce a finer control required for the animation']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
16134,
16229
]
],
"text": "['this transformation of the modulation factor no longer satisfies the area conservation property']",
"type": "own_claim",
"id": "T117"
},
{
"offsets": [
[
16258,
16316
]
],
"text": "['a modulation map, which further violates area conservation']",
"type": "own_claim",
"id": "T118"
},
{
"offsets": [
[
16237,
16246
]
],
"text": "['Section 4']",
"type": "data",
"id": "T119"
},
{
"offsets": [
[
16320,
16388
]
],
"text": "['Another very important property of our algorithm is that it is local']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
16413,
16488
]
],
"text": "['wrinkling effects caused by deformations are confined to the deformed areas']",
"type": "own_claim",
"id": "T121"
},
{
"offsets": [
[
16490,
16535
]
],
"text": "['This is crucial to obtain realistic wrinkling']",
"type": "own_claim",
"id": "T122"
},
{
"offsets": [
[
16550,
16683
]
],
"text": "['a garment wrinkles around the shoulder of an animated character as she lifts her arm, while it is stretched on the corresponding side']",
"type": "data",
"id": "T123"
},
{
"offsets": [
[
16853,
16951
]
],
"text": "['Wrinkling coefficients are sensitive to the wrinkle function and therefore to the wrinkle patterns']",
"type": "own_claim",
"id": "T124"
},
{
"offsets": [
[
16953,
17045
]
],
"text": "['Wrinkling coefficients for two different patterns on the same triangle will generally differ']",
"type": "own_claim",
"id": "T125"
},
{
"offsets": [
[
17058,
17144
]
],
"text": "['the same deformation applied to a triangle will yield two different modulation factors']",
"type": "own_claim",
"id": "T126"
},
{
"offsets": [
[
17253,
17304
]
],
"text": "['the wrinkling patterns are orthogonal to each other']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
17312,
17468
]
],
"text": "['a deformation in the orthogonal direction of one pattern will result in a smaller modulation factor as compared to a modulation factor for the other pattern']",
"type": "own_claim",
"id": "T128"
},
{
"offsets": [
[
17486,
17554
]
],
"text": "['the direction of the deformation “favors one pattern over the other”']",
"type": "own_claim",
"id": "T129"
},
{
"offsets": [
[
17556,
17628
]
],
"text": "['This property can be used for developing multi-fold wrinkling techniques']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
17778,
17833
]
],
"text": "['The numerical computation of the formulation is trivial']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
18144,
18173
]
],
"text": "['they differ in implementation']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
[
18112,
18142
]
],
"text": "['referred to as the same entity']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
18175,
18244
]
],
"text": "['The wrinkle pattern is gray scale texture image defining the bump map']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
18355,
18392
]
],
"text": "['The normalization factor is important']",
"type": "own_claim",
"id": "T135"
},
{
"offsets": [
[
18396,
18451
]
],
"text": "['the formulation assumes real distances for the bump map']",
"type": "data",
"id": "T136"
},
{
"offsets": [
[
18494,
18590
]
],
"text": "['The factor should be some fraction of the overall dimensions of the average triangle of the mesh']",
"type": "own_claim",
"id": "T137"
},
{
"offsets": [
[
18592,
18649
]
],
"text": "['The wrinkle function is a continuous real valued function']",
"type": "own_claim",
"id": "T138"
},
{
"offsets": [
[
18651,
18708
]
],
"text": "['which is a spline approximation of the normalized texture']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
18725,
18734
]
],
"text": "['next item']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
18912,
18952
]
],
"text": "['the wrinkling pattern needs to be smooth']",
"type": "own_claim",
"id": "T141"
},
{
"offsets": [
[
18859,
18910
]
],
"text": "['For the reasonable numerical accuracy and stability']",
"type": "data",
"id": "T142"
},
{
"offsets": [
[
19150,
19179
]
],
"text": "['solution to equation 4 exists']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
19183,
19216
]
],
"text": "['the input pattern is not constant']",
"type": "data",
"id": "T144"
},
{
"offsets": [
[
19137,
19147
]
],
"text": "['Appendix A']",
"type": "data",
"id": "T145"
},
{
"offsets": [
[
19250,
19303
]
],
"text": "['one needs to watch for the invalidity of the solution']",
"type": "own_claim",
"id": "T146"
},
{
"offsets": [
[
19221,
19248
]
],
"text": "['the pattern is user defined']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
19367,
19380
]
],
"text": "['eliminates it']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
19442,
19501
]
],
"text": "['The modulation factor varies significantly across triangles']",
"type": "own_claim",
"id": "T149"
},
{
"offsets": [
[
19665,
19760
]
],
"text": "['the modulation factors are linearly interpolated across triangles to give smooth Modulation Map']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
19763,
19771
]
],
"text": "['Figure 6']",
"type": "data",
"id": "T151"
},
{
"offsets": [
[
20010,
20104
]
],
"text": "['The final bump/displacement map is the product of the modulation map and the wrinkling pattern']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
20264,
20318
]
],
"text": "['Animating a single wrinkle pattern is not satisfactory']",
"type": "own_claim",
"id": "T153"
},
{
"offsets": [
[
20333,
20342
]
],
"text": "['for cloth']",
"type": "data",
"id": "T154"
},
{
"offsets": [
[
20345,
20357
]
],
"text": "['In real-life']",
"type": "data",
"id": "T155"
},
{
"offsets": [
[
20359,
20423
]
],
"text": "['the wrinkles are not mere modulations of a fixed wrinkle pattern']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
20433,
20481
]
],
"text": "['the pattern changes according to the deformation']",
"type": "own_claim",
"id": "T157"
},
{
"offsets": [
[
20586,
20685
]
],
"text": "['two different wrinkle patterns give different wrinkling coefficients for the same triangle geometry']",
"type": "own_claim",
"id": "T158"
},
{
"offsets": [
[
20573,
20584
]
],
"text": "['Section 3.3']",
"type": "data",
"id": "T159"
},
{
"offsets": [
[
20767,
20807
]
],
"text": "['the modulation factors will be different']",
"type": "own_claim",
"id": "T160"
},
{
"offsets": [
[
20694,
20734
]
],
"text": "['for the same deformation of the triangle']",
"type": "data",
"id": "T161"
},
{
"offsets": [
[
20809,
20904
]
],
"text": "['It all depends on how the wrinkle pattern is oriented with respect to the deformation direction']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
20990,
21035
]
],
"text": "['corresponding modulation factor will be small']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
20909,
20961
]
],
"text": "['a pattern is orthogonal to the deformation direction']",
"type": "own_claim",
"id": "T164"
},
{
"offsets": [
[
21053,
21119
]
],
"text": "['the direction of the deformation favors one pattern over the other']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
21553,
21603
]
],
"text": "['the methodology is developed for multiple patterns']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
21605,
21661
]
],
"text": "['The wrinkle patterns chosen are orthogonal to each other']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
21674,
21682
]
],
"text": "['Figure 8']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
21767,
21857
]
],
"text": "['it may compute different values for the modulation factor for each of the wrinkle patterns']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
21860,
21868
]
],
"text": "['Figure 8']",
"type": "data",
"id": "T170"
},
{
"offsets": [
[
21688,
21742
]
],
"text": "['the marked triangle undergoes a series of deformations']",
"type": "data",
"id": "T171"
},
{
"offsets": [
[
21744,
21752
]
],
"text": "['Figure 7']",
"type": "data",
"id": "T172"
},
{
"offsets": [
[
21972,
22005
]
],
"text": "['both the modulation factors are 1']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
21951,
21968
]
],
"text": "['For Stages 1 &']",
"type": "data",
"id": "T174"
},
{
"offsets": [
[
22009,
22028
]
],
"text": "['cloth is undeformed']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
22060,
22152
]
],
"text": "['the modulation factors differ significantly, depending upon the direction of the deformation']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
22039,
22056
]
],
"text": "['for Stages 2 &']",
"type": "data",
"id": "T177"
},
{
"offsets": [
[
22243,
22334
]
],
"text": "['the corresponding wrinkle pattern is well oriented towards the direction of the deformation']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
22154,
22192
]
],
"text": "['The relatively small modulation factor']",
"type": "data",
"id": "T179"
},
{
"offsets": [
[
22472,
22518
]
],
"text": "['will give a change of one pattern to the other']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
22394,
22443
]
],
"text": "['This selective application of the wrinkle pattern']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
22522,
22555
]
],
"text": "['the deformation direction changes']",
"type": "data",
"id": "T182"
},
{
"offsets": [
[
22566,
22640
]
],
"text": "['in the animation a sudden switch of the pattern is not temporally coherent']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
22645,
22673
]
],
"text": "['is visually quite disturbing']",
"type": "own_claim",
"id": "T184"
},
{
"offsets": [
[
22965,
23004
]
],
"text": "['there is a sudden switch in the pattern']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
22919,
22963
]
],
"text": "['the modulation factors cross the dotted line']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
23059,
23169
]
],
"text": "['a user definable variance around the mean value of the wrinkling coefficients, which defines a transition zone']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
23238,
23311
]
],
"text": "['There will be a smooth transition between wrinkling patterns in this zone']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
23415,
23495
]
],
"text": "['This is in fact a simple power function with an appropriate scaling and clipping']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
23557,
23616
]
],
"text": "['M F 1 will be smaller than (1 − variance)(M F 1 + M F 2 )/2']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
23621,
23679
]
],
"text": "['M F 2 will be bigger than (1 + variance)(M F 1 + M F 2 )/2']",
"type": "own_claim",
"id": "T190"
},
{
"offsets": [
[
23500,
23532
]
],
"text": "['M F 1 is much smaller than M F 2']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
23545,
23553
]
],
"text": "['Figure 7']",
"type": "data",
"id": "T193"
},
{
"offsets": [
[
23681,
23737
]
],
"text": "['This gives maximum weight (W 1 = 1,W 2 = 0) to pattern 1']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
23768,
23798
]
],
"text": "['M F 1 and M F 2 are comparable']",
"type": "data",
"id": "T195"
},
{
"offsets": [
[
23800,
23841
]
],
"text": "['the two patterns will be blended smoothly']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
23739,
23761
]
],
"text": "['In the transition zone']",
"type": "data",
"id": "T197"
},
{
"offsets": [
[
23843,
23921
]
],
"text": "['The user definable power n is representative of the tightness of the transitio']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
23927,
23962
]
],
"text": "['n = ∞ is a sudden switch of pattern']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
[
23964,
23972
]
],
"text": "['Figure 7']",
"type": "data",
"id": "T200"
},
{
"offsets": [
[
25318,
25407
]
],
"text": "['the Geometric Wrinkle formulation is developed keeping in mind a general deformable model']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
25750,
25810
]
],
"text": "['it is worthwhile to study the strain patterns in the garment']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
25694,
25748
]
],
"text": "['the user can design the wrinkles according to her wish']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
25829,
25874
]
],
"text": "['inappropriately placed wrinkles in the region']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
25905,
25936
]
],
"text": "['will not animate satisfactorily']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
25875,
25904
]
],
"text": "['where there is no deformation']",
"type": "data",
"id": "T206"
},
{
"offsets": [
[
25951,
26014
]
],
"text": "['the pattern should be orthogonal to the deformations in general']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
26032,
26043
]
],
"text": "['section 4.1']",
"type": "data",
"id": "T208"
},
{
"offsets": [
[
26045,
26095
]
],
"text": "['Figure 13 is a strain pattern for the second frame']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
26103,
26112
]
],
"text": "['Figure 12']",
"type": "data",
"id": "T210"
},
{
"offsets": [
[
26115,
26183
]
],
"text": "['Dark triangles are triangles with compression and depict the regions']",
"type": "own_claim",
"id": "T211"
},
{
"offsets": [
[
26184,
26211
]
],
"text": "['where wrinkles might appear']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
[
26500,
26559
]
],
"text": "['Each pattern represents a distinct direction of deformation']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
26571,
26603
]
],
"text": "['they are considerably orthogonal']",
"type": "own_claim",
"id": "T214"
},
{
"offsets": [
[
26901,
26983
]
],
"text": "['The frames on the left side correspond to the animation without Geometric Wrinkles']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
26996,
27044
]
],
"text": "['there are very few wrinkles in the second figure']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
27048,
27092
]
],
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http://dx.doi.org/10.1145/311535.311549
<Title>Computational Fluid Dynamics in a Traditional Animation Environment</Title>
Patrick Witting
DreamWorks Feature Animation and Squeaky Cat
<Abstract>This paper presents a system that uses computational fluid dynamics to produce smoke, water, and other effects for traditionallyanimated films. The system was used in over twenty scenes in the animated feature film The Prince of Egypt. Animators use images and animation sequences to drive two-dimensional numerical simulations of the time-dependent compressible Navier-Stokes equations. For instance, images can be used to initialize temperature fields which cause dynamic buoyancy-driven vortices to evolve. In addition to being image-driven, the system is unique in allowing for compressibility of the fluid, and in its use of partial differential equations for texture mapping.</Abstract>
Keywords: animation, animation systems, applications, fluid simulations, natural phenomena, numerical analysis, physically based animation, physically based modeling, scientific visualization, texture mapping
<H1>1 Introduction</H1>
<H2>1.1 Motivation</H2>
<H2>1.2 Modeling Fluid Behavior for the Sciences 1.4 Contributions of this Paper</H2>
Compared to computer graphics, the equations of fluid motion and solution methods for them have a long history. Equations expressing conservation of mass, momentum, and energy, often referred to as the Navier-Stokes equations, have been around since the early 1800’s. Sir Horace Lamb’s Hydrodynamics
[ 11 ]
, from 1932, is still regarded as one of the best sources for fundamental theorems, equations, and solutions in fluid mechanics. The equations of motion cannot be solved analytically, except in simplified situations, and therefore need to be solved numerically. Numerical integration methods for systems of equations predate the modern computer as well, and John von Neumann envisioned using the computer to solve the equations of motion for weather prediction in the 1940’s. Today, the use of computers to solve the Navier-Stokes equations is widespread, with descriptions of particular models and their solutions filling the pages of journals such as Journal of Fluid Mechanics and Journal of the Atmospheric Sciences. Although computational fluid dynamics is a fairly mature subject, the emphasis so far has been on accurately simulating physical situations for scientific purposes, rather than creating images and animations as the end goal, which has different concerns and motivations. One simple example of this is the use of artificial compressibility, employed in the equation set presented in section 3, as a means of speeding up the calculations. For scientific work, the non-physical compressibility effects introduced need to be rigorously justified, whereas for the creation of imagery and animation, the guiding standard is how the images look. When the emphasis is on the look of the final images, there are new sets of concerns about how to control and modify the simulation dynamics, and what and how to render. These concerns move us into the territory of computer graphics, with the highly practical production environment driving the process forward.
<H2>1.3 Modeling Fluid Behavior for Computer Graphics</H2>
Previous work in the graphics literature
[ 2 , 4 , 6 , 7 , 9 , 12 , 13 , 14 , 16 , 19 ]
has modeled various aspects of fluid behavior with an emphasis on efficiency and controllability issues. Some of this work makes use of existing velocity fields or allows users to create their own in a variety of ways, rather than have a simulation determine the velocity field. The emphasis in this paper is on the use of the full NavierStokes equations to solve for the dynamic velocity and temperature fields numerically. Kass and Miller
[ 9 ]
solve the shallow water equations, which reduce the Navier-Stokes equations down to solving for an evolving height field for the surface of a shallow body of liquid. Yaeger, Upson, and Myers
[ 19 ]
, used two-dimensional timedependent vorticity equations to model the atmosphere of Jupiter. The strongest advocacy for use of the full Navier-Stokes equations so far in the graphics literature is from Foster and Metaxas [ 7 ]
, who solve the three-dimensional equations of motion to model smoke. There may be no right or wrong answer as to what level of physical modeling is appropriate, in general, but there is usually a decision making process based on the imagery needed to guide this choice. Creative control and the level of realism desired are two of the main concerns. The decision making process is well illustrated in
[ 19 ]
, where the end goal, creating animations of Jupiter’s atmosphere for the film 2010, guided aspects from the equations being solved to their final rendering method. This paper is of that same style, describing a system built at DreamWorks to support the use of fluid dynamics simulations in the creation of special effects for the animated feature film The Prince of Egypt. Some of the unique features of the system described in this paper include: a compressible version of the equations of motion; the use of images and animations for controlling the dynamics; fast accurate texture mapping features; and finally, a complete production system. The compressible formulation, unlike any in the graphics literature, allows for the modeling of compressible effects, such as shock waves, and also provides a mechanism for speeding up flow calculations by an order of magnitude or more. Another unique feature of the system is the use of images and animations as input devices, which allows animators to control initial conditions, source terms, and movable internal boundaries in an easy and flexible way. The inclusion of texture mapping differential equations, another new concept developed here, makes it possible to precalculate particle paths on a fixed grid which can be used in a straight-forward manner at render time. The system also provides fast turn around time. Fourth order accuracy allows animators to use coarser grids, thus saving time. The use of two-dimensional simulations, the compressible formulation, and coarser grids, results in fast, useful simulations. Simulations performed on a 100 by 100 grid are detailed enough for film work and can be calculated at a rate of one frame per second. Additional production components make the overall process efficient for the animator.
Computer graphics simulations of fluid behavior are in demand in filmmaking for depicting gases, liquids, smoke, dust, fire, and other natural phenomena. Methods for creating these simulations vary widely, depending on the requirements for realism, controllability, rendering style, and complexity. This paper describes a system, which utilizes numerical simulations of the full equations of fluid dynamics, that is being used at DreamWorks Feature Animation Studios in the production of traditionally-animated films. The system employs techniques from both the scientific and computer graphics communities in order to be both efficient and accessible to animators.
Of the many ways of incorporating simulation into the creation of fluids animations, one end of the spectrum in a traditional animation environment is to use no simulation at all, and draw every frame of the animation. This approach gives a wide range of flexibility and control, but is a tedious process with realistic limits on the complexity that can be achieved. At the other extreme, there are many advantages to numerically solving the full equations of motion for fluids, usually referred to as the Navier-Stokes equations, to create an- imations of fluid behavior. With simple user set-ups, the physically accurate equations take over, generating lots of high quality animation, rich in complexity and guaranteed realistic motion. Figure 1 shows the results of a buoyancy-driven simulation created by simply interpreting the luminance of an image supplied by the user as the initial condition for the temperature field. The fluid inside the letters is colder and more dense than the surrounding fluid, causing it to sink. This is typical of the type of simulation that was used to generate smoke for The Prince of Egypt, where contours of temperature were rendered from a simulation driven by buoyant instabilities. This and other examples are discussed in more detail in section 5.
3763 Lockerbie Lane Glendale, CA 91208 [email protected]
Figure 1: Temperature distribution after 0, 100 and 400 time steps.
<H1>2 Design Goals</H1>
Desirable characteristics of a useful production system which incorporates fluid dynamics simulations include the following:
Simulating a Variety of Flow Situations: The equations being solved and the solution method should be capable of modeling a wide variety of flow situations, i.e. shear flow instabilities (Kelvin-Helmholtz), vortex motions, buoyant instabilities (Rayleigh-Taylor), Coriolis effects, gravity waves, compressible effects, etc. In addition, arbitrary forcing functions, or source terms, would be desirable to make many more situations possible, even those without any physical justification. Users should have easy access to setting up the various flow situations. Control Throughout the Process: The biggest difference between simulation systems for scientific purposes and simulation systems for production purposes is the level of control required in production work. Ideally, animators would control many aspects of the simulation dynamics and be able to incorporate the results into the final scene in a variety of ways. Speed: Speed is always a consideration in production work, because it usually translates into more iterations of the creative design cycle and a better final result. Pipeline: The overall process must make sense within the context of the production environment. The system should be able to make use of other scene elements, produce scene elements in the most convenient formats, and should be part of an efficient work flow. A Variety of Rendering Styles: The rendering style plays an important role in the overall process. A wide variety of rendering styles increases the expressive power of scene elements and their interpretation.
<H1>3 The Model</H1>
The equation set used was derived for a meteorology application, the study of clouds
[ 10 , 18 ]. The equation set presented in section 3.2 is a simplification of that system which meets the needs discussed in section 2 in a variety of ways discussed throughout.
<H2>3.1 Important Aspects of the Equation Set</H2>
Because this formulation of the equations of motion will be unfamiliar to many readers, this section has been included to characterize the equation set in a qualitative manner.
Conservation of Mass, Momentum, Energy: The system of five equations and five unknowns is used to express conservation of mass, conservation of the 3 components of momentum, and conservation of energy. Along with the equation of state, which is an equation for one thermodynamic quantity as a function of two others, this forms a complete description of the fluid, i.e. the velocity and thermodynamic state of the fluid at any point. Given appropriate initial conditions and boundary conditions, the equations can be used to advance the solution forward in time. At the boundaries, a well-posed problem can be formed by specifying information for all the variables except the pressure, where the solution needs to be calculated [ 8 ]. Compressibility: One of the most important aspects of the equation set is that there is no assumption of incompressibility. Not only does this mean that compressibility effects can be modeled, but the equations can be solved much faster. When an incompressible formulation is used, there is an elliptical partial differential equation involved, corresponding to an “infinite” speed of propagation of pressure waves. This typically translates into solving a large matrix equation, usually by iterative techniques, to ensure the pressure field is consistent with the velocity field. This is usually a time consuming part of the solution method and does not scale well as grid resolution is increased. Using the compressible formulation means that calculation times for each time step are essentially linear in the number of grid points. Pressure Equation: Because of the lengthy derivation, the pressure equation is presented as is. In summary, conservation of mass is expressed in the compressible equations by the mathematical statement that changes in density for a parcel of fluid are the result of divergence in the velocity field. Buoyancy: Some systems of equations make an assumption that the fluid has the same density everywhere, which simplifies the equation set at the expense of not modeling buoyancy effects. The equations being used here do not make that assumption and buoyancyeffects dominate the dynamics in most of the examples presented. Potential Temperature: Potential temperature is used in meteorology as the appropriate measure of static stability, instead of density, temperature, or other variables which are not conserved in the atmosphere. For instance, a situation of having a colder fluid on top of a hotter fluid is not necessarily an unstable arrangement, due to the stratified hydrostatic pressure in the atmosphere. This concept is defined in most meteorology texts [ 3 ]. Throughout this paper “temperature” is often used in place of “potential temperature” for ease of reading. Forcing: The equations also allow for arbitrary forcing functions to each of the equations, except the pressure equation, corresponding to localized source terms for momentum and
energy. These forcing functions can be analytical functions of the other variables, such as coriolis or buoyancy terms, or can come from other sources, such as images and animations. Diffusion: Each of the equations includes a diffusion term, which has the effect of damping out the high frequency waves. These terms have many interpretations, from molecular diffusion, to turbulence modeling, to numerical stability devices. Most ODE solvers (ordinary differential equation), including the fourth order Runge-Kutta scheme employed here, require some level of diffusion to avoid nonlinear instabilities. Passive Scalar: As discussed later, the system can also be augmented with additional equations, for things such as passive scalars which advect with the flow. Equations are derived for including texture mapping information, so that particle trajectories don’t need to be computed via integration later.
<H2>3.2 Equation Set</H2>
The equations being solved are essentially those in
[ 10 ]. The subgrid scale model is replaced by diffusion terms with constant diffusion coefficients, and the rain processes and coriolis terms are neglected. Also, the coefficient for the sound speed is multiplied by a constant, introducing artificial compressibility, so that the time step requirement is less severe. The primary variables being advanced forward in time are u; v; and w , which are the velocity components in the x;y; and z directions, respectively, the pressure perturbation variable, , defined in equation 9, and the potential temperature, , defined in equation 8. The meteorology convention of using z as the up direction is used here.
(1) (2) (3) (4)
(5) Dt D is the material derivative operator @t @ + u @x @ + v @y @ + w @z @ , and ation of is gravity, the Laplacian and operator are diffusion @x @ 2 2 + coefficients, @y @ 2 2 + @z @ 2 2 . c g p is is the the accelerspecific heat at constant pressure, c is the speed of sound, and f u ; f v ; f w ; and f are forcing functions, or source terms for their respective equations. Base state variables, denoted by overbars, are time-invariant functions of z , the vertical coordinate. The equation of state is the perfect gas law,
6
p = RT; R
constant, and T is the temperature. Using p 0 as a reference pressure, a non-dimensional pressure, , is defined by
(7)
and a potential temperature, , by
(8)
Defining a pressure perturbation variable by
9
= + ;
we assume the base state profiles obey the hydrostatic relationship
(10)
which reflects that the hydrostatic pressure of a parcel of air is caused by the weight of a column of air above it. A two-dimensional version of the above equations can be derived by assuming that in one of the horizontal directions there is no flow and no change in any of the variables. Taking y to be the flowless direction, equation 2 is no longer needed, and simplifications are made to equation 4 and to the material derivative and Laplacian operators to account for zero derivatives in the y direction. In addition to the basic equations of fluid motion, equations can be appended to the system which may or may not have feedback into the basic equations. Equation 11 is the prototypical passive scalar equation, which models an arbitrary scalar being advected along with the fluid, and optionally diffusing through the non-negative diffusion coefficient .
(11)
Derivations of the equations of motion from first principles can be found in many textbooks for the interested reader
[ 3 , 11 , 15 , 17 ].
<H3>3.2.1 Texture Mapping Equations</H3>
Figure 2: Texture mapping after 0 and 400 time steps.
A convenient way to record the flow field history is through the dynamic evolution of texture map information. The idea is to initialize passive scalar variables with the original positions of the fluid parcels. These variables would obey equation 11, and let you know the original location of the parcel at any stage in the simulation, at the fixed grid locations. This Eulerian description is particularly useful in the rendering phase, since the texture mapping coordinate information is evenly spaced in the output image space. This technique is shown in figure 2 for the same simulation used to produce figure 1 . Suppose we are running a two-dimensional simulation on a recttal angular texture domain map of variable, physical x dimensions , with initial L x condition by L z . Define x x; a horizonz; 0 = x=L z x; x and z; 0 a vertical = z=L z texture . If both map of variable, these variables z , with obey initial equation condition 11, texture then at map a later coordinates time, t , at x time x; z;t t =0 and for z the x; parcel z;t will at location contain x; the z at time t , that is, they tell where the parcel of fluid “came from.” When implementing periodic boundary conditions, it is more desirable to keep track of displacement offsets from x and z because Defining of the discontinuity p x = x , of x=L x and x and z p as z = you z cross , z=L periodic z , we arrive boundaries. at the following equations
(12) (13)
<H2>3.3 Solution Method</H2>
The solution method for solving the system of equations is the fourth order Runge-Kutta scheme, using fourth order centered finite differencing for spatial derivatives on a regular grid with equal grid spacing. At boundary points and one point away, one-sided differencing is used. This solution method is briefly described below: Ordinary differential equation solvers, such as the Runge-Kutta methods, solve the vector equation
(14)
The equation set 1 through 5 can be written in this form for the solution vector y = u v w T by moving the advective terms in the material derivatives over to the right hand side of their respective equations. The advective terms are those not involving partial derivatives with respect to time. The equations will now look like equation 14 where the prime in equation 14 denotes differentiation with respect to time. The right hand side of the equations become f y . The solution vector is initialized with values at the regularly spaced grid locations, then advanced forward in time according to the time integration scheme. This involves evaluating the function f y at each of the grid points, making use of the solution vector in a stencil of grid points surrounding the grid point being evaluated. First and second derivative terms are replaced by their fourth order finite difference approximations, which can be found in
[ 1 ]. The overall method is globally fourth order accurate in space and time, provided that the initial conditions, boundary conditions, and forcing functions are sufficiently smooth. The fourth order accuracy is not required for production purposes, but the effort in achieving this added accuracy is not significant, and the increased accuracy allows for the use of coarser grids. For instance, comparing Runge-Kutta fourth order with Euler’s method, four function evaluations per time step are required for Runge-Kutta compared with one for Euler, but this is almost offset by the time steps which can be 2.82 times larger, according to equation 15. The time step limitation for stability for the advection problem, i.e. negligible diffusion, is
(15)
where t is the time step, x is the grid spacing, n is the number of space dimensions, c is the speed of the fastest moving wave in the system, and m is a factor that accounts for the spatial differencing method. For fourth order centered first derivatives, this factor turns out to be 1.372, compared with 1.0 for second order centered first derivatives. Numerical methods for fluid dynamics can be found in a variety of places
[ 5 , 8 ], and an extensive book list and summary of available codes can be found at http://chemengineer.miningco.com/msub74.htm .
<H1>4 The Production System</H1>
This section describes the actual system built, which reflects the design goals of section 2, makes use of the model described in section 3, and also takes into account additional considerations specific to the traditional animation environment and the needs of The Prince of Egypt. In a traditional animation studio, most artwork and animation is two dimensional; the illusion of depth comes from the drawn or painted perspective, along with the camera moves and techniques available in the compositing software. Many simulation and rendering techniques were used in the visual development stage of the film. Test animation resulted from three-dimensional simulations with temperature being visualized via volume rendering, two-dimensional simulations creating velocity fields used for line integral convolution of source imagery, as well as other techniques. By far, the biggest success was twodimensional simulations of buoyant instabilities, where the temperature field was visualized as smoke. The plan was to use this technique to create “magical smoke” for the sequence Playing with the Big Boys, and the process was streamlined with this in mind.
<H2>4.1 Design Decisions</H2>
The components described in sections 4.2-4.4 were built to support two-dimensional simulations which use images and animations as input. The simulations output information at regular intervals which is later used in the compositor for rendering. Some of the advantages of these decisions are described below.
Control Through Layering: Animators can build up libraries of elements produced by simulations, all of which can be easily repositioned, scaled, and even put into perspective within the compositor. The bottom of figure 3 shows two layers and how they were integrated into the final image above. The top element was scaled and had animating transforms to match to the motion of one of the magician’s hands. The lower element had an animating transform to react to the sliding of one of his feet. Individual layers allow artists to make independent decisions for colors, opacities, rendering parameters, and transforms. Speed: Two-dimensional simulations allow for good interactivity in creating elements for later use. Some of the lower resolution final elements used in the film were created in under two minutes, and even the highest resolution simulations could be set up using the information gathered in simulations taking only a few minutes. Deferred Rendering: The texture mapping differential equations developed in section 3.2.1 and periodic output from the simulations allow for deferred rendering, using only a small fraction of the disk space required to save final images. Deferred rendering means that no rendering decisions need to be made at simulation time, and no simulation time is required at render time. This allows for a flexible system, where simulations can be run with a specific flow situation and final element in mind, such as
Figure 3: Reactionary elements created by simple transformations.
rising smoke. Artists choose rendering parameters later, e.g. to alter final timing or to animate contour levels that make the smoke slowly dissipate. At render time, a library of potentially useful simulations is already built up, and rendering involves little more than appropriate resampling (see section 4.4).
<H2>4.2 Setting Up and Running Simulations</H2>
Although the code is capable of handling more general situations, such as analytically defined forcing functions, gravitational fields, and diffusion coefficients, only a subset of the functionality is available via the user interface. Images define the initial conditions for velocity and temperature. Scalar variables on the interface aid the software in interpreting the images, e.g. assigning values to the black and white limits of the images. Similarly, images and animations are employed to apply forcing terms to the momentum and energy equations. In addition, two images are used to optionally assign profiles to the horizontal velocity and the temperature as functions of z . This makes it easy to set up shear flows and stratified layers of density. Figure 4 shows the interface for starting simulations. Using the simulation starting interface, animators can set other parameters such as the resolution, boundary condition types, output frequency, etc., and can monitor simulations in the viewer described below. If a simulation is evolving unsatisfactorily, an animator can quickly restart it using modified images or parameter settings. Before the simulation is run, the system performs a preprocessing step on the images, essentially resampling them and slightly smoothing them for the appropriate simulation resolution, and enforcing periodic conditions if needed. It also calculates the initial pressure field from the temperature field, ensuring that the hydrostatic relationship is satisfied for vertical columns of fluid. Figure 5 shows the input image summary before the preprocessing steps. Figures 4 through 6 are taken from example 2 discussed in section 5.
Figure 4: Simulation starter.
<H2>4.3 Previewing Simulations</H2>
As the simulations are running, or afterward, animators can preview and optionally render the results to disk via the interface shown in figure 6 . This previewer is a simple mapping of the temperature values to the luminance of the black and white images. More rendering options described below are available in the compositor.
<H2>4.4 Rendering Simulations</H2>
The compositor is a graph-based system (DAG) where rendering operations are “nodes” in the graph.
Temperature Contours: Two image generation nodes are provided in the compositor for rendering the temperature field, with temperature being mapped in a linear fashion to transparency. Values outside the linear range are clamped to “clear” or “solid.” One node maps the results of simulations done on a rectangle with periodic sides onto a circle, as in middle of figure 8, and the other renders the rectangular temperature field. All of the parameters, such as the timing and threshold values, have animation curves. The rendering process involves reading the data from disk at the simulation resolution and performing resampling with a two-pass, one-dimensional cubic convolution kernel. It is important to do periodic extensions before resampling to avoid seams at the periodic boundaries, and to do thresholding after resampling to avoid stair-step effects for magnification near the threshold values. Volume Rendering: Volume rendering of the thresholded temperature field was supported for three-dimensional simu-
Figure 5: Input summary before preprocessing.
Figure 6: Simulation viewer.
lations in the visual development phase, but not in the production system. Texture Mapping: As described in section 3.2.1 and seen in figure 2 , texture mapping is supported in the compositor. Inputs to this node are an image to be distorted, a simulation number, a reference time, and a current time. The image is distorted based on the flow field evolution between the reference time and the current time, using the texture mapping data for those two times. Image Smearing: Another rendering option supported in the compositor is the smearing of an image via line integral convolution using two-dimensional flow fields provided by the simulation. A single smearing uses one static flow field and a time range for the integration, provided by the user. Each output pixel receives its color from the colors visited along a flow integration path passing through the output pixel between the two specified times.
<H1>5 Examples</H1>
The example times quoted below are for a single processor SGI O2 with R10K floating point chip and processor chip. Calculation times are given for simulation time steps. Simulation time steps and simulation time between final frames are roughly equal, for comparison purposes, using the following logic: According to equation 15, if a Mach number of 0.4 is used and the largest possible stable time step is used, then the fluid speed will travel the distance of about one grid point per time step. Unless the grid is extremely large, structures moving by one grid point corresponds to a reasonable speed for an animation. For render times, the quote is for producing 640 by 480 images.
Figure 7: top) Temperature field. middle) Composition in scene. bottom) Final scene.
<H2>5.1 Example 1 Image Used for Initial Temperature</H2>
In the first example, an image defines the initial temperature distribution and drives the dynamics of the simulation. The lettering in “SI99RAPH” is colder than the surrounding fluid, which causes it to sink. Conservation of mass dictates that there be areas of return flow as the cold fluid sinks, creating vortices. There is enough variation in the initial distribution such that the nonlinear equations result in pleasing graphic shapes and interesting dynamics. This simulation was run on a 400 by 300 grid, with periodic sides. Figure 1 shows the temperature distribution at the start of the simulation and at two later times. Calculation time between time steps is 19.8 seconds, which include the texture mapping calculations. Render time for frames such as figure 1 is 3.16 seconds per frame. As described in section 3.2.1, texture mapping information can be calculated along with the simulation to provide rendering information. Particle advection through the dynamically evolving velocity field is thus precalculated, eliminating the need to calculate particle trajectories at render time. Figure 2 shows the result of advecting the colors in an image along with the fluid for the simulation used to produce figure 1 . An average render time for distortions such as those depicted in figure 2 is 9.8 seconds per frame.
<H2>5.2 Example 2 Constant Heat Flux from Below</H2>
The second example simulates heat being introduced at the bottom of the domain creating “magical smoke” (see figure 7 ). The initial temperature distribution is a random noise pattern with an overall average temperature which is essentially constant except in a narrow layer near the bottom, where it is hotter. The images used for defining the initial conditions are shown in figure 5 and the other input values are the same as those shown in figure 4 . The only images that are not scaled by zero, are the images used to define the unstable profile and the random perturbations in the initial temperature. The simulation is performed on a 960 by 321 grid, with the rendering aspect ratio adjusted to make the shapes look taller and thinner than the actual simulation, which would otherwise promote rising plumes with essentially round circulation patterns. One time step calculation takes 36.3 seconds, and one rendered frame such as at the top of figure 7 takes 4.7 seconds to render.
<H2>5.3 Example 3 Periodic Boundary Conditions in Action</H2>
In figure 8 , “magical blood” is created by a simulation driven by a random forcing function in the temperature equation, defined by one of the input images. Using the circular rendering option and a periodic simulation domain creates a seamless texture mapping with the appearance of blood emanating from the center of the bowl. The final composite shows the circular shape being repositioned in perspective, registered to the bowl. Everything can be defined and rendered in one pass within the compositing package, including the animating perspective transformation. The simulation resolution is 150 by 151. Time step calculation time is 2.7 seconds per time step and rendering time is 1.57 seconds per frame.
<H1>6 Summary</H1>
This paper presents a complete production system which enables animators to access the beauty and realism embodied in the physically accurate equations of motion, the Navier-Stokes equations. With this system, animators can express themselves by controlling the simulation dynamics through a familiar user interface—the use of images and animations. Texture mapping features allow deferred rendering of flow distortions, with no need to recompute particle trajectories through a time-evolving velocity field. A compressible formulation and two-dimensional simulations allow for quick turnaround time in the creative cycle of creating/modifying simulations and applying the results within the compositor to the final scene. While this production system emphasizes the needs of a traditional animation environment, many of the concepts apply outside this context as well. All of the equations, including the texture mapping equations, extend to three dimensions. One of the most useful ideas presented here for three-dimensional simulations is the implementation of an artificial speed of sound through the compressible formulation of the equations. Atmospheric researchers often use the compressible formulation because of its computational advantages over the incompressible formulation, even when using the actual speed of sound for pressure waves. For computer graphics purposes, an artificial speed of sound of an order of magnitude less than the actual one is often justified, and provides a mechanism for dramatic speed increases.
Figure 8: top) Temperature on periodic rectangular domain. middle) Circular domain remapping. bottom) Final scene.
<H1>References</H1>
[1] Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables. Dover, 1974.
[2] Cassidy J. Curtis, Sean E. Anderson, Joshua E. Seims, Kurt W. Fleischer, and David H. Salesin. Computer-Generated Watercolor. In Computer Graphics, pages 421–430. ACM SIGGRAPH, 1997.
[3] John A. Dutton. The Ceaseless Wind. Dover, 1986.
[4] David S. Ebert and Richard E. Parent. Rendering and Animation of Gaseous Phenomena by Combining Fast Volume and Scanline A-buffer Techniques. In Computer Graphics, volume 24(4), pages 357–366. ACM SIGGRAPH, 1990.
[5] C.A.J. Fletcher. Computational Techniques for Fluid Dynamics. Springer, 1990.
[6] Nick Foster and Dimitris Metaxas. Realistic Animation of Liquids. In Graphical Models and Image Proc., volume 58(5), pages 471–483, 1996.
[7] Nick Foster and Dimitris Metaxas. Modeling the Motion of a Hot, Turbulent Gas. In Computer Graphics, pages 181–188. ACM SIGGRAPH, 1997.
[8] Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger. Time Dependent Problems and Difference Methods. Wiley, 1995.
[9] Michael Kass and Gavin Miller. Rapid, Stable Fluid Dynamics for Computer Graphics. In Computer Graphics, volume 24(4), pages 49–57. ACM SIGGRAPH, 1990.
[10] Joseph B. Klemp and Robert B. Wilhelmson. The Simulation of Three-Dimensional Convective Storm Dynamics. Journal of the Atmospheric Sciences, 35:1070–1096, 1978.
[11] Sir Horace Lamb. Hydrodynamics. Dover, 1932.
[12] Karl Sims. Particle Animation and Rendering Using Data Parallel Computation. In Computer Graphics, volume 24(4), pages 405–413. ACM SIGGRAPH, 1990.
[13] Jos Stam and Eugene Fiume. Turbulent Wind Fields for Gaseous Phenomena. In Computer Graphics, pages 369–376. ACM SIGGRAPH, 1993.
[14] Jos Stam and Eugene Fiume. Depicting Fire and Other Gaseous Phenomena Using Diffusion Processes. In Computer Graphics, pages 129–136. ACM SIGGRAPH, 1995.
[15] Philip A. Thompson. Compressible-Fluid Dynamics. Rensselaer Polytechnic Institute Press, 1988.
[16] Jakub Wejchert and David Haumann. Animation Aerodynamics. In Computer Graphics, volume 25(4), pages 19–22. ACM SIGGRAPH, 1991.
[17] Frank M. White. Viscous Fluid Flow. McGraw-Hill, Inc., 1991.
[18] Patrick Witting. Numerical Investigation of Stratus Cloud Layer Breakup by Cloud Top Instabilities. PhD thesis, Stanford University, 1995.
[19] Larry Yaeger, Craig Upson, and Robert Myers. Combining Physical and Visual Simulation Creation of the Planet Jupiter for the Film “2010”. In Computer Graphics, volume 20(4), pages 85–93. ACM SIGGRAPH, 1986.
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A05 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A05_S06_Building_Effcient,_Accurate_Character_Skins_from_Examples_CITATION_PURPOSE_M_v1.xml">
4d7508619e6c2edec7350f85a067113fef474375b28592cf953cecd2f37b95ff
3wx9
http://dx.doi.org/10.1145/1201775.882308
To appear at SIGGRAPH 2003
<Title>Building Efficient, Accurate Character Skins from Examples</Title>
Alex Mohr ∗ Michael Gleicher University of Wisconsin
Madison
Figure 1: Several still frames of our character from Figure 9 in a new animation sequence.
<Abstract>Good character animation requires convincing skin deformations including subtleties and details like muscle bulges. Such effects are typically created in commercial animation packages which provide very general and powerful tools. While these systems are convenient and flexible for artists, the generality often leads to characters that are slow to compute or that require a substantial amount of memory and thus cannot be used in interactive systems. Instead, interactive systems restrict artists to a specific character deformation model which is fast and memory efficient but is notoriously difficult to author and can suffer from many deformation artifacts. This paper presents an automated framework that allows character artists to use the full complement of tools in high-end systems to create characters for interactive systems. Our method starts with an arbitrarily rigged character in an animation system. A set of examples is exported, consisting of skeleton configurations paired with the deformed geometry as static meshes. Using these examples, we fit the parameters of a deformation model that best approximates the original data yet remains fast to compute and compact in memory.</Abstract>
<H2>Keywords: </H2>Interactive, Skin, Approximation
<H1>1 Introduction</H1>
To be believable, animated characters must deform in plausible ways as they move. It is possible to accomplish this by having an artist sculpt an entire character mesh by hand for every frame of an animation sequence, but this is impractical. Instead, animators typi- cally manipulate an underlying hierarchical skeleton. The character mesh geometry must then be attached to the underlying skeleton so that as the skeleton deforms, the mesh also deforms appropriately. This attachment of model geometry to an underlying skeleton is called a “skin” and can be viewed as a function that maps from the skeletal parameters to a deformation field. There are two fundamental aspects of skin creation—authoring and computation. Skin authoring refers to how artists use tool sets to describe the behavior of skin geometry as the skeleton moves. Skin computation refers to the method by which the deformed mesh geometry is evaluated for display at some skeleton configuration. For high-end applications, the authoring methods drive skin creation while for interactive systems, computation methods dominate. For high-end applications such as film, the visual fidelity of characters is paramount, so artists require flexibility and control in skin authoring. Hence, there are many different ways to create characters using commercial tools. One technique involves modeling skin substructure such as muscles and tendons to drive the skin geometry
[Wilhelms and Gelder 1997; Scheepers et al. 1997]. Many deformers which drive skins by linking their control points to the skeletal parameters with custom expressions or scripts are also available.
Some examples include FFD lattices
[Sederberg and Parry 1986] or Wires
[Singh and Fiume 1998]. High-end characters often use a combination of these techniques—different tools are appropriate for different parts of the character. This generality and control means that the computation aspect of high-end characters is highly customizable, tightly coupled to authoring, and potentially unbounded. In fact, high-end tools allow authors to continually develop new skin computation models through custom scripts, expressions and complex deformers. In contrast, interactive systems require fast computation and small memory size for characters. Thus, the character computation model is fixed and artists must restrict their tool set to author characters in direct support of it. The most common skin computation model in games and interactive systems goes by many names including SSD, enveloping, smooth skinning, and linear blend skinning. This technique assigns a set of influencing joints and blending weights to each vertex in the character. The skin is computed by transforming each vertex by a weighted combination of the joints’ local coordinate frames. This technique is discussed in more detail in Section 3. While fast to evaluate and compact in memory, this method is notorious not only for its authoring difficulty, but also for its undesirable deformation artifacts. However, this method is widely used since these characters can be used with arbitrary amounts of animation data and can be posed at runtime. A different character computation mechanism previously used in interactive systems is called mesh animation. Mesh animation works by storing a large number of deformed models as static meshes—one for each frame of animation. These static models are then either displayed directly or are linearly interpolated at runtime. Mesh animation is interesting since it decouples skin authoring from runtime skin computation, allowing artists to use any tools they want to author characters. Unfortunately mesh animation is only appropriate when the required animation sequences are short and are known a priori. As games and interactive applications use larger amounts of animation, storing every frame becomes prohibitive. This technique is also incapable of generating new poses at runtime; for example, to place the character’s hand exactly on a door knob or to make footfalls land precisely on stairs. Due to these limitations, mesh animation is losing popularity. In this paper, we present an automated method to build character skins that are fast to compute and compactly represented from a set of examples. This technique allows artists to use any skin authoring tools they like while producing characters that meet the performance demands and work with the computation models used in interactive systems. We present a framework for extending linear blend skinning that allows us to capture these detailed skin deformations. We show how we can fit the parameters of our skinning model using a sampling of an arbitrarily rigged character’s deformations. The rest of this paper is organized as follows. After a review of related work, we describe the simple linear blend skinning model and its strengths and weaknesses. The limitations of this approach lead us to a discussion of our framework for extending this skinning model. Next we describe how we fit the parameters of our skinning model using a sampling of the original character deformations. Finally we present results and applications of our technique.
∗ {amohr, gleicher}@cs.wisc.edu , http://www.cs.wisc.edu/graphics
∗
1
To appear at SIGGRAPH 2003
<H2>1.1 System Overview</H2>
Building a skin with our system involves two major steps. We begin with a character rigged in an animation package such as Maya. We then sample this character’s skin deformations by exporting the character’s geometry in several poses. Next we fit the parameters of our underlying skinning model using this sampled data. We wish to obtain a good sampling of the character’s skin deformations to fit our underlying model with. To do this, we pose the character to exercise all the joints fully and include its extreme poses. This step does not require a trained animator since these poses are only intended to exercise the degrees of freedom of the character and need not correspond to a realistic motion. Once this is done, the poses are sampled regularly at k times. This sampling can be very simple to obtain from the user’s perspective—in our case, users must simply invoke a script we have implemented in Maya. Each sample consists of the skeleton configuration and the corresponding deformed skin geometry as a static mesh. We call a paired skeleton configuration and static mesh an example. Using this set of examples, our system first determines the set of joints that should influence each vertex, and then solves a bilinear least-squares problem to fit the parameters of the underlying skinning model. As mentioned earlier, the skinning model we use is an extension of the standard linear blend skinning model. Our extension adds extra joints to the character that are simply related to the existing joints. These new joints are designed in such a way to capture richer deformations than the standard linear blend skinning model. Our system is configured to add these extra joints automatically to characters, but we allow users to fine tune the specific set of extra joints if they wish.
<H1>2 Related Work</H1>
Character skin deformations are fundamental to character animation and have been addressed for some time in the literature. Catmull [1972]
introduced one of the first skeleton-driven techniques—rigid skinning to a hierarchically structured articulated figure. A 2D skeletal bilinear deformation method was presented by
Burtnyk and Wein [1976]. An early 3D skeleton-driven technique that went beyond rigid skinning was presented by MagnenatThalmann, et al. [1988]. Their technique used custom programmed algorithms to deform character meshes based on the nature of particular joints. More recently, novel skinning methods that start with a simple skin and use sparse data interpolation to correct errors between it and a set of examples have been introduced. Three examples, Pose Space Deformation, Shape by Example, and EigenSkin
[Lewis et al. 2000; Sloan et al. 2001; Kry et al. 2002] use radial basis interpolation of corrections to linear blend skins. Another recent work applies these techniques to range scan data
[Allen et al. 2002]. These techniques are similar to ours in that they take examples as input. The results of these approaches are quite good, and unlike our technique, they can handle skin deformations that depend on abstract parameters rather than only skeleton configurations. However, these methods are not appropriate for interactive characters since they require storing potentially large amounts of example data for runtime interpolation. In contrast, our method discards all example data after the fitting process so the size of our runtime structures does not scale with the number of inputs. Other authors have used physical simulation for interactive deformations, especially secondary animation
[James and Pai 2002; Capell et al. 2002]. Our method cannot capture these secondary deformations directly; however, a technique such as DyRT [James and Pai 2002]
can be applied to the characters we generate to add secondary animation. There has been some recent work on fitting skinning models. One method solves for joint centers and vertex weights for a scanned arm
[Nebel and Sibiryakov 2002]
but the Multi-Weight Enveloping technique [Wang and Phillips 2002]
, or MWE, is most similar to our approach.
MWE extends linear blend skinning by giving each vertex one weight to each coefficient of each influencing joint’s transformation matrix instead of one weight per influencing joint. They then find these weights by solving a linear leastsquares problem using a set of examples as input. While on the surface Multi-Weight Enveloping and our technique seem very similar, they are in fact different in a fundamental way. Both MWE and our technique use an extension of linear blend skinning as an underlying deformation model. However, MWE extends linear blend skinning by adding more vertex weights to the model while in contrast, our method adds more joints. MWE uses a large number of weights per vertex (12 per influencing joint). This introduces the possibility of rank deficient matrices in the least-squares solutions
[Wang and Phillips 2002], especially since the matrix coefficients are usually highly correlated. This can lead to overfitting, which MWE must take measures to avoid. In contrast, since the number of weights per vertex in one of our skins remains relatively small (1 per influencing joint) and our extra joints are explicitly designed to be very different from existing joints, our technique requires no special provisions to avoid overfitting. Even so, our method can detect and handle small amounts of overfitting if it occurs as explained in Section 5.2. Another consequence of having one weight per entry in the joint transformation matrices is that MWE skins are not as easily accelerated by graphics hardware as skins created using our method. Finally, since our skins are computed in the same manner as linear blend skins, existing software infrastructure can make use of them with little or no changes.
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<H1>3 Linear Blend Skinning</H1>
The traditional interactive skinning model goes by many names. Lewis et. al call it Skeleton Subspace Deformation or SSD, Maya calls it “smooth skinning” and we call it linear blend skinning. This technique is widely used for interactive applications. An excellent description of this method is found in
Lewis et al. [2000]. The linear blend skinning algorithm works by first placing a hierarchical skeleton inside a static model of a character, typically in some neutral pose. This initial character pose is referred to as “dress pose”. Then, each vertex is assigned a set of influencing joints and a blending weight for each influence. Computing the deformation in some pose involves rigidly transforming each dress pose vertex by all of its influencing joints. Then the blending weights are used to combine these rigidly transformed positions. The deformed vertex position at some skeletal configuration c, v c is computed as
1
n v c = ∑ w i M i,c M i,d −1 v d i=1
where w i are the weights (usually affine or convex), v d is the dresspose location of some vertex v, M i,c is the transformation matrix associated with the ith joint in configuration c and M i,d −1 is the inverse of the dress-pose matrix associated with the ith influence. (Taken together, M i,d −1 v d represents the location of v d in the local coordinate frame of the ith influence.) Note that a deformed vertex position in the dress pose configuration c = d is the same as the provided dress pose vertex ( v d = v d ) if the weights are affine. This skinning algorithm is notorious for its failings. It cannot represent complex deformations and suffers from characteristic artifacts such as the “candy-wrapper” collapse effect on wrists and collapsing around bending joints as shown in Figure 2 . The artifacts occur because vertices are transformed by linearly interpolated matrices. If the interpolated matrices are dissimilar as in a rotation of nearly 180 degrees, the interpolated transformation is degenerate, so the geometry must collapse. In addition to these deformation problems, linear blend skins are very difficult to author
[Lewis et al. 2000]. Despite its failings, this skinning algorithm is very fast and widely supported by commercial applications so it remains popular especially in games and virtual environments.
<H1>4 Extending Linear Blend Skinning</H1>
The linear blend skinning model is not sufficient to capture deformations well as shown in Figure 3 . The problem in this particular case is that as the twist approaches 180 degrees, the linearly blended matrix becomes degenerate and collapses the skin geometry. Linearly blended transformations tend to collapse the more different they are. The resulting loss of volume can also be observed around hinge joints such as the knee and elbow as shown in Figure 2 . We observe that we can help avoid the collapse problem by avoiding blending transformations that are so dissimilar. We can accomplish this by adding extra transformations that properly interpolates without collapsing. In the case of the twisting wrists, we can add an extra joint that interpolates the rotation angle correctly and does not collapse. In fact, artists sometimes do this by hand to help avoid wrist collapses. More generally, we observe that any deformation effect could be obtained by adding joints that deform appropriately to capture that deformation effect. For example, to capture muscle bulges, we can add joints that scale up when the muscle should bulge, and scale down when the muscle relaxes. For wrinkles, we could add several joints that move and scale in concert to capture the wrinkles. In
Figure 2: Common problems with linear blend skinning: the bent arm on the left demonstrates shrinkage around bent joints such as the elbow and knee while the twisted wrist on the right demonstrates the “candy-wrapper” collapse effect. These artifacts are caused by blending dissimilar transformations.
Figure 3: Top Row: Three examples of a twisting box driven by a nonlinear de-
former. Middle Row: Solved linear blend skin using one joint. Bottom Row: Our result with just one additional joint that half interpolates the twist rotation.
fact, in the limit we could add as many transformations as vertices and capture all deformations exactly. Unfortunately, adding so many extra joints is impractical. First, adding such a large number of joints would severely impact the performance of our resulting skins. Worse, even if we could find these transformations for the input examples, it is unclear how to determine the general relationships of these transformations to the skeletal parameters in all poses. Without knowledge of this relationship, our scheme would only be able to reproduce the input frames and would not work well in new poses. Instead, we extend the traditional linear blend skinning model by adding a relatively small number of joints that are simply related to the original skeletal parameters and fit using them. We choose these extra joints by both examining the places where the standard linear blend model fails and by examining extra character deformations that we would like to capture. We then add joints that we believe will help resolve these artifacts. Finally, we fit the parameters of our skinning model using this extended skeleton. The key to our success is that since vertices choose weighted sums of transformations, if any linear scaling of an added joint is beneficial it may be used. Thus the additional joints need not be exact. We emphasize that this is a framework for obtaining better deformations and the joints we choose to add are based on our observations of characters. Different characters with different deformations may require a different set of additional joints. However, once some set of these joints is determined, the skin may be solved using our fitting algorithm without change.
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To appear at SIGGRAPH 2003
blend skin approximation. Bottom Row: Our result using one additional joint.
Figure 4: Top Row: Original examples of a twisting wrist. Middle Row: Linear
Figure 5: Linear blend skinning alone is incapable of capturing correct creasing
around elbows. At the left is an example of a bent elbow. In the middle is the linear blend skin approximation. notice the interpenetration. In contrast, our method avoids the interpenetration.
<H2>4.1 Additional Joints</H2>
To help solve the collapsing geometry problem, our system can automatically add joints that properly interpolate rotations without collapsing. This is done by examining the rotation of a joint relative to the dress pose and computing the new joint as the halfway spherical linear interpolation [Shoemake 1985] of this rotation, located at the same position in space. More joints with evenly distributed interpolation parameters could be added to sample this rotation space even better; however, in our experience just a single interpolated rotation is sufficient. Figure 4 demonstrates the improvements gained by simply adding a single interpolated rotation joint in the twisting case. Figure 5 shows the improvements for the bent elbow case. Another type of effect not easily captured by the simple linear blend model is bulging and denting of skins caused by muscles, tendons, or other substructure. These particular effects cannot be captured since the joints employed in animating a character do not typically scale up and down as would be necessary to approximate these effects. We have observed that for many characters, the substructure deformation effects from muscles and tendons are often simply related to the angles between joints. For example, a bicep bulge is small when the elbow is near full extension while the bugle is large when the elbow is near full flexion. The effect is similar for other muscles in the body. To capture these effects, our system can add several joints that scale up and down based on the angle between particular joints. We add these scaling joints as follows. First we choose a joint in the original skeleton that will drive the scaling parameters of the new joints. Once this driver is chosen, there are two sets of joints that we add. The first set is “upstream” of the driver and lies in the middle of the bone connecting the driver to its parent, the second set is “downstream” and lies in the middle of the bones connecting the driver to its children. All upstream joints are oriented in the same way, with one axis aligned with the bone as shown in Figure 6 . We use four upstream joints. Two of them scale up about two axes orthogonal to the bone and a corresponding pair scale down about the two axes orthogonal to the bone. The scale parameters of these joints are set based on the angle of the bone connecting the driver to its parent and the bone connecting the driver to its child. If the driver has multiple children, a vector that is the sum of the bones connecting the driver to its children is used to measure the angle. Downstream joints are similar. We use four downstream joints on each bone connecting the driver to its children that scale just as the upstream joints do. The scale parameters are computed as follows. For joints that scale up, the scale parameter s is s = 1 + k b 1 · b 2 + 1 2 b 1 b 2 where b 1 and b 2 are the bone vectors used to measure the angle at the driver joint and k is the maximum scale factor when the angle between b 1 and b 2 is zero. For joints that scale down, the scale parameter is simply s −1 . The value for k may be chosen by the user but in our experience, we have found that 8 works well for our examples. Again, since vertices may take any scaling of these new joints, a conservative large value is fine. For example, if a vertex in fact needed a joint that scaled by 2 instead of 8, it could be assigned a weight of 4 1 .
J1 v1 θ v2 J3 J4-J7 J2 Driver
Figure 6: Our method adds extra joints to characters to help better approximate deformations. Here J4 through J7 are automatically added upstream joints that scale depending on the angle θ. As θ decreases, J4 scales up in the direction v1 which is orthogonal to the bone connecting J1 and J2. Meanwhile, J5 scales up in the direction v2, orthogonal to both the bone and v1. J6 and J7 operate similarly, but scale down as θ increases rather than up. Downstream joints are very similar except that these joints are positioned on the bone from J2 to J3.
<H1>5 Fitting the Skinning Model</H1>
Once our system has augmented the input skeleton, we use a fitting procedure to set the parameters of the underlying skinning model to match the example data well. As mentioned earlier, the input to the fitting process is a set of examples. An example is simply a static character mesh paired with a skeleton. This static mesh is deformed according to the skeleton configuration, but it is not attached to the skeleton in any way. For our results, our examples were generated by exporting rigged objects from Maya, but they could have been sculpted by hand or come from another program. A linear blend skin computes a deformed vertex as described earlier in Equation 1. Examining this skinning model, only the M i are predetermined. These are the coordinate frames associated with all the joints in the character. That means for each vertex, we are able to choose the set of influencing joints, influence weights (w i ) and the dress pose vertex position (v d ). We would like to choose the influence sets, weights and dress pose vertex positions that best approximate the examples and generalize well to new poses.
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To appear at SIGGRAPH 2003
Examples Compactness of Local Coordinate Point Clouds Upper Arm Lower Arm
Figure 7: Top: A set of three examples of a deforming arm mesh with a bulging
bicep. A particular point on the arm mesh is highlighted in each example. Bottom Left: Each example rotated and aligned so that the upper arm bones coincide. The highlighted points form a cloud in the local coordinate frame of the upper arm. Even though the bicep bulges significantly, this cloud is compact. Lower Right: A similar point cloud but relative to the forearm. This cloud is far less compact than the former, making the forearm a poorer choice for an influence.
<H2>5.1 Finding Influence Sets</H2>
We determine influence sets first for several reasons. Ideally, the influence sets would fall out naturally from the weight solving procedure (irrelevant joints would have a weight of zero) but this does not happen in practice because our samplings are necessarily not exhaustive. Also, the more joints that a vertex depends on, the slower the skin can be to compute and current hardware only supports a limited number of influences per vertex. Thus, we would like to select a small set of good influences. Also, choosing the influence sets appropriately lets us bound the size of the problems we must solve to determine the weights as discussed in Section 5.2. This makes the solving process faster. In most recent research, influence set determination has been left to users
[Lewis et al. 2000; Wang and Phillips 2002; Sloan et al. 2001]. The task is typically accomplished by “painting” the regions of influence for each joint over the mesh. While less difficult than painting the weights themselves [Lewis et al. 2000], it is a tedious process. In contrast, our system automatically determines the influence sets for each vertex using a heuristic algorithm. We observe that vertices in a character skin typically transform nearly rigidly with respect to some joint. For instance, vertices on the forearm roughly follow the forearm. We believe that for most characters, their skin is most heavily influenced by those joints that they are bound to. Even though a point on the bicep is not truly rigid as an arm moves (due to muscle bulge), we believe that these points remain mostly rigidly attached to the upper arm, and therefore should be influenced by it. Using this observation, we measure how rigidly a vertex transforms with every joint over all examples and use the most rigidly transforming joints for the influence set. For a single vertex, a rigidity score for a joint is computed as follows. For each example, the local coordinate position of the vertex is computed as M i,e −1 v e where M i,e is the coordinate frame associated with the ith joint in the eth example and v e is the global coordinate position of the vertex on the eth example. The collection of these local coordinate positions over all examples forms a point cloud as shown in Figure 7 . The more compact this point cloud, the more rigid we believe the vertex-joint relationship to be. We measure the compactness of this point cloud by taking its diameter (the maximum distance between any two points in the cloud). We have found that the simple O(n 2 ) algorithm that compares each point to every other to be fast enough for our purposes but this diameter may be computed more quickly. An O(n log n) time algorithm is possible. See [Malandain and Boissonnat 2002]
for faster methods. Once the compactness measures for all joints are computed for a vertex, the smallest k are chosen as the influence set for that vertex. It may be tempting to use a threshold scheme to choose influence sets but we have found this problematic. It is unclear how to pick a good threshold because as the rigidity scores get larger, they become less meaningful. For instance, it may happen as an artifact of the particular input examples that points on the left shoulder move much more rigidly relative to the right leg rather than the left leg but both choices make no sense for influences. Since larger rigidity scores are not particularly meaningful, it is nearly impossible to pick a meaningful threshold value. As in other linear blend skinning systems, influence sets need only be determined conservatively [Wang and Phillips 2002] so we allow users to choose k if desired. In our experience, we have found that between three and eight influences works well, depending on the complexity of the character.
<H2>5.2 Solving for Weights and Vertices</H2>
Once the influence sets have been determined, only the weights and dress pose vertex positions remain (w i and v d ). We would like to find the best vertices and weights that minimize the least-squares difference between the skin and the examples at all the example skeleton configurations. That is
n 2 min ∑ v e i − v e i i=1
for all examples where v e i is the input vertex position from the ith example and v e i is the deformed vertex computed by the skinning model at the ith example configuration. n v e = ∑ w i M i,e M i,d −1 v d i=1 This problem is bilinear in the weights and vertices. We use an alternation technique to solve the optimization. This works by first fixing the first variable and solving a linear least-squares problem to find the second, then fixing the second and solving a linear leastsquares problem for the first. This process is then repeated until it converges. This technique is commonly used and is described in
[Freeman and Tenenbaum 1997]. We start by solving for weights since we have no good guess for them but we know that the initial dress pose vertices are ideal. Next we hold the weights fixed and solve for vertex positions. This process typically converges after one or two iterations. As mentioned in Section 2, we have found that since we are solving for a small numbers of weights using large numbers of examples, our systems are often well conditioned and do not suffer from overfitting if the input data is well sampled. Thus we do not have to take special precautions to avoid overfitting as in
[Wang and Phillips 2002], although we include tests for robustness. For clarity, we present the matrices we solve via least-squares in block form. First we introduce some notation: T i,e = M i,e M i,d −1 . In order to ensure that the resulting weights are affine, we set w 1 = 1 − ∑ i=2 n w i , and solve for w 2 through w n .
(T (T 2,e 2,e 1 k − − . . . T T 1,e 1,e k 1 )v )v d d ··· ··· .. . (T (T n,e n,e 1 k − − . . . T T 1,e 1,e k 1 )v )v d d w w w . . . 2 n 3 = v v e e 1 k − − T T . . . 1,e 1,e 1 k v v d d
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To appear at SIGGRAPH 2003
The matrix used to solve for vertex positions is as follows. ∑ ∑ i=1 n i=1 n w w . . . i i T T i,e i,e 1 k v d = v v . . . e e 1 k To handle homogeneous coordinates, the translation parts of the ∑ i=1 n w i T i,e k matrices are subtracted from the v e on the right hand side. We solve these least-squares problems using the singular value decomposition. This lets us detect when our matrices are rank deficient, leading to overfitting. We detect this by comparing the ratio of the largest singular value to the smallest, and issuing a warning if there are any singular values below some fraction of this ratio. To recover, we zero these singular values and continue with the fitting process. If overfitting is a problem, provisions such as those taken in [Wang and Phillips 2002] could also be used. However, in all the examples in this paper, no singular values were zeroed.
<H2>5.3 Handling Normals</H2>
It is not only important for the geometry in a skin approximation to be accurate, but also important for normals to be well approximated. If they are not, lighting calculations will not produce good results. We assume that normals are specified per vertex. It may seem that just transforming a dress pose normal by the inverse transpose of the corresponding vertex’s transformation matrix would be correct. To be more explicit, n ̄ c = ( ∑ w i M i,c M i,d −1 ) −T n d ( ∑ w i M i,c M i,d −1 ) −T n d While this is technically valid for local neighborhoods of smooth surfaces
[Turkowski 1990], we do not have a smooth surface. Instead we have single points that are computed independently. Computing the normals in this manner can give undesirable results when the blended transformations are not pure rotations. Interactive systems typically approximate normal calculations as
2
n ̄ c = ∑ w i M i,c −T M i,d T n d ∑ w i M i,c −T M i,d T n d
since it is often faster to calculate the joint matrices and their inverses incrementally by composition because at each step, rotations and scales have special inverse forms. This alleviates the need for a general inversion operation. In EigenSkin [Kry et al. 2002]
, normals are treated as second skinning problem and are computed independently. In our system, we take the model used in existing systems as in Equation 2 and include normals in our optimization process. To do this, we simply add more terms to the objective function to include the differences between normal vectors. We allow users to scale normals if they wish to change their relative influence on the least-squares solution.
<H1>6 Results</H1>
The simple linear blend skinning model commonly used in video games and other interactive applications is very fast and compact but cannot capture the high quality deformations that make convincing characters. Our framework for extending the linear blend model allows us to capture much more interesting deformations while retaining its efficiency. The most egregious deformation problems of linear blend skinning are solved by our approach. Figures 3 and 4 show how our system can fix collapsing twists by adding just a single extra joint. Collapsing and interpenetrations around hinge joints are also fixed using our method as shown in Figure 5 . In addition to solving these problems with linear blend skinning, our extension framework can capture other more subtle and detailed deformations required for convincing characters. Figure 8 demonstrates how our method can capture the bulges in the biceps and triceps of a character’s arm. While the particular extra joints we have chosen to add to our characters may not be capable of capturing the full deformation for any character, different extra joints that do capture the desired deformations may be added and solved using our technique. To demonstrate that our technique can be used on more than just simple arms and legs, Figure 9 shows a rigged upper body and its approximation by our system. This figure also shows this character in new poses from an animation sequence, demonstrating that our resulting skins generalize well to new poses. Our solution procedure is generally very fast. None of the examples shown here took more than five minutes to solve on a modern personal computer. The slowest was the upper body model which has more than 6000 vertices, 50 examples, and 5 influences per vertex. The computation time for each vertex depends on the number of influences and the number of examples. Also, since each vertex is solved independently, our algorithm is trivial to parallelize.
Figure 8: Top: Examples of a muscular arm flexing. Middle: Linear blend skin approximation. Note the lack of bicep bulge. Bottom: Results using our method.
<H2>6.1 Applications</H2>
The ability to generate compactly represented, fast to evaluate, high quality skin approximations from a set of examples is very useful. Applications range from building characters for video games and virtual environments to high-end animation previewing. Many current interactive systems such as video games only support linear blend skinned characters. Aside from the deformation problems associated with using this model, authoring these skins is notoriously difficult. Determining the blending weights and influence sets is left to the skin author to set directly. None of the more intuitive or useful deformer primitives provided by animation systems may be used. Using our method, character authors may use any tools they like to author characters. All our system requires is a set of examples which is used to compute the appropriate influence sets and blending weights automatically. This frees the author from setting them manually. It is important to note that since our characters are a straightforward extension to linear blend skinning, many existing interactive systems already have the software infrastructure to sup- port them. In addition, since our skins are computed in the same manner as existing linear blend skins, they are already accelerated by current graphics hardware. Another application of our system is to map a character originally attached to one skeleton onto a different underlying skeleton. We call this process skin retargeting. Skin retargeting is useful if a particular interactive system requires characters to have a specific skeleton. For instance, a video game may have an optimized engine for characters with a particular skeleton topology. Ordinarily, if a character was created for a different skeleton, the character would have to be re-rigged manually to work on the new skeleton topology. However, this can be accomplished much more easily with our system. One just exports a set of example meshes deformed by the original skeleton but paired with corresponding poses of the new skeleton. Our system sees this as any other set of data and solves for the proper influence sets and blending weights. Another application of our technique is targeted at high-end animation. High-end characters often have such complex deformations that they cannot be computed interactively. Thus, animators typically work with low fidelity versions that only roughly suggest the actual shape of the character. Using our method, interactive characters could be built that allow animators to interact with much better approximations of the deformed characters.
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To appear at SIGGRAPH 2003
Figure 9: Top Row: Examples of an upper body rigged in Maya. Middle Row: Best linear blend skin. Note the circled problem areas. Bottom Row: Our results.
<H2>6.2 Discussion</H2>
In this paper, we have presented a method for building fast to evaluate, compact representations that produce accurate approximations of deforming characters. The characters may be rigged using any available tool since our system only requires static deformed meshes paired with skeletal configurations as input. While our technique works well for a wide variety of character skins, it has limitations. For instance, character deformations in our model are only driven by the skeleton’s joint parameters. Our method cannot capture deformations that are driven by abstract parameters such as “happiness” as in
[Lewis et al. 2000; Sloan et al. 2001]. Our system also cannot accurately reproduce deformations that are not representable as linear combinations of the transformations expressed in our skeletons. For instance, the scaling joints presented in this paper can only fully capture deformations that are well approximated by a scaling that is linearly related to the cosine of the angle between two bones. This assumption may be violated by a character whose muscle bulges only when its arm is fully bent. The scaling joints also assume that only the angle between joints is important, so bending the shoulder forward is treated the same as bending it up. Even though not all deformations can be captured using the extra joints presented here, new joints may be added to capture any important deformation, and our influence set and vertex weight solving framework may be applied without change. Despite these limitations, our method produces high-quality yet fast and compact skinned characters that work with existing game engines, graphics hardware and other runtime systems.
<H1>Acknowledgments</H1>
We would like to thank Luke Tokheim for help with rigging and J. P. Lewis and Karan Singh for their generosity with characters. We would also like to thank the UW Graphics Group, especially Andrew Selle and Hyun Joon Shin for their help with video production. This research is supported by NSF grants CCR-9984506 and CCR-0204372, and equipment donations from Intel.
<H1>References</H1>
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C APELL , S., G REEN , S., C URLESS , B., D UCHAMP , T., AND P OPOVI Ć , Z. 2002. Interactive skeleton-driven dynamic deformations. ACM Transactions on Graphics 21, 3, 586–593.
C ATMULL , E. E. 1972. A system for computer generated movies. In Proc. ACM Annual Conf. August, 422–431.
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K RY , P. G., J AMES , D. L., AND P AI , D. K. 2002. Eigenskin: Real time large deformation character skinning in hardware. In ACM SIGGRAPH Symposium on Computer Animation, 153–160.
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7
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3032,
3053
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],
"text": "['high-end applications']",
"type": "data",
"id": "T9"
},
{
"offsets": [
[
3107,
3126
]
],
"text": "['interactive systems']",
"type": "data",
"id": "T10"
},
{
"offsets": [
[
3128,
3156
]
],
"text": "['computation methods dominate']",
"type": "background_claim",
"id": "T11"
},
{
"offsets": [
[
3162,
3196
]
],
"text": "['high-end applications such as film']",
"type": "data",
"id": "T12"
},
{
"offsets": [
[
3198,
3244
]
],
"text": "['the visual fidelity of characters is paramount']",
"type": "background_claim",
"id": "T13"
},
{
"offsets": [
[
3249,
3306
]
],
"text": "['artists require flexibility and control in skin authoring']",
"type": "background_claim",
"id": "T14"
},
{
"offsets": [
[
3315,
3388
]
],
"text": "['there are many different ways to create characters using commercial tools']",
"type": "background_claim",
"id": "T15"
},
{
"offsets": [
[
3390,
3439
]
],
"text": "['One technique involves modeling skin substructure']",
"type": "background_claim",
"id": "T16"
},
{
"offsets": [
[
3448,
3455
]
],
"text": "['muscles']",
"type": "data",
"id": "T17"
},
{
"offsets": [
[
3460,
3467
]
],
"text": "['tendons']",
"type": "data",
"id": "T18"
},
{
"offsets": [
[
3468,
3494
]
],
"text": "['to drive the skin geometry']",
"type": "background_claim",
"id": "T19"
},
{
"offsets": [
[
3497,
3521
]
],
"text": "['Wilhelms and Gelder 1997']",
"type": "data",
"id": "T20"
},
{
"offsets": [
[
3523,
3544
]
],
"text": "['Scheepers et al. 1997']",
"type": "data",
"id": "T21"
},
{
"offsets": [
[
3547,
3692
]
],
"text": "['Many deformers which drive skins by linking their control points to the skeletal parameters with custom expressions or scripts are also available']",
"type": "background_claim",
"id": "T22"
},
{
"offsets": [
[
3695,
3729
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],
"text": "['Some examples include FFD lattices']",
"type": "background_claim",
"id": "T23"
},
{
"offsets": [
[
3732,
3756
]
],
"text": "['Sederberg and Parry 1986']",
"type": "data",
"id": "T24"
},
{
"offsets": [
[
3761,
3766
]
],
"text": "['Wires']",
"type": "background_claim",
"id": "T25"
},
{
"offsets": [
[
3769,
3789
]
],
"text": "['Singh and Fiume 1998']",
"type": "data",
"id": "T26"
},
{
"offsets": [
[
3792,
3855
]
],
"text": "['High-end characters often use a combination of these techniques']",
"type": "background_claim",
"id": "T27"
},
{
"offsets": [
[
3856,
3924
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],
"text": "['different tools are appropriate for different parts of the character']",
"type": "background_claim",
"id": "T28"
},
{
"offsets": [
[
4101,
4238
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],
"text": "['high-end tools allow authors to continually develop new skin computation models through custom scripts, expressions and complex deformers']",
"type": "background_claim",
"id": "T29"
},
{
"offsets": [
[
4253,
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]
],
"text": "['interactive systems require fast computation and small memory size for characters']",
"type": "background_claim",
"id": "T30"
},
{
"offsets": [
[
3926,
4033
]
],
"text": "['This generality and control means that the computation aspect of high-end characters is highly customizable']",
"type": "background_claim",
"id": "T31"
},
{
"offsets": [
[
4035,
4063
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],
"text": "['tightly coupled to authoring']",
"type": "background_claim",
"id": "T32"
},
{
"offsets": [
[
4069,
4090
]
],
"text": "['potentially unbounded']",
"type": "background_claim",
"id": "T33"
},
{
"offsets": [
[
4342,
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],
"text": "['the character computation model is fixed']",
"type": "background_claim",
"id": "T34"
},
{
"offsets": [
[
4387,
4468
]
],
"text": "['artists must restrict their tool set to author characters in direct support of it']",
"type": "background_claim",
"id": "T35"
},
{
"offsets": [
[
4470,
4560
]
],
"text": "['The most common skin computation model in games and interactive systems goes by many names']",
"type": "background_claim",
"id": "T36"
},
{
"offsets": [
[
4571,
4574
]
],
"text": "['SSD']",
"type": "data",
"id": "T37"
},
{
"offsets": [
[
4576,
4586
]
],
"text": "['enveloping']",
"type": "data",
"id": "T38"
},
{
"offsets": [
[
4588,
4603
]
],
"text": "['smooth skinning']",
"type": "data",
"id": "T39"
},
{
"offsets": [
[
4609,
4630
]
],
"text": "['linear blend skinning']",
"type": "data",
"id": "T40"
},
{
"offsets": [
[
4632,
4735
]
],
"text": "['This technique assigns a set of influencing joints and blending weights to each vertex in the character']",
"type": "background_claim",
"id": "T41"
},
{
"offsets": [
[
4737,
4850
]
],
"text": "['The skin is computed by transforming each vertex by a weighted combination of the joints’ local coordinate frames']",
"type": "background_claim",
"id": "T42"
},
{
"offsets": [
[
4955,
5017
]
],
"text": "['this method is notorious not only for its authoring difficulty']",
"type": "background_claim",
"id": "T43"
},
{
"offsets": [
[
4915,
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],
"text": "['fast to evaluate and compact in memory']",
"type": "background_claim",
"id": "T44"
},
{
"offsets": [
[
5023,
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]
],
"text": "['also for its undesirable deformation artifacts']",
"type": "background_claim",
"id": "T45"
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{
"offsets": [
[
5080,
5106
]
],
"text": "['this method is widely used']",
"type": "background_claim",
"id": "T46"
},
{
"offsets": [
[
5113,
5182
]
],
"text": "['these characters can be used with arbitrary amounts of animation data']",
"type": "background_claim",
"id": "T47"
},
{
"offsets": [
[
5187,
5210
]
],
"text": "['can be posed at runtime']",
"type": "background_claim",
"id": "T48"
},
{
"offsets": [
[
5321,
5403
]
],
"text": "['Mesh animation works by storing a large number of deformed models as static meshes']",
"type": "background_claim",
"id": "T49"
},
{
"offsets": [
[
5212,
5319
]
],
"text": "['A different character computation mechanism previously used in interactive systems is called mesh animation']",
"type": "background_claim",
"id": "T50"
},
{
"offsets": [
[
5437,
5531
]
],
"text": "['These static models are then either displayed directly or are linearly interpolated at runtime']",
"type": "background_claim",
"id": "T51"
},
{
"offsets": [
[
5533,
5562
]
],
"text": "['Mesh animation is interesting']",
"type": "background_claim",
"id": "T52"
},
{
"offsets": [
[
5569,
5626
]
],
"text": "['it decouples skin authoring from runtime skin computation']",
"type": "background_claim",
"id": "T53"
},
{
"offsets": [
[
5628,
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],
"text": "['allowing artists to use any tools they want to author characters']",
"type": "background_claim",
"id": "T54"
},
{
"offsets": [
[
5748,
5790
]
],
"text": "['the required animation sequences are short']",
"type": "data",
"id": "T55"
},
{
"offsets": [
[
5795,
5813
]
],
"text": "['are known a priori']",
"type": "data",
"id": "T56"
},
{
"offsets": [
[
5708,
5742
]
],
"text": "['mesh animation is only appropriate']",
"type": "background_claim",
"id": "T57"
},
{
"offsets": [
[
5818,
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]
],
"text": "['games and interactive applications use larger amounts of animation']",
"type": "background_claim",
"id": "T58"
},
{
"offsets": [
[
5886,
5925
]
],
"text": "['storing every frame becomes prohibitive']",
"type": "background_claim",
"id": "T59"
},
{
"offsets": [
[
5927,
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]
],
"text": "['This technique is also incapable of generating new poses at runtime']",
"type": "background_claim",
"id": "T60"
},
{
"offsets": [
[
6009,
6061
]
],
"text": "['to place the character’s hand exactly on a door knob']",
"type": "data",
"id": "T61"
},
{
"offsets": [
[
6065,
6107
]
],
"text": "['to make footfalls land precisely on stairs']",
"type": "data",
"id": "T62"
},
{
"offsets": [
[
6135,
6170
]
],
"text": "['mesh animation is losing popularity']",
"type": "background_claim",
"id": "T63"
},
{
"offsets": [
[
6187,
6316
]
],
"text": "['we present an automated method to build character skins that are fast to compute and compactly represented from a set of examples']",
"type": "own_claim",
"id": "T64"
},
{
"offsets": [
[
6318,
6515
]
],
"text": "['This technique allows artists to use any skin authoring tools they like while producing characters that meet the performance demands and work with the computation models used in interactive systems']",
"type": "own_claim",
"id": "T65"
},
{
"offsets": [
[
6517,
6634
]
],
"text": "['We present a framework for extending linear blend skinning that allows us to capture these detailed skin deformations']",
"type": "own_claim",
"id": "T66"
},
{
"offsets": [
[
7918,
7963
]
],
"text": "['This step does not require a trained animator']",
"type": "own_claim",
"id": "T67"
},
{
"offsets": [
[
7970,
8097
]
],
"text": "['these poses are only intended to exercise the degrees of freedom of the character and need not correspond to a realistic motion']",
"type": "own_claim",
"id": "T68"
},
{
"offsets": [
[
8162,
8232
]
],
"text": "['This sampling can be very simple to obtain from the user’s perspective']",
"type": "own_claim",
"id": "T69"
},
{
"offsets": [
[
8246,
8307
]
],
"text": "['users must simply invoke a script we have implemented in Maya']",
"type": "own_claim",
"id": "T70"
},
{
"offsets": [
[
8233,
8244
]
],
"text": "['in our case']",
"type": "data",
"id": "T71"
},
{
"offsets": [
[
8911,
9031
]
],
"text": "['These new joints are designed in such a way to capture richer deformations than the standard linear blend skinning model']",
"type": "own_claim",
"id": "T72"
},
{
"offsets": [
[
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9111
]
],
"text": "['Our system is configured to add these extra joints automatically to characters']",
"type": "own_claim",
"id": "T73"
},
{
"offsets": [
[
9117,
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],
"text": "['we allow users to fine tune the specific set of extra joints']",
"type": "own_claim",
"id": "T74"
},
{
"offsets": [
[
9246,
9368
]
],
"text": "['Character skin deformations are fundamental to character animation and have been addressed for some time in the literature']",
"type": "background_claim",
"id": "T75"
},
{
"offsets": [
[
9370,
9384
]
],
"text": "['Catmull [1972]']",
"type": "data",
"id": "T76"
},
{
"offsets": [
[
9386,
9440
]
],
"text": "['introduced one of the first skeleton-driven techniques']",
"type": "background_claim",
"id": "T77"
},
{
"offsets": [
[
9507,
9562
]
],
"text": "['A 2D skeletal bilinear deformation method was presented']",
"type": "background_claim",
"id": "T78"
},
{
"offsets": [
[
9567,
9590
]
],
"text": "['Burtnyk and Wein [1976]']",
"type": "data",
"id": "T79"
},
{
"offsets": [
[
9592,
9675
]
],
"text": "['An early 3D skeleton-driven technique that went beyond rigid skinning was presented']",
"type": "background_claim",
"id": "T80"
},
{
"offsets": [
[
9679,
9710
]
],
"text": "['MagnenatThalmann, et al. [1988]']",
"type": "data",
"id": "T81"
},
{
"offsets": [
[
9712,
9829
]
],
"text": "['Their technique used custom programmed algorithms to deform character meshes based on the nature of particular joints']",
"type": "background_claim",
"id": "T82"
},
{
"offsets": [
[
9846,
10004
]
],
"text": "['novel skinning methods that start with a simple skin and use sparse data interpolation to correct errors between it and a set of examples have been introduced']",
"type": "background_claim",
"id": "T83"
},
{
"offsets": [
[
10135,
10202
]
],
"text": "['use radial basis interpolation of corrections to linear blend skins']",
"type": "background_claim",
"id": "T84"
},
{
"offsets": [
[
10080,
10097
]
],
"text": "['Lewis et al. 2000']",
"type": "data",
"id": "T86"
},
{
"offsets": [
[
10099,
10116
]
],
"text": "['Sloan et al. 2001']",
"type": "data",
"id": "T87"
},
{
"offsets": [
[
10118,
10133
]
],
"text": "['Kry et al. 2002']",
"type": "data",
"id": "T88"
},
{
"offsets": [
[
10022,
10077
]
],
"text": "['Pose Space Deformation, Shape by Example, and EigenSkin']",
"type": "background_claim",
"id": "T85"
},
{
"offsets": [
[
10204,
10267
]
],
"text": "['Another recent work applies these techniques to range scan data']",
"type": "background_claim",
"id": "T89"
},
{
"offsets": [
[
10270,
10287
]
],
"text": "['Allen et al. 2002']",
"type": "data",
"id": "T90"
},
{
"offsets": [
[
10290,
10326
]
],
"text": "['These techniques are similar to ours']",
"type": "own_claim",
"id": "T91"
},
{
"offsets": [
[
10335,
10362
]
],
"text": "['they take examples as input']",
"type": "background_claim",
"id": "T92"
},
{
"offsets": [
[
10364,
10410
]
],
"text": "['The results of these approaches are quite good']",
"type": "background_claim",
"id": "T93"
},
{
"offsets": [
[
10438,
10547
]
],
"text": "['they can handle skin deformations that depend on abstract parameters rather than only skeleton configurations']",
"type": "background_claim",
"id": "T94"
},
{
"offsets": [
[
10558,
10618
]
],
"text": "['these methods are not appropriate for interactive characters']",
"type": "background_claim",
"id": "T95"
},
{
"offsets": [
[
10625,
10713
]
],
"text": "['they require storing potentially large amounts of example data for runtime interpolation']",
"type": "background_claim",
"id": "T96"
},
{
"offsets": [
[
10728,
10790
]
],
"text": "['our method discards all example data after the fitting process']",
"type": "own_claim",
"id": "T97"
},
{
"offsets": [
[
10794,
10869
]
],
"text": "['the size of our runtime structures does not scale with the number of inputs']",
"type": "own_claim",
"id": "T98"
},
{
"offsets": [
[
10871,
10943
]
],
"text": "['Other authors have used physical simulation for interactive deformations']",
"type": "background_claim",
"id": "T99"
},
{
"offsets": [
[
10978,
10996
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],
"text": "['James and Pai 2002']",
"type": "data",
"id": "T100"
},
{
"offsets": [
[
10998,
11016
]
],
"text": "['Capell et al. 2002']",
"type": "data",
"id": "T101"
},
{
"offsets": [
[
11019,
11082
]
],
"text": "['Our method cannot capture these secondary deformations directly']",
"type": "own_claim",
"id": "T102"
},
{
"offsets": [
[
11093,
11117
]
],
"text": "['a technique such as DyRT']",
"type": "own_claim",
"id": "T103"
},
{
"offsets": [
[
11119,
11137
]
],
"text": "['James and Pai 2002']",
"type": "data",
"id": "T104"
},
{
"offsets": [
[
11140,
11211
]
],
"text": "['can be applied to the characters we generate to add secondary animation']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
11213,
11271
]
],
"text": "['There has been some recent work on fitting skinning models']",
"type": "background_claim",
"id": "T106"
},
{
"offsets": [
[
11273,
11345
]
],
"text": "['One method solves for joint centers and vertex weights for a scanned arm']",
"type": "background_claim",
"id": "T107"
},
{
"offsets": [
[
11348,
11373
]
],
"text": "['Nebel and Sibiryakov 2002']",
"type": "data",
"id": "T108"
},
{
"offsets": [
[
11380,
11417
]
],
"text": "['the Multi-Weight Enveloping technique']",
"type": "own_claim",
"id": "T109"
},
{
"offsets": [
[
11453,
11484
]
],
"text": "['is most similar to our approach']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
11419,
11441
]
],
"text": "['Wang and Phillips 2002']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
11487,
11667
]
],
"text": "['MWE extends linear blend skinning by giving each vertex one weight to each coefficient of each influencing joint’s transformation matrix instead of one weight per influencing joint']",
"type": "background_claim",
"id": "T112"
},
{
"offsets": [
[
11669,
11771
]
],
"text": "['They then find these weights by solving a linear leastsquares problem using a set of examples as input']",
"type": "background_claim",
"id": "T113"
},
{
"offsets": [
[
11779,
11853
]
],
"text": "['on the surface Multi-Weight Enveloping and our technique seem very similar']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
11855,
11863
]
],
"text": "['they are']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
11872,
11902
]
],
"text": "['different in a fundamental way']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
11904,
12007
]
],
"text": "['Both MWE and our technique use an extension of linear blend skinning as an underlying deformation model']",
"type": "own_claim",
"id": "T117"
},
{
"offsets": [
[
12018,
12094
]
],
"text": "['MWE extends linear blend skinning by adding more vertex weights to the model']",
"type": "background_claim",
"id": "T118"
},
{
"offsets": [
[
12114,
12141
]
],
"text": "['our method adds more joints']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
12143,
12188
]
],
"text": "['MWE uses a large number of weights per vertex']",
"type": "background_claim",
"id": "T120"
},
{
"offsets": [
[
12190,
12214
]
],
"text": "['12 per influencing joint']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
12217,
12306
]
],
"text": "['This introduces the possibility of rank deficient matrices in the least-squares solutions']",
"type": "background_claim",
"id": "T122"
},
{
"offsets": [
[
12309,
12331
]
],
"text": "['Wang and Phillips 2002']",
"type": "data",
"id": "T123"
},
{
"offsets": [
[
12351,
12404
]
],
"text": "['the matrix coefficients are usually highly correlated']",
"type": "background_claim",
"id": "T124"
},
{
"offsets": [
[
12406,
12434
]
],
"text": "['This can lead to overfitting']",
"type": "background_claim",
"id": "T125"
},
{
"offsets": [
[
12442,
12473
]
],
"text": "['MWE must take measures to avoid']",
"type": "background_claim",
"id": "T126"
},
{
"offsets": [
[
12494,
12571
]
],
"text": "['the number of weights per vertex in one of our skins remains relatively small']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
12573,
12596
]
],
"text": "['1 per influencing joint']",
"type": "data",
"id": "T128"
},
{
"offsets": [
[
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],
"text": "['our extra joints are explicitly designed to be very different from existing joints']",
"type": "own_claim",
"id": "T129"
},
{
"offsets": [
[
12686,
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],
"text": "['our technique requires no special provisions to avoid overfitting']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
12762,
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],
"text": "['our method can detect and handle small amounts of overfitting']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
12853,
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],
"text": "['Section 5.2']",
"type": "data",
"id": "T132"
},
{
"offsets": [
[
12962,
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],
"text": "['MWE skins are not as easily accelerated by graphics hardware as skins created using our method']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
12889,
12953
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],
"text": "['having one weight per entry in the joint transformation matrices']",
"type": "data",
"id": "T134"
},
{
"offsets": [
[
13073,
13136
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],
"text": "['our skins are computed in the same manner as linear blend skins']",
"type": "own_claim",
"id": "T135"
},
{
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[
13138,
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],
"text": "['existing software infrastructure can make use of them with little or no changes']",
"type": "own_claim",
"id": "T136"
},
{
"offsets": [
[
13519,
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],
"text": "['This technique is widely used for interactive applications']",
"type": "background_claim",
"id": "T137"
},
{
"offsets": [
[
13579,
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],
"text": "['An excellent description of this method is found']",
"type": "background_claim",
"id": "T138"
},
{
"offsets": [
[
13632,
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],
"text": "['Lewis et al. [2000]']",
"type": "data",
"id": "T139"
},
{
"offsets": [
[
14740,
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],
"text": "['a deformed vertex position in the dress pose configuration c = d is the same as the provided dress pose vertex ( v d = v d )']",
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"id": "T140"
},
{
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[
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],
"text": "['the weights are affine']",
"type": "data",
"id": "T141"
},
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"offsets": [
[
14892,
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],
"text": "['This skinning algorithm is notorious for its failings']",
"type": "background_claim",
"id": "T142"
},
{
"offsets": [
[
14947,
14987
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],
"text": "['It cannot represent complex deformations']",
"type": "background_claim",
"id": "T143"
},
{
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14992,
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],
"text": "['suffers from characteristic artifacts']",
"type": "background_claim",
"id": "T144"
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"offsets": [
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15038,
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],
"text": "['the “candy-wrapper” collapse effect on wrists']",
"type": "data",
"id": "T145"
},
{
"offsets": [
[
15088,
15120
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],
"text": "['collapsing around bending joints']",
"type": "data",
"id": "T146"
},
{
"offsets": [
[
15133,
15141
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],
"text": "['Figure 2']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
15144,
15163
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],
"text": "['The artifacts occur']",
"type": "background_claim",
"id": "T148"
},
{
"offsets": [
[
15172,
15230
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],
"text": "['vertices are transformed by linearly interpolated matrices']",
"type": "background_claim",
"id": "T149"
},
{
"offsets": [
[
15235,
15314
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],
"text": "['the interpolated matrices are dissimilar as in a rotation of nearly 180 degrees']",
"type": "data",
"id": "T150"
},
{
"offsets": [
[
15316,
15361
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],
"text": "['the interpolated transformation is degenerate']",
"type": "background_claim",
"id": "T151"
},
{
"offsets": [
[
15366,
15392
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],
"text": "['the geometry must collapse']",
"type": "background_claim",
"id": "T152"
},
{
"offsets": [
[
15437,
15484
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],
"text": "['linear blend skins are very difficult to author']",
"type": "background_claim",
"id": "T153"
},
{
"offsets": [
[
15486,
15505
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],
"text": "['[Lewis et al. 2000]']",
"type": "data",
"id": "T154"
},
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"offsets": [
[
15529,
15565
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],
"text": "['this skinning algorithm is very fast']",
"type": "background_claim",
"id": "T155"
},
{
"offsets": [
[
15570,
15613
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],
"text": "['widely supported by commercial applications']",
"type": "background_claim",
"id": "T156"
},
{
"offsets": [
[
15617,
15680
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],
"text": "['it remains popular especially in games and virtual environments']",
"type": "background_claim",
"id": "T157"
},
{
"offsets": [
[
15755,
15833
]
],
"text": "['The linear blend skinning model is not sufficient to capture deformations well']",
"type": "background_claim",
"id": "T158"
},
{
"offsets": [
[
15846,
15854
]
],
"text": "['Figure 3']",
"type": "data",
"id": "T159"
},
{
"offsets": [
[
15938,
16016
]
],
"text": "['the linearly blended matrix becomes degenerate and collapses the skin geometry']",
"type": "background_claim",
"id": "T160"
},
{
"offsets": [
[
15904,
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],
"text": "['the twist approaches 180 degrees']",
"type": "data",
"id": "T161"
},
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"offsets": [
[
15857,
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],
"text": "['The problem']",
"type": "background_claim",
"id": "T162"
},
{
"offsets": [
[
15893,
15900
]
],
"text": "['is that']",
"type": "background_claim",
"id": "T163"
},
{
"offsets": [
[
16018,
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]
],
"text": "['Linearly blended transformations tend to collapse the more different they are']",
"type": "background_claim",
"id": "T164"
},
{
"offsets": [
[
16097,
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],
"text": "['The resulting loss of volume can also be observed around hinge joints']",
"type": "background_claim",
"id": "T165"
},
{
"offsets": [
[
16175,
16183
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],
"text": "['the knee']",
"type": "data",
"id": "T166"
},
{
"offsets": [
[
16188,
16193
]
],
"text": "['elbow']",
"type": "data",
"id": "T167"
},
{
"offsets": [
[
16206,
16214
]
],
"text": "['Figure 2']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
16233,
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],
"text": "['we can help avoid the collapse problem by avoiding blending transformations that are so dissimilar']",
"type": "own_claim",
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{
"offsets": [
[
16333,
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],
"text": "['We can accomplish this by adding extra transformations that properly interpolates without collapsing']",
"type": "own_claim",
"id": "T170"
},
{
"offsets": [
[
16471,
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],
"text": "['we can add an extra joint that interpolates the rotation angle correctly and does not collapse']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
16450,
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],
"text": "['the twisting wrists']",
"type": "data",
"id": "T172"
},
{
"offsets": [
[
16576,
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],
"text": "['artists sometimes do this by hand to help avoid wrist collapses']",
"type": "background_claim",
"id": "T173"
},
{
"offsets": [
[
16673,
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],
"text": "['any deformation effect could be obtained by adding joints that deform appropriately to capture that deformation effect']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
16832,
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],
"text": "['we can add joints that scale up when the muscle should bulge, and scale down when the muscle relaxes']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
16948,
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],
"text": "['we could add several joints that move and scale in concert to capture the wrinkles']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
17687,
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],
"text": "['we could add as many transformations as vertices and capture all deformations exactly']",
"type": "own_claim",
"id": "T177"
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{
"offsets": [
[
17789,
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],
"text": "['adding so many extra joints is impractical']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
17840,
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],
"text": "['adding such a large number of joints would severely impact the performance of our resulting skins']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
17954,
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],
"text": "['we could find these transformations for the input examples']",
"type": "data",
"id": "T180"
},
{
"offsets": [
[
18014,
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],
"text": "['it is unclear how to determine the general relationships of these transformations to the skeletal parameters in all poses']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
18177,
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],
"text": "['our scheme would only be able to reproduce the input frames and would not work well in new poses']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
18137,
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],
"text": "['Without knowledge of this relationship']",
"type": "data",
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"offsets": [
[
18284,
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],
"text": "['we extend the traditional linear blend skinning model by adding a relatively small number of joints that are simply related to the original skeletal parameters and fit using them']",
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"id": "T184"
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{
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[
18464,
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],
"text": "['We choose these extra joints by both examining the places where the standard linear blend model fails and by examining extra character deformations that we would like to capture']",
"type": "own_claim",
"id": "T185"
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"offsets": [
[
18643,
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],
"text": "['We then add joints that we believe will help resolve these artifacts']",
"type": "own_claim",
"id": "T186"
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{
"offsets": [
[
18833,
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],
"text": "['vertices choose weighted sums of transformations']",
"type": "own_claim",
"id": "T187"
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{
"offsets": [
[
18886,
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],
"text": "['any linear scaling of an added joint is beneficial']",
"type": "data",
"id": "T188"
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18937,
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],
"text": "['it may be used']",
"type": "own_claim",
"id": "T189"
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{
"offsets": [
[
18953,
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],
"text": "['Thus the additional joints need not be exact']",
"type": "own_claim",
"id": "T190"
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{
"offsets": [
[
18722,
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],
"text": "['we fit the parameters of our skinning model using this extended skeleton']",
"type": "own_claim",
"id": "T191"
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{
"offsets": [
[
19017,
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],
"text": "['this is a framework for obtaining better deformations']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
19075,
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],
"text": "['the joints we choose to add are based on our observations of characters']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
19148,
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],
"text": "['Different characters with different deformations may require a different set of additional joints']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
19301,
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],
"text": "['the skin may be solved using our fitting algorithm without change']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
19261,
19299
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],
"text": "['some set of these joints is determined']",
"type": "data",
"id": "T196"
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{
"offsets": [
[
20062,
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],
"text": "['our system can automatically add joints that properly interpolate rotations without collapsing']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
[
20382,
20499
]
],
"text": "['More joints with evenly distributed interpolation parameters could be added to sample this rotation space even better']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
20528,
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],
"text": "['just a single interpolated rotation is sufficient']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
[
20158,
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],
"text": "['This is done by examining the rotation of a joint relative to the dress pose and computing the new joint as the halfway spherical linear interpolation']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
20325,
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],
"text": "['of this rotation, located at the same position in space']",
"type": "own_claim",
"id": "T201"
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{
"offsets": [
[
20310,
20323
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],
"text": "['Shoemake 1985']",
"type": "data",
"id": "T202"
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{
"offsets": [
[
20605,
20699
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],
"text": "['improvements gained by simply adding a single interpolated rotation joint in the twisting case']",
"type": "own_claim",
"id": "T203"
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{
"offsets": [
[
20579,
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],
"text": "['Figure 4']",
"type": "data",
"id": "T204"
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"offsets": [
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20720,
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],
"text": "['improvements for the bent elbow case']",
"type": "own_claim",
"id": "T205"
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{
"offsets": [
[
20701,
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],
"text": "['Figure 5']",
"type": "data",
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{
"offsets": [
[
20758,
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],
"text": "['Another type of effect not easily captured by the simple linear blend model is bulging and denting of skins caused by muscles, tendons, or other substructure']",
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"id": "T207"
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{
"offsets": [
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20917,
20960
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],
"text": "['These particular effects cannot be captured']",
"type": "background_claim",
"id": "T208"
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{
"offsets": [
[
20967,
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],
"text": "['the joints employed in animating a character do not typically scale up and down as would be necessary to approximate these effects']",
"type": "background_claim",
"id": "T209"
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{
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[
21142,
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"text": "['the substructure deformation effects from muscles and tendons are often simply related to the angles between joints']",
"type": "own_claim",
"id": "T210"
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{
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[
21125,
21140
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],
"text": "['many characters']",
"type": "data",
"id": "T211"
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[
21300,
21332
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],
"text": "['the elbow is near full extension']",
"type": "data",
"id": "T212"
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"offsets": [
[
21272,
21294
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],
"text": "['a bicep bulge is small']",
"type": "own_claim",
"id": "T213"
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{
"offsets": [
[
21339,
21357
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],
"text": "['the bugle is large']",
"type": "own_claim",
"id": "T214"
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21363,
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],
"text": "['the elbow is near full flexion']",
"type": "data",
"id": "T215"
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"offsets": [
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21395,
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],
"text": "['The effect is similar for other muscles in the body']",
"type": "own_claim",
"id": "T216"
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{
"offsets": [
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21474,
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],
"text": "['our system can add several joints that scale up and down based on the angle between particular joints']",
"type": "own_claim",
"id": "T217"
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{
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22014,
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"text": "['All upstream joints are oriented in the same way, with one axis aligned with the bone']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
22112,
22120
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],
"text": "['Figure 6']",
"type": "data",
"id": "T219"
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{
"offsets": [
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22454,
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],
"text": "['the driver has multiple children']",
"type": "data",
"id": "T220"
},
{
"offsets": [
[
22488,
22592
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],
"text": "['a vector that is the sum of the bones connecting the driver to its children is used to measure the angle']",
"type": "own_claim",
"id": "T221"
},
{
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22594,
22623
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],
"text": "['Downstream joints are similar']",
"type": "own_claim",
"id": "T222"
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22625,
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],
"text": "['We use four downstream joints on each bone connecting the driver to its children that scale just as the upstream joints do']",
"type": "own_claim",
"id": "T223"
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{
"offsets": [
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22821,
22964
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],
"text": "['the scale parameter s is s = 1 + k b 1 · b 2 + 1 2 b 1 b 2 where b 1 and b 2 are the bone vectors used to measure the angle at the driver joint']",
"type": "own_claim",
"id": "T224"
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{
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22969,
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"text": "['k is the maximum scale factor']",
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{
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23004,
23041
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],
"text": "['the angle between b 1 and b 2 is zero']",
"type": "data",
"id": "T226"
},
{
"offsets": [
[
22799,
22819
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],
"text": "['joints that scale up']",
"type": "data",
"id": "T227"
},
{
"offsets": [
[
23047,
23069
]
],
"text": "['joints that scale down']",
"type": "data",
"id": "T228"
},
{
"offsets": [
[
23071,
23105
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],
"text": "['the scale parameter is simply s −1']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
23108,
23149
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],
"text": "['The value for k may be chosen by the user']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
23192,
23221
]
],
"text": "['8 works well for our examples']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
23236,
23285
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],
"text": "['vertices may take any scaling of these new joints']",
"type": "own_claim",
"id": "T232"
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{
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[
23287,
23321
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],
"text": "['a conservative large value is fine']",
"type": "own_claim",
"id": "T233"
},
{
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[
23339,
23347
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],
"text": "['a vertex']",
"type": "data",
"id": "T234"
},
{
"offsets": [
[
23356,
23400
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],
"text": "['needed a joint that scaled by 2 instead of 8']",
"type": "data",
"id": "T235"
},
{
"offsets": [
[
23402,
23438
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],
"text": "['it could be assigned a weight of 4 1']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
24176,
24288
]
],
"text": "['we use a fitting procedure to set the parameters of the underlying skinning model to match the example data well']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
24131,
24174
]
],
"text": "['our system has augmented the input skeleton']",
"type": "data",
"id": "T238"
},
{
"offsets": [
[
24312,
24365
]
],
"text": "['the input to the fitting process is a set of examples']",
"type": "own_claim",
"id": "T239"
},
{
"offsets": [
[
24436,
24504
]
],
"text": "['This static mesh is deformed according to the skeleton configuration']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
24510,
24555
]
],
"text": "['it is not attached to the skeleton in any way']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
24574,
24639
]
],
"text": "['our examples were generated by exporting rigged objects from Maya']",
"type": "own_claim",
"id": "T242"
},
{
"offsets": [
[
24645,
24711
]
],
"text": "['they could have been sculpted by hand or come from another program']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
24951,
25087
]
],
"text": "['for each vertex, we are able to choose the set of influencing joints, influence weights (w i ) and the dress pose vertex position (v d )']",
"type": "own_claim",
"id": "T244"
},
{
"offsets": [
[
24827,
24857
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],
"text": "['only the M i are predetermined']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
25089,
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],
"text": "['We would like to choose the influence sets, weights and dress pose vertex positions that best approximate the examples and generalize well to new poses']",
"type": "own_claim",
"id": "T246"
},
{
"offsets": [
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],
"text": "['We determine influence sets first']",
"type": "own_claim",
"id": "T247"
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{
"offsets": [
[
26078,
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],
"text": "['Ideally, the influence sets would fall out naturally from the weight solving procedure']",
"type": "own_claim",
"id": "T248"
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{
"offsets": [
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26217,
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"text": "['this does not happen in practice']",
"type": "own_claim",
"id": "T249"
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{
"offsets": [
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26258,
26302
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],
"text": "['our samplings are necessarily not exhaustive']",
"type": "own_claim",
"id": "T250"
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{
"offsets": [
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26310,
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],
"text": "['the more joints that a vertex depends on, the slower the skin can be to compute']",
"type": "own_claim",
"id": "T251"
},
{
"offsets": [
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26394,
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],
"text": "['current hardware only supports a limited number of influences per vertex']",
"type": "own_claim",
"id": "T252"
},
{
"offsets": [
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],
"text": "['we would like to select a small set of good influences']",
"type": "own_claim",
"id": "T253"
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{
"offsets": [
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"text": "['choosing the influence sets appropriately lets us bound the size of the problems we must solve to determine the weights']",
"type": "own_claim",
"id": "T254"
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{
"offsets": [
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],
"text": "['Section 5.2']",
"type": "data",
"id": "T255"
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"offsets": [
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],
"text": "['This makes the solving process faster']",
"type": "own_claim",
"id": "T256"
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{
"offsets": [
[
26724,
26799
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],
"text": "['In most recent research, influence set determination has been left to users']",
"type": "background_claim",
"id": "T257"
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{
"offsets": [
[
26802,
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],
"text": "['Lewis et al. 2000']",
"type": "data",
"id": "T258"
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{
"offsets": [
[
26821,
26843
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],
"text": "['Wang and Phillips 2002']",
"type": "data",
"id": "T259"
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{
"offsets": [
[
26845,
26862
]
],
"text": "['Sloan et al. 2001']",
"type": "data",
"id": "T260"
},
{
"offsets": [
[
26865,
26967
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],
"text": "['The task is typically accomplished by “painting” the regions of influence for each joint over the mesh']",
"type": "background_claim",
"id": "T261"
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{
"offsets": [
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],
"text": "['less difficult than painting the weights themselves']",
"type": "background_claim",
"id": "T262"
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{
"offsets": [
[
27048,
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],
"text": "['it is a tedious process']",
"type": "background_claim",
"id": "T263"
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{
"offsets": [
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27028,
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],
"text": "['Lewis et al. 2000']",
"type": "data",
"id": "T264"
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"offsets": [
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27086,
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],
"text": "['our system automatically determines the influence sets for each vertex using a heuristic algorithm']",
"type": "own_claim",
"id": "T265"
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{
"offsets": [
[
27202,
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],
"text": "['vertices in a character skin typically transform nearly rigidly with respect to some joint']",
"type": "own_claim",
"id": "T266"
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"offsets": [
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],
"text": "['vertices on the forearm roughly follow the forearm']",
"type": "own_claim",
"id": "T267"
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{
"offsets": [
[
27397,
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],
"text": "['their skin is most heavily influenced by those joints that they are bound to']",
"type": "own_claim",
"id": "T268"
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{
"offsets": [
[
27380,
27395
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],
"text": "['most characters']",
"type": "data",
"id": "T269"
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{
"offsets": [
[
27487,
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],
"text": "['a point on the bicep is not truly rigid as an arm moves']",
"type": "own_claim",
"id": "T270"
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{
"offsets": [
[
27551,
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],
"text": "['muscle bulge']",
"type": "data",
"id": "T271"
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"offsets": [
[
27582,
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],
"text": "['these points remain mostly rigidly attached to the upper arm']",
"type": "own_claim",
"id": "T272"
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{
"offsets": [
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27658,
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],
"text": "['should be influenced by it']",
"type": "own_claim",
"id": "T273"
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{
"offsets": [
[
27710,
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],
"text": "['we measure how rigidly a vertex transforms with every joint over all examples and use the most rigidly transforming joints for the influence set']",
"type": "own_claim",
"id": "T274"
},
{
"offsets": [
[
28181,
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],
"text": "['The collection of these local coordinate positions over all examples forms a point cloud']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
28282,
28290
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],
"text": "['Figure 7']",
"type": "data",
"id": "T276"
},
{
"offsets": [
[
28293,
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],
"text": "['The more compact this point cloud, the more rigid we believe the vertex-joint relationship to be']",
"type": "own_claim",
"id": "T277"
},
{
"offsets": [
[
28540,
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],
"text": "['the simple O(n 2 ) algorithm that compares each point to every other to be fast enough for our purposes']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
28648,
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]
],
"text": "['this diameter may be computed more quickly']",
"type": "own_claim",
"id": "T279"
},
{
"offsets": [
[
28692,
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]
],
"text": "['An O(n log n) time algorithm is possible']",
"type": "own_claim",
"id": "T280"
},
{
"offsets": [
[
28775,
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],
"text": "['faster methods']",
"type": "background_claim",
"id": "T281"
},
{
"offsets": [
[
28739,
28768
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],
"text": "['Malandain and Boissonnat 2002']",
"type": "data",
"id": "T282"
},
{
"offsets": [
[
28927,
28996
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],
"text": "['It may be tempting to use a threshold scheme to choose influence sets']",
"type": "own_claim",
"id": "T283"
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{
"offsets": [
[
29001,
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],
"text": "['we have found this problematic']",
"type": "own_claim",
"id": "T284"
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{
"offsets": [
[
29033,
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],
"text": "['It is unclear how to pick a good threshold']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
29087,
29117
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],
"text": "['the rigidity scores get larger']",
"type": "data",
"id": "T286"
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{
"offsets": [
[
29119,
29146
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],
"text": "['they become less meaningful']",
"type": "own_claim",
"id": "T287"
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{
"offsets": [
[
29162,
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],
"text": "['it may happen as an artifact of the particular input examples that points on the left shoulder move much more rigidly relative to the right leg rather than the left leg']",
"type": "own_claim",
"id": "T288"
},
{
"offsets": [
[
29335,
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],
"text": "['both choices make no sense for influences']",
"type": "own_claim",
"id": "T289"
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{
"offsets": [
[
29384,
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],
"text": "['larger rigidity scores are not particularly meaningful']",
"type": "own_claim",
"id": "T290"
},
{
"offsets": [
[
29440,
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],
"text": "['it is nearly impossible to pick a meaningful threshold value']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
29502,
29598
]
],
"text": "['As in other linear blend skinning systems, influence sets need only be determined conservatively']",
"type": "own_claim",
"id": "T292"
},
{
"offsets": [
[
29600,
29622
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],
"text": "['Wang and Phillips 2002']",
"type": "data",
"id": "T293"
},
{
"offsets": [
[
29627,
29664
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],
"text": "['we allow users to choose k if desired']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
29704,
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],
"text": "['between three and eight influences works well']",
"type": "own_claim",
"id": "T295"
},
{
"offsets": [
[
29991,
30162
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],
"text": "['We would like to find the best vertices and weights that minimize the least-squares difference between the skin and the examples at all the example skeleton configurations']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
30164,
30171
],
[
30172,
30207
]
],
"text": "['That is', 'n 2 min ∑ v e i − v e i i=1']",
"type": "own_claim",
"id": "T297"
},
{
"offsets": [
[
30239,
30294
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],
"text": "['v e i is the input vertex position from the ith example']",
"type": "data",
"id": "T298"
},
{
"offsets": [
[
30299,
30391
]
],
"text": "['v e i is the deformed vertex computed by the skinning model at the ith example configuration']",
"type": "data",
"id": "T299"
},
{
"offsets": [
[
30430,
30482
]
],
"text": "['This problem is bilinear in the weights and vertices']",
"type": "own_claim",
"id": "T300"
},
{
"offsets": [
[
30484,
30541
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],
"text": "['We use an alternation technique to solve the optimization']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
30543,
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],
"text": "['This works by first fixing the first variable and solving a linear least-squares problem to find the second, then fixing the second and solving a linear leastsquares problem for the first']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
30782,
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],
"text": "['This technique is commonly used and is described']",
"type": "own_claim",
"id": "T303"
},
{
"offsets": [
[
30732,
30780
]
],
"text": "['This process is then repeated until it converges']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
30836,
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],
"text": "['Freeman and Tenenbaum 1997']",
"type": "data",
"id": "T305"
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{
"offsets": [
[
30865,
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],
"text": "['We start by solving for weights']",
"type": "own_claim",
"id": "T306"
},
{
"offsets": [
[
30903,
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],
"text": "['we have no good guess for them']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
30938,
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],
"text": "['we know that the initial dress pose vertices are ideal']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
31057,
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],
"text": "['This process typically converges after one or two iterations']",
"type": "own_claim",
"id": "T309"
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{
"offsets": [
[
31171,
31248
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],
"text": "['we are solving for a small numbers of weights using large numbers of examples']",
"type": "own_claim",
"id": "T310"
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{
"offsets": [
[
31250,
31323
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],
"text": "['our systems are often well conditioned and do not suffer from overfitting']",
"type": "own_claim",
"id": "T311"
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{
"offsets": [
[
31327,
31357
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],
"text": "['the input data is well sampled']",
"type": "data",
"id": "T312"
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{
"offsets": [
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31135,
31144
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],
"text": "['Section 2']",
"type": "data",
"id": "T313"
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"offsets": [
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31364,
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"text": "['we do not have to take special precautions to avoid overfitting']",
"type": "own_claim",
"id": "T314"
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{
"offsets": [
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31470,
31501
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"text": "['we include tests for robustness']",
"type": "own_claim",
"id": "T315"
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"offsets": [
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32628,
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],
"text": "['there are any singular values below some fraction of this ratio']",
"type": "data",
"id": "T316"
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"offsets": [
[
32518,
32601
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],
"text": "['We detect this by comparing the ratio of the largest singular value to the smallest']",
"type": "own_claim",
"id": "T317"
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{
"offsets": [
[
32607,
32624
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],
"text": "['issuing a warning']",
"type": "own_claim",
"id": "T318"
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{
"offsets": [
[
32436,
32516
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],
"text": "['This lets us detect when our matrices are rank deficient, leading to overfitting']",
"type": "own_claim",
"id": "T319"
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{
"offsets": [
[
32777,
32801
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],
"text": "['overfitting is a problem']",
"type": "data",
"id": "T320"
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{
"offsets": [
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32803,
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],
"text": "['provisions such as those taken']",
"type": "own_claim",
"id": "T321"
},
{
"offsets": [
[
32862,
32880
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],
"text": "['could also be used']",
"type": "own_claim",
"id": "T322"
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{
"offsets": [
[
32838,
32860
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],
"text": "['Wang and Phillips 2002']",
"type": "data",
"id": "T323"
},
{
"offsets": [
[
32891,
32956
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],
"text": "['in all the examples in this paper, no singular values were zeroed']",
"type": "own_claim",
"id": "T324"
},
{
"offsets": [
[
33018,
33098
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],
"text": "['It is not only important for the geometry in a skin approximation to be accurate']",
"type": "own_claim",
"id": "T325"
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{
"offsets": [
[
33109,
33154
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],
"text": "['important for normals to be well approximated']",
"type": "own_claim",
"id": "T326"
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{
"offsets": [
[
33159,
33171
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],
"text": "['they are not']",
"type": "data",
"id": "T327"
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{
"offsets": [
[
33173,
33224
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],
"text": "['lighting calculations will not produce good results']",
"type": "own_claim",
"id": "T328"
},
{
"offsets": [
[
33275,
33423
]
],
"text": "['It may seem that just transforming a dress pose normal by the inverse transpose of the corresponding vertex’s transformation matrix would be correct']",
"type": "own_claim",
"id": "T329"
},
{
"offsets": [
[
33525,
33593
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],
"text": "['this is technically valid for local neighborhoods of smooth surfaces']",
"type": "own_claim",
"id": "T330"
},
{
"offsets": [
[
33596,
33610
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],
"text": "['Turkowski 1990']",
"type": "data",
"id": "T331"
},
{
"offsets": [
[
33613,
33644
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],
"text": "['we do not have a smooth surface']",
"type": "own_claim",
"id": "T332"
},
{
"offsets": [
[
33709,
33774
]
],
"text": "['Computing the normals in this manner can give undesirable results']",
"type": "own_claim",
"id": "T333"
},
{
"offsets": [
[
33646,
33707
]
],
"text": "['Instead we have single points that are computed independently']",
"type": "own_claim",
"id": "T334"
},
{
"offsets": [
[
33780,
33830
]
],
"text": "['the blended transformations are not pure rotations']",
"type": "data",
"id": "T335"
},
{
"offsets": [
[
33832,
33896
]
],
"text": "['Interactive systems typically approximate normal calculations as']",
"type": "background_claim",
"id": "T336"
},
{
"offsets": [
[
33928,
33990
]
],
"text": "['n ̄ c = ∑ w i M i,c −T M i,d T n d ∑ w i M i,c −T M i,d T n d']",
"type": "background_claim",
"id": "T337"
},
{
"offsets": [
[
34014,
34112
]
],
"text": "['it is often faster to calculate the joint matrices and their inverses incrementally by composition']",
"type": "background_claim",
"id": "T338"
},
{
"offsets": [
[
34121,
34182
]
],
"text": "['at each step, rotations and scales have special inverse forms']",
"type": "background_claim",
"id": "T339"
},
{
"offsets": [
[
34184,
34242
]
],
"text": "['This alleviates the need for a general inversion operation']",
"type": "background_claim",
"id": "T340"
},
{
"offsets": [
[
34244,
34256
]
],
"text": "['In EigenSkin']",
"type": "background_claim",
"id": "T341"
},
{
"offsets": [
[
34258,
34273
]
],
"text": "['Kry et al. 2002']",
"type": "data",
"id": "T342"
},
{
"offsets": [
[
34277,
34354
]
],
"text": "['normals are treated as second skinning problem and are computed independently']",
"type": "background_claim",
"id": "T343"
},
{
"offsets": [
[
34371,
34478
]
],
"text": "['we take the model used in existing systems as in Equation 2 and include normals in our optimization process']",
"type": "own_claim",
"id": "T344"
},
{
"offsets": [
[
34492,
34592
]
],
"text": "['we simply add more terms to the objective function to include the differences between normal vectors']",
"type": "own_claim",
"id": "T345"
},
{
"offsets": [
[
34594,
34625
]
],
"text": "['We allow users to scale normals']",
"type": "own_claim",
"id": "T346"
},
{
"offsets": [
[
34629,
34703
]
],
"text": "['they wish to change their relative influence on the least-squares solution']",
"type": "data",
"id": "T347"
},
{
"offsets": [
[
34754,
34881
]
],
"text": "['The simple linear blend skinning model commonly used in video games and other interactive applications is very fast and compact']",
"type": "background_claim",
"id": "T348"
},
{
"offsets": [
[
34886,
34962
]
],
"text": "['cannot capture the high quality deformations that make convincing characters']",
"type": "background_claim",
"id": "T349"
},
{
"offsets": [
[
34964,
35101
]
],
"text": "['Our framework for extending the linear blend model allows us to capture much more interesting deformations while retaining its efficiency']",
"type": "own_claim",
"id": "T350"
},
{
"offsets": [
[
35103,
35194
]
],
"text": "['The most egregious deformation problems of linear blend skinning are solved by our approach']",
"type": "own_claim",
"id": "T351"
},
{
"offsets": [
[
35221,
35293
]
],
"text": "['our system can fix collapsing twists by adding just a single extra joint']",
"type": "own_claim",
"id": "T352"
},
{
"offsets": [
[
35196,
35211
]
],
"text": "['Figures 3 and 4']",
"type": "data",
"id": "T353"
},
{
"offsets": [
[
35295,
35379
]
],
"text": "['Collapsing and interpenetrations around hinge joints are also fixed using our method']",
"type": "own_claim",
"id": "T354"
},
{
"offsets": [
[
35392,
35400
]
],
"text": "['Figure 5']",
"type": "data",
"id": "T355"
},
{
"offsets": [
[
35469,
35583
]
],
"text": "['our extension framework can capture other more subtle and detailed deformations required for convincing characters']",
"type": "own_claim",
"id": "T356"
},
{
"offsets": [
[
35611,
35691
]
],
"text": "['our method can capture the bulges in the biceps and triceps of a character’s arm']",
"type": "own_claim",
"id": "T357"
},
{
"offsets": [
[
35585,
35593
]
],
"text": "['Figure 8']",
"type": "data",
"id": "T358"
},
{
"offsets": [
[
35699,
35837
]
],
"text": "['the particular extra joints we have chosen to add to our characters may not be capable of capturing the full deformation for any character']",
"type": "own_claim",
"id": "T359"
},
{
"offsets": [
[
35839,
35946
]
],
"text": "['different extra joints that do capture the desired deformations may be added and solved using our technique']",
"type": "own_claim",
"id": "T360"
},
{
"offsets": [
[
35968,
36032
]
],
"text": "['our technique can be used on more than just simple arms and legs']",
"type": "own_claim",
"id": "T361"
},
{
"offsets": [
[
36034,
36042
]
],
"text": "['Figure 9']",
"type": "data",
"id": "T362"
},
{
"offsets": [
[
36204,
36252
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"text": "['our resulting skins generalize well to new poses']",
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"text": "['Our solution procedure is generally very fast']",
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"text": "['None of the examples shown here took more than five minutes to solve on a modern personal computer']",
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"text": "['The computation time for each vertex depends on the number of influences and the number of examples']",
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"text": "['each vertex is solved independently']",
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"text": "['our algorithm is trivial to parallelize']",
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"text": "['The ability to generate compactly represented, fast to evaluate, high quality skin approximations from a set of examples is very useful']",
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"text": "['Applications range from building characters for video games and virtual environments to high-end animation previewing']",
"type": "background_claim",
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"text": "['Many current interactive systems']",
"type": "background_claim",
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"text": "['only support linear blend skinned characters']",
"type": "background_claim",
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"text": "['video games']",
"type": "data",
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"text": "['authoring these skins is notoriously difficult']",
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"text": "['deformation problems associated with using this model']",
"type": "background_claim",
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"text": "['Determining the blending weights and influence sets is left to the skin author to set directly']",
"type": "background_claim",
"id": "T376"
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"text": "['None of the more intuitive or useful deformer primitives provided by animation systems may be used']",
"type": "background_claim",
"id": "T377"
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"text": "['Using our method, character authors may use any tools they like to author characters']",
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"text": "['All our system requires is a set of examples which is used to compute the appropriate influence sets and blending weights automatically']",
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"text": "['This frees the author from setting them manually']",
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"text": "['our characters are a straightforward extension to linear blend skinning']",
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"text": "['many existing interactive systems already have the software infrastructure to sup- port them']",
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"text": "['our skins are computed in the same manner as existing linear blend skins']",
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"text": "['they are already accelerated by current graphics hardware']",
"type": "own_claim",
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"text": "['Another application of our system is to map a character originally attached to one skeleton onto a different underlying skeleton']",
"type": "own_claim",
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"text": "['Skin retargeting is useful']",
"type": "own_claim",
"id": "T386"
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"text": "['a particular interactive system requires characters to have a specific skeleton']",
"type": "data",
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"text": "['a video game may have an optimized engine for characters with a particular skeleton topology']",
"type": "background_claim",
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"text": "['a character was created for a different skeleton']",
"type": "data",
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"text": "['the character would have to be re-rigged manually to work on the new skeleton topology']",
"type": "background_claim",
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"text": "['this can be accomplished much more easily with our system']",
"type": "own_claim",
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"text": "['Our system sees this as any other set of data and solves for the proper influence sets and blending weights']",
"type": "own_claim",
"id": "T392"
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"text": "['Another application of our technique is targeted at high-end animation']",
"type": "own_claim",
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"text": "['High-end characters often have such complex deformations that they cannot be computed interactively']",
"type": "background_claim",
"id": "T394"
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"text": "['animators typically work with low fidelity versions that only roughly suggest the actual shape of the character']",
"type": "background_claim",
"id": "T395"
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"text": "['Using our method, interactive characters could be built that allow animators to interact with much better approximations of the deformed characters']",
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"text": "['we have presented a method for building fast to evaluate, compact representations that produce accurate approximations of deforming characters']",
"type": "own_claim",
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"text": "['The characters may be rigged using any available tool']",
"type": "own_claim",
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"text": "['our system only requires static deformed meshes paired with skeletal configurations as input']",
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"text": "['our technique works well for a wide variety of character skins']",
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"text": "['it has limitations']",
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"text": "['character deformations in our model are only driven by the skeleton’s joint parameters']",
"type": "own_claim",
"id": "T402"
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"text": "['Our method cannot capture deformations that are driven by abstract parameters']",
"type": "own_claim",
"id": "T403"
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"text": "['“happiness”']",
"type": "data",
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"text": "['Lewis et al. 2000']",
"type": "data",
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"text": "['Sloan et al. 2001']",
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"text": "['Our system also cannot accurately reproduce deformations that are not representable as linear combinations of the transformations expressed in our skeletons']",
"type": "own_claim",
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"text": "['the scaling joints presented in this paper can only fully capture deformations that are well approximated by a scaling that is linearly related to the cosine of the angle between two bones']",
"type": "own_claim",
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"text": "['This assumption may be violated by a character whose muscle bulges only when its arm is fully bent']",
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"text": "['new joints may be added to capture any important deformation']",
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"text": "['our method produces high-quality yet fast and compact skinned characters that work with existing game engines, graphics hardware and other runtime systems']",
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A12 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A12_M02_Constraint-based_Motion_Optimization_Using_A_Statistical_Dynamic_Model_CITATION_PURPOSE_M_v1.xml">
85466c5c30862095d9d4d95af8bc76dbccf21092676aa1dbb628d1d721ff1e79
3vt6
http://dx.doi.org/10.1145/1276377.1276387
<Title>Constraint-based Motion Optimization Using A Statistical Dynamic Model</Title>
Jinxiang Chai ∗ Texas A&M University
∗
∗ e-mail: [email protected] † e-mail: [email protected]
Figure 1: Motions computed from spatial-temporal constraints.
<Abstract>In this paper, we present a technique for generating animation from a variety of user-defined constraints. We pose constraint-based motion synthesis as a maximum a posterior (MAP) problem and develop an optimization framework that generates natural motion satisfying user constraints. The system automatically learns a statistical dynamic model from motion capture data and then enforces it as a motion prior. This motion prior, together with user-defined constraints, comprises a trajectory optimization problem. Solving this problem in the low-dimensional space yields optimal natural motion that achieves the goals specified by the user. We demonstrate the effectiveness of this approach by generating whole-body and facial motion from a variety of spatial-temporal constraints. CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—animation; I.3.6 [Computer Graphics]: Methodology and Techniques—interaction techniques Keywords: human body animation, facial animation, motion control, statistical dynamic models, spatial-temporal constraints, constraint-based motion synthesis, motion capture data</Abstract>
Jessica K. Hodgins † Carnegie Mellon University
†
<H1>1 Introduction</H1>
Our objective in this paper is to design an animation system that allows users to easily create natural-looking character animation by specifying spatial-temporal constraints throughout the motion. For
example, a naive user might use a performance animation system to control the trajectories of the end-positions of the limbs of a character. A more skilled user might specify a small set of poses at key time instants. The system then automatically finds a motion that best satisfies those constraints. An ideal motion synthesis system should allow users to specify a variety of constraints either at isolated points or across the entire motion in order to accommodate users with different skill levels. One appealing solution to this problem is physically based optimization [Witkin and Kass 1988]
, which allows the user to specify various constraints throughout the motion and relies on optimization to compute the physically valid motion that best satisfies these constraints. Unfortunately, correct physics does not ensure that the motion will appear natural for characters with many degrees of freedom. Like physically based optimization, we formulate the problem as a trajectory optimization and consider the entire motion simultaneously. Instead of using the physical laws to generate physically correct animation, we rely on statistical models of human motion to generate a statistically plausible motion. Our approach allows the user to generate a wide range of human body and facial animation by specifying spatial-temporal constraints throughout the motion. The system automatically learns a statistical dynamic model from motion capture data and then enforces this model as a motion prior. The statistical dynamic model plays a role similar to that played by the dynamics in physically based optimization because it constrains the motion to only part of the space of possible human motions. The statistical dynamic model, however, is usually lower dimensional than the dynamics model, making the optimization more efficient, less likely to be subject to local minima, and more likely to produce natural motion. We demonstrate the effectiveness of this approach in two domains: human body animation and facial animation. We show that the system can generate natural-looking animation from key-frame constraints, key-trajectory constraints, and a combination of these two constraints. For example, the user can generate a walking animation from a small set of key frames and foot contact constraints ( figure 1 top). The user can also specify a small set of key trajectories for the root, hands and feet positions to generate a realistic jumping motion ( figure 1 bottom). The user can fine tune the animation by incrementally modifying the constraints. For example, the user can create a slightly different jumping motion by adjusting the positions of both hands at the top of the jump. The system can generate motions for a character whose skeletal model is markedly different from those of the subjects in the database. We also show that the system can use a statistical dynamic model learned from a normal walking sequence to create new motion such as walking on a slope. The quality of the final animation produced by our system depends on the motion priors derived from the motion capture database and the number of user-defined constraints. We, therefore, evaluate how the database influences the final motion and how increasing or decreasing the number of user-defined constraints influences the final animation. We also compare alternative techniques for generating animation from user-defined constraints such as linear interpolation, trajectory-based inverse kinematics, and inverse kinematics in a PCA subspace.
<H1>2 Background</H1>
In this paper, we construct statistical models from motion capture data and then combine these models with trajectory optimization to generate a motion that satisfies user-defined constraints. Consequently, we discuss related work in constraint-based trajectory optimization and data-driven animation with an emphasis on statistical models.
<H2>2.1 Constraint-based Trajectory Optimization</H2>
Trajectory optimization methods, which were first introduced to the graphics community by
Witkin and Kass [1988]
, provide a powerful framework for generating character animation from user-specified constraints, physics constraints, and an objective function that measures the performance of a generated motion. Extending this approach to generate natural motion for a full human character has proved to be hard because the system is high dimensional, the physics constraints make it highly nonlinear, and defining an objective function that reliably measures the naturalness of human motion is difficult. Much of the difficulty in solving this problem appears to result from the physics constraints because optimization without physics is effective for editing
[Gleicher 1998]. Therefore, one way to make the problem tractable is to simplify the governing physical laws. Both Liu and Popović [2002] and Abe and his colleagues [2004] showed that many dynamic effects can be preserved by enforcing patterns of linear and angular momentum during the motion. Reformulating the dynamics to avoid directly computing the torques also provides a significant performance improvement
[Fang and Pollard 2003]. Reducing the number of degrees of freedom to be optimized can also create tractable problems. For example, Popović and Witkin [1999]
showed that significant changes to motion capture data can be made by manually reducing the degrees of freedom to those most important for the task. Safonova and her colleagues [2004]
demonstrated that an efficient optimization can be achieved in a behavior-specific, low-dimensional space without simplifying the dynamics. More recently, Liu and her colleagues [2005]
introduced a novel optimization framework— Nonlinear Inverse Optimization—for optimizing appropriate parameters of the objective function from a small set of motion examples and then used the estimated parameters to synthesize a new locomotion. Our work also uses a trajectory optimization framework but replaces the physical dynamic model with a statistical dynamic model computed from a motion capture database.
<H2>2.2 Data-driven Motion Synthesis</H2>
Our approach is also part of an alternative set of techniques that relies on motion data to constrain the search to natural looking motions. For example, motion graphs can be used to resequence whole-body or facial motions (see, for example,
[Arikan and Forsyth 2002; Kovar et al. 2002; Lee et al. 2002; Zhang et al. 2004]. These systems cannot match poses or satisfy such kinematic constraints as end effector constraints unless the motion database happens to contain a motion that satisfies those constraints. Motion interpolation, on the other hand, does allow isolated constraints to be satisfied (for example,
[Rose et al. 1998; Kovar and Gleicher 2004; Mukai and Kuriyama 2005]). However, interpolation across a complete behavior does not have enough degrees of freedom to allow the specification of full pose constraints or end effector constraints across multiple frames. Recently, interpolation and motion graphs have been combined to obtain some of the advantages of each approach [Safonova and Hodgins 2007]. Statistical models of human motion have also been used for motion synthesis.
A number of researchers have used variants of Hidden Markov Models (HMMs) to statistically represent human motion: either full-body movements
[Molina Tanco and Hilton 2000; Brand and Hertzmann 2000; Galata et al. 2001] or speechdriven facial expressions
[Bregler et al. 1997; Brand 1999].
HMMs learned from human motion data have been used to interpolate key frames
[Molina Tanco and Hilton 2000; Galata et al. 2001], synthesize a new style of motion
[Brand and Hertzmann 2000], and generate facial expressions from speech signals
[Bregler et al. 1997; Brand 1999]. Grzeszczuk and his colleagues[1998]
developed a neural network approximation of dynamics based on simulated data and use it to animate dynamic models such as fish and lunar landers. Urtasun and her colleagues[2006]
learned linear motion models from pre-aligned motion data via Principal Component Analysis (PCA) and used them to track 3D human body movements from video by performing nonlinear optimization over a small sliding temporal window. Switching linear dynamic system (SLDS) have also been used to model human motion. Pavlović and his colleagues [2000]
present results for human motion synthesis, classification, and visual tracking using learned SLDS models. Li and his colleagues [2002]
used SLDS to synthesize and edit disco dancing motion. Our approach is also to learn a statistical dynamic model from human motion capture data; however, the dynamic behavior of our model is controlled by a continuous control state rather than a discrete hidden state as in HMMs and SLDS. This property led us to formulate the motion synthesis problem as a trajectory optimization problem. More importantly, our system allows the user to specify a variety of spatial-temporal constraints such as end effector constraints throughout the motion, a capability that has not been demonstrated by previous approaches. A number of researchers have developed statistical models for human poses and used them to solve the inverse kinematics problem. Grochow and colleagues [2004]
applied a global nonlinear dimensionality reduction technique, Gaussian Process Latent Variable Model, to human motion data and then used the learned statistical pose model to compute poses from a small set of user-defined constraints. Another solution for data-driven inverse kinematics is to interpolate a small set of preexisting examples using constraints. This idea has been used to compute human body poses [Rose et al. 2001] and facial expressions
[Zhang et al. 2004]
from kinematic constraints at a single frame. These models lack temporal information and therefore cannot be used to generate an animation from sparse constraints such as key frames. Local statistical models are sufficient if the user provides continuous control signals (the performance animation problem). Chai and colleagues [2003]
presented a real-time vision-based performance animation system that transforms a small set of automatically tracked facial features into facial animation by interpolating examples in a database at run time. They also used a series of local statistical pose models constructed at run time to reconstruct full-body motion from continuous, low-dimensional control signals obtained from video cameras
[Chai and Hodgins 2005]. The statistical dynamic model used in this paper was motivated by the dynamic model used for video textures by
Soatto and his colleagues [2001]. They showed that a sequence of images of such moving scenes as sea-waves, smoke, and whirlwinds can be modeled by second-order linear dynamic systems. They applied the learned dynamic systems to synthesize an “infinite length” texture sequence by sampling noise from a known Gaussian distribution. We extend the model to learn an efficient and low-dimensional representation of human motion and use it to generate an animation that achieves the goal specified by the user.
<H1>3 Overview</H1>
The key idea behind our approach is that motion priors learned from prerecorded motion data can be used to create natural human motion that matches constraints specified by the user. The combination of the motion prior and the user’s constraints provides sufficient information to produce motion with a natural appearance. The human body motion capture database (about 15 minutes) includes data of locomotion (jumping, running, walking, and hopping) and interacting with the environment (standing up/sitting down, reaching/picking up/placing an object). The facial expression database (about 9 minutes) includes six basic facial expressions (happiness, surprise, disgust, fear, anger, sadness) and three facial movements related to everyday life (speaking, eating, and snoring). The motion was captured with a Vicon motion capture system of 12 MX-40 cameras
[Vicon Systems 2004]
with 41 markers for full-body movements and 92 markers for facial expressions. The motion was captured at 120Hz and then downsampled to 30Hz. We denote the set of motion capture data in the database as y 1:N = [y 1 , ..., y N ], where y n , n = 1, ..., N, is the measurement of the character’s configuration in the nth frame. In facial animation, y n is the 3D positions of all vertices on the face model. In human body animation, y n is the position and orientation of the root and the joint angles. We preprocess the motion capture data by applying Principal Component Analysis (PCA)
[Bishop 1996]
to the motion capture data and obtain a reduced subspace representation for y n :
1
y n = C · x n + D
where the vector x n ∈ R d x is a low-dimensional representation of the character configuration y n ∈ R d y . The matrix C is constructed from the eigenvectors corresponding to the largest eigenvalues of the covariance matrix of the data, and D is the mean of all example data, D = (Σ N n=1 y n )/N . The dimensionality of the system state, d x , can be automatically determined by choosing the d x for which the singular values drop below a threshold. The constraints defined by the user are represented by E = {e j |j = 1, ..., J}. The goal of our constraint-based motion synthesis problem is to create an animation, H, based on the constraints, E. We formulate the constraint-based motion synthesis in a maximum a posterior (MAP) framework and consider the entire motion simultaneously. From Bayes’ theorem, the goal of MAP is to infer the
most likely motion, H, given the user-defined constraints, E:
2
p (E|H) p (H) arg max H p(H|E) = arg max H p (E) ∝ arg max H p(E|H)p(H)
where p(E)is the normalizing constant that ensures that the posterior distribution on the left-hand side is a valid probability density and integrates to one. In our implementation, we minimize the negative log of p(H|E), yielding the following optimization for motion H: ˆ
3
H ˆ = arg min H − ln p(E|H) − ln p(H)
where the first term measures how well a motion sequence matches the user-specified constraints and the second term measures a priori likelihood of the motion sequence using the knowledge embedded in human motion data. The system contains three major components: Motion prior. The system first automatically learns a statistical dynamic model from motion capture data. This model is then used to compute the motion prior, − ln p(H). User-defined Constraints. The user defines various forms of constraints, E, throughout the motion, which are then used to compute the likelihood term, − ln p(E|H). The constraints could be any kinematic constraints such as position, orientation, or the distance between two points on the character. They could be specified either at isolated points (key frames) or across the whole motion (key trajectories). Motion optimization. The system uses trajectory optimization to automatically find an animation H ˆ that best satisfies the userspecified constraints while matching the statistical properties of the motion capture data: H ˆ = arg min H − ln p(E|H) − ln p(H). We describe the statistical model in the next section and then present the three components in detail in section 5.
<H1>4 Motion Analysis</H1>
We use an m-order linear time-invariant system to describe the dynamical behavior of the captured motion in the low-dimensional space
[Ljung 1999]: m
4
x n = A i x n−i + Bu n i=1
where m is the order of the linear dynamic model. x n ∈ R d x and u n ∈ R d u are the system state and control input, and d u is the dimensionality of the control input u n . This formulation is similar to the linear time-invariant control system commonly adopted in the control community
[Palm 1999]. However, the matrix B is not unique because the control input u t is unknown. Therefore, any non-singular transformation of the matrix B represents the motion because BT and T −1 u n are also consistent with the dynamic model. To remove this ambiguity, we assume that the matrix B is an orthogonal matrix. Given the low-dimensional representation of the original motion capture data, x 1:N = [x 1 , ..., x N ], we want to identify the statespace model, including system matrices {A i |i = 1, ..., m}, B, and the corresponding control input u m+1:N = [u m+1 , ..., u N ]. The matrices {A i |i = 1, ..., m} are dependent on the distribution of u n . To eliminate the ambiguity of the matrices A i , we seek to
(a)
Figure 2: The average reconstruction error of the linear time-invariant system computed by cross-validation techniques: (a) The average per frame reconstruction error for the walking test data as a function of the order of the dynamic system (m) and the number of dimensions of the control input (d u ); (b) The average per frame reconstruction error of the facial test data as a function of the order of the dynamic system (m) and the number of dimensions of the control input (d u ).
find the {A i |i = 1, ..., m} that minimize the sum of the squared control input u n :
5
A ˆ 1 , ..., A ˆ m = arg min A 1 ,...,A m n u n 2
The matrices A i can thus be uniquely found by computing the leastsquare solution:
A ˆ 1 , ..., A ˆ m = arg min A 1 ,...,Am n=m+1 N x n − i=1 m A i x n−i 2 (6) We use the estimated matrices {A i |i = 1, ..., m} to compute the control input term:
7
z n = x n − i=1 m A ˆ i x n−i , n = m + 1, ..., N
We form a d x × (N − m) matrix by stacking the estimated control inputs z n : z m+1 ... z N = B· u m+1 ... u N (8) Z U
Equation (8) shows that without noise, the rank of the matrix Z is d u . Therefore, we can automatically determine the dimensionality of the control input u n by computing the rank of matrix Z. When noise corrupts the motion capture data, the data matrix Z will not be exactly of rank d u . However, we can perform singular value decomposition (SVD) on the data matrix Z such that Z = W SV T , and then get the best possible rank d u approximation of the data matrix, factoring it into two matrices: B ˆ = W and U ˆ = SV T , where B ˆ is a d x × d u matrix and U ˆ is a d u × (T − m) matrix. The dimensionality of the control input (d u ) can be automatically determined by choosing the d u for which the singular values drop below a threshold. Functionally, a statistical dynamic model is similar to a physical dynamic model. For example, given initial values of the system state
4
(x 1:m = [x 1 , ..., x m ]), the linear dynamic model in Equation
can be used to generate an animation (x m+1:T = [x m+1 , ..., x T ]) by sequentially choosing an appropriate value for the control input (u m+1:T = [u m+1 , ..., u T ]), just as joint torques would be used to advance a physical model through time. The main advantage
(b)
of using a statistical dynamic model for animation is that the dimensionality of the control input in a statistical dynamic model is usually much lower than a physical dynamic model. Therefore, the statistical dynamic model might achieve faster convergence and be less subject to local minima. The number of dimensions of the control input, d u , characterizes the complexity of our dynamic model. Figure 2 plots the reconstruction error of a walking test data set and a facial test data set as a function of the order of the dynamic system (m) and the number of dimensions of the control input, d u . The walk data set is from multiple subjects and contains different styles. The facial expression data are from the same subject and contain a variety of facial expressions such as “happy” and “sad.” The average reconstruction error is the L 2 distance between the original test motion and the motion reconstructed from the linear time-invariant system and computed by cross-validation techniques. We observe that the reconstruction error of the statistical model decreases as both the order of dynamic system and the number of dimensions of the control input increases. If we choose d u as “zero” (simply dropping off the control term), our model becomes the linear dynamic model used by
Soatto and colleagues [2001] and has the largest reconstruction error. If d u is equal to the number of dimensions of the system state d x , the model can be used to represent an arbitrary motion sequence with zero error. In practice, human motion is highly coordinated, and the dimensionality of the control input for accurate motion representation, d u , is often much lower than the dimensionality of the system state, d x . For the examples reported here, we set the dynamic order to three and the dimensionality of control input to four for human body animation (the reconstruction error is about 0.7 degrees/joint per frame); we set the dynamic order to two and the dimensionality of control input to one for facial movement (the reconstruction error is about 0.1 mm/vertex per frame).
<H1>5 Constraint-based Motion Synthesis</H1>
Constraint-based motion synthesis provides the user with intuitive control over the resulting motion: the user specifies a desired motion with various forms of constraints, such as key frames, end effector target positions, or joint angle values; the system then auto-
(a)
(b)
Figure 3: Various form of spatial-temporal constraints: (a) key-frame constraints for creating full-body animation; (b) key-trajectory constraints where the user selects six points on the character and then specifies their 3D trajectories across the motion (from a performance animation interface); (c) the user picks six points on the face and their screen-space position constraints at some moment in time; (d) the user defines a distance between two facial points (the width of the mouth) and controls the distance throughout the motion.
matically finds the animation that best satisfies the user-specified constraints while matching the spatial-temporal properties of the motion capture data. This section first derives the likelihood term, − ln p(E|H), based on the user-defined constraints, E. We then model the motion prior term, − ln p(H), using the learned statistical dynamic model. Finally, we discuss how to optimize motion by combining both terms: H ˆ = arg min H − ln p(E|H) − ln p(H). Like physically based optimization
[Witkin and Kass 1988], we represent the system state x t and the control signal u t independently. The motion to be synthesized is therefore represented as a sequence of system states and control inputs H = (x 1 , ..., x T , ..., u m+1 , ..., u T ). The system allows the user to specify various forms of kinematic constraints E = {e j |j = 1, ..., J} throughout the motion or at isolated points in the motion. For facial animation, the user can specify the positions or orientations of any points on the face, or the distance between any two points. For whole-body animation, the user can specify the positions or orientations of any points on the body, or joint angle values for any joints. Rather than requiring that constraints be specified in 3D, it is often more intuitive to specify where the projection of a point on the character should be located. Therefore, the system also allows the user to specify the 2D projections of any 3D point on a user-defined screen space. This approach could be used for rotoscoping a video, or for a single camera performance animation. The system allows the user to sketch out the motion in greater or lesser detail. For example, a novice user might want to control the paths of specific joints or paths over a period of time using a performance animation system while a more skilled user might prefer using key frame constraints. Spatially, the constraints could provide either an exact configuration such as a full-body pose or a small subset of the joint angles or end-positions. Temporally, the constraints could be instantaneous constraints for a particular frame, multiple-frame constraints, or continuous constraints over a period of time. User-defined constraints can be linear or nonlinear. Linear constraints can be used to define joint angle constraints in human body animation and positions in facial animation. The most common nonlinear constraints in human body animation might be end effector constraints, for example, foot contact constraints. In facial animation, nonlinear constraints can be used to specify the distance between two points on the face or 2D projections of 3D facial points. Figure 3 illustrates the user-defined constraints that were used to generate human body animation and facial animation.
<H2>5.1 User-defined Constraints</H2>
Mathematically, we can model the likelihood term, − ln p(E|H), as
(c)
(d)
follows:
E constraints = − ln p(E|H) ∼ j=1 J β e j − f j (y 1 , ..., y T ) 2 ∼ j=1 J β e j − f j (Cx 1 + D, ..., Cx T + D) 2 (9) where the function f j is usually a forward kinematics function and the parameter β is a constant specifying the importance of the constraints. The likelihood term evaluates how well the synthesized motion matches the constraints specified by the user. A good match between the motion and the user-defined constraints results in a low energy solution.
<H2>5.2 Motion Priors</H2>
Many motions might satisfy the user-defined constraints. For example, when the user specifies a small set of key frames or key trajectories, the number of constraints is not sufficient to completely determine the whole motion sequence, x 1:T . To remove ambiguities, we would like to constrain the generated motion to lie in the space of natural human motions by imposing a prior on the generated motion:
10
E prior = − ln p(H)
10
= − ln p(x 1:T , u m+1:T )
Based on the statistical dynamic equation (Equation 4), the current system state x t only depends on the previous system states x t−m:t−1 and the current control input u t . We have p(H) = p(x 1:T , u m+1:T ) T = t=m+1 p(x t |x t−1:t−m , u t ) · p(x 1:m , u m+1:T ) (11) We assume that the likelihood of the first term on the right side of Equation 11 is measured by the deviation of the statistical dynamic equation (Equation 4). We have the corresponding energy term E prior dynamic = − ln T t=m+1 p(x t |x t−1:t−m , u t ) ∼ −α T t=m+1 x t − i=1 m A i x t−i − Bu t 2 (12) where α is a tuning parameter. Conceptually, the dynamic prior can be thought as dimensionality reduction of the motion in a spatialtemporal domain. It significantly reduces the dimensionality of the motion from the space of x 1:T to the space of the initial state x 1:m and the control input u m+1:T . The second term on the right side of Equation 11 computes the prior for the initial state, x 1:m , and control input, u m+1:T . We assume that both the initial state, x 1:m , and the control input, u t , are independent and identically distributed. The energy term for the second term on the right side of Equation 11 can be simplified as follows:
Figure 4: The horizontal axis shows the iteration number and the vertical axis shows the value of the objective function for wholebody optimization The three colored curves show the evolution of the objective function values with three different initial guesses. The optimization converges within 100 iterations.
13
E prior control = − ln p(x 1:m , u m+1:T ) m T = − t=1 ln p(x t ) − t=m+1 ln p(u t )
We model the control input (u t ) as a mixture with K component Gaussian densities
[Bishop 1996]:
14
K p(u t ) = Σ k=1 π k N(u t ; φ k , Λ k )
where K is the number of Gaussian density models and π k is a mixing parameter that corresponds to the prior probability that u t was generated by the kth component. The function N(u t ; φ j , Λ j ) denotes the multivariate normal density function with mean φ j and covariance matrix Λ j . The parameters of the Gaussian mixture models (π k , φ k , Λ k ) are automatically estimated using an Expectation-Maximization (EM) algorithm
[Bishop 1996]. The training data are the values of control inputs { u n } computed from the original motion capture data ({y n |n = 1, ..., N }) (see section 4). The density function of the initial states, p(x t ), t = 1, ..., m, is also modeled as a mixture of multivariate Gaussian distributions whose parameters are learned from motion data, x 1:N , using the EM algorithm. Note that we choose weak priors (static models) to model the priors for both initial states and control inputs so as not to restrict the type of motions the algorithm can generate.
<H2>5.3 Motion Optimization</H2>
After combining the user-defined constraints and the motion prior, the constraint-based motion synthesis problem becomes the following unconstrained motion optimization problem:
15
arg min x , u E constraint + E prior dynamic + E prior control
where x and u are the concatenation of the system states x t and the concatenation of the control signals u t over the entire motion. We follow a standard approach of representing x t and u t using cubic B-splines. We solve the optimization problem using sequential quadratic programming (SQP)
[Bazaraa et al. 1993]
, where each iteration solves a quadratic programming subproblem. The Jacobian matrix and the Hessian matrix of the energy function are symbolically evaluated at each iteration. We choose all initial values using random values between 0 and 1 except that a linear interpolation of the user-specified keyframe constraints is used for initialization. We found that the optimization procedure always converges quickly (usually less than 100 iterations and less than 30 seconds). Typically, the objective function values decrease rapidly in the early iterations and then level off as they approach the optimal value. Figure 4 shows the objective function values for three different initial guesses. Our optimization framework can also be applied to the problem of generating human body motions for a character whose skeletal model is markedly different from the subjects in the database. User-defined constraints for motion retargeting can either be directly computed from the source motion or specified by the user. In our experiment, we extract foot positions from a source walking motion and then use it to generate a walking sequence for a new character. We also add one term in the objective function that measures the difference between the source motion and retargeted motion:
16
E dif f = t=1 T y t source − Cx t − D 2
where y t source is the source pose at frame t.
<H1>6 Results</H1>
We test our system by generating both human body animation and facial animation from various forms of user-defined constraints. We also evaluate the performance of our algorithm in terms of the motion priors and user-defined constraints. We learn the statistical model for each individual behavior and use it to generate individual behavior based on user-defined constraints. Two kinds of constraints were used to generate most of the examples in this paper: key-frame constraints and key-trajectory constraints. We can also combine these two constraints. For example, a jumping motion can be created by specifying a start pose and the positions of both feet and root throughout the motion. The accompanying video demonstrates the effectiveness of our system for generating a number of individual behaviors, including walking, running, and jumping. Our behavior-specific statistical motion model is capable of generating a rich variety of actions. For example, we can use a small set of key frames and foot contacts to generate normal walking, climbing over an obstacle, a baby walking, and mickey-mouse style walking. Figure 5 shows sample frames of the results. Our system can also synthesize motion that transitions from one behavior to another by using the statistical model learned from transition data. In the accompanying video, we demonstrate that the user can generate a transition from walking to jumping, from walking to sitting down, and from walking to picking up an object (figure 6). The accompanying video also shows that the system can generate motions for characters with skeletal dimensions different from those in the database. Figure 7 shows sample frames of the results. We also show that we can use motion priors learned from a small sequence of a normal walking motion (about 100 frames) to create walking on a slope and walking with small steps. The user can refine the animation by incrementally modifying the constraints. For example, the user can create a slightly different jumping motion by adjusting the positions of both hands at the top of the jump. Figure 8 shows sample frames of the results.
<H2>6.1 Full-body Animation</H2>
Figure 5: Animation generated by a small set of key frames: (top) baby walking; (middle) running; (bottom) stylized walking.
Figure 6: Our system can generate a transition from walking to jumping using a small set of key frames.
Figure 7: Motion generality: (top) The system generates motion for a character whose skeletal dimensions are different from the subjects in the database; (bottom) The system modifies a normal walking motion to create a new motion–walking on a slope.
Figure 8: The user can fine tune an animation by incrementally adding constraints: (top) jumping generated by the user using five key trajectories (both hands, both feet, and root); (bottom) a slightly different jumping motion generated after adjusting the positions of the hands at the top of the jump.
<H2>6.2 Facial Animation</H2>
The system learns a single statistical model from the whole facial motion capture database and then uses it to create facial animation with a variety of spatial-temporal constraints. The following examples are illustrated in the accompanying video: Combination of keyframe and trajectory constraints. The user can generate realistic facial animation by combining sparse keyframe constraints (three key frames) and sparse trajectory constraints (one trajectory). Sparse screen constraints. The user selects six points on the face and specifies the 2D projections on the screen space at three key instants. This type of constraint could be extracted by rotoscoping. Trajectory constraints. The user can achieve detailed control over facial movement by specifying the trajectories of a small set of 3D facial points. The user can also use trajectories of a small set of high-level facial features (the mouth width and height and the openness of the eyes) to generate facial animation.
<H2>6.3 Evaluation</H2>
The quality of the final animation depends on the motion priors and the user-defined constraints. We, therefore, have designed a number of experiments to evaluate the performance of our algorithm: The importance of the motion priors. We evaluate the importance of motion priors by comparing our method against alternative constraint-based motion synthesis methods. The first method is a simple linear interpolation of key frames. The second method is trajectory-based inverse kinematics that minimizes the velocity changes of the motion in the original configuration space, y t , without any priors. The third method is a simple data-driven inverse kinematics algorithm that minimizes the velocity changes of the motion in a reduced PCA space, x t . We compare the methods using key-frame constraints and key-trajectory constraints. We keep the constraints constant and use a cubic spline to represent the motion. The results of this comparison are shown in the video. Without the use of the statistical dynamic model, the system can not generate natural motions unless the user specifies a very detailed set of constraints across the entire motion. Motion priors from different databases. We evaluate how the database influences the final motion by keeping the user-defined constraints constant. We have experimented with both key-frame and key-trajectory constraints. For key-frame constraints, the user defined a sparse set of walking constraints and used them to generate walking motion from the priors learned from a number of different databases. We compare the results for a database of general locomotion, running, hopping, jumping and walking. The accompanying video shows that we can generate a good walking motion with a walking database. The quality of the animation becomes worse when we use a large and general locomotion database to generate walking. As would be expected, the system fails to generate a good walking motion if the motion prior is learned from running, hopping, or jumping data. We have tested the creation of jumping motion from key-trajectory jumping constraints when the prior is learned from a database of jumping, general locomotion, or walking. Similarly, the prior from a walking database fails to generate a good jumping motion because of the mismatch between the prior and the user-defined constraints. Different numbers of constraints. With an appropriate database, we compare the quality of motions generated by different numbers of constraints. More specifically, we take one motion sequence out of the database and use it as a testing sequence. We then compare the animations created by key frames that are spaced increasingly far apart in time. We also compare the results by decreasing the number of key trajectories. The accompanying video shows that results become worse when we decrease the number of the userdefined constraints. For example, the numerical error increases steadily (0.94, 1.06, 1.81 degrees per joint per frame) when the number of constraints is decreased (6, 4, 2 key frames). We observe a noticeable foot sliding artifact on one foot when two key trajectories (root and one foot) are used to create a walking motion.
<H1>7 Discussion</H1>
We have presented an approach for generating both full-body movement and facial expression from spatial-temporal constraints while matching the statistical properties of a database of captured motion. The system automatically learns a low-dimensional linear dynamic model from motion capture data and then enforces this as spatial-temporal priors to generate the motion. The statistical dynamic equations, together with an automatically derived objective function and user-defined constraints, comprise a trajectory optimization problem. Solving this optimization problem in the lowdimensional space yields optimal, natural motion that achieves the goals specified by the user. The system achieves a degree of generality beyond the motion capture data. For example, we have generated a motion using constraints that cannot be satisfied directly by any motion in the database and found that the quality of the reconstructed motion was acceptable. Our video also demonstrates that the system can generate motion for characters whose skeletal models differ significantly from those in the database. However, we have not yet attempted to assess how far the user’s constraints can stray from the motions in the database before the quality of the resulting animation declines to an unacceptable level. This statistically based optimization approach complements a physically based optimization approach and offers a few potential advantages. First, using a low-dimensional statistical dynamic model for the constrained optimization might achieve faster convergence and be less subject to local minima. Second, our approach can generate slow and even stylized motions that have proven particularly difficult for physically based optimization. Third, the optimization does not require physical models. Building anatomically accurate physical models for facial animation or whole-body motion remains challenging. There are two limitations of our approach: an appropriate database must be available and the user cannot specify such dynamic constraints as ground reaction forces or character mass. The main focus of this paper has been an exploration of the use of prior knowledge in motion capture data to generate natural motion that best satisfies user-defined constraints. Another important issue for building any interactive animation system is to design an intuitive interface to specify the desired motion. In our experiments, most of keyframe constraints were modified from example poses in the database. Foot contact constraints were specified by the user directly. Key trajectory constraints were extracted from a performance interface using two video cameras
[Chai and Hodgins 2005]. Alternatively, the user could rely on commercial animation software such as Maya to specify constraints. This process is timeconsuming even for a professional artist; it is more difficult for a naive user to specify such constraints. One of immediate directions for future work is, therefore, to design intuitive interfaces that allow the user to specify spatial-temporal constraints quickly and easily.
<H1>Acknowledgements</H1>
The authors would like to thank Moshe Mahler for his help in modeling and rendering the images for this paper and Autodesk for the donation of Maya software. Partial support for this research was provided by NSF IIS-0326322.
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"text": "['x n = A i x n−i + Bu n i=1']",
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"text": "['Ljung 1999']",
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"text": "['This formulation is similar to the linear time-invariant control system commonly adopted in the control community']",
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"text": "['BT and T −1 u n are also consistent with the dynamic model']",
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"text": "['the low-dimensional representation of the original motion capture data, x 1:N = [x 1 , ..., x N ]']",
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"text": "['To eliminate the ambiguity of the matrices A i , we seek to']",
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"text": "['The matrices A i can']",
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"text": "['Equation (8)']",
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"text": "['This approach could be used for rotoscoping a video, or for a single camera performance animation']",
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"text": "['The system allows the user to sketch out the motion in greater or lesser detail']",
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"text": "['Linear constraints can be used to define joint angle constraints in human body animation and positions in facial animation']",
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"text": "['In facial animation, nonlinear constraints can be used to specify the distance between two points on the face or 2D projections of 3D facial points']",
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"text": "['E constraints = − ln p(E|H) ∼ j=1 J β e j − f j (y 1 , ..., y T ) 2 ∼ j=1 J β e j − f j (Cx 1 + D, ..., Cx T + D) 2']",
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"text": "['A good match between the motion and the user-defined constraints results in a low energy solution']",
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"text": "['the number of constraints is not sufficient to completely determine the whole motion sequence, x 1:T']",
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"text": "['To remove ambiguities, we would like to constrain the generated motion to lie in the space of natural human motions by imposing a prior on the generated motion']",
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"text": "['= − ln p(x 1:T , u m+1:T )']",
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"text": "['the current system state x t only depends on the previous system states x t−m:t−1 and the current control input u t']",
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"text": "['Based on the statistical dynamic equation']",
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"text": "['Equation 4']",
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"text": "['We have p(H) = p(x 1:T , u m+1:T ) T = t=m+1 p(x t |x t−1:t−m , u t ) · p(x 1:m , u m+1:T )']",
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"text": "['the likelihood of the first term on the right side of Equation 11 is measured by the deviation of the statistical dynamic equation']",
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"text": "['We have the corresponding energy term E prior dynamic = − ln T t=m+1 p(x t |x t−1:t−m , u t ) ∼ −α T t=m+1 x t − i=1 m A i x t−i − Bu t 2']",
"type": "own_claim",
"id": "T205"
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"text": "['Conceptually, the dynamic prior can be thought as dimensionality reduction of the motion in a spatialtemporal domain']",
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"text": "['It significantly reduces the dimensionality of the motion from the space of x 1:T to the space of the initial state x 1:m and the control input u m+1:T']",
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"id": "T207"
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"text": "['The energy term for the second term on the right side of Equation 11 can be simplified as follows']",
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"id": "T208"
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{
"offsets": [
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"text": "['E prior control = − ln p(x 1:m , u m+1:T ) m T = − t=1 ln p(x t ) − t=m+1 ln p(u t )']",
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"id": "T209"
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"text": "['We model the control input (u t ) as a mixture with K component Gaussian densities']",
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"text": "['K p(u t ) = Σ k=1 π k N(u t ; φ k , Λ k )']",
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"id": "T211"
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"text": "['Bishop 1996']",
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"text": "['he function N(u t ; φ j , Λ j ) denotes the multivariate normal density function with mean φ j and covariance matrix Λ j']",
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"text": "['The parameters of the Gaussian mixture models']",
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"text": "['are automatically estimated using an Expectation-Maximization']",
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"text": "['algorithm']",
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"text": "['Bishop 1996']",
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"text": "['The density function of the initial states, p(x t ), t = 1, ..., m, is also modeled as a mixture of multivariate Gaussian distributions whose parameters are learned from motion data, x 1:N , using the EM algorithm']",
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"text": "['The training data are the values of control inputs { u n } computed from the original motion capture data ({y n |n = 1, ..., N })']",
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"text": "['section 4']",
"type": "data",
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"text": "['we choose weak priors']",
"type": "own_claim",
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"offsets": [
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"text": "['to model the priors for both initial states and control inputs so as not to restrict the type of motions the algorithm can generate']",
"type": "own_claim",
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"offsets": [
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"text": "['After combining the user-defined constraints and the motion prior, the constraint-based motion synthesis problem becomes the following unconstrained motion optimization problem']",
"type": "own_claim",
"id": "T223"
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"text": "['arg min x , u E constraint + E prior dynamic + E prior control']",
"type": "own_claim",
"id": "T224"
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"offsets": [
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"text": "['We follow a standard approach of representing x t and u t using cubic B-splines']",
"type": "own_claim",
"id": "T225"
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"text": "['We solve the optimization problem using sequential quadratic programming']",
"type": "own_claim",
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"text": "['Bazaraa et al. 1993']",
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"text": "['where each iteration solves a quadratic programming subproblem']",
"type": "own_claim",
"id": "T228"
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"text": "['the optimization procedure always converges quickly']",
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"text": "['usually less than 100 iterations and less than 30 seconds']",
"type": "own_claim",
"id": "T230"
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"text": "['Typically, the objective function values decrease rapidly in the early iterations and then level off as they approach the optimal value']",
"type": "own_claim",
"id": "T231"
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"offsets": [
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"text": "['Figure 4']",
"type": "data",
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"offsets": [
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"text": "['Our optimization framework can also be applied to the problem of generating human body motions for a character whose skeletal model is markedly different from the subjects in the database']",
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"id": "T233"
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"text": "['User-defined constraints for motion retargeting can either be directly computed from the source motion or specified by the user']",
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"text": "['We also add one term in the objective function that measures the difference between the source motion and retargeted motion']",
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"id": "T235"
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"offsets": [
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"text": "['E dif f = t=1 T y t source − Cx t − D 2']",
"type": "own_claim",
"id": "T236"
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{
"offsets": [
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"text": "['Two kinds of constraints were used to generate most of the examples in this paper']",
"type": "own_claim",
"id": "T237"
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"text": "['key-frame constraints']",
"type": "own_claim",
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"text": "['key-trajectory constraints']",
"type": "own_claim",
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"text": "['We can also combine these two constraints']",
"type": "own_claim",
"id": "T240"
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"text": "['a jumping motion can be created by specifying a start pose and the positions of both feet and root throughout the motion']",
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"text": "['Our behavior-specific statistical motion model is capable of generating a rich variety of actions']",
"type": "own_claim",
"id": "T242"
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"text": "['we can use a small set of key frames and foot contacts to generate normal walking, climbing over an obstacle, a baby walking, and mickey-mouse style walking']",
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"text": "['Figure 5']",
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"text": "['Our system can also synthesize motion that transitions from one behavior to another by using the statistical model learned from transition data']",
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"text": "['the user can generate a transition from walking to jumping, from walking to sitting down, and from walking to picking up an object']",
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"text": "['figure 6']",
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"text": "['video']",
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"id": "T248"
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"offsets": [
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"text": "['the system can generate motions for characters with skeletal dimensions different from those in the database']",
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"text": "['Figure 7']",
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"offsets": [
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"text": "['we can use motion priors learned from a small sequence of a normal walking motion']",
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"text": "['The user can refine the animation by incrementally modifying the constraints']",
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"id": "T253"
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"text": "['the user can create a slightly different jumping motion by adjusting the positions of both hands at the top of the jump']",
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"text": "['Figure 8']",
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"text": "['The system learns a single statistical model from the whole facial motion capture database and then uses it to create facial animation with a variety of spatial-temporal constraints']",
"type": "own_claim",
"id": "T256"
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"text": "['The user can generate realistic facial animation by combining sparse keyframe constraints']",
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"text": "['and sparse trajectory constraints']",
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"text": "['The user selects six points on the face and specifies the 2D projections on the screen space at three key instants']",
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"text": "['This type of constraint could be extracted by rotoscoping']",
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"id": "T260"
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"text": "['The user can achieve detailed control over facial movement by specifying the trajectories of a small set of 3D facial points']",
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"text": "['The user can also use trajectories of a small set of high-level facial features']",
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"text": "['to generate facial animation']",
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"id": "T263"
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"text": "['The quality of the final animation depends on the motion priors and the user-defined constraints']",
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"id": "T265"
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"text": "['We evaluate the importance of motion priors by comparing our method against alternative constraint-based motion synthesis methods']",
"type": "own_claim",
"id": "T266"
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"text": "['The first method is a simple linear interpolation of key frames']",
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"id": "T267"
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"text": "['The second method is trajectory-based inverse kinematics that minimizes the velocity changes of the motion in the original configuration space, y t , without any priors']",
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"id": "T268"
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"offsets": [
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"text": "['The third method is a simple data-driven inverse kinematics algorithm that minimizes the velocity changes of the motion in a reduced PCA space, x t']",
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"text": "['Without the use of the statistical dynamic model, the system can not generate natural motions unless the user specifies a very detailed set of constraints across the entire motion']",
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"text": "['We evaluate how the database influences the final motion by keeping the user-defined constraints constant']",
"type": "own_claim",
"id": "T271"
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"text": "['we can generate a good walking motion with a walking database']",
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"text": "['For key-frame constraints, the user defined a sparse set of walking constraints and used them to generate walking motion from the priors learned from a number of different databases']",
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"text": "['The quality of the animation becomes worse when we use a large and general locomotion database to generate walking']",
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"text": "['the motion prior is learned from running, hopping, or jumping data']",
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"text": "['the prior from a walking database fails to generate a good jumping motion']",
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"text": "['results become worse when we decrease the number of the userdefined constraints']",
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"text": "['the numerical error increases steadily']",
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"text": "['0.94, 1.06, 1.81 degrees per joint per frame']",
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"text": "['We observe a noticeable foot sliding artifact on one foot']",
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"text": "['two key trajectories']",
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"text": "['are used to create a walking motion']",
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"text": "['We have presented an approach for generating both full-body movement and facial expression from spatial-temporal constraints while matching the statistical properties of a database of captured motion']",
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"text": "['The system automatically learns a low-dimensional linear dynamic model from motion capture data and then enforces this as spatial-temporal priors to generate the motion']",
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"text": "['The statistical dynamic equations, together with an automatically derived objective function and user-defined constraints, comprise a trajectory optimization problem']",
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"text": "['Solving this optimization problem in the lowdimensional space yields optimal, natural motion that achieves the goals specified by the user']",
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"text": "['The system achieves a degree of generality beyond the motion capture data']",
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"id": "T292"
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"text": "['we have generated a motion using constraints that cannot be satisfied directly by any motion in the database and found that the quality of the reconstructed motion was acceptable']",
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"text": "['the system can generate motion for characters whose skeletal models differ significantly from those in the database']",
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"text": "['video']",
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"id": "T295"
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"offsets": [
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"text": "['we have not yet attempted to assess how far the user’s constraints can stray from the motions in the database before the quality of the resulting animation declines to an unacceptable level']",
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"text": "['This statistically based optimization approach complements a physically based optimization approach and offers a few potential advantages']",
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"id": "T297"
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{
"offsets": [
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"text": "['using a low-dimensional statistical dynamic model for the constrained optimization might achieve faster convergence and be less subject to local minima']",
"type": "own_claim",
"id": "T298"
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"offsets": [
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"text": "['our approach can generate slow and even stylized motions that have proven particularly difficult for physically based optimization']",
"type": "own_claim",
"id": "T299"
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"text": "['the optimization does not require physical models']",
"type": "own_claim",
"id": "T300"
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{
"offsets": [
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"text": "['Building anatomically accurate physical models for facial animation or whole-body motion remains challenging']",
"type": "own_claim",
"id": "T301"
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{
"offsets": [
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"text": "['There are two limitations of our approach']",
"type": "own_claim",
"id": "T302"
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"text": "['an appropriate database must be available']",
"type": "own_claim",
"id": "T303"
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"offsets": [
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"text": "['the user cannot specify such dynamic constraints as ground reaction forces or character mass']",
"type": "own_claim",
"id": "T304"
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{
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"text": "['The main focus of this paper has been an exploration of the use of prior knowledge in motion capture data to generate natural motion that best satisfies user-defined constraints']",
"type": "own_claim",
"id": "T305"
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{
"offsets": [
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"text": "['Another important issue for building any interactive animation system is to design an intuitive interface to specify the desired motion']",
"type": "own_claim",
"id": "T306"
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{
"offsets": [
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"text": "['In our experiments, most of keyframe constraints were modified from example poses in the database']",
"type": "own_claim",
"id": "T307"
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{
"offsets": [
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"text": "['Foot contact constraints were specified by the user directly']",
"type": "own_claim",
"id": "T308"
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{
"offsets": [
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],
"text": "['Key trajectory constraints were extracted from a performance interface using two video cameras']",
"type": "own_claim",
"id": "T309"
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{
"offsets": [
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],
"text": "['Chai and Hodgins 2005']",
"type": "data",
"id": "T310"
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"offsets": [
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],
"text": "['the user could rely on commercial animation software such as Maya to specify constraints']",
"type": "own_claim",
"id": "T311"
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A30 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A30_F10_Matching_Fluid_Simulation_Elements_to_Surface_Geometry_and_Topology_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1833349.1778784
<Title>Matching Fluid Simulation Elements to Surface Geometry and Topology</Title>
Tyson Brochu ∗ Christopher Batty ∗ Robert Bridson ∗ University of British Columbia University of British Columbia University of British Columbia
∗
∗ e-mail: {tbrochu|batty|rbridson}@cs.ubc.ca
Figure 1: Sphere Splash. Coupling an explicit surface tracker to a Voronoi simulation mesh built from pressure points sampled in a geometry-aware fashion lets us capture very fine details in this sphere splash animation that uses only 314K tetrahedra.
<Abstract>We introduce an Eulerian liquid simulation framework based on the Voronoi diagram of a potentially unorganized collection of pressure samples. Constructing the simulation mesh in this way allows us to place samples anywhere in the computational domain; we exploit this by choosing samples that accurately capture the geometry and topology of the liquid surface. When combined with highresolution explicit surface tracking this allows us to simulate nearly arbitrarily thin features, while eliminating noise and other artifacts that arise when there is a resolution mismatch between the simulation and the surface—and allowing a precise inclusion of surface tension based directly on and at the same resolution as the surface mesh.</Abstract>
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation Keywords: fluid simulation, liquids, meshes, surface tension
<H1>1 Introduction</H1>
One of the most visually compelling aspects of liquids is the variety of complex thin sheets and droplets that arise during splashing. However, these remain among the most difficult features to simulate plausibly and accurately with existing techniques. Such detailed behaviour is extremely computationally expensive to resolve because of the tremendous grid resolution required for both the fluid solver and the surface tracking mechanism. Recent advances in explicit surface tracking with triangle meshes
[Wojtan et al. 2009; Brochu and Bridson 2009; Müller 2009]
have made feasible the geometric representation and manipulation of small features, without the loss of detail exhibited by implicit surface methods. However, when the surface is coupled to a standard Eulerian simulator, the liquid volume must first be resampled onto the simulation mesh or grid to provide geometric information for boundary conditions. As this resampling process typically destroys small details, they are invisible to the fluid solver and cannot be advanced appropriately. This can lead to a variety of visible artifacts including lingering surface noise, liquid behaving as if it were connected when it is not (and vice versa), and thin features simply halting in mid-air because the simulator fails to see them
[Bargteil et al. 2006; Kim et al. 2009]. When combined with surface tension forces, noisy sub-mesh details can also severely hamper stability if they are not artificially smoothed out. We will address these problems by constructing a simulator that “sees” every detail in the explicit liquid surface. We carefully generate pressure sample points near the liquid surface, build a Voronoi simulation mesh from these points and a background lattice, and apply a ghost fluid/finite volume pressure discretization which captures the precise position of the liquid interface. We couple this with a semi-Lagrangian advection scheme and a new approach to surface tension, arriving at a complete liquid simulator. In summary, our key contribution is coupling an explicit surface tracker to a Voronoi-based liquid simulator with: • a pressure sample placement strategy that captures the complete liquid surface geometry, • an accurate surface tension model combining mesh-based curvature estimates and ghost fluid boundary conditions, • embedded free surface and solid boundary conditions adapted to Voronoi cells, avoiding the need for more onerous conforming tetrahedral mesh generation, • and a new velocity interpolant over unstructured meshes. The practical benefits of such a system include: • improved animation of detailed liquid features, including very thin sheets, tendrils and droplets, • elimination of noise in explicit surface tracking without nonphysical smoothing, • more detailed and less damped surface tension effects, • and faster semi-Lagrangian advection on unstructured meshes without increased dissipation.
<H1>2 Related Work</H1>
<H2>2.1 Unstructured Mesh Fluids</H2>
Unstructured and semi-structured meshes have a long history in computational fluid dynamics, and have gained traction in computer animation as well. An important reason for their popularity is that careful control of mesh geometry can simplify the discretization or improve accuracy. For example, conforming the simulation mesh to solid walls makes the no-flow boundary condition trivial, and adaptivity can be easily introduced by grading mesh elements as desired. Past work in graphics has extensively explored finite volume methods for tetrahedral meshes
[Feldman et al. 2005a; Feldman et al. 2005b; Klingner et al. 2006; Chentanez et al. 2006; Elcott et al. 2007; Wendt et al. 2007; Chentanez et al. 2007], and now many of the features of standard grid-based solvers are supported on tetrahedra, including free surfaces and implicit coupling to dynamic solids.
Batty et al. [2010] augmented this approach with embedded boundaries
[Enright et al. 2003; Batty et al. 2007]
, improving free surface accuracy and reducing remeshing complexity. Our method extends these advantages to Voronoi meshes. In a related approach,
Sin et al. [2009]
developed a particle method which solves a finite volume pressure projection on the Voronoi diagram of the liquid particles. An advantage of this approach is that the pressure degrees of freedom are directly tied to the number of particles, so there can never be a resolution mismatch between surface geometry and simulator. This idea motivates our work. Franklin & Lee [2010]
subdivide polyhedra into tetrahedra for interpolation similar to our method, but our method is simpler due to use of the Voronoi diagram.
<H2>2.2 Surface tracking</H2>
Implicit surfaces have long been used to capture liquid geometry in animation; this family of schemes includes level set (LS) methods
[Enright et al. 2002a], volume-of-fluid (VOF)
[Mihalef et al. 2006; Mullen et al. 2007], and semi-Lagrangian contouring (SLC)
[Bargteil et al. 2006]. Implicit approaches naturally yield smooth surfaces and seamlessly handle topological change. However, the resolution of the underlying grid imposes a severe limit on the smallest representable feature, beyond which geometry either vanishes (LS, SLC) or artificially coalesces into grid-scale “flotsam and jetsam” (VOF). Ensuring temporal coherence and avoiding visual artifacts due to the use of regular grids can also be problematic. The shortcomings of implicit schemes have spurred interest in explicit methods, i.e. “front tracking”
[Glimm et al. 1998]. Here the surface is represented explicitly as a triangle mesh, whose vertices are moved with the fluid velocity field. The greatest challenge is handling topological change, due to mesh tangling that may occur during merging and splitting. One solution is to determine problematic regions, switch to an implicit surface to repair the tangles there, then stitch back in a new consistent mesh patch
[Du et al. 2006; Wojtan et al. 2009]. Müller [2009]
takes a similar grid-based approach to untangling, rebuilding a consistent mesh using marching-cubeslike stencils. Unfortunately these methods still are subject, in complex regions, to a resolution limited by the voxel grid. Another approach is to work strictly on the triangle mesh itself, using “mesh surgery” for repairs. While this is difficult in general, Brochu & Bridson [2009]
recently showed that the problem can be simplified using ideas from cloth animation, enforcing the invariant that the surface remain intersection-free. Topological operations are only allowed when safe, while robust collision processing is used as a last resort to avoid tangles, i.e. the surface is minimally perturbed to avoid problems. We use this method in the presented examples, though note that other front tracking methods could easily be used instead—for example, recent work by
Campen & Kobbelt [2010]
suggests that the need for collision processing could be obviated with exact Boolean operations.
Figure 2: Explicit Surface Tracking. Our method exploits the El Topo explicit mesh tracking software to capture thin features.
<H2>2.3 Surface Resolution vs. Simulation Resolution</H2>
A prime focus of our work is matching the surface mesh resolution to that of the liquid solver. Most level set-based solvers use one level set sample per pressure grid cell, conservatively avoiding resolution inconsistencies (e.g.
[Foster and Fedkiw 2001; Enright et al. 2002b]). Goktekin et al. [2004]
experimented with a doubleresolution level set, trading better volume conservation for other artifacts. Bargteil et al. [2006]
similarly coupled an octree contouring method to a uniform grid fluid solver and explicitly discussed potential artifacts due to resolution mismatch, such as erroneously preserving surface noise and the solver interpreting disconnected fluid regions as connected. Kim et al. [2009]
coupled a high resolution particle level set to a low resolution ghost fluid-based liquid solver, but ensured that pressure projection captured all liquid geometry by resampling an inflated level set at the pressure grid resolution—however, this can exacerbate other artifacts, since liquid components behave as if half a cell-width larger than they appear. Kim et al. also introduced extra surface smoothing to prevent retention of small-scale noise. Mismatched resolutions have been found useful for deformable solids, particularly as surface details are expected to generally persist, unlike in liquids. For example, Wojtan & Turk [2008]
used a surface mesh coupled to a lower resolution finite element solver; forcing the simulation mesh to have the same topology, if not resolution, as the embedded surface mesh may improve realism [Teran et al. 2005; Nesme et al. 2009].
<H2>2.4 Surface Tension Models</H2>
Approaches to surface tension generally fall into two categories: those which apply surface tension as a body force in a region around the interface via smeared delta functions
[Brackbill et al. 1992; Hong and Kim 2003; Zheng et al. 2006; Wojtan et al. 2009], and those which apply surface tension discontinuously at the interface, typically as a boundary condition in the pressure projection step. The latter is exemplified by the ghost fluid method and related approaches
[Enright et al. 2003; Hong and Kim 2005; Hong et al. 2007]
, and has been shown to provide more realistic results. Surface tension models can also be compared in terms of how the force itself is approximated. In level set schemes, finite differences are often used to estimate mean curvature, though this can be quite inaccurate without careful modification (e.g.
[Shin 2007]) and cannot capture small details.
If a surface mesh is available, a more accurate approach is either to use mesh-based curvature operators (e.g.
[Meyer et al. 2002b]), or as proposed recently, to model a physical tension directly in the surface mesh geometry
[Perot and Nallapati 2003; Brochu 2006; Wojtan and Turk 2008]. We take the best of each, computing an accurate force from the surface mesh and incorporating it precisely at the surface with the ghost fluid method. We also remedy a shortcoming of existing mesh-based approaches: that surface details below the simulation resolution add energy but cannot be correctly evolved by the solver; without correct feedback from the physics this noise tends to worsen and destroy stability. Wojtan & Turk [2008]
handle this with Laplacian smoothing to eliminate small features: note, however, this non-physical operation is dissipative rather than conservative. By instead combining our surface tension model with a geometry-aware sampling, we ensure all relevant details are properly resolved. This yields accurate and comparatively stable surface tension effects without artificial smoothing.
<H1>3 Algorithm Outline</H1>
We simulate inviscid liquids with semi-Lagrangian advection and an embedded-boundary finite volume pressure projection. We generally follow the tetrahedral scheme of
Batty et al. [2010] with modifications to use specially designed Voronoi meshes instead. Like Sin et al. [2009], we place pressure samples on the vertices of a Delaunay tetrahedral mesh, corresponding to the sites of the dual Voronoi diagram (figures 3(a) and 3(b)). Normal components of velocity lie on the faces of the Voronoi cells, so that the velocity sample is parallel to the line segment connecting the pressure samples in the Delaunay mesh. This configuration requires a slightly different velocity reconstruction compared to previous methods, but semi-Lagrangian advection is otherwise straightforward. For front tracking, we used Brochu & Bridson’s
El Topo code [2009]
, in particular using its triangle mesh surface to determine the location of pressure samples for our Voronoi simulation mesh. Purely explicit front tracking algorithms generally use mesh refinement and coarsening to maintain a high quality discretization as the surface deforms. El Topo uses a sequence of edge subdivision, collapse and flipping operations, combined with null-space Laplacian smoothing. While these operations change mesh connectivity, they are designed to be geometry-preserving.
For example, the smoothing moves vertices only in the null space of the local quadric metric tensor
[Garland and Heckbert 1997], as suggested by
Jiao [2007]. If the vertex lies on a locally smooth patch it is moved in the plane tangent to the surface, but if on a ridge or corner it is moved only along this line. Therefore, sharp features are preserved, allowing the present paper’s algorithm to handle them physically.
El Topo.
The solver runs through the following stages each time step: 1. Advect the explicit surface with 2. Generate a new simulation mesh as the Voronoi diagram of a lattice with extra samples near the liquid surface (section 5). 3. Advect velocities onto the new mesh with semi-Lagrangian advection (section 6). 4. Add external forces—typically just gravity. 5. Solve for the embedded-boundary pressure projection on the Voronoi mesh, including surface tension forces (section 4).
(a) (b) (c) (d)
Figure 3: Embedded boundaries on Voronoi/Delaunay meshes. Pressure samples are shown as green circles. (a) Delaunay triangulation. (b) Voronoi diagram dual to the Delaunay triangulation (velocity components for the central cell are shown as red arrows). (c) Computation of ghost fluid weights on the edges of the triangulation. (d) Computation of non-solid weights on the faces of the Voronoi diagram. In 2D, Voronoi faces are simply line segments, so solid weights are just fractions of segment lengths. In 3D, Voronoi faces are convex polygons, so determining non-solid weights involves computing polygon areas.
<H1>4 Embedded Boundaries on Voronoi Meshes</H1>
We use finite volumes on a Voronoi mesh for the pressure projection step, similar to
Sin et al. [2009]. However, rather than applying boundary conditions as they describe, we adapt the embedded boundary methods of
Batty et al. [2010] to Voronoi meshes. Conveniently, the duality/orthogonality relationship between Voronoi and Delaunay meshes lets the accuracy benefits of the method carry over. Figure 3 illustrates our mesh configuration, and the computation of the required weights, as discussed below. We solve the resulting symmetric positive definite linear system using incomplete Cholesky-preconditioned conjugate gradients. To enforce embedded solid boundary conditions, we need to estimate the partial unobstructed area of each element face ( figure 3(d) ). Batty et al. [2010]
used marching triangles cases for computing tetrahedra face fractions from signed distance values on the vertices. However, in the Voronoi setting, the faces are arbitrary convex planar polygons rather than triangles. To handle this, we temporarily place an extra vertex at the face centroid, and use it to triangulate the face. We then use signed distance estimates at the vertices to compute each sub-triangle’s partial area, and sum them to determine the partial area for the complete face. The embedded (ghost fluid) free surface condition uses signed distance estimates at pressure samples to estimate the surface position; these are now located at Voronoi sites rather than tetrahedra circumcenters, but the method is otherwise unchanged ( figure 3(c) ). A slight improvement can be achieved by casting rays to find the exact position of the surface mesh between pressure samples. In some cases this is much more accurate than the estimate derived from signed distances, but in practice we found it made minimal visual difference. To actually compute the liquid signed distance field on the tetrahedral mesh, we compute exact geometric distance for a narrow band of tetrahedra near the surface, then use a graph- based propagation of closest triangle indices to roughly fill in the rest of the mesh. This family of redistancing schemes is described by
Bridson [2008], and is easily adapted to tetrahedra.
Figure 4: Surface Tension. Our accurate surface tension model captures capillary waves even on relatively low resolution meshes. From left to right: A cube in zero gravity begins to collapse due to surface tension, inverts to become an octahedron, and continues to oscillate rapidly before settling down to a sphere.
<H2>4.1 Surface Tension</H2>
To incorporate surface tension, we follow Enright et al. [2003] in setting the free surface pressure p fs = p air + γκ fs , where p air is the constant air pressure, γ is the surface tension coefficient and κ fs is the mean curvature of the surface. Rather than using level set finite differences, we compute curvature directly from the surface mesh to accurately capture high-frequency features. We chose the operator of
Meyer et al. [2002b]
because it provides high quality estimates using just the one-ring of triangles surrounding each vertex, but others could work too. Curvature is evaluated at the intersection point between the the triangle mesh surface and the line joining an interior pressure sample to an exterior one. Often this intersection point will coincide with a surface mesh vertex due to our choice of sampling scheme; where it does not, we use simple linear interpolation between the vertices of the surface triangle mesh. This method appears highly accurate, and leads to much less damping than that of
Wojtan et al. [2009].
<H1>5 Mesh Generation</H1>
An advantage of a Voronoi-based discretization is the freedom to explicitly choose pressure sample locations, which is critical for accurate ghost fluid free surface conditions as the signed distance at these samples communicate the surface geometry to the solver. We can visualize the solver’s “knowledge” by contouring this level set: figures 5 and 6 illustrate how uniform sampling may fail. Careful pressure sample placement with respect to the surface helps in three important ways. First, we can inform the solver of all local geometric extrema, allowing the physics to act upon them correctly. This eliminates the accumulation of erroneous surface noise without requiring non-physical smoothing; this is especially vital for surface tension where spurious noise affects the curvature estimates and induces disastrously large yet futile compensating velocities that destabilize the simulation. Second, we can ensure that the solver sees the correct surface topology so that the physics responds to merging or splitting only when the surface mesh itself merges or splits. Lastly, grid-scale features often disappear and reappear in regular grid sampling, from the perspective of the solver, as the surface translates through the grid. By specifically placing points inside such small features, we ensure they cannot be missed. Comparison to Adaptive Lattices: The brute-force approach to these issues is to locally refine using octree grids or graded BCC lattice tetrahedra to capture smaller features. However, this scales poorly since many of the extra samples yield little benefit, while incurring memory and computational overhead. Furthermore, there remains no guarantee that features below the smallest grid cell size will be captured. By choosing sample points to precisely capture the geometry rather than naıvely increasing sample density, we can guarantee sampling of features which would require potentially orders of magnitude more samples with pure adaptive lattices. Comparison to Conforming Tetrahedra: While the tetrahedral method of
Chentanez et al. [2007] also builds a volumetric mesh that attempts to respect the liquid surface, it matches boundary faces rather than positioning pressure samples. This is considerably more difficult than non-conforming Delaunay tetrahedralization, and generally requires more Steiner points, worse-shaped tetrahedra, and/or the loss of the Delaunay property. Since our method uses embedded boundary conditions, we do not require conforming elements. (Note that this advantage is shared by the method of Batty et al. [2010].) Moreover, the position of pressure samples plays a more important role in free surface conditions than the position of element faces. As accuracy requires that tetrahedral schemes store pressures at circumcenters
[Klingner et al. 2006; Batty et al. 2010], and since circumcenters often lie outside their associated tetrahedra, even filling a thin feature with conforming tetrahedra provides no guarantee that its interior will be sampled at all.
Figure 5: Left: Even with the ghost fluid method, regular sampling may miss surface details which do not align with the simulation mesh, such as this wave crest. Right: Adaptive samples (shown in red) placed on either side of each mesh vertex ensure that all geometric detail is resolved by the simulation.
<H2>5.1 Pressure sample placement strategy</H2>
We begin by choosing a characteristic length scale for the simulation, ∆x, and configure El Topo to try to maintain triangle edge 1 3 lengths in the range [ 2 ∆x, 2 ∆x]. To resolve all surface details with our volumetric mesh, we need to place pressure samples so that they capture the surface’s local geometric extrema, i.e. around surface mesh vertices. In particular, we try to ensure that one edge of the Delaunay triangulation passes through each surface vertex, with one sample inside and one outside. Therefore we take the inward and outward normal at each surface vertex (averaged from the incident surface triangles), and attempt to place a pressure sample 1 a short distance along each. We placed outward samples at 2 ∆x 1 and inner samples at 4 ∆x, though other ratios would work as well. As a result, surface mesh normal directions will often align exactly with a velocity sample in the simulation mesh; this lends additional accuracy to the vertex’s normal motion, and to the incorporation of the normal force due to surface tension calculated at the vertex. This placement may miss very thin sheets or other fine structures: to robustly sample such features, we check line segments of length ∆x from each surface vertex in both offset directions for intersection with the rest of the surface mesh. If we find any triangle closer than ∆x, we store the distance d to the closest intersection, and use d in place of ∆x in the offset distance calculations above (see figure 7 ). We further reject new pressure samples which are too close to an existing sample by some epsilon, which would cause a very short edge in the final mesh. If the distance between the surface vertex and the first intersection 1 is below some threshold (e.g. 20 ∆x) at which we consider the two surfaces to have effectively collided, and the proposed sample is an air sample, we also discard it. This is necessary because the divergence constraint is not enforced on air cells, so they can act as liquid sinks
[Losasso et al. 2006] and destroy liquid volume until the geometry finally merges. Unfortunately, merging in this scenario can often take several time steps to resolve because the interpolated velocity in the air gap still averages to zero, thereby preventing surface geometry from actually intersecting and flagging a collision. By not placing a sample point in these very small gaps, our simulator treats the two liquid bodies as merged and prevents volume loss; the geometric merge is usually then processed within a few timesteps. (With regular sampling, merging will depend on where grid points happen to fall with respect to the surface; hence the physics can respond as if merged when the surfaces are still as much as ∆x apart, as in figure 9 . This generates non-physical air bubbles which linger for many timesteps before they self-collide and are eliminated.) After placing the surface-adapted pressure samples, we complete the sampling of the domain by adding regularly-spaced points from a BCC lattice with cell size 2∆x, again rejecting samples which fall too near existing samples—of course, a graded octree or any other strategy could also be used to fill the domain. All samples are then run through a Delaunay mesh generator such as TetGen
[Si 2006]. Figure 8 illustrates in 2D how this sampling approach is able to capture thin features such as splashes. Further experimentation with relative mesh spacing parameters could yield improved results.
Figure 6: Left: The input surface geometry. Centre: The resulting surface after resampling onto a regular lattice simulation mesh. Note the spurious topology change, rounding of sharp features, aliasing of high frequency details, and the complete disappearance of one small fluid component due to poor placement relative to the mesh samples. Right: The resampled surface after adding geometry-aware sample points to the simulation mesh; the result is much more consistent with the input. (Mesh sample locations are indicated by points, coloured blue when inside, red when outside.)
Figure 7: Sampling Thin Features. A pressure sample is seeded along the outward normal direction from a surface vertex (black square). The initial proposed pressure location (empty black circle) would land in the wrong component and potentially fail to resolve the intervening air gap. We instead place the final pressure sample (filled black circle) midway between the starting vertex and the first intersection point (red X).
Figure 8: Simulating Thin Features. A 2D example of a thin feature simulated with our method. The zoom on the right illustrates the sample placement with respect to surface vertices, and the resulting Voronoi mesh. Notice that even the very sharp tip contains a pressure sample, as indicated by the surrounding Voronoi cell.
Figure 9: Merging. Left: Regular sampling erroneously identifies a topology change, causing a premature reaction in both liquid bodies. Right: Geometry-aware sampling responds correctly.
<H1>6 Interpolation and Advection</H1>
Velocity interpolation methods for unstructured meshes typically proceed in two steps
[Klingner et al. 2006; Elcott et al. 2007; Batty et al. 2010]. First, a full velocity vector is reconstructed at selected mesh locations using a least-squares fit to the nearby velocity components. Then barycentric or generalized barycentric interpolation between those locations interpolates velocity over the full domain. Given such an interpolant, advection of velocities and geometry is straightforward. We follow this general framework, with two modifications. In previous work, face normal components on tetrahedra were used to reconstruct velocities at circumcenters (Voronoi vertices). In our configuration, velocity components instead lie along the tetrahedra edges (Voronoi faces) so we perform the least squares fit on this data instead.
We could then apply the usual generalized barycentric interpolant over Voronoi cells, but this is expensive
[Chentanez et al. 2007] and requires special case handling to avoid degeneracies
[Meyer et al. 2002a]. A simple and fast alternative discussed by Klingner et al. and Chentanez et al. is to first interpolate velocities to Voronoi sites (tetrahedra vertices) and apply standard (and fast) barycentric interpolation over each tetrahedron. However, the interpolation onto tetrahedra vertices discards any local extrema at the Voronoi vertices, thereby severely over-smoothing the velocity field in practice, damping out interesting flow behavior. Rather than discard extrema at Voronoi vertices, we use a slightly refined tetrahedral mesh that includes them. We conceptually tetrahedralize the Voronoi cells themselves by placing additional vertices at Voronoi face centroids and Voronoi sites (see figure 10 ). Velocities for each of these new points need to be computed; while previous work used the generalized barycentric interpolant for this transfer step, we found that simply averaging the velocities of the surrounding ring or cell of Voronoi vertices is quicker and equally effective. For maximum fidelity at the face centroids, we also replace the normal component of the averaged full velocity with the exact normal component already stored at the face. Simple and efficient barycentric interpolations can then be applied on the resulting smaller tetrahedra. Because the sharper, more accurate velocities at the Voronoi vertices are retained and merely augmented with additional data, this is far less dissipative, yielding results that closely match generalized barycentric interpolation (see figure 11 ). Lastly, note that reconstructions should only use face velocities which were assigned valid data by the pressure projection, and thus we can only reconstruct reasonable velocities inside the fluid. We therefore extrapolate velocities outwards from the fluid using a breadth-first graph propagation: each unknown point in a layer is set by averaging all adjacent known points from previous layers, repeating until we have a sufficiently large band of velocities surrounding the surface. This simple method, suggested in the context of cloth-fluid coupling by
[Guendelman et al. 2005], sufficed for all our animations.
In summary, the steps of our interpolation scheme are: 1. Reconstruct full velocity vectors at Voronoi vertices using least squares. 2. Assign full velocity vectors to Voronoi sites and faces using simple averaging from neighboring vertices. 3. Subdivide the Voronoi cells into sub-tetrahedra using the sites and face centroids (see figure 10 ). 4. Apply a simple graph-based extrapolation of velocities to fill in velocities near the liquid.
Figure 10: Rather than interpolating velocity over Voronoi regions directly, we tetrahedralize them and use simple barycentric interpolation. Left: A 2D Voronoi cell with standard dual Delaunay mesh overlaid. Centre: The same cell subdivided into smaller triangles that include the Voronoi vertices. Right: In 3D, each Voronoi face is triangulated using its centroid, and joined to its Voronoi site to build a tetrahedralization.
(a) (b)
(c) (d)
Figure 11: a) Initial conditions for the collapse of a liquid block due to surface tension in zero gravity. (b) Naıve barycentric interpolation on tetrahedra generates very little detail. (c) Generalized barycentric interpolation over Voronoi cells retains interesting small scale structure. (d) Applying barycentric interpolation over our refined tetrahedra produces qualitatively consistent results.
5. To interpolate at a point, locate the sub-tetrahedron containing the point and apply basic barycentric interpolation from its four associated data points (i.e. one site, one face centroid, and two Voronoi vertices).
One potential issue, not unique to our method, is that despite enforcing a lower bound on the distance between pressure samples, our unstructured sampling can cause sliver tetrahedra in the unmodified Delaunay tetrahedralization. While we found this posed little problem for the pressure projection, it can cause the least squares velocity reconstructions to be ill-conditioned due to nearly co-planar face normals. This can be readily resolved by requesting that the mesh generator add Steiner points to enforce fairly lax quality bounds; because our embedded pressure projection does not require the mesh generator to match boundaries, this is relatively inexpensive and effective. If mesh quality cannot be improved sufficiently, using additional nearby velocity samples in the reconstruction can ameliorate this at the cost of a smoother result.
Figure 12: Surface noise. (a) A perturbation is introduced into a smooth surface. (b) On a regular tetrahedral mesh, the sub-meshresolution noise causes instability. (c, d) With adaptively-placed samples, the surface noise is accurately captured by the fluid solver and initially causes ripples before steadily settling.
<H1>7 Results</H1>
<H2>7.1 Sampling</H2>
The issues that arise from regular, non-geometry-aware pressure sampling are common and consistent across Cartesian grids, octrees, Voronoi meshes, and tetrahedral meshes. We will therefore use Voronoi meshes throughout, and simply compare our geometryaware sampling against naıve regular sampling. Surface Noise: As discussed above, regularly-spaced pressure samples can miss fine surface details, resulting in surface noise which is never physically smoothed out. Figure 12 illustrates that our sampling approach successfully resolves and corrects such small surface details. In contrast, regular samples cannot fully capture the initial surface perturbation, so it cannot be rectified. Though the ghost fluid method on regular samples does detect some differences in surface height, this actually exacerbates the problem because noisy sub-mesh details will appear to the simulator as rapid discontinuous changes in surface position over time, inducing noisy responses in the fluid velocity. Topology Mismatch: Another visible artifact of using mismatched surface and simulation resolutions is topological inconsistencies. For example, a surface with two disjoint volumes of liquid may appear to the solver as one volume, resulting in a premature response. Figure 9 shows a liquid drop impacting a still surface. With regular sampling, the droplet begins to influence the static liquid before the surfaces are actually joined. Because our adaptively-placed samples match the topology of the surface tracker, they easily correct this spurious motion. Figure 1 also features such a topological merge, along with many splitting and tearing operations, with timings listed in table 1 . Thin Features: To illustrate our method’s ability to animate thin features, figure 13 shows a scene in which we drop a small sphere of liquid onto the ground. Thin sheets rapidly develop as the fluid spreads out across the floor. With regular pressure samples, sheets of this kind often end up between samples, effectively disappearing from the solver. Our sampling ensures that almost arbitrarily thin sheets of liquid remain visible to the solver, and as such, interesting rippling and splashing motion still occurs. Our method also resolves thin sheets and small surface details generated by large splashes, as shown in figure 1 . To counteract gradual volume drift, we do add a corrective motion-in-the-normaldirection
[Brochu 2006; Müller 2009]
, which further aids in pre- serving thin sheets. Our video also includes an example of a column of liquid being released into a still pool. Although we are using only first-order semi-Lagrangian advection, the liquid motion remains lively and active throughout. We suspect that because our method retains sharp wave peaks and splashes rather than continually eroding them, their extra kinetic and gravitational potential energy is retained in the simulation, accounting for this reduced dissipation. Table 1 gives timings for our 3D examples. All figures are averages per frame and all timings are in seconds. These simulations used no more than 320K tetrahedra each, whereas recent tetrahedra-based free surface methods used up to 4 times more tetrahedra to achieve a similar level of detail.
Figure 13: Thin Sheet. Seeding pressure samples directly inside the fluid volume allows us to capture almost arbitrarily thin sheets.
<H2>7.2 Surface Tension</H2>
Figure 4 illustrates the action of our surface tension model on a low resolution cube in zero gravity. Rather than quickly collapsing into a sphere, a cascade of detailed capillary waves propagate along the surface, causing it to oscillate rapidly. It initially inverts almost completely into an octahedron (the geometric dual of a cube), and continues to oscillate for many subsequent frames. To illustrate the benefits of our sampling approach in the context of surface tension, we launch an identical simulation using the same time steps on a regular mesh. Because this mesh cannot respond and correct high frequency sub-mesh details present in the curvature estimates, the simulation becomes unstable almost immediately. Applying an excessively strict timestep restriction only brings the simulation to a halt as the surface noise introduces increasingly sharp features. Inspired by an example from the work of
Wojtan & Turk [2008], we run another zero gravity simulation on a rectangular block (see figure 11 ). Because our simulation does not use diffusive Laplacian mesh smoothing and applies accurate mesh-based surface tension forces discontinuously at the interface, we retain substantially greater detail in the resulting capillary wave motion.
<H2>7.3 Interpolation</H2>
We revisit our surface tension block example to compare different interpolation schemes. As seen in figure 11 , our barycentric method is substantially less damped than the naıve barycentric interpolation approach, and matches the more complex generalized barycentric interpolant.
Statistic
Thin sheet
Liquid column
Sphere Splash
# tetrahedra
141,701
197,911
313,587
Velocity reconstruction (s)
3
8
18
Surface tracking (s)
7
37
26
Remeshing (s)
15
39
69
Velocity advection (s)
7
18
15
Redistancing (s)
5
22
42
Pressure solve (s)
0.29
1.8
0.45
Total simulation time (s)
37
127
171
Statistic Thin sheet Liquid column Sphere Splash # tetrahedra 141,701 197,911 313,587 Velocity reconstruction (s) 3 8 18 Surface tracking (s) 7 37 26 Remeshing (s) 15 39 69 Velocity advection (s) 7 18 15 Redistancing (s) 5 22 42 Pressure solve (s) 0.29 1.8 0.45 Total simulation time (s) 37 127 171
Table 1: Simulation statistics for 3D examples (all statistics are per-frame values, averaged over all frames).
<H1>8 Discussion and Limitations</H1>
Our implementation is not heavily optimized, and we defer various potential performance gains to future work. Obvious optimizations include: reducing the number of tetrahedra through smarter sampling, improving the broad phase algorithm for point-location queries, and streamlining the construction of mesh data structures. More fundamentally, our Voronoi simulator is in many ways dual to a tetrahedral scheme, and for a given mesh the number of velocity samples is identical; we believe that approximately comparable costs are therefore reasonable to expect. The main contribution of this paper is the coupling of simulation elements to an existing explicit surface tracking method, and not the explicit surface tracking itself. Therefore, not all artifacts due to surface tracking are addressed. For example, El Topo delays handling some very difficult collisions for a few timesteps until the topological operations can be safely processed, which occasionally yields visible lingering surface noise. (Reducing the time step size can help by introducing fewer and simpler collisions, and more aggressive simplification can also be enabled by tuning the volume change tolerance that El Topo uses to decide whether to accept a given simplification.) Likewise, despite the use of featurepreserving mesh improvement, some popping artifacts due to onthe-fly remeshing are still visible in our animations. We chose El Topo because its resolution is not constrained to a regular grid and it is therefore able to showcase very thin features; nevertheless our method could adapt to any of the front tracking methods mentioned in section 2.2. Surface tension was only used for examples in subsections 7.2 and 7.3. Our goal in many of the other examples was to highlight the ability to track thin sheets, whereas surface tension would break these sheets into droplets. Moreover, explicit surface tension schemes, such as the ghost-fluid-based method used in this paper, 3 suffer from a stringent O(∆x 2 ) time step restriction for stability, which is particularly costly when small scale capillary waves are not erroneously damped out. Pursuing a more efficient, fully implicit surface tension model is a promising future direction.
<H1>9 Conclusions and Future Work</H1>
We have shown that with careful placement of pressure samples, our Voronoi mesh-based fluid solver makes it possible for explicit surface tracking to achieve its full potential in capturing small scale liquid features. In addition, we adapted embedded boundary pressure projection techniques to Voronoi meshes, introduced a simple improvement to barycentric velocity interpolation for Voronoi/Delaunay meshes, and extended the ghost fluid surface tension model with mesh-based curvature in order to capture complex capillary waves with minimal damping. Several directions for future work remain. For example, it may be possible to enhance our sampling scheme in various ways, perhaps by exploiting curvature adaptivity, topological information, or measures of vorticity and velocity variation. Likewise, improvements to front tracking would be welcome, such as curvature-driven adaptivity, or greater robustness and efficiency. Lastly, many common extensions to basic inviscid liquid simulation rely on regular grids, and would need to be adapted to accomodate our approach.
<H1>10 Acknowledgements</H1>
This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. We would like to thank our anonymous reviewers for their helpful comments and suggestions.
<H1>References</H1>
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"text": "['beyond which geometry either vanishes']",
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7267,
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"text": "['or artificially coalesces into grid-scale “flotsam and jetsam”']",
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"text": "['can also be problematic']",
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"text": "['the use of regular grids']",
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"text": "['The shortcomings of implicit schemes have spurred interest in explicit methods']",
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"text": "['“front tracking”']",
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7556,
7573
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"text": "['Glimm et al. 1998']",
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"text": "['the surface is represented explicitly as a triangle mesh']",
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"text": "['Here']",
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7748
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"text": "['The greatest challenge is handling topological change']",
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"text": "['mesh tangling']",
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"text": "['that may occur during merging and splitting']",
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"text": "['Du et al. 2006']",
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8009
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"text": "['Wojtan et al. 2009']",
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"text": "['One solution is to determine problematic regions, switch to an implicit surface to repair the tangles there, then stitch back in a new consistent mesh patch']",
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"text": "['Müller [2009]']",
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8156,
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"text": "['these methods still are subject']",
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8209,
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"text": "['to a resolution limited by the voxel grid']",
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8189,
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"text": "['in complex regions']",
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"text": "['Another approach is to work strictly on the triangle mesh itself, using “mesh surgery” for repairs']",
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8358,
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"text": "['this is difficult']",
"type": "background_claim",
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8388,
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"text": "['Brochu & Bridson [2009]']",
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"text": "['recently showed that the problem can be simplified using ideas from cloth animation']",
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"text": "['enforcing the invariant']",
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"text": "['the surface remain intersection-free']",
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"text": "['Topological operations are only allowed when safe']",
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"text": "['robust collision processing is used as a last resort to avoid tangles']",
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"text": "['the surface is minimally perturbed to avoid problems']",
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"text": "['other front tracking methods could easily be used instead']",
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"text": "['Campen & Kobbelt [2010]']",
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"text": "['the need for collision processing could be obviated with exact Boolean operations']",
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"text": "['A prime focus of our work is matching the surface mesh resolution to that of the liquid solver']",
"type": "own_claim",
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"text": "['Most level set-based solvers use one level set sample per pressure grid cell, conservatively avoiding resolution inconsistencies']",
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"text": "['Foster and Fedkiw 2001']",
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9559,
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"text": "['Enright et al. 2002b']",
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9583,
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"text": "['Goktekin et al. [2004]']",
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"text": "['experimented with a doubleresolution level set, trading better volume conservation for other artifacts']",
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"text": "['Bargteil et al. [2006]']",
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9735,
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"text": "['similarly coupled an octree contouring method to a uniform grid fluid solver and explicitly discussed potential artifacts']",
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9864,
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"text": "['resolution mismatch']",
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"text": "['erroneously preserving surface noise and the solver interpreting disconnected fluid regions as connected']",
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"text": "['Kim et al. [2009]']",
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10018,
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"text": "['coupled a high resolution particle level set to a low resolution ghost fluid-based liquid solver']",
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"text": "['ensured that pressure projection captured all liquid geometry by resampling an inflated level set at the pressure grid resolution']",
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"text": "['this can exacerbate other artifacts']",
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"text": "['liquid components behave as if half a cell-width larger than they appear']",
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10376,
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"text": "['Kim et al.']",
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"text": "['also introduced extra surface smoothing to prevent retention of small-scale noise']",
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"text": "['Mismatched resolutions have been found useful for deformable solids']",
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"text": "['surface details are expected to generally persist']",
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"text": "['in liquids']",
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10638,
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"text": "['Wojtan & Turk [2008]']",
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"text": "['the embedded surface mesh may improve realism']",
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"text": "['Teran et al. 2005']",
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"text": "['Nesme et al. 2009']",
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"text": "['Approaches to surface tension generally fall into two categories']",
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"text": "['those which apply surface tension as a body force in a region around the interface via smeared delta functions']",
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"text": "['those which apply surface tension discontinuously at the interface, typically as a boundary condition in the pressure projection step']",
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"text": "['Brackbill et al. 1992']",
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11176,
11193
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"text": "['Hong and Kim 2003']",
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11195,
11212
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"text": "['Zheng et al. 2006']",
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11214,
11232
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"text": "['Wojtan et al. 2009']",
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11374,
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"text": "['The latter is exemplified by the ghost fluid method and related approaches']",
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"text": "['Enright et al. 2003']",
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11472,
11489
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"text": "['Hong and Kim 2005']",
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11491,
11507
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"text": "['Hong et al. 2007']",
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"text": "['has been shown to provide more realistic results']",
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"text": "['Surface tension models can also be compared in terms of how the force itself is approximated']",
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"text": "['finite differences are often used to estimate mean curvature']",
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"text": "['this can be quite inaccurate without careful modification']",
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11816,
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"text": "['Shin 2007']",
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"text": "['cannot capture small details']",
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11659,
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"text": "['In level set schemes']",
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"text": "['a more accurate approach is either to use mesh-based curvature operators']",
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"text": "['If a surface mesh is available']",
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11976,
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"text": "['Meyer et al. 2002b']",
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12023,
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"text": "['to model a physical tension directly in the surface mesh geometry']",
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"text": "['Perot and Nallapati 2003']",
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12117,
12128
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"text": "['Brochu 2006']",
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12130,
12150
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"text": "['Wojtan and Turk 2008']",
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"text": "['this noise tends to worsen and destroy stability']",
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"text": "['without correct feedback from the physics']",
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"text": "['Wojtan & Turk [2008]']",
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"text": "['that surface details below the simulation resolution add energy']",
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"text": "['cannot be correctly evolved by the solver']",
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"text": "['this non-physical operation is dissipative rather than conservative']",
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"text": "['all relevant details are properly resolved']",
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12747,
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"text": "['By instead combining our surface tension model with a geometry-aware sampling']",
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"text": "['This yields accurate and comparatively stable surface tension effects without artificial smoothing']",
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"text": "['We generally follow the tetrahedral scheme of']",
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13235,
13302
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"text": "['with modifications to use specially designed Voronoi meshes instead']",
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13215,
13234
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],
"text": "['Batty et al. [2010]']",
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"offsets": [
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13304,
13326
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],
"text": "['Like Sin et al. [2009]']",
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"offsets": [
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13328,
13456
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],
"text": "['we place pressure samples on the vertices of a Delaunay tetrahedral mesh, corresponding to the sites of the dual Voronoi diagram']",
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"offsets": [
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13458,
13470
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"text": "['figures 3(a)']",
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"text": "['3(b)']",
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"text": "['Normal components of velocity lie on the faces of the Voronoi cells']",
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13559,
13663
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"text": "['the velocity sample is parallel to the line segment connecting the pressure samples in the Delaunay mesh']",
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13665,
13737
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],
"text": "['This configuration requires a slightly different velocity reconstruction']",
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13750,
13766
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"text": "['previous methods']",
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13772,
13800
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"text": "['semi-Lagrangian advection is']",
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{
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13811,
13826
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"text": "['straightforward']",
"type": "background_claim",
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13880,
13892
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],
"text": "['El Topo code']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
13848,
13855
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],
"text": "['we used']",
"type": "own_claim",
"id": "T186"
},
{
"offsets": [
[
13828,
13846
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],
"text": "['For front tracking']",
"type": "data",
"id": "T188"
},
{
"offsets": [
[
13856,
13878
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],
"text": "['Brochu & Bridson’s']",
"type": "data",
"id": "T189"
},
{
"offsets": [
[
13894,
13898
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],
"text": "['2009']",
"type": "data",
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"offsets": [
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14027,
14178
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],
"text": "['Purely explicit front tracking algorithms generally use mesh refinement and coarsening to maintain a high quality discretization as the surface deforms']",
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"id": "T191"
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{
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14180,
14303
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],
"text": "['El Topo uses a sequence of edge subdivision, collapse and flipping operations, combined with null-space Laplacian smoothing']",
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{
"offsets": [
[
14354,
14397
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],
"text": "['they are designed to be geometry-preserving']",
"type": "background_claim",
"id": "T193"
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{
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[
14311,
14352
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],
"text": "['these operations change mesh connectivity']",
"type": "background_claim",
"id": "T194"
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14413,
14499
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"text": "['the smoothing moves vertices only in the null space of the local quadric metric tensor']",
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"text": "['Garland and Heckbert 1997']",
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"text": "['Jiao [2007]']",
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"text": "['it is moved in the plane tangent to the surface']",
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"text": "['the vertex lies on a locally smooth patch']",
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"text": "['on a ridge or corner']",
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"text": "['it is moved only along this line']",
"type": "own_claim",
"id": "T201"
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14727,
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"text": "['sharp features are preserved, allowing the present paper’s algorithm to handle them physically']",
"type": "own_claim",
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"text": "['The solver runs through the following stages each time step']",
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"text": "['We use finite volumes on a Voronoi mesh for the pressure projection step, similar to']",
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{
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16168,
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"text": "['Sin et al.']",
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16179,
16185
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"text": "['[2009]']",
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"text": "['we adapt the embedded boundary methods of']",
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"text": "['applying boundary conditions as they describe']",
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"text": "['Batty et al.']",
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16311,
16317
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"text": "['[2010]']",
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"text": "['to Voronoi meshes']",
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"text": "['the duality/orthogonality relationship between Voronoi and Delaunay meshes lets the accuracy benefits of the method carry over']",
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"text": "['To enforce embedded solid boundary conditions, we need to estimate the partial unobstructed area of each element face']",
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"text": "['figure 3(d)']",
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16851,
16863
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"text": "['Batty et al.']",
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16864,
16870
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"text": "['[2010]']",
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"text": "['used marching triangles cases for computing tetrahedra face fractions from signed distance values on the vertices']",
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"text": "['the faces are arbitrary convex planar polygons rather than triangles']",
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"text": "['in the Voronoi setting']",
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"text": "['these are now located at Voronoi sites']",
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"id": "T220"
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"offsets": [
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17552,
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"text": "['tetrahedra circumcenters']",
"type": "own_claim",
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"text": "['the method is']",
"type": "own_claim",
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17606,
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"text": "['unchanged']",
"type": "own_claim",
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17618,
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"text": "['figure 3(c)']",
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17633,
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"text": "['A slight improvement can be achieved by casting rays to find the exact position of the surface mesh between pressure samples']",
"type": "own_claim",
"id": "T225"
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{
"offsets": [
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17799
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],
"text": "['this is much more accurate']",
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17805,
17847
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"text": "['the estimate derived from signed distances']",
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17759,
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"text": "['In some cases']",
"type": "data",
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"offsets": [
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17865,
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"text": "['we found it made minimal visual difference']",
"type": "own_claim",
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"text": "['in practice']",
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"offsets": [
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18229
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"text": "['This family of redistancing schemes is described by']",
"type": "own_claim",
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18251,
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"text": "['is easily adapted to tetrahedra']",
"type": "own_claim",
"id": "T232"
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18231,
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"text": "['Bridson [2008]']",
"type": "data",
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18698,
18739
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"text": "['To incorporate surface tension, we follow']",
"type": "own_claim",
"id": "T234"
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"offsets": [
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18762,
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],
"text": "['in setting the free surface pressure p fs = p air + γκ fs']",
"type": "own_claim",
"id": "T235"
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{
"offsets": [
[
18740,
18754
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],
"text": "['Enright et al.']",
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[
18755,
18761
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],
"text": "['[2003]']",
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"text": "['We chose the operator of']",
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19151,
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],
"text": "['it provides high quality estimates using just the one-ring of triangles surrounding each vertex']",
"type": "own_claim",
"id": "T239"
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{
"offsets": [
[
19121,
19133
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],
"text": "['Meyer et al.']",
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{
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19134,
19141
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],
"text": "['[2002b]']",
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19252,
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"text": "['others could work too']",
"type": "own_claim",
"id": "T242"
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{
"offsets": [
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19275,
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"text": "['Curvature is evaluated at the intersection point between the the triangle mesh surface and the line joining an interior pressure sample to an exterior one']",
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"text": "['Often this intersection point will coincide with a surface mesh vertex due to our choice of sampling scheme']",
"type": "own_claim",
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"text": "['This method appears highly accurate, and leads to much less damping']",
"type": "own_claim",
"id": "T245"
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{
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19718,
19725
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],
"text": "['that of']",
"type": "own_claim",
"id": "T246"
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{
"offsets": [
[
19727,
19740
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],
"text": "['Wojtan et al.']",
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19741,
19747
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"text": "['[2009]']",
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19923
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"text": "['An advantage of a Voronoi-based discretization is the freedom to explicitly choose pressure sample locations']",
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"text": "['which is critical for accurate ghost fluid free surface conditions']",
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19995,
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"text": "['the signed distance at these samples communicate the surface geometry to the solver']",
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20080,
20150
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"text": "['We can visualize the solver’s “knowledge” by contouring this level set']",
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"id": "T252"
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{
"offsets": [
[
20183,
20208
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"text": "['uniform sampling may fail']",
"type": "own_claim",
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20152,
20161
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"text": "['figures 5']",
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20166,
20167
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"text": "['6']",
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20210,
20301
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"text": "['Careful pressure sample placement with respect to the surface helps in three important ways']",
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"id": "T256"
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{
"offsets": [
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20310,
20414
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],
"text": "['we can inform the solver of all local geometric extrema, allowing the physics to act upon them correctly']",
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"id": "T257"
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{
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20416,
20516
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"text": "['This eliminates the accumulation of erroneous surface noise without requiring non-physical smoothing']",
"type": "own_claim",
"id": "T258"
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"offsets": [
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20518,
20562
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"text": "['this is especially vital for surface tension']",
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20569,
20713
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"text": "['spurious noise affects the curvature estimates and induces disastrously large yet futile compensating velocities that destabilize the simulation']",
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"text": "['the physics responds to merging or splitting only when the surface mesh itself merges or splits']",
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20900,
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"text": "['grid-scale features often disappear and reappear in regular grid sampling']",
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"text": "['the surface translates through the grid']",
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21055,
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"text": "['By specifically placing points inside such small features']",
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21180,
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"text": "['The brute-force approach to these issues is to locally refine using octree grids or graded BCC lattice tetrahedra to capture smaller features']",
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"text": "['this scales poorly']",
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21357,
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"text": "['many of the extra samples yield little benefit']",
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"text": "['incurring memory and computational overhead']",
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21469,
21560
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"text": "['there remains no guarantee that features below the smallest grid cell size will be captured']",
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21895,
21968
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"text": "['also builds a volumetric mesh that attempts to respect the liquid surface']",
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"text": "['the tetrahedral method of']",
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21871,
21887
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],
"text": "['Chentanez et al.']",
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21888,
21894
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"text": "['[2007]']",
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22038,
22121
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"text": "['This is considerably more difficult than non-conforming Delaunay tetrahedralization']",
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"offsets": [
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22137,
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"text": "['requires more Steiner points, worse-shaped tetrahedra, and/or the loss of the Delaunay property']",
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22377
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"text": "['this advantage is shared by the method of']",
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22378,
22397
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"text": "['Batty et al. [2010]']",
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"offsets": [
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22410,
22532
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"text": "['the position of pressure samples plays a more important role in free surface conditions than the position of element faces']",
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21970,
22036
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"text": "['it matches boundary faces rather than positioning pressure samples']",
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22286,
22323
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"text": "['we do not require conforming elements']",
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22240,
22284
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"text": "['our method uses embedded boundary conditions']",
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22534,
22612
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"text": "['As accuracy requires that tetrahedral schemes store pressures at circumcenters']",
"type": "own_claim",
"id": "T284"
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{
"offsets": [
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22728,
22845
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"text": "['even filling a thin feature with conforming tetrahedra provides no guarantee that its interior will be sampled at all']",
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"text": "['circumcenters often lie outside their associated tetrahedra']",
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{
"offsets": [
[
22615,
22635
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],
"text": "['Klingner et al. 2006']",
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[
22637,
22654
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"text": "['Batty et al. 2010']",
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"offsets": [
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23486,
23578
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],
"text": "['we need to place pressure samples so that they capture the surface’s local geometric extrema']",
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"id": "T289"
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{
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23585,
23613
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],
"text": "['around surface mesh vertices']",
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{
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24175,
24281
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],
"text": "['this lends additional accuracy to the vertex’s normal motion, and to the incorporation of the normal force']",
"type": "own_claim",
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24289,
24329
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"text": "['surface tension calculated at the vertex']",
"type": "data",
"id": "T292"
},
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"offsets": [
[
24331,
24396
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],
"text": "['This placement may miss very thin sheets or other fine structures']",
"type": "own_claim",
"id": "T293"
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"offsets": [
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24611,
24702
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],
"text": "['we store the distance d to the closest intersection, and use d in place of ∆x in the offset']",
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24736,
24744
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],
"text": "['figure 7']",
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24904,
24996
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],
"text": "['the distance between the surface vertex and the first intersection 1 is below some threshold']",
"type": "data",
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},
{
"offsets": [
[
25082,
25118
]
],
"text": "['the proposed sample is an air sample']",
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{
"offsets": [
[
25120,
25138
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],
"text": "['we also discard it']",
"type": "own_claim",
"id": "T298"
},
{
"offsets": [
[
25140,
25157
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],
"text": "['This is necessary']",
"type": "own_claim",
"id": "T299"
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{
"offsets": [
[
25166,
25220
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],
"text": "['the divergence constraint is not enforced on air cells']",
"type": "data",
"id": "T300"
},
{
"offsets": [
[
25225,
25253
]
],
"text": "['they can act as liquid sinks']",
"type": "own_claim",
"id": "T301"
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{
"offsets": [
[
25277,
25336
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],
"text": "['and destroy liquid volume until the geometry finally merges']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
25256,
25275
]
],
"text": "['Losasso et al. 2006']",
"type": "data",
"id": "T303"
},
{
"offsets": [
[
25353,
25422
]
],
"text": "['merging in this scenario can often take several time steps to resolve']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
25431,
25494
]
],
"text": "['the interpolated velocity in the air gap still averages to zero']",
"type": "data",
"id": "T305"
},
{
"offsets": [
[
25504,
25583
]
],
"text": "['preventing surface geometry from actually intersecting and flagging a collision']",
"type": "own_claim",
"id": "T306"
},
{
"offsets": [
[
25641,
25718
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],
"text": "['our simulator treats the two liquid bodies as merged and prevents volume loss']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
25720,
25788
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],
"text": "['the geometric merge is usually then processed within a few timesteps']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
25585,
25639
]
],
"text": "['By not placing a sample point in these very small gaps']",
"type": "data",
"id": "T309"
},
{
"offsets": [
[
25905,
25941
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],
"text": "['the physics can respond as if merged']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
25947,
25989
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],
"text": "['the surfaces are still as much as ∆x apart']",
"type": "data",
"id": "T311"
},
{
"offsets": [
[
25997,
26005
]
],
"text": "['figure 9']",
"type": "data",
"id": "T312"
},
{
"offsets": [
[
26362,
26437
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],
"text": "['a graded octree or any other strategy could also be used to fill the domain']",
"type": "own_claim",
"id": "T313"
},
{
"offsets": [
[
26439,
26512
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],
"text": "['All samples are then run through a Delaunay mesh generator such as TetGen']",
"type": "own_claim",
"id": "T314"
},
{
"offsets": [
[
26515,
26522
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"text": "['Si 2006']",
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"id": "T315"
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"text": "['Velocity interpolation methods for unstructured meshes typically proceed in two steps']",
"type": "background_claim",
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"text": "['Klingner et al. 2006']",
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28607,
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"text": "['Elcott et al. 2007']",
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28627,
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"text": "['Batty et al. 2010']",
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"text": "['advection of velocities and geometry is straightforward']",
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28908,
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"text": "['Given such an interpolant']",
"type": "data",
"id": "T321"
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{
"offsets": [
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29068,
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"text": "['face normal components on tetrahedra were used to reconstruct velocities at circumcenters']",
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"id": "T322"
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{
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29050,
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"text": "['In previous work']",
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"offsets": [
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29334,
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"text": "['We could then apply the usual generalized barycentric interpolant over Voronoi cells']",
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"text": "['this is expensive']",
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"text": "['and requires special case handling to avoid degeneracies']",
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"text": "['Chentanez et al. 2007']",
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"text": "['Meyer et al. 2002a']",
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"text": "['A simple and fast alternative discussed by']",
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"text": "['and apply standard']",
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"text": "['barycentric interpolation over each tetrahedron']",
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"text": "['Klingner et al. and Chentanez et al.']",
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"text": "['the interpolation onto tetrahedra vertices discards any local extrema at the Voronoi vertices']",
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"text": "['severely over-smoothing the velocity field in practice, damping out interesting flow behavior']",
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"text": "['Velocities for each of these new points need to be computed']",
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30319,
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"text": "['previous work used the generalized barycentric interpolant for this transfer step']",
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"text": "['we found that simply averaging the velocities of the surrounding ring or cell of Voronoi vertices is quicker and equally effective']",
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"text": "['Simple and efficient barycentric interpolations can then be applied on the resulting smaller tetrahedra']",
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"text": "['we also replace the normal component of the averaged full velocity with the exact normal component already stored at the face']",
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"text": "['more accurate velocities at the Voronoi vertices are retained and merely augmented with additional data']",
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"text": "['the sharper']",
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"text": "['this is far less dissipative, yielding results that closely match generalized barycentric interpolation']",
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"text": "['figure 11']",
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"text": "['reconstructions should only use face velocities which were assigned valid data by the pressure projection']",
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"text": "['This simple method']",
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"text": "['Guendelman et al. 2005']",
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"text": "['One potential issue']",
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"text": "['nearly co-planar face normals']",
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"text": "['This can be readily resolved by requesting that the mesh generator add Steiner points to enforce fairly lax quality bounds']",
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"text": "['using additional nearby velocity samples in the reconstruction can ameliorate this at the cost of a smoother result']",
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"text": "['Figure 12']",
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"text": "['noisy sub-mesh details will appear to the simulator as rapid discontinuous changes in surface position over time, inducing noisy responses in the fluid velocity']",
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"text": "['Figure 1']",
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36560,
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36953,
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"text": "['To counteract gradual volume drift, we do add a corrective motion-in-the-normaldirection']",
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"text": "['Brochu 2006']",
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37068,
37079
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"text": "['Müller 2009']",
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"text": "['their extra kinetic and gravitational potential energy is retained in the simulation']",
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"text": "['accounting for this reduced dissipation']",
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37368,
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"text": "['our method retains sharp wave peaks and splashes rather than continually eroding them']",
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"text": "['These simulations used no more than 320K tetrahedra each']",
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"text": "['recent tetrahedra-based free surface methods used up to 4 times more tetrahedra to achieve a similar level of detail']",
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"text": "['this mesh cannot respond and correct high frequency sub-mesh details present in the curvature estimates']",
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"text": "['the simulation becomes unstable almost immediately']",
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38943,
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"text": "['the surface noise introduces increasingly sharp features']",
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39068,
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"text": "['Inspired by an example from the work of']",
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39135,
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"text": "['figure 11']",
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39156,
39306
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"text": "['our simulation does not use diffusive Laplacian mesh smoothing and applies accurate mesh-based surface tension forces discontinuously at the interface']",
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"text": "['we retain substantially greater detail in the resulting capillary wave motion']",
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"text": "['our barycentric method is substantially less damped than the naıve barycentric interpolation approach']",
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"text": "['matches the more complex generalized barycentric interpolant']",
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"text": "['figure 11']",
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42254
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"text": "['for a given mesh']",
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"text": "['reasonable to expect']",
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"text": "['The main contribution of this paper is the coupling of simulation elements to an existing explicit surface tracking method, and not the explicit surface tracking itself']",
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"text": "['not all artifacts']",
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"text": "['surface tracking']",
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"text": "['are addressed']",
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"text": "['El Topo delays handling some very difficult collisions for a few timesteps until the topological operations can be safely processed']",
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"text": "['which occasionally yields visible lingering surface noise']",
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"text": "['Reducing the time step size can help by introducing fewer and simpler collisions']",
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"text": "['more aggressive simplification can also be enabled by tuning the volume change tolerance']",
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"text": "['are still visible in our animations']",
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"text": "['some popping artifacts']",
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"text": "['onthe-fly remeshing']",
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"text": "['its resolution is not constrained to a regular grid']",
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"text": "['it is']",
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"text": "['able to showcase very thin features']",
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"text": "['our method could adapt to any of the front tracking methods']",
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"text": "['section 2.2']",
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"text": "['Our goal in many of the other examples was to highlight the ability to track thin sheets']",
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A21 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A21_F01_A_Method_for_Animating_Viscoelastic_Fluids_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1015706.1015746
Computer Graphics Proceedings, Annual Conference Series, 2004
<Title>A Method for Animating Viscoelastic Fluids</Title>
Tolga G. Goktekin Adam W. Bargteil James F. O’Brien University of California
Berkeley
<Abstract>This paper describes a technique for animating the behavior of viscoelastic fluids, such as mucus, liquid soap, pudding, toothpaste, or clay, that exhibit a combination of both fluid and solid characteristics. The technique builds upon prior Eulerian methods for animating incompressible fluids with free surfaces by including additional elastic terms in the basic Navier-Stokes equations. The elastic terms are computed by integrating and advecting strain-rate throughout the fluid. Transition from elastic resistance to viscous flow is controlled by von Mises’s yield condition, and subsequent behavior is then governed by a quasi-linear plasticity model.</Abstract>
Keywords: Natural phenomena, physically based animation, computational fluid dynamics, viscoelastic fluids, elastoplastic, viscous fluids, goop, glop. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation; I.6.8 [Simulation and Modeling]: Types of Simulation—Animation.
<H1>1 Introduction</H1>
In this paper we describe a technique for animating the behavior of materials, such as the one shown in Figure 1 , that exhibit a combination of both fluid and solid characteristics. Often referred to as viscoelastic fluids, these materials initially respond to strain elastically like a solid, but when subjected to increasingly large stresses they flow like a fluid. A tremendous variety of materials exhibit this type of behavior, and a few common examples include: mucus, egg white, dough, gelatin, unset cement, liquid acrylic, toothpaste, gels, clay, and liquid soap. Like a solid, these materials can bounce and jiggle, but they will also flow like a fluid. For some of these materials, such as egg white or clay, the combination of elastic and fluid behavior is quite apparent. For others, such as liquid soap, the elastic behavior manifests less obviously as predominately fluid behavior that differs subtly from that of a simply viscous fluid. The technique we present builds on prior Eulerian methods for animating incompressible fluids with free surfaces. As evidenced by their widespread use, these methods can efficiently produce results that are realistic enough for applications in the demanding visual effects industry. Our method computes viscoelastic fluid behavior by supplementing the basic Navier-Stokes equations with additional terms for elastic body forces. These terms can be
E-mail: {goktekin,adamb,job}@eecs.berkeley.edu . From the ACM SIGGRAPH 2004 conference proceedings. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ACM SIGGRAPH 2004, Los Angels, CA c Copyright ACM 2004
Figure 1:
A sequence of images showing a splash in a viscoelastic fluid.
readily computed on rectilinear grids using a staggered discretization scheme, and the use of an Eulerian formulation easily accommodates modeling large flows. These elastic terms require computing the material strain throughout the fluid. Because the fluid simulations do not make use of an explicit reference configuration, strain is computed by integrating strain-rate and advecting the results. The transition from elastic resistance to viscous flow is controlled by von Mises’s yield condition, and subsequent behavior is then governed by a quasi-linear plasticity model.
<H1>2 Fluids, Solids, and the In-between</H1>
While the mechanics describing the behaviors exhibited by solids and fluids may seem distinct, they are actually quite similar. First, both resist changes to their volume. The physical reasons for why they conserve volume may differ, but the mathematical expressions capturing the behaviors are essentially the same. Furthermore, many fluid methods assume incompressibility and most solid methods assume that volume changes will be negligible. Second, the internal damping force for a solid and the viscous force for a fluid are not just similar, they are identical. One can easily start with the expression for strain-rate-proportional damping and derive the viscous terms of the Navier-Stokes equations. The key difference between an ideal solid and an ideal fluid is the presence or absence of an elastic term that attempts to restore the material to its original shape. Many materials cannot be classified clearly as a solid or fluid. These materials elastically resist deformation up to a certain point after which they begin to flow. In fact, even materials that are clearly solids behave this way: they act elastically over some regime and then begin to flow plastically (or perhaps fracture). Similarly, some materials that are clearly fluids can withstand very small strains elastically. The continuous variable that spans the space between solid and fluid materials is this limit on how much stress can be tolerated before flow occurs. Other properties such as damping/viscosity, density, and elastic stiffness are largely orthogonal. When the elastic limit is set to a high value, the material behaves like a solid, when it is zero the material behaves like a fluid, and intermediate values correspond to materials like mucus, liquid soap, toothpaste, or clay. These intermediate materials are often referred to as viscoelastic fluids or as elastoplastic solids, depending on whether their behavior is closer to that of an ideal fluid or ideal solid.
1
ACM SIGGRAPH 2004, Los Angels, CA, August, 8–12, 2004
<H1>3 Background</H1>
In the field of computer graphics, the technique described in
[Carlson et al., 2002]
is perhaps closest in intent to the method we describe here. Like us, they were interested in modeling materials with properties intermediate between solids and fluids using an Eulerian grid-based fluid simulation method. However, they opt to map the continuum between fluids and solids to varying viscosity. In their system a solid is simply a fluid with very high viscosity. This approach ignores the elastic behavior demonstrated by many materials. Nevertheless, they do generate nice results for highly viscous fluids, and they describe an implicit integration method for coping with stability issues arising from very high viscosities. Other graphics researchers have used particle-based methods for modeling highly viscous fluids and for modeling fluids with some form of elasticity. In [Terzopoulos et al., 1989]
the authors modeled melting thermoelastic materials. The particles exerted cohesive, viscous, and volume-preserving forces on their neighbors. While solid, each particle was connected to a fixed set of neighbors using elastic springs. As the material would become more fluid-like, the springs would weaken, and eventually disappear. By varying the elastic properties of the materials, this method could model a range of behaviors, but without plasticity, it could not model materials, like clay, that flow into a new configuration and then resist changes from that configuration.
Similar approaches using different particle formulations have appeared in
[Desbrun and Gascuel, 1995], [Desbrun and Cani, 1996], [Cani and Desbrun, 1997], and
[Stora et al., 1999].
The method appearing in
[Desbrun and Gascuel, 1995]
used elastic forces with dynamically determined neighbors to allow behavior that is similar to plastic flow.
Perhaps the greatest limitation on the level of realism achievable by these particle methods was the relatively small number of particles used. However, as processor speeds have increased, particle-based methods have been able to achieve increasingly impressive results. Compelling real-time results for modestly sized systems appear in
[Müller et al., 2003]
, and [Premo ze et al., 2003]
demonstrates off-line results that are comparable to the current best grid-based methods.
Although both of these recent methods focus on strictly liquid behavior, they could be extended along lines similar to what we propose here. Some methods for modeling solids have dealt with limited amounts of plastic flow. Both [Terzopoulos and Fleischer, 1988a] and [Terzopoulos and Fleischer, 1988b]
describe transition to plastic flow based on von Mises’s yield condition, and [O’Brien et al., 2002]
used a similar plasticity model for ductile fracture behavior.
We use the same yield condition of von Mises, but we do not assume that plastic flow occurs instantaneously. Instead we use a more complex model that accommodates phenomena such as creep. Additionally, these prior methods used Lagrangian meshes with largely fixed topology, and so they would have encountered “tangling” difficulties, such as inverting elements, for large amounts of plastic flow. Another, rather interesting, approach to combining solid and fluid behaviors appears in
[Nixon and Lobb, 2002]. They surround a fluid simulation with an elastic membrane. The result is an object that behaves somewhat like a water balloon. Our work builds directly on previous grid-based, Eulerian methods for animating fluids with free surfaces.
Details on these methods can be found in
[Foster and Metaxas, 1996], [Stam, 1999], [Foster and Fedkiw, 2001]
, and [Enright et al., 2002].
In particular, our work essentially extends
[Enright et al., 2002]
to include the behavior of viscoelastic fluids.
Outside the graphics field, viscoelastic materials have been studied extensively.
We refer the reader to the texts
[Fung, 1965], [Han and Reddy, 1999], and
[ Bird et al., 1987 ]
for detailed descriptions of several different models for viscoelastic and elastoplastic materials. The general approaches we use for introducing elastic forces into the Navier-Stokes equations, and integrating and advecting strain are not completely novel.
Some recent examples of fluid simulation outside the graphics literature that involve elastic forces include
[Gerritsma, 1996], [Tomé et al., 2002], and
[ Bonito et al., 2003 ].
A detailed analysis of two-dimensional simulations of viscoelastic fluids on staggered rectilinear grids appears in
[Gerritsma, 1996]. The three-dimensional method we use for storing rank-two tensor quantities on a staggered grid is a generalization of their two-dimensional method. In [ Bonito et al., 2003 ]
a combination of rectilinear grids and finite elements are used with a volume-of-fluid method to model three-dimensional fluids with elastic properties. They do not include plasticity and they store all quantities at cell centers. The marker-andcell based method in
[Tomé et al., 2002]
is another example solving viscoelastic free-surface flows. They address issues relating to elastic-stress based boundary conditions at rigidbody and free-surfaces. Although they use a staggered grid for the velocity field, they still store their tensor values at cell centers.
<H1>4 Simulation Methods</H1>
The framework we use for fluid simulation is based on the method described in
[Enright et al., 2002]. This framework consists of two primary components which work together to produce useful results. The first is a rectilinear grid that stores the values that define the fluid’s state. The values on the grid change as forces act on the fluid, and they also change as the fluid moves through the space delineated by the grid. The second component is a function whose levelset at zero locates the boundaries of the fluid. The function is represented using a combination of particles and values defined on a second rectilinear grid. The particles and grid values evolve based on the motion of the fluid. A full description of this type of system is beyond the scope of this paper, so we focus on the changes we make to accommodate viscoelastic behavior.
We suggest the following references for a more complete description of this simulation methodology:
[Foster and Metaxas, 1996], [Stam, 1999], [Foster and Fedkiw, 2001], and
[Enright et al., 2002].
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Computer Graphics Proceedings, Annual Conference Series, 2004
Figure 2: This figure demonstrates some effects achieved by varying α and γ. Each image shows a cube of material that has been dropped onto a hard surface inside an invisible box. The rightmost images show a low viscosity simple fluid and an extremely viscous simple fluid for comparison. The viscoelastic examples have the same viscosity as the low viscosity fluid. Observe that for appropriate values of α and γ the viscoelastic fluid actually bounces.
<H2>4.1 Governing Equations</H2>
Behavior of the viscoelastic fluid is governed by a modified version of the Navier-Stokes equations that includes an additional term for elastic stress. At any point in the fluid the fluid’s acceleration is determined by
1
∂u = − u T u − p + μ v T u + μ e T + f ∂t ρ ρ ρ ρ
where u is the fluid’s velocity at a point in space, p its pressure, ρ its density, μ v its coefficient of viscosity, μ e its elastic modulus, the elastic strain tensor at that point in space, and f includes any other forces such as gravity or vorticity confinement. The symbol denotes the vector of difT ferential operators = [∂/∂x, ∂/∂y, ∂/∂z] , and we have implicitly assumed that μ v and μ e are constant throughout the material. By omitting elastic and viscous terms relating to dilation, we have also assumed that the fluid is incompressible. Because the fluid is incompressible, the velocity field within the fluid volume is subject to the constraint that
2
T u =0 .
This condition is enforced by adjusting the pressure field at each integration step. Additionally, we do not use the first term of Equation (1) (the advection term) directly. Instead, we use a semi-Lagrangian method to advect field values. We refer the reader to
[Stam, 1999] and [Foster and Fedkiw, 2001]
for a discussion on efficiently modeling the above equations.
<H2>4.2 Elastic, Plastic, and Total Strain</H2>
The fourth term of Equation (1) computes acceleration due to elastic forces and it requires knowing the elastic strain throughout the fluid. If we had an explicit deformation function then we could use its spatial derivatives to compute strain. However, with the Eulerian formulation we are using there is no deformation function available. Furthermore, the large deformation and flow experienced by the material makes tracking deformation impractical. Instead we compute strain by integrating strain-rate. Because we do not wish to model a perfectly elastic material, we also require rules concerning how the elastic strain changes due to plastic yielding. We first separate the total strain into an elastic and a plastic component so that
Tot Elc Plc = + . (3)
Where Tot , Elc , and Plc , are respectively the total, elastic, and plastic strains. (Outside this section we denote elastic strain as simply .) The strain rate is the time derivative of the total strain, so the total strain at some time t is
4
t Tot Tot Tot = 0 + ̇ dt 0
where 0 Tot is the total strain at time t = 0 and ̇ Tot is the strain rate given by
5
Tot T T T ̇ = u + u /2 .
Similarly, the plastic strain is given by integrating plastic flow. t
6
Plc Plc Plc = 0 + ̇ dt 0
We use von Mises’s criterion for determining when plastic flow should occur. This criterion depends on the elastic These images, along with the ones shown in Figure 1, show how splash behavior is affected by elasticity. The upper-left image shows a fluid example. As expected, the Worthington column moves in the direction of the impact. In the viscoelastic examples, the column rises vertically (upper-right) or actually rises back toward the impact direction (lower-left). The lower-right image shows a fluid with both elasticity and high viscosity.
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ACM SIGGRAPH 2004, Los Angels, CA, August, 8–12, 2004
Figure 3:
strain deviation, , which is the elastic strain with any dilation removed 1
7
Tr Elc = Elc − I . 3
So long as the magnitude (Frobenius norm) of the strain deviation remains below the yield point, γ, no plastic flow occurs. When the limit is exceeded, flow occurs at a rate proportional to the amount the limit has been exceeded by. So the flow rate for plastic strain is
8
̇ Plc = α max 0, − γ ,
where α is the material’s elastic decay rate, which determines the rate of plastic flow. Assuming that the initial total and plastic strains are both zero, we can combine the above to compute the time derivative of the elastic strain which takes into account changes to both total and plastic strains ̇ Elc = u T + u T T /2 − α max 0, − γ (9) The images shown in Figure 2 illustrate some effects generated by varying α and γ. Note that Equation (9) does not take into account the movement of the material through space. Like velocity or any other fluid property, the elastic strain must be advected with the fluid. We use the same semi-Lagrangian advection scheme that we use for the fluid velocities, and we update the elastic strain using Equation (9) after our advection step.
<H2>4.3 Grid Issues</H2>
The well known staggered grid method, originally described by
[Harlow and Welch, 1965]
, elegantly avoids problems that 1 This distinction is a bit pedantic here, because the fluid is incompressible and so the dilation should always be zero. Examples of fluid being sprayed into a container. The way different fluids flow or pile in the container varies significantly. plague methods that store collocated pressure and velocity values. However in addition to scalars (e.g. pressure) and rank-one tensors (e.g. velocity), we also need to store the elastic strain, a rank-two tensor, on the simulation grid. Just as velocity components are stored separately on faces, the different components of the strain tensor are stored at different locations. The diagonal entries are stored at the cell centers. The off-diagonal entries are stored at the center of edges perpendicular to the component directions. For example, the xy components are stored on the edges parallel to the z axis. This approach is a generalization of the 2D method described in
[Gerritsma, 1996]
, and they describe its merits in detail.
We use a particle-level-set method for tracking the fluid’s free surface as described in
[Enright et al., 2002], but with the substantially faster, though less accurate, method detailed in
[Enright et al., 2004]. The authors note that the method is susceptible to volume loss, and we found this behavior to be problematic for some of our examples that involve fixed, small amounts of fluid. We were able to ameliorate this problem somewhat by using a level-set grid with twice the fluid grid’s resolution, and that is staggered with respect to the fluid grid. This scheme places level-set grid centers on the cell centers, face centers, edge centers, and nodes of the fluid grid. In addition to helping to prevent volume loss by locating level-set values where velocity boundary constraints are enforced, the higher resolution also benefits the rendered surfaces.
Figure 4:
<H1>5 Results and Discussion</H1>
We have implemented this method for modeling viscoelastic behavior and used it to generate several example animations. Most of these examples were selected to illustrate some interesting aspect of viscoelastic fluid behavior. All of the examples shown in this paper also appear on the accompanying video, which also contains additional examples. Figures 1 and 3 show several splashes that are generated when a fluid sphere is hurled into a tank containing the same material. The motion of the pure fluid example differs substantially from the viscoelastic examples. Additionally, the surfaces of the viscoelastic examples retain evidence of the impact even after motion has stopped. Figure 4 shows jets of different fluids sprayed into a closed container. Again, the behavior of simple and viscoelastic fluids differ substantially.
4
Computer Graphics Proceedings, Annual Conference Series, 2004
Figure 5: A sequence of images showing a viscoelastic fluid draining from a tank. The stream spirals in a fashion characteristic of viscoelastic fluids.
Figure 6:
These additional examples of viscoelastic fluids draining from a tank show a range of different behaviors.
Figures 5 and 6 illustrate some of the variation in viscoelastic fluid behaviors. In Figure 5 , fluid draining from a tank forms a thin stream that forms a spiral pattern as it piles. The examples in Figure 6 billow up around the downward stream, create a folding pattern, and break apart into gobs. A simply viscous fluid would merely flow out to fill the container. The drip examples in Figures 7 and 8 show the behavior of a gob of material that has been stuck to the underside of a horizontal surface. The effect of shape memory created by the elastic forces can be seen in these images. Some of the images in Figure 2 show similar behavior that occurs when cubes of different materials are dropped onto a hard surface. The examples with high yield strain, i.e. large γ, behave like deformable solids and bounce. As shown in Figure 9 , we can also model highly deformable, sticky objects that interact with each other. When the spheres collide, their level-set surfaces merge so that they adhere. The fluid retains its momentum, generating the resulting spinning and stretching motion. Close examination shows that the spheres slightly anticipate their collision. This error occurs because the surfaces begin to interact through shared ghost cells. The images in Figure 10 are still images from an animation we produced using this simulation method. These examples demonstrate viscoelastic fluids interacting with interesting geometry. All of the images were rendered with a standard Newtoniteration based ray marching algorithm implemented in the open source renderer Pixie developed by Okan Arikan. While ray marching produces nice results we think they might be improved using the method described in
[Heckbert, 1987]. Some of our examples suffer from noticeable volume loss. This occurs because, while the particle level-set method does a nice job modeling moderately thick volumes of fluids, very thin surfaces, or strands, still have a tendency to vanish. These effects are particularly noticeable visually when the fluid is moving in orderly fashion, as opposed to splashing about chaotically. It is difficult to say if this behavior is a deficiency in our implementation or a limitation of the surface tracking method. The speed of this simulation method is approximately the same with and without the addition of elastic forces. For example, one of the falling cube examples on a 40 3 grid requires about half an hour of computation per second of animation on a single 3 GHz Pentium 4 processor. We are using an explicit integration method for the viscous and elastic forces, so very high viscous or elastic coefficients would probably cause stability problems and force smaller time steps. If it became an issue, that difficulty could be ameliorated somewhat with an implicit integration scheme. The material can be made to adhere to or slip off of boundary surfaces by adjusting the velocity or pressure constraints enforced along closed boundaries. However, in our current implementation all fluids will stick to each other because different surface components merge when they collide. For the fluids we show in our examples, this behavior is a desirable feature. However, for non-sticky materials, like cold gelatin, it would be undesirable. To a large extent, our method for incorporating elastoplastic terms does not depend on the underlying fluid simulation method, and one could easily adapt the method to other fluid simulation techniques such as smoothed-particle hydrodynamics. Furthermore, we found that once we already had a working fluid simulation, adding the elastoplastic terms was fairly easy. Finally, we note that while the method we present can model a wide range of phenomena, many real materials can demonstrate behaviors not captured by this model. Biological fluids, such as blood, can exhibit many interesting effects that arise from their microscopic structure. Even relatively simple polymer suspensions can demonstrate behavior that can only be roughly captured with this model. Quoting from
[ Bird et al., 1987 ]
: A fluid that’s macromolecular Is really quite weird — in particular The abnormal stresses The fluid possesses Give rise to effects quite spectacular.
<H1>Acknowledgments</H1>
We thank the other members of the Berkeley Graphics Group for their helpful criticism and comments. We especially thank Okan Arikan for his help with rendering our images. This work was supported in part by NSF CCR02-04377, California MICRO 02-055, and by generous support from Pixar Animation Studios, Intel Corporation, Sony Computer Entertainment America, the Okawa Foundation, and the Alfred P. Sloan Foundation.
<H1>References</H1>
Bird, R. B., et al. 1987. Dynamics of Polymeric Liquids. John Wiley & Sons, New York.
Bonito, A., Picasso, M., and Laso, M. 2003. Numerical simulation of 3D viscoelastic flows with free surfaces. Submitted to Journal Non-Newtonian Fluid Mechanics. Preprint.
Cani, M.-P., and Desbrun, M. 1997. Animation of deformable models using implicit
surfaces. IEEE Transactions on Visualization and Computer Graphics 3, 1 (Jan.), 39–50.
Carlson, M., Mucha, P. J., III, R. B. V. H., and Turk, G. 2002. Melting and flowing. In the ACM SIGGRAPH 2002 Symposium on Computer Animation, 167–174.
Desbrun, M., and Cani, M.-P. 1996. Smoothed particles: A new paradigm for animating
highly deformable bodies. In Computer Animation and Simulation 1996, 61–76.
Desbrun, M., and Gascuel, M.-P. 1995. Animating soft substances with implicit surfaces. In the Proceedings of SIGGRAPH 95, 287–290.
Enright, D. P., Marschner, S. R., and Fedkiw, R. P. 2002. Animation and rendering of
complex water surfaces. In the Proceedings of ACM SIGGRAPH 2002, 736–744.
Enright, D., Losasso, F., and Fedkiw, R. 2004. A fast and accurate semi-Lagrangian
particle level set method. Computers and Structures.
Foster, N., and Fedkiw, R. 2001. Practical animation of liquids. In the Proceedings of ACM SIGGRAPH 2001, 23–30.
Foster, N., and Metaxas, D. 1996. Realistic animation of liquids. In Graphics Interface 1996, 204–212.
Fung, Y. C. 1965. Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs, N.J.
Gerritsma, M. I. 1996. Time dependent numerical simulations of a viscoelastic fluid on a staggered grid. PhD thesis, Rijksuniversiteit Groningen.
Han, W., and Reddy, B. D. 1999. Plasticity: Mathematical Theory and Numerical Analysis. Interdisciplinary Applied Mathematics. Springer-Verlag, New York.
Harlow, F., and Welch, J. 1965. Numerical calculation of timedependent viscous
incompressible flow of fluid with a free surface. The Physics of Fluids 8, 2182–2189.
Heckbert, P. S. 1987. Ray tracing Jell-O brand gelatin. In the Proceedings of ACM SIGGRAPH 87, 73–74.
M ̈uller, M., Charypar, D., and Gross, M. 2003. Particle-based fluid simulation for
interactive applications. In the ACM SIGGRAPH 2003 Symposium on Computer
Animation, 154–159.
Nixon, D., and Lobb, R. 2002. A fluid-based soft-object model. IEEE Computer Graphics &
Applications 22, 4 (July), 68–75.
O’Brien, J. F., Bargteil, A. W., and Hodgins, J. K. 2002. Graphical modeling and animation of ductile fracture. In the Proceedings of ACM SIGGRAPH 2002, 291–294.
Premoˇze, S., Tasdizen, T., Bigler, J., Lefohn, A., and Whitaker,R. 2003. Particle-based simulation of fluids. Computer Graphics Forum 22, 3 (Sept.), 401–410. Stam, J. 1999. Stable fluids. In the Proceedings of ACM SIGGRAPH 99, 121–128.
Stora, D., Agliati, P.-O., Cani, M.-P., Neyret, F., and Gascuel, J.D. 1999. Animating lava flows. In Graphics Interface 99, 203–210.
Terzopoulos, D., and Fleischer, K. 1988. Deformable models. The Visual Computer 4, 306 331.
Terzopoulos, D., and Fleischer, K. 1988. Modeling inelastic deformation: Viscoelasticity, plasticity, fracture. In the Proceedings of ACM SIGGRAPH 88, 269–278.
Terzopoulos, D., Platt, J., and Fleischer, K. 1989. Heating and melting deformable models (from goop to glop). In Graphics Interface 1989, 219–226.
Tom ́e, M., Mangiavacchi, N., Cuminato, J., Castelo, A., and McKee, S. 2002. A finite difference technique for simulating unsteady viscoelastic free surface flows. Journal of Non-Newtonian Fluid Mechanics 106, 61–106.
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Figure 7:
A sequence of images showing a viscoelastic fluid dripping off of a surface to which it has adhered.
Figure 8:
Other drip examples. The first is highly elastic and falls off as a single blob. The second does as well, but splats because it is less rigid. The final example demonstrates a characteristic swirling behavior.
Figure 9:
Figure 10:
Images from the animation Gratuitous Goop, which was produced using the methods described in this paper and appeared in the SIGGRAPH 2004 Electronic Theater.
6
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"text": "['The transition from elastic resistance to viscous flow is controlled by von Mises’s yield condition']",
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4400,
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"text": "['the mechanics describing the behaviors exhibited by solids and fluids may seem distinct']",
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4642,
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"text": "['they are actually quite similar']",
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4682,
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4719,
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"text": "['The physical reasons for why they conserve volume may differ']",
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4785,
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"text": "['the mathematical expressions capturing the behaviors are essentially the same']",
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4925,
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"text": "['the internal damping force for a solid and the viscous force for a fluid are not just similar, they are identical']",
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"text": "['One can easily start with the expression for strain-rate-proportional damping and derive the viscous terms of the Navier-Stokes equations']",
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"text": "['The key difference between an ideal solid and an ideal fluid is the presence or absence of an elastic term that attempts to restore the material to its original shape']",
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"text": "['Many materials cannot be classified clearly as a solid or fluid']",
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5486,
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"text": "['These materials elastically resist deformation up to a certain point after which they begin to flow']",
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5759,
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"text": "['some materials that are clearly fluids can withstand very small strains elastically']",
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5844,
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"text": "['The continuous variable that spans the space between solid and fluid materials is this limit on how much stress can be tolerated before flow occurs']",
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"text": "['Other properties such as damping/viscosity, density, and elastic stiffness are largely orthogonal']",
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6097,
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6179,
6189
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"text": "['it is zero']",
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6190,
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"text": "['the material behaves like a fluid']",
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6229,
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"text": "['intermediate values correspond to materials like mucus, liquid soap, toothpaste, or clay']",
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6319,
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"text": "['These intermediate materials are often referred to as viscoelastic fluids or as elastoplastic solids, depending on whether their behavior is closer to that of an ideal fluid or ideal solid']",
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6622,
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"type": "background_claim",
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6709,
6768
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"text": "['is perhaps closest in intent to the method we describe here']",
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6686,
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"text": "['Carlson et al., 2002']",
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"offsets": [
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6779,
6929
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"text": "['they were interested in modeling materials with properties intermediate between solids and fluids using an Eulerian grid-based fluid simulation method']",
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6940,
7016
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"text": "['they opt to map the continuum between fluids and solids to varying viscosity']",
"type": "background_claim",
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7018,
7084
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"text": "['In their system a solid is simply a fluid with very high viscosity']",
"type": "background_claim",
"id": "T66"
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"offsets": [
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7086,
7159
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"text": "['This approach ignores the elastic behavior demonstrated by many materials']",
"type": "background_claim",
"id": "T67"
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"offsets": [
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7175,
7230
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"text": "['they do generate nice results for highly viscous fluids']",
"type": "background_claim",
"id": "T68"
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"offsets": [
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7236,
7348
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],
"text": "['they describe an implicit integration method for coping with stability issues arising from very high viscosities']",
"type": "background_claim",
"id": "T69"
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{
"offsets": [
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7350,
7497
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"text": "['Other graphics researchers have used particle-based methods for modeling highly viscous fluids and for modeling fluids with some form of elasticity']",
"type": "background_claim",
"id": "T70"
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"offsets": [
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7530,
7581
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"text": "['the authors modeled melting thermoelastic materials']",
"type": "background_claim",
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"offsets": [
[
7503,
7527
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],
"text": "['Terzopoulos et al., 1989']",
"type": "data",
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"offsets": [
[
7863,
7959
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],
"text": "['By varying the elastic properties of the materials, this method could model a range of behaviors']",
"type": "background_claim",
"id": "T73"
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{
"offsets": [
[
7965,
8013
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],
"text": "['without plasticity, it could not model materials']",
"type": "background_claim",
"id": "T74"
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"offsets": [
[
8026,
8108
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"text": "['that flow into a new configuration and then resist changes from that configuration']",
"type": "background_claim",
"id": "T75"
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{
"offsets": [
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8020,
8024
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],
"text": "['clay']",
"type": "data",
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"offsets": [
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8111,
8181
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"text": "['Similar approaches using different particle formulations have appeared']",
"type": "background_claim",
"id": "T77"
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{
"offsets": [
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8187,
8212
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],
"text": "['Desbrun and Gascuel, 1995']",
"type": "data",
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8216,
8238
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"text": "['Desbrun and Cani, 1996']",
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8242,
8264
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"text": "['Cani and Desbrun, 1997']",
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"id": "T80"
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[
8273,
8291
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],
"text": "['Stora et al., 1999']",
"type": "data",
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"offsets": [
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8295,
8305
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],
"text": "['The method']",
"type": "background_claim",
"id": "T82"
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{
"offsets": [
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8349,
8456
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],
"text": "['used elastic forces with dynamically determined neighbors to allow behavior that is similar to plastic flow']",
"type": "background_claim",
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{
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8321,
8346
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"text": "['Desbrun and Gascuel, 1995']",
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8459,
8601
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"text": "['Perhaps the greatest limitation on the level of realism achievable by these particle methods was the relatively small number of particles used']",
"type": "background_claim",
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{
"offsets": [
[
8612,
8728
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"text": "['as processor speeds have increased, particle-based methods have been able to achieve increasingly impressive results']",
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8730,
8792
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"text": "['Compelling real-time results for modestly sized systems appear']",
"type": "background_claim",
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"offsets": [
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8798,
8817
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"text": "['Müller et al., 2003']",
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8850,
8938
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"text": "['demonstrates off-line results that are comparable to the current best grid-based methods']",
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8826,
8847
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"text": "['Premo ze et al., 2003']",
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8950,
9012
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"text": "['both of these recent methods focus on strictly liquid behavior']",
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9014,
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"type": "own_claim",
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9082,
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"text": "['Some methods for modeling solids have dealt with limited amounts of plastic flow']",
"type": "background_claim",
"id": "T93"
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{
"offsets": [
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9244,
9316
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"text": "['describe transition to plastic flow based on von Mises’s yield condition']",
"type": "background_claim",
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9170,
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"text": "['Terzopoulos and Fleischer, 1988a']",
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9209,
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"text": "['Terzopoulos and Fleischer, 1988b']",
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"text": "['used a similar plasticity model for ductile fracture behavior']",
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9323,
9343
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"text": "['O’Brien et al., 2002']",
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9410,
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"text": "['these prior methods used Lagrangian meshes with largely fixed topology']",
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"text": "['they would have encountered “tangling” difficulties']",
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9770
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"text": "['inverting elements']",
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9776,
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"text": "['large amounts of plastic flow']",
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"offsets": [
[
9807,
9894
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"text": "['Another, rather interesting, approach to combining solid and fluid behaviors appears in']",
"type": "background_claim",
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{
"offsets": [
[
9897,
9917
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"text": "['Nixon and Lobb, 2002']",
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"offsets": [
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9979,
10045
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"text": "['The result is an object that behaves somewhat like a water balloon']",
"type": "background_claim",
"id": "T108"
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10047,
10152
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"text": "['Our work builds directly on previous grid-based, Eulerian methods for animating fluids with free surfaces']",
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10155,
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"text": "['Details on these methods can be found in']",
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"offsets": [
[
10198,
10222
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"text": "['Foster and Metaxas, 1996']",
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"offsets": [
[
10226,
10236
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"text": "['Stam, 1999']",
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10240,
10263
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"text": "['Foster and Fedkiw, 2001']",
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"offsets": [
[
10272,
10292
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"text": "['Enright et al., 2002']",
"type": "data",
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"offsets": [
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10312,
10340
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"text": "['our work essentially extends']",
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10366,
10412
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"text": "['to include the behavior of viscoelastic fluids']",
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10343,
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"text": "['Enright et al., 2002']",
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10415,
10495
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"text": "['Outside the graphics field, viscoelastic materials have been studied extensively']",
"type": "background_claim",
"id": "T118"
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{
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10597,
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"text": "['for detailed descriptions of several different models for viscoelastic and elastoplastic materials']",
"type": "background_claim",
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"offsets": [
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10576,
10593
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"text": "['Bird et al., 1987']",
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10547,
10566
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],
"text": "['Han and Reddy, 1999']",
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"offsets": [
[
10533,
10543
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],
"text": "['Fung, 1965']",
"type": "data",
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"offsets": [
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10697,
10853
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"text": "['The general approaches we use for introducing elastic forces into the Navier-Stokes equations, and integrating and advecting strain are not completely novel']",
"type": "own_claim",
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"offsets": [
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10880,
10964
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"text": "['fluid simulation outside the graphics literature that involve elastic forces include']",
"type": "background_claim",
"id": "T124"
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{
"offsets": [
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10967,
10982
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"text": "['Gerritsma, 1996']",
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10986,
11003
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"text": "['Tomé et al., 2002']",
"type": "data",
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11013,
11032
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"text": "['Bonito et al., 2003']",
"type": "data",
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"offsets": [
[
11037,
11152
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"text": "['A detailed analysis of two-dimensional simulations of viscoelastic fluids on staggered rectilinear grids appears in']",
"type": "background_claim",
"id": "T128"
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{
"offsets": [
[
11155,
11170
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],
"text": "['Gerritsma, 1996']",
"type": "data",
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"offsets": [
[
11173,
11319
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"text": "['The three-dimensional method we use for storing rank-two tensor quantities on a staggered grid is a generalization of their two-dimensional method']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
11349,
11500
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],
"text": "['a combination of rectilinear grids and finite elements are used with a volume-of-fluid method to model three-dimensional fluids with elastic properties']",
"type": "background_claim",
"id": "T131"
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{
"offsets": [
[
11326,
11345
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],
"text": "['Bonito et al., 2003']",
"type": "data",
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},
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"offsets": [
[
11502,
11578
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],
"text": "['They do not include plasticity and they store all quantities at cell centers']",
"type": "background_claim",
"id": "T133"
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{
"offsets": [
[
11580,
11611
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],
"text": "['The marker-andcell based method']",
"type": "background_claim",
"id": "T134"
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{
"offsets": [
[
11637,
11695
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],
"text": "['is another example solving viscoelastic free-surface flows']",
"type": "background_claim",
"id": "T135"
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"offsets": [
[
11617,
11634
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"text": "['Tomé et al., 2002']",
"type": "data",
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[
11697,
11800
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"text": "['They address issues relating to elastic-stress based boundary conditions at rigidbody and free-surfaces']",
"type": "background_claim",
"id": "T137"
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{
"offsets": [
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11811,
11859
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],
"text": "['they use a staggered grid for the velocity field']",
"type": "background_claim",
"id": "T138"
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{
"offsets": [
[
11861,
11913
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],
"text": "['they still store their tensor values at cell centers']",
"type": "background_claim",
"id": "T139"
},
{
"offsets": [
[
11975,
12052
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],
"text": "['The framework we use for fluid simulation is based on the method described in']",
"type": "own_claim",
"id": "T140"
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{
"offsets": [
[
12055,
12075
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"text": "['Enright et al., 2002']",
"type": "data",
"id": "T141"
},
{
"offsets": [
[
12078,
12173
]
],
"text": "['This framework consists of two primary components which work together to produce useful results']",
"type": "background_claim",
"id": "T142"
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{
"offsets": [
[
12175,
12259
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],
"text": "['The first is a rectilinear grid that stores the values that define the fluid’s state']",
"type": "background_claim",
"id": "T143"
},
{
"offsets": [
[
12401,
12494
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],
"text": "['The second component is a function whose levelset at zero locates the boundaries of the fluid']",
"type": "background_claim",
"id": "T146"
},
{
"offsets": [
[
12757,
12825
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],
"text": "['we focus on the changes we make to accommodate viscoelastic behavior']",
"type": "own_claim",
"id": "T149"
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{
"offsets": [
[
12828,
12926
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],
"text": "['We suggest the following references for a more complete description of this simulation methodology']",
"type": "background_claim",
"id": "T150"
},
{
"offsets": [
[
12930,
12954
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],
"text": "['Foster and Metaxas, 1996']",
"type": "data",
"id": "T151"
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{
"offsets": [
[
12958,
12968
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],
"text": "['Stam, 1999']",
"type": "data",
"id": "T152"
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{
"offsets": [
[
12972,
12995
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],
"text": "['Foster and Fedkiw, 2001']",
"type": "data",
"id": "T153"
},
{
"offsets": [
[
13004,
13024
]
],
"text": "['Enright et al., 2002']",
"type": "data",
"id": "T154"
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[
12677,
12752
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"text": "['A full description of this type of system is beyond the scope of this paper']",
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"text": "['we also require rules concerning how the elastic strain changes due to plastic yielding']",
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"text": "['of the strain deviation remains below the yield point']",
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"text": "['the limit is exceeded']",
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"text": "['Equation (9) does not take into account the movement of the material through space']",
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"text": "['Harlow and Welch, 1965']",
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"text": "['in addition to scalars']",
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"text": "['a rank-two tensor']",
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"text": "['velocity']",
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"text": "['velocity components are stored separately on faces']",
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19410,
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19710,
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19733,
19767
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19770,
19842
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19888,
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19861,
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"text": "['Enright et al., 2002']",
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19962,
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"text": "['Enright et al., 2004']",
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19985,
20047
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"text": "['The authors note that the method is susceptible to volume loss']",
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"text": "['we found this behavior to be problematic for some of our examples that involve fixed, small amounts of fluid']",
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"text": "['We were able to ameliorate this problem somewhat by using a level-set grid with twice the fluid grid’s resolution, and that is staggered with respect to the fluid grid']",
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"text": "['The motion of the pure fluid example differs substantially from the viscoelastic examples']",
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"text": "['the surfaces of the viscoelastic examples retain evidence of the impact even after motion has stopped']",
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"text": "['several splashes that are generated when a fluid sphere is hurled into a tank containing the same material']",
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"text": "['show similar behavior that occurs when cubes of different materials are dropped onto a hard surface']",
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A03 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A03_S04_Automatic_Rigging_and_Animation_of_3D_Characters_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1275808.1276467
To appear in the ACM SIGGRAPH conference proceedings
<Title>Automatic Rigging and Animation of 3D Characters</Title>
Ilya Baran ∗ Jovan Popović † Computer Science
Artificial Intelligence Laboratory Massachusetts Institute of Technology
†
∗ e-mail: [email protected] † e-mail: [email protected]
<Abstract>Animating an articulated 3D character currently requires manual rigging to specify its internal skeletal structure and to define how the input motion deforms its surface. We present a method for animating characters automatically. Given a static character mesh and a generic skeleton, our method adapts the skeleton to the character and attaches it to the surface, allowing skeletal motion data to animate the character. Because a single skeleton can be used with a wide range of characters, our method, in conjunction with a library of motions for a few skeletons, enables a user-friendly animation system for novices and children. Our prototype implementation, called Pinocchio, typically takes under a minute to rig a character on a modern midrange PC.</Abstract>
<H2>CR Categories: </H2>I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation Keywords: Animation, Deformations, Geometric Modeling <H2>Keywords: </H2>Animation, Deformations, Geometric Modeling
<H1>1 Introduction</H1>
Modeling in 3D is becoming much easier than before. User-friendly systems such as Teddy
[Igarashi et al. 1999]
and Cosmic Blobs ( http://www.cosmicblobs.com/) have made the creation of 3D characters accessible to novices and children. Bringing these static shapes to life, however, is still not easy. In a conventional skeletal animation package, the user must rig the character manually. This requires placing the skeleton joints inside the character and specifying which parts of the surface are attached to which bone. The tedium of this process makes simple character animation more difficult than it could be. We envision a system that eliminates this tedium to make animation more accessible for children, educators, researchers, and other non-expert animators. For example, a child should be able to model a unicorn, click the “Quadruped Gallop” button, and watch the unicorn start galloping. To support this functionality, we need a method (as shown in Figure 1 ) that takes a character, a skeleton, and a motion of that skeleton as input, and outputs the moving character. The missing portion is the rigging: motion transfer has been addressed in prior work
[Gleicher 2001]. Our algorithm consists of two main steps: skeleton embedding and skin attachment. Skeleton embedding computes the joint positions of the skeleton inside the character by minimizing a penalty function. To make the optimization problem computationally feasible, we first embed the skeleton into a discretization of the character’s interior and then refine this embedding using continuous optimization. The skin attachment is computed by assigning bone weights based on the proximity of the embedded bones smoothed by a diffusion equilibrium equation over the character’s surface. Our design decisions relied on three criteria, which we also used to evaluate our system:
Figure 1: The automatic rigging method presented in this paper allowed us to implement an easy-to-use animation system, which we called Pinocchio. In this example, the triangle mesh of a jolly cartoon character is brought to life by embedding a skeleton inside it and applying a walking motion to the initially static shape.
• Generality: A single skeleton is applicable to a wide variety of characters: for example, our method can use a generic biped skeleton to rig an anatomically correct human model, an anthropomorphic robot, and even something that has very little resemblance to a human. • Quality: The resulting animation quality is comparable to that of modern video games. • Performance: The automatic rigging usually takes under one minute on an everyday PC.
A key design challenge is constructing a penalty function that penalizes undesirable embeddings and generalizes well to new characters. For this, we designed a maximum-margin supervised learning method to combine a set of hand-constructed penalty functions. To ensure an honest evaluation and avoid overfitting, we tested our algorithm on 16 characters that we did not see or use during development. Our algorithm computed a good rig for all but 3 of these characters. For each of the remaining cases, one joint placement hint corrected the problem. We simplify the problem by making the following assumptions. The character mesh must be the boundary of a connected volume. The character must be given in approximately the same orientation and pose as the skeleton. Lastly, the character must be proportioned roughly like the given skeleton. We introduce several new techniques to solve the automatic rigging problem: • A maximum-margin method for learning the weights of a linear combination of penalty functions based on examples, as an alternative to hand-tuning (Section 3.3). • An A ∗ -like heuristic to accelerate the search for an optimal skeleton embedding over an exponential search space (Section 3.4). • Use of Laplace’s diffusion equation to generate weights for attaching mesh vertices to the skeleton using linear blend skinning (Section 4). This method could also be useful in existing 3D packages. Our prototype system, called Pinocchio, rigs the given character using our algorithm. It then transfers a motion to the character using online motion retargetting
[Choi and Ko 2000] to eliminate footskate by constraining the feet trajectories of the character to the feet trajectories of the given motion.
1
To appear in the ACM SIGGRAPH conference proceedings
<H1>2 Related Work</H1>
Character Animation Most prior research in character animation, especially in 3D, has focused on professional animators; very little work is targeted at novice users. Recent exceptions include Motion Doodles
[Thorne et al. 2004] as well as the work of Igarashi et al. on spatial keyframing [2005b] and as-rigid-as-possible shape manipulation [2005a]. These approaches focus on simplifying animation control, rather than simplifying the definition of the articulation of the character. In particular, a spatial keyframing system expects an articulated character as input, and as-rigid-as-possible shape manipulation, besides being 2D, relies on the constraints to provide articulation information. The Motion Doodles system has the ability to infer the articulation of a 2D character, but their approach relies on very strong assumptions about how the character is presented. Skeleton Extraction Although most skeleton-based prior work on automatic rigging focused on skeleton extraction, for our problem, we advocate skeleton embedding. A few approaches to the skeleton extraction problem are representative. Teichmann and Teller [1998] extract a skeleton by simplifying the Voronoi skeleton with a small amount of user assistance. Liu et al. [2003]
use repulsive force fields to find a skeleton. In their paper, Katz and Tal [2003] describe a surface partitioning algorithm and suggest skeleton extraction as an application. The technique in Wade [2000]
is most similar to our own: like us, they approximate the medial surface by finding discontinuities in the distance field, but they use it to construct a skeleton tree. For the purpose of automatically animating a character, however, skeleton embedding is much more suitable than extraction. For example, the user may have motion data for a quadruped skeleton, but for a complicated quadruped character, the extracted skeleton is likely to have a different topology. The anatomically appropriate skeleton generation by Wade [2000]
ameliorates this problem by techniques such as identifying appendages and fitting appendage templates, but the overall topology of the resulting skeleton may still vary. For example, for the character in Figure 1 , ears may be mistaken for arms. Another advantage of embedding over extraction is that the given skeleton provides information about the expected structure of the character, which may be difficult to obtain from just the geometry. So although we could use an existing skeleton extraction algorithm and embed our skeleton into the extracted one, the results would likely be undesirable. For example, the legs of the character in Figure 1 would be too short if a skeleton extraction algorithm were used. Template Fitting Animating user-provided data by fitting a template has been successful in cases when the model is fairly similar to the template. Most of the work has been focused on human models, making use of human anatomy specifics, e.g.
[Moccozet et al. 2004]. For segmenting and animating simple 3D models of characters and inanimate objects,
Anderson et al. [2000] fit voxel-based volumetric templates to the data. Skinning Almost any system for mesh deformation (whether surface based [Lipman et al. 2005; Yu et al. 2004] or volume based [Zhou et al. 2005]
) can be adapted for skeleton-based deformation. Teichmann and Teller [1998] propose a spring-based method. Unfortunately, at present, these methods are unsuitable for real-time animation of even moderate size meshes. Because of its simplicity and efficiency (and simple GPU implementation), and despite its quality shortcomings, linear blend skinning (LBS), also known as skeleton subspace deformation, remains the most popular method used in practice. Most real-time skinning work, e.g. [Kry et al. 2002; Wang et al. 2007]
, has focused on improving on LBS by inferring the character articulation from multiple example meshes. However, such techniques are unsuitable for our problem because we only have a single mesh. Instead, we must infer articulation by using the given skeleton as an encoding of the likely modes of deformation, not just as an animation control structure. To our knowledge, the problem of finding bone weights for LBS from a single mesh and a skeleton has not been sufficiently addressed in the literature. Previous methods are either mesh resolution dependent
[Katz and Tal 2003]
or the weights do not vary smoothly along the surface [Wade 2000]
, causing artifacts on highresolution meshes.
Some commercial packages use proprietary methods to assign default weights. For example, Autodesk Maya 7 assigns weights based solely on the vertex proximity to the bone, ignoring the mesh structure, which results in serious artifacts when the mesh intersects the Voronoi diagram faces between logically distant bones.
<H1>3 Skeleton Embedding</H1>
Skeleton embedding resizes and positions the given skeleton to fit inside the character. This can be formulated as an optimization problem: “compute the joint positions such that the resulting skeleton fits inside the character as nicely as possible and looks like the given skeleton as much as possible.” For a skeleton with s joints (by “joints,” we mean vertices of the skeleton tree, including leaves), this is a 3s-dimensional problem with a complicated objective function. Solving such a problem directly using continuous optimization is infeasible. Pinocchio therefore discretizes the problem by constructing a graph whose vertices represent potential joint positions and whose edges are potential bone segments. This is challenging because the graph must have few vertices and edges, and yet capture all potential bone paths within the character. The graph is constructed by packing spheres centered on the approximate medial surface into the character and by connecting sphere centers with graph edges. Pinocchio then finds the optimal embedding of the skeleton into this graph with respect to a discrete penalty function. It uses the discrete solution as a starting point for continuous optimization. To help with optimization, the given skeleton can have a little extra information in the form of joint attributes: for example, joints that should be approximately symmetric should be marked as such; also some joints can be marked as “feet,” indicating that they should be placed near the bottom of the character. We describe the attributes Pinocchio uses in a supplemental document[Baran and Popović 2007a]. These attributes are specific to the skeleton but are independent of the character shape and do not reduce the generality of the skeletons.
2
To appear in the ACM SIGGRAPH conference proceedings
Figure 2: Approximate Medial Sur-
Figure 3: Packed Spheres
face
<H2>3.1 Discretization</H2>
Before any other computation, Pinocchio rescales the character to fit inside an axis-aligned unit cube. As a result, all of the tolerances are relative to the size of the character. Distance Field To approximate the medial surface and to facilitate other computations, Pinocchio computes a trilinearly interpolated adaptively sampled signed distance field on an octree
[Frisken et al. 2000]. It constructs a kd-tree to evaluate the exact signed distance to the surface from an arbitrary point. It then constructs the distance field from the top down, starting with a single octree cell and splitting a cell until the exact distance is within a tolerance τ of the interpolated distance. We found that τ = 0.003 provides a good compromise between accuracy and efficiency for our purposes. Because only negative distances (i.e. from points inside the character) are important, Pinocchio does not split cells that are guaranteed not to intersect the character’s interior. Approximate Medial Surface Pinocchio uses the adaptive distance field to compute a sample of points approximately on the medial surface ( Figure 2 ). The medial surface is the set of C 1 discontinuities of the distance field. Within a single cell of our octree, the interpolated distance field is guaranteed to be C 1 , so it is necessary to look at only the cell boundaries. Pinocchio therefore traverses the octree and for each cell, looks at a grid (of spacing τ ) of points on each face of the cell. It then computes the gradient vectors for the cells adjacent to each grid point—if the angle between two of them is 120 ◦ or greater, it adds the point to the medial surface sample. We impose the 120 ◦ condition because we do not want the “noisy” parts of the medial surface—we want the points where skeleton joints are likely to lie. For the same reason, Pinocchio filters out the sampled points that are too close to the character surface (within 2τ ). Wade discusses a similar condition in Chapter 4 of his thesis [2000]. Sphere Packing To pick out the graph vertices from the medial surface, Pinocchio packs spheres into the character as follows: it sorts the medial surface points by their distance to the surface (those that are farthest from the surface are first). Then it processes these points in order and if a point is outside all previously added spheres, adds the sphere centered at that point whose radius is the distance to the surface. In other words, the largest spheres are added first, and no sphere contains the center of another sphere ( Figure 3 ). Although the procedure described above takes O(nb) time in the worst case (where n is the number of points, and b is the final number of spheres inserted), worst case behavior is rarely seen because most points are processed while there is a small number of large
Figure 4: Constructed Graph Figure 5: The original and reduced quadruped skeleton
spheres. In fact, this step typically takes less than 1% of the time of the entire algorithm.
Graph Construction The final discretization step constructs the edges of the graph by connecting some pairs of sphere centers (Figure 4). Pinocchio adds an edge between two sphere centers if the spheres intersect. We would also like to add edges between spheres that do not intersect if that edge is well inside the surface and if that edge is “essential.” For example, the neck and left shoulder spheres of the character in Figure 3 are disjoint, but there should still be an edge between them. The precise condition Pinocchio uses is that the distance from any point of the edge to the surface must be at least half of the radius of the smaller sphere, and the closest sphere centers to the midpoint of the edge must be the edge endpoints. The latter condition is equivalent to the requirement that additional edges must be in the Gabriel graph of the sphere centers (see e.g.
[Jaromczyk and Toussaint 1992]). While other conditions can be formulated, we found that the Gabriel graph provides a good balance between sparsity and connectedness. Pinocchio precomputes the shortest paths between all pairs of vertices in this graph to speed up penalty function evaluation.
<H2>3.2 Reduced Skeleton G = (V, E)</H2>
The discretization stage constructs a geometric graph into which Pinocchio needs to embed the given skeleton in an optimal way. The skeleton is given as a rooted tree on s joints. To reduce the degrees of freedom, for the discrete embedding, Pinocchio works with a reduced skeleton, in which all bone chains have been merged (all degree two joints, such as knees, eliminated), as shown in Figure 5 . The reduced skeleton thus has only r joints. This works because once Pinocchio knows where the endpoints of a bone chain are in V , it can compute the intermediate joints by taking the shortest path between the endpoints and splitting it in accordance with the proportions of the unreduced skeleton. For the humanoid skeleton we use, for example, s = 18, but r = 7; without a reduced skeleton, the optimization problem would typically be intractable. Therefore, the discrete skeleton embedding problem is to find the embedding of the reduced skeleton into G, represented by an rtuple v = (v 1 , . . . , v r ) of vertices in V , which minimizes a penalty function f (v) that is designed to penalize differences in the embedded skeleton from the given skeleton.
<H2>3.3 Discrete Penalty Function</H2>
The discrete penalty function has great impact on the generality and quality of the results. A good embedding should have the proportions, bone orientations, and size similar to the given skeleton. The paths representing the bone chains should be disjoint, if possible. Joints of the skeleton may be marked as “feet,” in which case they should be close to the bottom of the character. Designing a penalty function that satisfies all of these requirements simultaneously is difficult. Instead we found it easier to design penalties independently and then rely on learning a proper weighting for a global penalty that combines each term. The Setup We represent the penalty function f as a linear combination of k “basis” penalty functions: f (v) = P k i=1 γ i b i (v). Pinocchio uses k = 9 basis penalty functions constructed by hand. They penalize short bones, improper orientation between joints, length differences in bones marked symmetric, bone chains sharing vertices, feet away from the bottom, zero-length bone chains, improper orientation of bones, degree-one joints not embedded at extreme vertices, and joints far along bone-chains but close in the graph [Baran and Popović 2007a]. We determine the weights Γ = (γ 1 , . . . , γ k ) semi-automatically via a new maximum margin approach inspired by support vector machines. Suppose that for a single character, we have several example embeddings, each marked “good” or “bad”. The basis penalty functions assign a feature vector b(v) = (b 1 (v), . . . , b k (v)) to each example embedding v. Let p 1 , . . . , p m be the k-dimensional feature vectors of the good embeddings and let q 1 , . . . , q n be the feature vectors of the bad embeddings. Maximum Margin To provide context for our approach, we review the relevant ideas from the theory of support vector machines. See Burges [1998]
for a much more complete tutorial. If our goal were to automatically classify new embeddings into “good” and “bad” ones, we could use a support vector machine to learn a maximum margin linear classifier. In its simplest form, a support vector machine finds the hyperplane that separates the p i ’s from the q i ’s and is as far away from them as possible. More precisely, if Γ is a k-dimensional vector with Γ = 1, the classification margin of the best hyperplane normal to Γ is 1 2 ` min n i=1 Γ T q i − max m i=1 Γ T p i ́ . Recalling that the total penalty of an embedding v is Γ T b(v), we can think of the maximum margin Γ as the one that best distinguishes between the best “bad” embedding and the worst “good” embedding in the training set. In our case, however, we do not need to classify embeddings, but rather find a Γ such that the embedding with the lowest penalty f (v) = Γ T b(v) is likely to be good. To this end, we want Γ to distinguish between the best “bad” embedding and the best “good” embedding, as illustrated in Figure 6 . We therefore wish to maximize the optimization margin (subject to Γ = 1), which we define as: n m min Γ T q i − min Γ T p i . i=1 i=1 Because we have different characters in our training set, and because the embedding quality is not necessarily comparable between different characters, we find the Γ that maximizes the minimum margin over all of the characters. Our approach is similar to margin-based linear structured classification
[Taskar et al. 2003], the problem of learning a classifier that to each problem instance (cf. character) assigns the discrete label (cf. embedding) that minimizes the dot product of a weights vector with basis functions of the problem instance and label. The key difference is that structured classification requires an explicit loss function (in our case, the knowledge of the quality of all possible skeleton embeddings for each character in the training set), whereas our approach only makes use of the loss function on the training labels and allows for the possibility of multiple correct labels. This possibility of multiple correct skeleton embeddings prevented us from formulating our margin maximization problem as a convex optimization problem. However, multiple correct skeleton embeddings are necessary for our problem in cases such as the hand joint being embedded into different fingers.
3
To appear in the ACM SIGGRAPH conference proceedings
Good embeddings (p i ’s): Bad embeddings (q i ’s): b 2 Best Γ Margin 0 b 1
Figure 6: Illustration of optimization margin: marked skeleton embeddings in the space of their penalties (b i ’s)
Learning Procedure The problem of finding the optimal Γ does not appear to be convex. However, an approximately optimal Γ is acceptable, and the search space dimension is sufficiently low (9 in our case) that it is feasible to use a continuous optimization method. We use the Nelder-Mead method
[Nelder and Mead 1965] starting from random Γ’s. We start with a cube [0, 1] k , pick random normalized Γ’s, and run Nelder-Mead from each of them. We then take the best Γ, use a slightly smaller cube around it, and repeat. To create our training set of embeddings, we pick a training set of characters, manually choose Γ, and use it to construct skeleton embeddings of the characters. For every character with a bad embedding, we manually tweak Γ until a good embedding is produced. We then find the maximum margin Γ as described above and use this new Γ to construct new skeleton embeddings. We manually classify the embeddings that we have not previously seen, augment our training set with them, and repeat the process. If Γ eventually stops changing, as happened on our training set, we use the found Γ. It is also possible that a positive margin Γ cannot be found, indicating that the chosen basis functions are probably inadequate for finding good embeddings for all characters in the training set. For training, we used 62 different characters (Cosmic Blobs models, free models from the web, scanned models, and Teddy models), and Γ was stable with about 400 embeddings. The weights we learned resulted in good embeddings for all of the characters in our training set; we could not accomplish this by manually tuning the weights. Examining the optimization results and the extremal embeddings also helped us design better basis penalty functions. Although this process of finding the weights is labor-intensive, it only needs to be done once. According to our tests, if the basis functions are carefully chosen, the overall penalty function generalizes well to both new characters and new skeletons. Therefore, a novice user will be able to use the system, and more advanced users will be able to design new skeletons without having to learn new weights.
<H2>3.4 Discrete Embedding</H2>
Computing a discrete embedding that minimizes a general penalty function is intractable because there are exponentially many embeddings. However, if it is easy to estimate a good lower bound on f from a partial embedding (of the first few joints), it is possible to use a branch-and-bound method. Pinocchio uses this idea: it maintains a priority queue of partial embeddings ordered by their lower bound estimates. At every step, it takes the best partial embedding from the queue, extends it in all possible ways with the next joint, and pushes the results back on the queue. The first full embedding extracted is guaranteed to be the optimal one. This is essentially the A* algorithm on the tree of possible embeddings. To speed up the process and conserve memory, if a partial embedding has a very high lower bound, it is rejected immediately and not inserted into the queue. Although this algorithm is still worst-case exponential, it is fast on most real problems with the skeletons we tested. We considered adapting an approximate graph matching algorithm, like
[Gold and Rangarajan 1996]
, which would work much faster and enable more complicated reduced skeletons. However, computing the exact optimum simplified penalty function design and debugging. The joints of the skeleton are given in order, which induces an order on the joints of the reduced skeleton. Referring to the joints by their indices (starting with the root at index 1), we define the parent function p R on the reduced skeleton, such that p R (i) (for 1 < i ≤ r) is the index of the parent of joint i. We require that the order in which the joints are given respects the parent relationship, i.e. p R (i) < i. Our penalty function (f ) can be expressed as the sum of independent functions of bone chain endpoints (f i ’s) and a term (f D ) that incorporates the dependence between different joint positions. The dependence between joints that have not been embedded can be ignored to obtain a lower bound on f . More precisely, f can be written as: r r f (v 1 , . . . , v r ) = X f i (v i , v p R (i) ) + X f D (v 1 , . . . , v i ). i=2 i=2 A lower bound when the first k joints are embedded is then: k k X f i (v i , v p R (i) ) + X f D (v 1 , . . . , v i ) + i=2 i=2 + X min f i (v i , v p R (i) ) v i ∈V {i>k|p R (i)≤k} If f D is small compared to the f i ’s, as is often the case for us, the lower bound is close to the true value of f . Because of this lower bound estimate, the order in which joints are embedded is very important to the performance of the optimization algorithm. High degree joints should be embedded first because they result in more terms in the rightmost sum of the lower bound, leading to a more accurate lower bound. For example, our biped skeleton has only two joints of degree greater than two, so after Pinocchio has embedded them, the lower bound estimate includes f i terms for all of the bone chains. Because there is no perfect penalty function, discrete embedding will occasionally produce undesirable results (see Model 13 in Figure 9). In such cases it is possible for the user to provide manual hints in the form of constraints for reduced skeleton joints. For example, such a hint might be that the left hand of the skeleton should be embedded at a particular vertex in G (or at one of several vertices). Embeddings that do not satisfy the constraints are simply not considered by the algorithm.
4
To appear in the ACM SIGGRAPH conference proceedings
<H2>3.5 Embedding Refinement</H2>
Pinocchio takes the optimal embedding of the reduced skeleton found by discrete optimization and reinserts the degree-two joints by splitting the shortest paths in G in proportion to the given skeleton. The resulting skeleton embedding should have the general shape we are looking for, but typically, it will not fit nicely inside the character. Also, smaller bones are likely to be incorrectly oriented because they were not important enough to influence the discrete optimization. Embedding refinement corrects these problems by minimizing a new continuous penalty function ( Figure 7 ). For the continuous optimization, we represent the embedding of the skeleton as an s-tuple of joint positions (q 1 , . . . , q s ) in R 3 . Because we are dealing with an unreduced skeleton, and discrete optimization has already found the correct general shape, the penalty function can be much simpler than the discrete penalty function. The continuous penalty function g that Pinocchio tries to minimize is the sum of penalty functions over the bones plus an asymmetry penalty: where p S is the parent function for the unreduced skeleton (analogous to p R ). Each g i penalizes bones that do not fit inside the surface nicely, bones that are too short, and bones that are oriented differently from the given skeleton: g i = α S g i S + α L g i L + α O g i O . Unlike the discrete case, we choose the α’s by hand because there are only four of them [Baran and Popović 2007a]. Any continuous optimization technique [Gill et al. 1989] should produce good results. Pinocchio uses a gradient descent method that takes advantage of the fact that there are relatively few interactions. As a subroutine, it uses a step-doubling line search: starting from a given point (in R 3s ), it takes steps in the given optimization direction, doubling step length until the penalty function increases. Pinocchio intersperses a line search in the gradient direction with line searches in the gradient direction projected onto individual bones. Repeating the process 10 times is usually sufficient for convergence.
Figure 7: The embedded skeleton after discrete embedding (blue) and the results of embedding refinement (dark red)
s g(q 1 , . . . , q s ) = α A g A (q 1 , . . . , q s ) + X g i (q i , q p S (i) ) i=2
<H1>4 Skin Attachment</H1>
The character and the embedded skeleton are disconnected until skin attachment specifies how to apply deformations of the skeleton to the character mesh. Although we could make use of one of the various mesh editing techniques for the actual mesh deformation, we choose to focus on the standard linear blend skinning (LBS) method because of its widespread use. If v j is the position of vertex j, T i is the transformation of the i th bone, and w j i is the weight of the i th bone for vertex j, LBS gives the position of the transformed vertex j as P i w j i T i (v j ). The attachment problem is finding bone weights w i for the vertices—how much each bone transform affects each vertex. There are several properties we desire of the weights. First of all, they should not depend on the mesh resolution. Second, for the results to look good, the weights need to vary smoothly along the surface. Finally, to avoid folding artifacts, the width of a transition between two bones meeting at a joint should be roughly proportional to the distance from the joint to the surface. Although a scheme that assigns bone weights purely based on proximity to bones can be made to satisfy these properties, such schemes will often fail because they ignore the character’s geometry: for example, part of the torso may become attached to an arm. Instead, we use the analogy to heat equilibrium to find the weights. Suppose we treat the character volume as an insulated heat-conducting body and force the temperature of bone i to be 1 while keeping the temperature of all of the other bones at 0. Then we can take the equilibrium temperature at each vertex on the surface as the weight of bone i at that vertex. Figure 8 illustrates this in two dimensions. Solving for heat equilibrium over a volume would require tessellating the volume and would be slow. Therefore, for simplicity, Pinocchio solves for equilibrium over the surface only, but at some vertices, it adds the heat transferred from the nearest bone. i The equilibrium over the surface for bone i is given by ∂w = ∂t ∆w i + H(p i − w i ) = 0, which can be written as
5
To appear in the ACM SIGGRAPH conference proceedings
Figure 8: Top: heat equilibrium for two bones. Bottom: the result of rotating the right bone with the heat-based attachment
1
−∆w i + Hw i = Hp i ,
where ∆ is the discrete surface Laplacian, calculated with the cotangent formula
[Meyer et al. 2003], p i is a vector with p i j = 1 if the nearest bone to vertex j is i and p i j = 0 otherwise, and H is the diagonal matrix with H jj being the heat contribution weight of the nearest bone to vertex j. Because ∆ has units of length −2 , so must H. Letting d(j) be the distance from vertex j to the nearest bone, Pinocchio uses H jj = c/d(j) 2 if the shortest line segment from the vertex to the bone is contained in the character volume and H jj = 0 if it is not. It uses the precomputed distance field to determine whether a line segment is entirely contained in the character volume. For c ≈ 0.22, this method gives weights with similar transitions to those computed by finding the equilibrium over the volume. Pinocchio uses c = 1 (corresponding to anisotropic heat diffusion) because the results look more natural. When k bones are equidistant from vertex j, heat contributions from all of them are used: p j is 1/k for all of them, and H jj = kc/d(j) 2 . Equation (1) is a sparse linear system, and the left hand side matrix −∆ + H does not depend on i, the bone we are interested in. Thus we can factor the system once and back-substitute to find the weights for each bone. Botsch et al. [2005] show how to use a sparse Cholesky solver to compute the factorization for this kind of system. Pinocchio uses the TAUCS [Toledo 2003]
library for this computation. Note also that the weights w i sum to 1 for each vertex: if we sum (1) over i, we get (−∆ + H) P i w i = H · 1, which yields P i w i = 1. It is possible to speed up this method slightly by finding vertices that are unambiguously attached to a single bone and forcing their weight to 1. An earlier variant of our algorithm did this, but the improvement was negligible, and this introduced occasional artifacts.
<H1>5 Results</H1>
We evaluate Pinocchio with respect to the three criteria stated in the introduction: generality, quality, and performance. To ensure an objective evaluation, we use inputs that were not used during development. To this end, once the development was complete, we tested Pinocchio on 16 biped Cosmic Blobs models that we had not previously tried.
Figure 10: A centaur pirate with a centaur skeleton embedded looks at a cat with a quadruped skeleton embedded
Figure 11: The human scan on the left is rigged by Pinocchio and is posed on the right by changing joint angles in the embedded skeleton. The well-known deficiencies of LBS can be seen in the right knee and hip areas.
<H2>5.1 Generality</H2>
Figure 9 shows our 16 test characters and the skeletons Pinocchio embedded. The skeleton was correctly embedded into 13 of these models (81% success). For Models 7, 10 and 13, a hint for a single joint was sufficient to produce a good embedding. These tests demonstrate the range of proportions that our method can tolerate: we have a well-proportioned human (Models 1–4, 8), large arms and tiny legs (6; in 10, this causes problems), and large legs and small arms (15; in 13, the small arms cause problems). For other characters we tested, skeletons were almost always correctly embedded into well-proportioned characters whose pose matched the given skeleton. Pinocchio was even able to transfer a biped walk onto a human hand, a cat on its hind legs, and a donut. The most common issues we ran into on other characters were: • The thinnest limb into which we may hope to embed a bone has a radius of 2τ . Characters with extremely thin limbs often fail because the the graph we extract is disconnected. Reducing τ , however, hurts performance. • Degree 2 joints such as knees and elbows are often positioned incorrectly within a limb. We do not know of a reliable way to identify the right locations for them: on some characters they are thicker than the rest of the limb, and on others they are thinner. Although most of our tests were done with the biped skeleton, we have also used other skeletons for other characters ( Figure 10 ).
<H2>5.2 Quality</H2>
Figure 11 shows the results of manually posing a human scan using our attachment. Our video [Baran and Popović 2007b]
demonstrates the quality of the animation produced by Pinocchio.
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To appear in the ACM SIGGRAPH conference proceedings
1. 2. 3.
7.
8. 9.
13.
14.
Figure 9: Test Results for Skeleton Embedding
Model
3
10
11
Mean
Number of Vertices
19,001
34,339
56,856
33,224
Discretization Time
10.3s
25.8s
68.2s
24.3s
Embedding Time
1.4s
29.1s
5.7s
5.2s
Attachment Time
0.9s
1.9s
3.2s
1.8s
Total Time
12.6s
56.8s
77.1s
31.3s
Model 3 10 11 Mean Number of Vertices 19,001 34,339 56,856 33,224 Discretization Time 10.3s 25.8s 68.2s 24.3s Embedding Time 1.4s 29.1s 5.7s 5.2s Attachment Time 0.9s 1.9s 3.2s 1.8s Total Time 12.6s 56.8s 77.1s 31.3s
Table 1: Timings for three representative models and the mean over our 16 character test set
The quality problems of our attachment are a combination of the deficiencies of our automated weights generation as well as those inherent in LBS. A common class of problems is caused by Pinocchio being oblivious to the material out of which the character is made: the animation of both a dress and a knight’s armor has an unrealistic, rubbery quality. Other problems occur at difficult areas, such as hips and the shoulder/neck region, where hand-tuned weights could be made superior to those found by our algorithm.
<H2>5.3 Performance</H2>
Table 1 shows the fastest and slowest timings of Pinocchio rigging the 16 models discussed in Section 5.1 on a 1.73 MHz Intel Core Duo with 1GB of RAM. Pinocchio is single-threaded so only one core was used. We did not run timing tests on denser models because someone wishing to create real-time animation is likely to keep the triangle count low. Also, because of our volume-based approach, once the distance field has been computed, subsequent discretization and embedding steps do not depend on the given mesh size. For the majority of models, the running time is dominated by the discretization stage, and that is dominated by computing the distance field. Embedding refinement takes about 1.2 seconds for all of these models, and the discrete optimization consumes the rest of the embedding time.
<H1>6 Conclusion and Future Work</H1>
We have presented the first method for automatically rigging an unfamiliar character for skeletal animation. In conjunction with ex-
4. 5. 6.
10. 11.
12.
15.
16.
isting techniques, it allows a user to go from a static mesh to an animated character quickly and effortlessly. We have shown that using this method, Pinocchio can animate a wide range of characters. We also believe that some of our techniques, such as finding LBS weights and using examples to learn the weights of a linear combination of penalty functions, can be useful in other contexts. We have several ideas for improving Pinocchio that we have not yet tried. Discretization could be improved by packing ellipsoids instead of spheres. Although this is more difficult, we believe it would greatly reduce the size of the graph, resulting in faster and higher quality discrete embeddings. Animation quality can be improved with a better skinning model
[Kavan and Zára ˇ 2005] (although possibly at the cost of performance). One approach would be to use a technique
[Wang et al. 2007]
that corrects LBS errors by using example meshes, which we could synthesize using slower, but more accurate deformation techniques. A more involved approach would be automatically building a tetrahedral mesh around the embedded skeleton and applying the dynamic deformation method of
Capell et al. [2002]. Combining retargetting with joint limits should eliminate some artifacts in the motion. A better retargetting scheme could be used to make animations more physically plausible and prevent global self-intersections. Finally, it would be nice to eliminate the assumption that the character must have a well-defined interior. Beyond Pinocchio’s current capabilities, an interesting problem is dealing with hand animation to give animated characters the ability to grasp objects, type, or speak sign language. The variety of types of hands makes this challenging (see, for example, Models 13, 5, 14, and 11 in Figure 9 ). Automatically rigging characters for facial animation is even more difficult, but a solution requiring a small amount of user assistance may succeed. Combined with a system for motion synthesis [Arikan et al. 2003], this would allow users to begin interacting with their creations.
<H1>7 Acknowledgments</H1>
We thank Yeuhi Abe and Eugene Hsu for help with motion capture. Thanks to Soonmin Bae, Inna Baran, Frédo Durand, Sylvain Paris, Ariel Shamir, Daniel Vlasic, and Robert Wang for their helpful feedback. Thanks to Emily Whiting for narrating the video. We thank Dragomir Anguelov for the human meshes. We would also like to thank Solidworks for the permission to use Cosmic Blobs models. This work was supported by a grant from Solidworks Corporation. The first author was also supported by an NSF Graduate Research Fellowship.
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8
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"text": "['The character must be given in approximately the same orientation and pose as the skeleton']",
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"text": "['the character must be proportioned roughly like the given skeleton']",
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"text": "['We introduce several new techniques to solve the automatic rigging problem']",
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"text": "['A maximum-margin method for learning the weights of a linear combination of penalty functions based on examples, as an alternative to hand-tuning']",
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"text": "['Section 3.3']",
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"text": "['An A ∗ -like heuristic to accelerate the search for an optimal skeleton embedding over an exponential search space']",
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"text": "['Use of Laplace’s diffusion equation to generate weights for attaching mesh vertices to the skeleton using linear blend skinning']",
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"text": "['This method could also be useful in existing 3D packages']",
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"text": "['Section 3.4']",
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"text": "['Our prototype system']",
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"text": "['rigs the given character using our algorithm']",
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"text": "['It then transfers a motion to the character using online motion retargetting']",
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"text": "['to eliminate footskate by constraining the feet trajectories of the character to the feet trajectories of the given motion']",
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"text": "['Choi and Ko 2000']",
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"text": "['very little work is targeted at novice users']",
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"text": "['Recent exceptions include Motion Doodles']",
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"text": "['Thorne et al. 2004']",
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"text": "['as well as the work of']",
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"text": "['on spatial keyframing']",
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"text": "['Igarashi et al']",
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],
"text": "['2005b']",
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"text": "['2005a']",
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"text": "['as-rigid-as-possible shape manipulation']",
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"id": "T58"
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"text": "['These approaches focus on simplifying animation control, rather than simplifying the definition of the articulation of the characte']",
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"text": "['their approach relies on very strong assumptions about how the character is presented']",
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"text": "['The Motion Doodles system has the ability to infer the articulation of a 2D character']",
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"text": "['a spatial keyframing system expects an articulated character as input, and as-rigid-as-possible shape manipulation']",
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"text": "['relies on the constraints to provide articulation information']",
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"text": "['Although most skeleton-based prior work on automatic rigging focused on skeleton extraction']",
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"text": "['we advocate skeleton embedding']",
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"text": "['our problem']",
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"text": "['skeleton embedding is much more suitable than extraction']",
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"text": "['A few approaches to the skeleton extraction problem are representative']",
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"text": "['extract a skeleton by simplifying the Voronoi skeleton with a small amount of user assistance']",
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"text": "['Teichmann and Teller [1998]']",
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"text": "['use repulsive force fields to find a skeleton']",
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"text": "['Liu et al. [2003]']",
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"text": "['Katz and Tal [2003]']",
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"text": "['describe a surface partitioning algorithm and suggest skeleton extraction as an application']",
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"text": "['Wade [2000]']",
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"text": "['The technique']",
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"text": "['is most similar to our own']",
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"text": "['they approximate the medial surface by finding discontinuities in the distance field']",
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"text": "['they use it to construct a skeleton tree']",
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"text": "['the purpose of automatically animating a character']",
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"text": "['the user may have motion data for a quadruped skeleton']",
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"text": "['a complicated quadruped character']",
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"text": "['the extracted skeleton is likely to have a different topology']",
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"text": "['The anatomically appropriate skeleton generation']",
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"text": "['ameliorates this problem by techniques such as identifying appendages and fitting appendage templates']",
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"text": "['Wade [2000]']",
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"text": "['the overall topology of the resulting skeleton may still vary']",
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"text": "['ears may be mistaken for arms']",
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"text": "['the character in Figure 1']",
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"text": "['Another advantage of embedding over extraction is that the given skeleton provides information about the expected structure of the character']",
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"text": "['may be difficult to obtain from just the geometry']",
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"text": "['we could use an existing skeleton extraction algorithm and embed our skeleton into the extracted one']",
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"text": "['the results would likely be undesirable']",
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"text": "['Figure 1']",
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"text": "['the legs of the character']",
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"id": "T95"
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"offsets": [
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"text": "['would be too short if a skeleton extraction algorithm were used']",
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"text": "['the model is fairly similar to the template']",
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"text": "['Animating user-provided data by fitting a template has been successful']",
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"text": "['Most of the work has been focused on human models, making use of human anatomy specifics']",
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"text": "['Moccozet et al. 2004']",
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"text": "['segmenting and animating simple 3D models of characters and inanimate objects']",
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"text": "['fit voxel-based volumetric templates to the data']",
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"text": "['Anderson et al. [2000]']",
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"text": "['Almost any system for mesh deformation']",
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"text": "['can be adapted for skeleton-based deformation']",
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"text": "['surface based']",
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"text": "['Lipman et al. 2005']",
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"text": "['Yu et al. 2004']",
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"text": "['Zhou et al. 2005']",
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"text": "['propose a spring-based method']",
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{
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"text": "['Teichmann and Teller [1998]']",
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"text": "['these methods are unsuitable for real-time animation of even moderate size meshes']",
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"text": "['efficiency']",
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"text": "['Kry et al. 2002']",
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"text": "['Wang et al. 2007']",
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"text": "['such techniques are unsuitable for our problem']",
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"text": "['we must infer articulation by using the given skeleton as an encoding of the likely modes of deformation, not just as an animation control structure']",
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"text": "['we only have a single mesh']",
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"text": "['the problem of finding bone weights for LBS from a single mesh and a skeleton has not been sufficiently addressed in the literature']",
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"text": "['Previous methods are either mesh resolution dependent']",
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"text": "['or the weights do not vary smoothly along the surface']",
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"text": "['Katz and Tal 2003']",
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"text": "['Wade 2000']",
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"text": "['causing artifacts on highresolution meshes']",
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"text": "['Some commercial packages use proprietary methods to assign default weights']",
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"text": "['Autodesk Maya 7 assigns weights based solely on the vertex proximity to the bone']",
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"text": "['results in serious artifacts when the mesh intersects the Voronoi diagram faces between logically distant bones']",
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"text": "['Other problems occur at difficult areas']",
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"text": "['hand-tuned weights could be made superior to those found by our algorithm']",
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"text": "['hips']",
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"text": "['the shoulder/neck region']",
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"text": "['Skeleton embedding resizes and positions the given skeleton to fit inside the character']",
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"text": "['This can be formulated as an optimization problem']",
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"text": "['compute the joint positions such that the resulting skeleton fits inside the character as nicely as possible and looks like the given skeleton as much as possible.']",
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],
"text": "['a skeleton with s joints']",
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],
"text": "['this is a 3s-dimensional problem with a complicated objective function']",
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"text": "['Solving such a problem directly using continuous optimization is infeasible']",
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"text": "['Pinocchio']",
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"text": "['discretizes the problem by constructing a graph whose vertices represent potential joint positions and whose edges are potential bone segments']",
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"text": "['the graph must have few vertices and edges']",
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"text": "['capture all potential bone paths within the character']",
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"text": "['This is challenging']",
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12188,
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"text": "['Pinocchio then finds the optimal embedding of the skeleton into this graph with respect to a discrete penalty function']",
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"text": "['These attributes are specific to the skeleton']",
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"text": "['are independent of the character shape and do not reduce the generality of the skeletons']",
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"text": "['Pinocchio rescales the character to fit inside an axis-aligned unit cube']",
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"text": "['Before any other computation']",
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"text": "['all of the tolerances are relative to the size of the character']",
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"text": "['Pinocchio computes a trilinearly interpolated adaptively sampled signed distance field on an octree']",
"type": "own_claim",
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"text": "['Frisken et al. 2000']",
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"text": "['It constructs a kd-tree to evaluate the exact signed distance to the surface from an arbitrary point']",
"type": "own_claim",
"id": "T159"
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"text": "['τ = 0.003 provides a good compromise between accuracy and efficiency for our purposes']",
"type": "own_claim",
"id": "T160"
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"offsets": [
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14010,
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"text": "['only negative distances (i.e. from points inside the character) are important']",
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14089,
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"text": "['Pinocchio does not split cells that are guaranteed not to intersect the character’s interior']",
"type": "own_claim",
"id": "T162"
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{
"offsets": [
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14210,
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],
"text": "['Pinocchio uses the adaptive distance field to compute a sample of points approximately on the medial surface']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
14321,
14329
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],
"text": "['Figure 2']",
"type": "data",
"id": "T164"
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{
"offsets": [
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14333,
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"text": "['The medial surface is the set of C 1 discontinuities of the distance field']",
"type": "own_claim",
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"text": "['the interpolated distance field is guaranteed to be C 1']",
"type": "own_claim",
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14409,
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"text": "['Within a single cell of our octree']",
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14506,
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],
"text": "['it is necessary to look at only the cell boundaries']",
"type": "own_claim",
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"text": "['Pinocchio']",
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14579,
14634
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],
"text": "['traverses the octree and for each cell, looks at a grid']",
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"text": "['of points on each face of the cell']",
"type": "own_claim",
"id": "T171"
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{
"offsets": [
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14869,
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],
"text": "['We impose the 120 ◦ condition']",
"type": "own_claim",
"id": "T172"
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],
"text": "['we do not want the “noisy” parts of the medial surface']",
"type": "data",
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],
"text": "['we want the points where skeleton joints are likely to lie']",
"type": "data",
"id": "T174"
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"offsets": [
[
15043,
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],
"text": "['Pinocchio filters out the sampled points that are too close to the character surface']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
15147,
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],
"text": "['discusses a similar condition']",
"type": "background_claim",
"id": "T176"
},
{
"offsets": [
[
15205,
15209
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],
"text": "['2000']",
"type": "data",
"id": "T177"
},
{
"offsets": [
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15180,
15189
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],
"text": "['Chapter 4']",
"type": "data",
"id": "T178"
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{
"offsets": [
[
15142,
15146
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],
"text": "['Wade']",
"type": "data",
"id": "T179"
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"offsets": [
[
15283,
15325
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],
"text": "['Pinocchio packs spheres into the character']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
15338,
15405
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],
"text": "['it sorts the medial surface points by their distance to the surface']",
"type": "own_claim",
"id": "T181"
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{
"offsets": [
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15507,
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],
"text": "['a point is outside all previously added spheres']",
"type": "data",
"id": "T182"
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{
"offsets": [
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15556,
15638
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],
"text": "['adds the sphere centered at that point whose radius is the distance to the surface']",
"type": "own_claim",
"id": "T183"
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"offsets": [
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15465,
15499
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"text": "['it processes these points in order']",
"type": "own_claim",
"id": "T184"
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{
"offsets": [
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15697,
15744
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],
"text": "['no sphere contains the center of another sphere']",
"type": "own_claim",
"id": "T185"
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{
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[
15747,
15755
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],
"text": "['Figure 3']",
"type": "data",
"id": "T186"
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"offsets": [
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15656,
15691
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"text": "['the largest spheres are added first']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
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15768,
15832
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],
"text": "['the procedure described above takes O(nb) time in the worst case']",
"type": "own_claim",
"id": "T188"
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{
"offsets": [
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15915,
15949
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],
"text": "['worst case behavior is rarely seen']",
"type": "own_claim",
"id": "T189"
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"text": "['most points are processed while there is a small number of large']",
"type": "own_claim",
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16175,
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"text": "['spheres']",
"type": "own_claim",
"id": "T191"
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{
"offsets": [
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16193,
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],
"text": "['this step typically takes less than 1% of the time of the entire algorithm']",
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"id": "T192"
},
{
"offsets": [
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16298,
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],
"text": "['The final discretization step constructs the edges of the graph by connecting some pairs of sphere centers']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
16406,
16414
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],
"text": "['Figure 4']",
"type": "data",
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"text": "['Pinocchio adds an edge between two sphere centers']",
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"id": "T195"
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"text": "['the spheres intersect']",
"type": "data",
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"offsets": [
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16493,
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],
"text": "['We would also like to add edges between spheres that do not intersect']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
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16602
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],
"text": "['that edge is well inside the surface']",
"type": "data",
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16610,
16635
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],
"text": "['that edge is “essential.”']",
"type": "data",
"id": "T199"
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16649,
16700
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],
"text": "['the neck and left shoulder spheres of the character']",
"type": "own_claim",
"id": "T200"
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16713,
16725
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],
"text": "['are disjoint']",
"type": "own_claim",
"id": "T201"
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"text": "['there should still be an edge between them']",
"type": "own_claim",
"id": "T202"
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],
"text": "['Figure 3']",
"type": "data",
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"offsets": [
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"text": "['We describe the attributes Pinocchio uses in a supplemental document']",
"type": "own_claim",
"id": "T204"
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{
"offsets": [
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12770,
12793
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],
"text": "['Baran and Popović 2007a']",
"type": "data",
"id": "T205"
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{
"offsets": [
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17021,
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"text": "['The latter condition is equivalent to the requirement that additional edges must be in the Gabriel graph of the sphere centers']",
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{
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17160,
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],
"text": "['Jaromczyk and Toussaint 1992']",
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"id": "T207"
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17248,
17324
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],
"text": "['the Gabriel graph provides a good balance between sparsity and connectedness']",
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{
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17198,
17232
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],
"text": "['other conditions can be formulated']",
"type": "own_claim",
"id": "T209"
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"offsets": [
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"text": "['Pinocchio works with a reduced skeleton']",
"type": "own_claim",
"id": "T210"
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"text": "['all bone chains have been merged']",
"type": "own_claim",
"id": "T211"
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{
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17920,
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],
"text": "['Figure 5']",
"type": "data",
"id": "T212"
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],
"text": "['The reduced skeleton']",
"type": "own_claim",
"id": "T213"
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{
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17957,
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"text": "['has only r joints']",
"type": "own_claim",
"id": "T214"
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{
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18063,
18229
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],
"text": "['it can compute the intermediate joints by taking the shortest path between the endpoints and splitting it in accordance with the proportions of the unreduced skeleton']",
"type": "own_claim",
"id": "T215"
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{
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17976,
17986
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],
"text": "['This works']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
18000,
18060
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],
"text": "['Pinocchio knows where the endpoints of a bone chain are in V']",
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{
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18297,
18380
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"text": "['without a reduced skeleton, the optimization problem would typically be intractable']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
18235,
18263
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],
"text": "['the humanoid skeleton we use']",
"type": "data",
"id": "T219"
},
{
"offsets": [
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18393,
18689
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],
"text": "['the discrete skeleton embedding problem is to find the embedding of the reduced skeleton into G, represented by an rtuple v = (v 1 , . . . , v r ) of vertices in V , which minimizes a penalty function f (v) that is designed to penalize differences in the embedded skeleton from the given skeleton']",
"type": "own_claim",
"id": "T220"
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{
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18768,
18859
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],
"text": "['The discrete penalty function has great impact on the generality and quality of the results']",
"type": "own_claim",
"id": "T221"
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{
"offsets": [
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18861,
18964
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"text": "['A good embedding should have the proportions, bone orientations, and size similar to the given skeleton']",
"type": "own_claim",
"id": "T222"
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{
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18966,
19023
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"text": "['The paths representing the bone chains should be disjoint']",
"type": "own_claim",
"id": "T223"
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19028,
19036
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],
"text": "['possible']",
"type": "data",
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19100,
19151
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"text": "['they should be close to the bottom of the character']",
"type": "own_claim",
"id": "T225"
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{
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19038,
19085
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],
"text": "['Joints of the skeleton may be marked as “feet,”']",
"type": "data",
"id": "T226"
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{
"offsets": [
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19153,
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],
"text": "['Designing a penalty function that satisfies all of these requirements simultaneously is difficult']",
"type": "own_claim",
"id": "T227"
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{
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[
19260,
19402
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"text": "['we found it easier to design penalties independently and then rely on learning a proper weighting for a global penalty that combines each term']",
"type": "own_claim",
"id": "T228"
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{
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19601,
19931
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"text": "['They penalize short bones, improper orientation between joints, length differences in bones marked symmetric, bone chains sharing vertices, feet away from the bottom, zero-length bone chains, improper orientation of bones, degree-one joints not embedded at extreme vertices, and joints far along bone-chains but close in the graph']",
"type": "own_claim",
"id": "T229"
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19933,
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"text": "['Baran and Popović 2007a']",
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"text": "['a much more complete tutorial']",
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"id": "T231"
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20599,
20612
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"text": "['Burges [1998]']",
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"text": "['we could use a support vector machine to learn a maximum margin linear classifier']",
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"id": "T233"
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{
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"text": "['our goal were to automatically classify new embeddings into “good” and “bad” ones']",
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"text": "['we can think of the maximum margin Γ as the one that best distinguishes between the best “bad” embedding and the worst “good” embedding in the training set']",
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"id": "T235"
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"text": "['the total penalty of an embedding v is Γ T b(v)']",
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"text": "['we do not need to classify embeddings']",
"type": "own_claim",
"id": "T237"
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{
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21428,
21529
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"text": "['rather find a Γ such that the embedding with the lowest penalty f (v) = Γ T b(v) is likely to be good']",
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"id": "T238"
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"text": "['our case']",
"type": "data",
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"text": "['we want Γ to distinguish between the best “bad” embedding and the best “good” embedding']",
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"id": "T240"
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"text": "['Figure 6']",
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"text": "['We']",
"type": "own_claim",
"id": "T242"
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"text": "['wish to maximize the optimization margin']",
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21804,
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"text": "['we have different characters in our training set']",
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"text": "['the embedding quality is not necessarily comparable between different characters']",
"type": "own_claim",
"id": "T245"
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{
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"text": "['we find the Γ that maximizes the minimum margin over all of the characters']",
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"id": "T246"
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{
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"text": "['Our approach is similar to margin-based linear structured classification']",
"type": "own_claim",
"id": "T247"
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{
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22099,
22117
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"text": "['Taskar et al. 2003']",
"type": "data",
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22353,
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"text": "['The key difference is that structured classification requires an explicit loss function']",
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"id": "T249"
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{
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22569,
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"text": "['our approach only makes use of the loss function on the training labels and allows for the possibility of multiple correct labels']",
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"id": "T250"
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{
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22862,
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"text": "['multiple correct skeleton embeddings are necessary for our problem']",
"type": "own_claim",
"id": "T251"
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{
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22946,
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],
"text": "['the hand joint being embedded into different fingers']",
"type": "data",
"id": "T252"
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{
"offsets": [
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22700,
22851
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"text": "['This possibility of multiple correct skeleton embeddings prevented us from formulating our margin maximization problem as a convex optimization problem']",
"type": "own_claim",
"id": "T253"
},
{
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23338,
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],
"text": "['The problem of finding the optimal Γ does not appear to be convex']",
"type": "own_claim",
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{
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23454
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"text": "['an approximately optimal Γ is acceptable']",
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"id": "T255"
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{
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],
"text": "['the search space dimension is sufficiently low']",
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"id": "T256"
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23509
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"text": "['9']",
"type": "data",
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"text": "['it is feasible to use a continuous optimization method']",
"type": "own_claim",
"id": "T258"
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{
"offsets": [
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24484
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],
"text": "['It is also possible that a positive margin Γ cannot be found']",
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"id": "T259"
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{
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24502,
24619
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],
"text": "['the chosen basis functions are probably inadequate for finding good embeddings for all characters in the training set']",
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"id": "T260"
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{
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24794,
24890
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],
"text": "['The weights we learned resulted in good embeddings for all of the characters in our training set']",
"type": "own_claim",
"id": "T261"
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{
"offsets": [
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24892,
24951
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],
"text": "['we could not accomplish this by manually tuning the weights']",
"type": "own_claim",
"id": "T262"
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{
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24953,
25068
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"text": "['Examining the optimization results and the extremal embeddings also helped us design better basis penalty functions']",
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"id": "T263"
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{
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],
"text": "['this process of finding the weights is labor-intensive']",
"type": "own_claim",
"id": "T264"
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{
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25164
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],
"text": "['it only needs to be done once']",
"type": "own_claim",
"id": "T265"
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{
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25235,
25321
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],
"text": "['the overall penalty function generalizes well to both new characters and new skeletons']",
"type": "own_claim",
"id": "T266"
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{
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25193,
25233
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],
"text": "['the basis functions are carefully chosen']",
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25179,
25188
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],
"text": "['our tests']",
"type": "data",
"id": "T268"
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{
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25378
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],
"text": "['a novice user will be able to use the system']",
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"id": "T269"
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],
"text": "['more advanced users will be able to design new skeletons without having to learn new weights']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
25548,
25635
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],
"text": "['Computing a discrete embedding that minimizes a general penalty function is intractable']",
"type": "own_claim",
"id": "T271"
},
{
"offsets": [
[
25644,
25683
]
],
"text": "['there are exponentially many embeddings']",
"type": "data",
"id": "T272"
},
{
"offsets": [
[
25796,
25843
]
],
"text": "['it is possible to use a branch-and-bound method']",
"type": "own_claim",
"id": "T273"
},
{
"offsets": [
[
25697,
25768
]
],
"text": "['it is easy to estimate a good lower bound on f from a partial embedding']",
"type": "data",
"id": "T274"
},
{
"offsets": [
[
25845,
25869
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],
"text": "['Pinocchio uses this idea']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
25871,
25961
]
],
"text": "['it maintains a priority queue of partial embeddings ordered by their lower bound estimates']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
26125,
26195
]
],
"text": "['The first full embedding extracted is guaranteed to be the optimal one']",
"type": "own_claim",
"id": "T277"
},
{
"offsets": [
[
26197,
26268
]
],
"text": "['This is essentially the A* algorithm on the tree of possible embeddings']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
26367,
26425
]
],
"text": "['it is rejected immediately and not inserted into the queue']",
"type": "own_claim",
"id": "T279"
},
{
"offsets": [
[
26318,
26365
]
],
"text": "['a partial embedding has a very high lower bound']",
"type": "data",
"id": "T280"
},
{
"offsets": [
[
26436,
26482
]
],
"text": "['this algorithm is still worst-case exponential']",
"type": "own_claim",
"id": "T281"
},
{
"offsets": [
[
26484,
26545
]
],
"text": "['it is fast on most real problems with the skeletons we tested']",
"type": "own_claim",
"id": "T282"
},
{
"offsets": [
[
26570,
26609
]
],
"text": "['an approximate graph matching algorithm']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
26646,
26720
]
],
"text": "['which would work much faster and enable more complicated reduced skeletons']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
26618,
26642
]
],
"text": "['Gold and Rangarajan 1996']",
"type": "data",
"id": "T285"
},
{
"offsets": [
[
26731,
26807
]
],
"text": "['computing the exact optimum simplified penalty function design and debugging']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
27065,
27129
]
],
"text": "['p R (i) (for 1 < i ≤ r) is the index of the parent of joint i']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
26996,
27053
]
],
"text": "['we define the parent function p R on the reduced skeleton']",
"type": "data",
"id": "T288"
},
{
"offsets": [
[
27560,
27663
]
],
"text": "['f can be written as: r r f (v 1 , . . . , v r ) = X f i (v i , v p R (i) ) + X f D (v 1 , . . . , v i )']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
27673,
27857
]
],
"text": "['A lower bound when the first k joints are embedded is then: k k X f i (v i , v p R (i) ) + X f D (v 1 , . . . , v i ) + i=2 i=2 + X min f i (v i , v p R (i) ) v i ∈V {i>k|p R (i)≤k}']",
"type": "own_claim",
"id": "T290"
},
{
"offsets": [
[
27927,
27974
]
],
"text": "['the lower bound is close to the true value of f']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
27858,
27896
]
],
"text": "['If f D is small compared to the f i ’s']",
"type": "data",
"id": "T292"
},
{
"offsets": [
[
28015,
28120
]
],
"text": "['the order in which joints are embedded is very important to the performance of the optimization algorithm']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
27988,
28013
]
],
"text": "['this lower bound estimate']",
"type": "data",
"id": "T294"
},
{
"offsets": [
[
28122,
28165
]
],
"text": "['High degree joints should be embedded first']",
"type": "own_claim",
"id": "T295"
},
{
"offsets": [
[
28174,
28239
]
],
"text": "['they result in more terms in the rightmost sum of the lower bound']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
28241,
28279
]
],
"text": "['leading to a more accurate lower bound']",
"type": "own_claim",
"id": "T297"
},
{
"offsets": [
[
28294,
28359
]
],
"text": "['our biped skeleton has only two joints of degree greater than two']",
"type": "data",
"id": "T298"
},
{
"offsets": [
[
28364,
28397
]
],
"text": "['after Pinocchio has embedded them']",
"type": "data",
"id": "T299"
},
{
"offsets": [
[
28399,
28469
]
],
"text": "['the lower bound estimate includes f i terms for all of the bone chains']",
"type": "own_claim",
"id": "T300"
},
{
"offsets": [
[
28479,
28515
]
],
"text": "['there is no perfect penalty function']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
28517,
28581
]
],
"text": "['discrete embedding will occasionally produce undesirable results']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
28587,
28607
]
],
"text": "['Model 13 in Figure 9']",
"type": "data",
"id": "T303"
},
{
"offsets": [
[
28624,
28730
]
],
"text": "['it is possible for the user to provide manual hints in the form of constraints for reduced skeleton joints']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
28745,
28847
]
],
"text": "['such a hint might be that the left hand of the skeleton should be embedded at a particular vertex in G']",
"type": "data",
"id": "T305"
},
{
"offsets": [
[
28881,
28970
]
],
"text": "['Embeddings that do not satisfy the constraints are simply not considered by the algorithm']",
"type": "own_claim",
"id": "T306"
},
{
"offsets": [
[
29322,
29403
]
],
"text": "['The resulting skeleton embedding should have the general shape we are looking for']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
29420,
29463
]
],
"text": "['it will not fit nicely inside the character']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
29471,
29522
]
],
"text": "['smaller bones are likely to be incorrectly oriented']",
"type": "own_claim",
"id": "T309"
},
{
"offsets": [
[
29531,
29600
]
],
"text": "['they were not important enough to influence the discrete optimization']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
29602,
29694
]
],
"text": "['Embedding refinement corrects these problems by minimizing a new continuous penalty function']",
"type": "own_claim",
"id": "T311"
},
{
"offsets": [
[
29697,
29705
]
],
"text": "['Figure 7']",
"type": "data",
"id": "T312"
},
{
"offsets": [
[
29856,
29897
]
],
"text": "['we are dealing with an unreduced skeleton']",
"type": "data",
"id": "T313"
},
{
"offsets": [
[
29903,
29968
]
],
"text": "['discrete optimization has already found the correct general shape']",
"type": "data",
"id": "T314"
},
{
"offsets": [
[
29970,
30045
]
],
"text": "['the penalty function can be much simpler than the discrete penalty function']",
"type": "own_claim",
"id": "T315"
},
{
"offsets": [
[
30496,
30521
]
],
"text": "['we choose the α’s by hand']",
"type": "own_claim",
"id": "T316"
},
{
"offsets": [
[
30530,
30557
]
],
"text": "['there are only four of them']",
"type": "own_claim",
"id": "T317"
},
{
"offsets": [
[
30559,
30582
]
],
"text": "['Baran and Popović 2007a']",
"type": "data",
"id": "T318"
},
{
"offsets": [
[
30585,
30622
]
],
"text": "['Any continuous optimization technique']",
"type": "own_claim",
"id": "T319"
},
{
"offsets": [
[
30642,
30669
]
],
"text": "['should produce good results']",
"type": "own_claim",
"id": "T320"
},
{
"offsets": [
[
30624,
30640
]
],
"text": "['Gill et al. 1989']",
"type": "data",
"id": "T321"
},
{
"offsets": [
[
31135,
31203
]
],
"text": "['Repeating the process 10 times is usually sufficient for convergence']",
"type": "own_claim",
"id": "T322"
},
{
"offsets": [
[
31692,
31787
]
],
"text": "['we could make use of one of the various mesh editing techniques for the actual mesh deformation']",
"type": "own_claim",
"id": "T323"
},
{
"offsets": [
[
31789,
31858
]
],
"text": "['we choose to focus on the standard linear blend skinning (LBS) method']",
"type": "own_claim",
"id": "T324"
},
{
"offsets": [
[
31870,
31888
]
],
"text": "['its widespread use']",
"type": "background_claim",
"id": "T325"
},
{
"offsets": [
[
32025,
32099
]
],
"text": "['LBS gives the position of the transformed vertex j as P i w j i T i (v j )']",
"type": "own_claim",
"id": "T326"
},
{
"offsets": [
[
31893,
31924
]
],
"text": "['v j is the position of vertex j']",
"type": "data",
"id": "T327"
},
{
"offsets": [
[
31926,
31968
]
],
"text": "['T i is the transformation of the i th bone']",
"type": "data",
"id": "T328"
},
{
"offsets": [
[
31974,
32023
]
],
"text": "['w j i is the weight of the i th bone for vertex j']",
"type": "data",
"id": "T329"
},
{
"offsets": [
[
32101,
32168
]
],
"text": "['The attachment problem is finding bone weights w i for the vertices']",
"type": "own_claim",
"id": "T330"
},
{
"offsets": [
[
32288,
32333
]
],
"text": "['they should not depend on the mesh resolution']",
"type": "own_claim",
"id": "T331"
},
{
"offsets": [
[
32219,
32272
]
],
"text": "['There are several properties we desire of the weights']",
"type": "own_claim",
"id": "T332"
},
{
"offsets": [
[
32373,
32424
]
],
"text": "['the weights need to vary smoothly along the surface']",
"type": "own_claim",
"id": "T333"
},
{
"offsets": [
[
32463,
32602
]
],
"text": "['the width of a transition between two bones meeting at a joint should be roughly proportional to the distance from the joint to the surface']",
"type": "own_claim",
"id": "T334"
},
{
"offsets": [
[
32435,
32461
]
],
"text": "['to avoid folding artifacts']",
"type": "data",
"id": "T335"
},
{
"offsets": [
[
32343,
32371
]
],
"text": "['for the results to look good']",
"type": "data",
"id": "T336"
},
{
"offsets": [
[
32613,
32722
]
],
"text": "['a scheme that assigns bone weights purely based on proximity to bones can be made to satisfy these properties']",
"type": "own_claim",
"id": "T337"
},
{
"offsets": [
[
32724,
32797
]
],
"text": "['such schemes will often fail because they ignore the character’s geometry']",
"type": "own_claim",
"id": "T338"
},
{
"offsets": [
[
32812,
32859
]
],
"text": "['part of the torso may become attached to an arm']",
"type": "own_claim",
"id": "T339"
},
{
"offsets": [
[
32870,
32928
]
],
"text": "['we use the analogy to heat equilibrium to find the weights']",
"type": "own_claim",
"id": "T340"
},
{
"offsets": [
[
33116,
33224
]
],
"text": "['we can take the equilibrium temperature at each vertex on the surface as the weight of bone i at that vertex']",
"type": "own_claim",
"id": "T341"
},
{
"offsets": [
[
32938,
33004
]
],
"text": "['we treat the character volume as an insulated heat-conducting body']",
"type": "data",
"id": "T342"
},
{
"offsets": [
[
33009,
33109
]
],
"text": "['force the temperature of bone i to be 1 while keeping the temperature of all of the other bones at 0']",
"type": "data",
"id": "T343"
},
{
"offsets": [
[
33226,
33234
]
],
"text": "['Figure 8']",
"type": "data",
"id": "T344"
},
{
"offsets": [
[
33271,
33369
]
],
"text": "['Solving for heat equilibrium over a volume would require tessellating the volume and would be slow']",
"type": "own_claim",
"id": "T345"
},
{
"offsets": [
[
33398,
33452
]
],
"text": "['Pinocchio solves for equilibrium over the surface only']",
"type": "own_claim",
"id": "T346"
},
{
"offsets": [
[
33476,
33526
]
],
"text": "['it adds the heat transferred from the nearest bone']",
"type": "own_claim",
"id": "T347"
},
{
"offsets": [
[
33458,
33474
]
],
"text": "['at some vertices']",
"type": "data",
"id": "T348"
},
{
"offsets": [
[
33530,
33618
]
],
"text": "['The equilibrium over the surface for bone i is given by ∂w = ∂t ∆w i + H(p i − w i ) = 0']",
"type": "own_claim",
"id": "T349"
},
{
"offsets": [
[
33626,
33643
]
],
"text": "['can be written as']",
"type": "own_claim",
"id": "T350"
},
{
"offsets": [
[
33909,
33928
]
],
"text": "['−∆w i + Hw i = Hp i']",
"type": "own_claim",
"id": "T351"
},
{
"offsets": [
[
33954,
34028
]
],
"text": "['∆ is the discrete surface Laplacian, calculated with the cotangent formula']",
"type": "own_claim",
"id": "T352"
},
{
"offsets": [
[
34031,
34048
]
],
"text": "['Meyer et al. 2003']",
"type": "data",
"id": "T353"
},
{
"offsets": [
[
34051,
34081
]
],
"text": "['p i is a vector with p i j = 1']",
"type": "own_claim",
"id": "T354"
},
{
"offsets": [
[
34085,
34118
]
],
"text": "['the nearest bone to vertex j is i']",
"type": "data",
"id": "T355"
},
{
"offsets": [
[
34123,
34132
]
],
"text": "['p i j = 0']",
"type": "own_claim",
"id": "T356"
},
{
"offsets": [
[
34133,
34142
]
],
"text": "['otherwise']",
"type": "data",
"id": "T357"
},
{
"offsets": [
[
34259,
34283
]
],
"text": "['∆ has units of length −2']",
"type": "data",
"id": "T358"
},
{
"offsets": [
[
34286,
34295
]
],
"text": "['so must H']",
"type": "own_claim",
"id": "T359"
},
{
"offsets": [
[
34361,
34391
]
],
"text": "['Pinocchio uses H jj = c/d(j) 2']",
"type": "own_claim",
"id": "T360"
},
{
"offsets": [
[
34395,
34485
]
],
"text": "['the shortest line segment from the vertex to the bone is contained in the character volume']",
"type": "data",
"id": "T361"
},
{
"offsets": [
[
34490,
34498
]
],
"text": "['H jj = 0']",
"type": "own_claim",
"id": "T362"
},
{
"offsets": [
[
34502,
34511
]
],
"text": "['it is not']",
"type": "data",
"id": "T363"
},
{
"offsets": [
[
34649,
34760
]
],
"text": "['this method gives weights with similar transitions to those computed by finding the equilibrium over the volume']",
"type": "own_claim",
"id": "T364"
},
{
"offsets": [
[
34639,
34647
]
],
"text": "['c ≈ 0.22']",
"type": "data",
"id": "T365"
},
{
"offsets": [
[
34837,
34866
]
],
"text": "['the results look more natural']",
"type": "own_claim",
"id": "T366"
},
{
"offsets": [
[
34762,
34782
]
],
"text": "['Pinocchio uses c = 1']",
"type": "own_claim",
"id": "T367"
},
{
"offsets": [
[
34912,
34956
]
],
"text": "['heat contributions from all of them are used']",
"type": "own_claim",
"id": "T368"
},
{
"offsets": [
[
34958,
34984
]
],
"text": "['p j is 1/k for all of them']",
"type": "own_claim",
"id": "T369"
},
{
"offsets": [
[
34990,
35006
]
],
"text": "['H jj = kc/d(j) 2']",
"type": "own_claim",
"id": "T370"
},
{
"offsets": [
[
34873,
34910
]
],
"text": "['k bones are equidistant from vertex j']",
"type": "data",
"id": "T371"
},
{
"offsets": [
[
35144,
35227
]
],
"text": "['we can factor the system once and back-substitute to find the weights for each bone']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
35022,
35047
]
],
"text": "['is a sparse linear system']",
"type": "own_claim",
"id": "T373"
},
{
"offsets": [
[
35053,
35106
]
],
"text": "['the left hand side matrix −∆ + H does not depend on i']",
"type": "own_claim",
"id": "T374"
},
{
"offsets": [
[
35009,
35021
]
],
"text": "['Equation (1)']",
"type": "data",
"id": "T375"
},
{
"offsets": [
[
35250,
35343
]
],
"text": "['show how to use a sparse Cholesky solver to compute the factorization for this kind of system']",
"type": "background_claim",
"id": "T376"
},
{
"offsets": [
[
35229,
35249
]
],
"text": "['Botsch et al. [2005]']",
"type": "data",
"id": "T377"
},
{
"offsets": [
[
35345,
35369
]
],
"text": "['Pinocchio uses the TAUCS']",
"type": "own_claim",
"id": "T378"
},
{
"offsets": [
[
35385,
35413
]
],
"text": "['library for this computation']",
"type": "own_claim",
"id": "T379"
},
{
"offsets": [
[
35371,
35382
]
],
"text": "['Toledo 2003']",
"type": "data",
"id": "T380"
},
{
"offsets": [
[
35430,
35470
]
],
"text": "['the weights w i sum to 1 for each vertex']",
"type": "own_claim",
"id": "T381"
},
{
"offsets": [
[
35494,
35525
]
],
"text": "['we get (−∆ + H) P i w i = H · 1']",
"type": "own_claim",
"id": "T382"
},
{
"offsets": [
[
35533,
35551
]
],
"text": "['yields P i w i = 1']",
"type": "own_claim",
"id": "T383"
},
{
"offsets": [
[
35475,
35492
]
],
"text": "['we sum (1) over i']",
"type": "data",
"id": "T384"
},
{
"offsets": [
[
35553,
35699
]
],
"text": "['It is possible to speed up this method slightly by finding vertices that are unambiguously attached to a single bone and forcing their weight to 1']",
"type": "own_claim",
"id": "T385"
},
{
"offsets": [
[
35701,
35745
]
],
"text": "['An earlier variant of our algorithm did this']",
"type": "own_claim",
"id": "T386"
},
{
"offsets": [
[
35751,
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"type": "own_claim",
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"text": "['Figure 9']",
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"text": "['Models 7, 10 and 13']",
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"type": "own_claim",
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"text": "['Models 1–4, 8']",
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"text": "['large arms and tiny legs']",
"type": "own_claim",
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"text": "['large legs and small arms']",
"type": "own_claim",
"id": "T397"
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"text": "['the small arms cause problems']",
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"text": "['13']",
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"text": "['this causes problems']",
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"text": "['skeletons were almost always correctly embedded into well-proportioned characters whose pose matched the given skeleton']",
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"text": "['other characters we tested']",
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"text": "['Pinocchio was even able to transfer a biped walk onto a human hand, a cat on its hind legs, and a donut']",
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"text": "['The thinnest limb into which we may hope to embed a bone has a radius of 2τ']",
"type": "own_claim",
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"text": "['Characters with extremely thin limbs often fail']",
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"text": "['the graph we extract is disconnected']",
"type": "own_claim",
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"text": "['Reducing τ']",
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"text": "['hurts performance']",
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"text": "['are often positioned incorrectly within a limb']",
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"text": "['knees']",
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"text": "['elbows']",
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"text": "['Figure 10']",
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"text": "['Our video']",
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"text": "['Baran and Popović 2007b']",
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"text": "['The quality problems of our attachment are a combination of the deficiencies of our automated weights generation as well as those inherent in LBS']",
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"text": "['A common class of problems is caused by Pinocchio being oblivious to the material out of which the character is made']",
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"text": "['the animation of both a dress and a knight’s armor has an unrealistic, rubbery quality']",
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"text": "['only one core was used']",
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"type": "own_claim",
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"text": "['We did not run timing tests on denser models']",
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"text": "['subsequent discretization and embedding steps do not depend on the given mesh size']",
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"text": "['the distance field has been computed']",
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"text": "['volume-based approach']",
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"text": "['that is dominated by computing the distance field']",
"type": "own_claim",
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"text": "['majority of models']",
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"text": "['Embedding refinement takes about 1.2 seconds for all of these models']",
"type": "own_claim",
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"text": "['the discrete optimization consumes the rest of the embedding time']",
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"text": "['We have presented the first method for automatically rigging an unfamiliar character for skeletal animation']",
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"text": "['it allows a user to go from a static mesh to an animated character quickly and effortlessly']",
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"text": "['some of our techniques']",
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"text": "['can be useful in other contexts']",
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42485,
42557
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"text": "['We have several ideas for improving Pinocchio that we have not yet tried']",
"type": "own_claim",
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{
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"text": "['Discretization could be improved by packing ellipsoids instead of spheres']",
"type": "own_claim",
"id": "T445"
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{
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"text": "['this is more difficult']",
"type": "own_claim",
"id": "T446"
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{
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42678,
42723
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"text": "['it would greatly reduce the size of the graph']",
"type": "own_claim",
"id": "T447"
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"text": "['resulting in faster and higher quality discrete embeddings']",
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"text": "['Animation quality can be improved with a better skinning model']",
"type": "own_claim",
"id": "T449"
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"text": "['Kavan and Zára ˇ 2005']",
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"text": "['possibly at the cost of performance']",
"type": "own_claim",
"id": "T451"
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"text": "['One approach would be to use a technique']",
"type": "own_claim",
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"text": "['Wang et al. 2007']",
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"text": "['that corrects LBS errors by using example meshes, which we could synthesize using slower, but more accurate deformation techniques']",
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"text": "['A more involved approach would be automatically building a tetrahedral mesh around the embedded skeleton and applying the dynamic deformation method']",
"type": "own_claim",
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"text": "['Capell et al. [2002]']",
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"text": "['Combining retargetting with joint limits should eliminate some artifacts in the motion']",
"type": "own_claim",
"id": "T457"
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"text": "['A better retargetting scheme could be used to make animations more physically plausible and prevent global self-intersections']",
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"text": "['it would be nice to eliminate the assumption that the character must have a well-defined interior']",
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"text": "['5']",
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"text": "['11 in Figure 9']",
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A38 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A38_C08_Estimating_Cloth_Simulation_Parameters_from_Video_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1186562.1015729
Eurographics/SIGGRAPH Symposium on Computer Animation (2003) D. Breen, M. Lin (Editors)
<Title>Estimating Cloth Simulation Parameters from Video</Title>
Kiran S. Bhat
Christopher D. Twigg
Jessica K. Hodgins
Pradeep K. Khosla
Zoran Popović
Steven M. Seitz
1 School of Computer Science, Carnegie Mellon University 2 Department of Computer Science and Engineering, University of Washington
<Abstract>Cloth simulations are notoriously difficult to tune due to the many parameters that must be adjusted to achieve the look of a particular fabric. In this paper, we present an algorithm for estimating the parameters of a cloth simulation from video data of real fabric. A perceptually motivated metric based on matching between folds is used to compare video of real cloth with simulation. This metric compares two video sequences of cloth and returns a number that measures the differences in their folds. Simulated annealing is used to minimize the frame by frame error between the metric for a given simulation and the real-world footage. To estimate all the cloth parameters, we identify simple static and dynamic calibration experiments that use small swatches of the fabric. To demonstrate the power of this approach, we use our algorithm to find the parameters for four different fabrics. We show the match between the video footage and simulated motion on the calibration experiments, on new video sequences for the swatches, and on a simulation of a full skirt.</Abstract>
<H1>1. Introduction</H1>
Several recent major movie releases have demonstrated that the motion of clothing adds greatly to the appearance of a virtual character. This effect is particularly compelling for scenes that include both real and synthetic actors such as those with Yoda and Anakin Skywalker in Episode II: Attack of the Clones. In such scenes, the virtual clothing must move and be rendered so that it blends in seamlessly with the motion and appearance of the real clothing in the scene. Realistic virtual clothing is possible now because of recent advances in cloth simulation techniques
4 , 9 , 5 , 37 , 6 . The motion of fabric is determined by resistance to bending, stretching, shearing, external forces, aerodynamic effects, friction, and collisions. Although with the right set of parameters good simulators produce very realistic looking motion, choosing parameters that will provide a particular appearance remains a time consuming task that requires the computation and viewing of many forward simulations. Some parameters can be chosen based on the animator’s intuition about the fabric—a knit fabric is more stretchy than a woven fabric such as linen, for example. But not all the parameters of a cloth simulator are intuitive or map directly to measurements that can made by a system such as the Kawabata system
22 . In our paper, we address this problem by using optimization to automatically determine these parameters from a sequence of video frames of the fabrics under consideration. The parameters are optimized on a set of static shots and motion clips of a small swatch of a particular fabric and then tested on a simulation of a full skirt made from that fabric. We designed the swatch tests to span the space of behaviors that we expect to see in the final sequences of motion with the skirt so that all parameters can be tuned appropriately. We use simulated annealing for the optimization step with an optimization function that assesses the extent to which the folds in the simulated and physical fabric match. This match is evaluated by means of a shape metric that uses projected light to detect surface orientation in real and simulated fabrics. The metric is tuned to be most sensitive along folds and to discount planar regions. We use the system to find the parameters for four different fabrics. We show the match between the video footage and the simulated motion on the calibration experiments, on new video sequences for the swatches, and on a simulation of a full skirt as shown in the image on the previous page.
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Bhat et al. / Estimating Cloth Simulation Parameters from Video
<H1>2. Related Work</H1>
Cloth modeling has a long history, dating back to work in the textile community from the mid-1930s by Peirce
27 . Work on cloth modeling in computer graphics has focused on developing dynamic simulation techniques that are both realistic and fast. Baraff and Witkin describe a cloth model that uses stiff springs with implicit time integration
4 . This model was subsequently adapted to reduce the over-damping due to implicit integration 9 . Explicit time integration approaches 18 use weaker springs for stretching and shearing, often explicitly limiting the amount of stretching
29 , 6 . Choi and Ko introduced a bending energy model that more accurately captures the fine creases and bends of cloth 9 . Lahey provides a comprehensive overview of cloth hysteresis models from the perspective of computational fabric mechanics
23 . Extensive work has also been done on modeling collisions and friction.
Cloth self-collision is handled either by untangling the cloth
37 , 39 , 3 or by preemptively avoiding collisions
30 , 20 , 6 . Various potential field methods have been used for general collision detection and response
33 , 32 . Despite this large body of work on cloth simulation models, little work has appeared in the computer graphics literature on estimating the parameters of these models so that they match the behavior of real fabrics. Cloth parameter estimation has been studied in the textile community (for an overview, see Breen and colleagues
17
), but such methods have not yet enjoyed wide-spread use in the computer graphics community. An important exception is the work by Breen
5 who used the Kawabata system 22 to measure bending, shearing, and tensile parameters by subjecting a swatch of fabric to a series of mechanical tests and measuring the force needed to deform it into a standard set of shapes. Although the Kawabata system can provide accurate measurements, these measurements are problematic for computer graphics cloth simulation problems for two reasons. First, there might not be a direct and simple mapping between the parameters for a particular cloth model and the Kawabata parameters. Second, the Kawabata system does not measure dynamic cloth parameters, e.g. air drag or damping, which are of key importance for moving cloth. One promising approach for modeling cloth parameters is to automatically search for parameters that match real, observed cloth. Jojic and Huang fit parameters of a particlebased cloth model to fit a range scan of real cloth in a static rest configuration, draped over a sphere
21 . More challenging still, they attacked the problem of measuring the 3D geometry of an object from the resting shape of a piece of cloth draped over it, a problem that we do not consider in this paper. However, Jojic and Huang did not treat the problem of measuring dynamic parameters or demonstrate accurate results across a range of fabric types. More distantly related are techniques for computing the geometry of cloth from images. Coarse estimates of the time-varying geometry of cloth can be computed using traditional stereo matching techniques by using two or more cameras and treating each time instant independently (see Scharstein and Szeliski
31 for an overview). More accurate results may be obtained by projecting structured light patterns on the cloth (see Zhang et al.
40 for an overview). Rather than computing shape at every time instant independent from the next, it can be advantageous to integrate images over time to improve accuracy.
Two examples of promising work along these lines are Carceroni and Kutulakos
8 and Torresani et al.
34
; both studies demonstrated reconstructions of moving cloth.
<H1>3. Cloth Model</H1>
Because our framework for estimating cloth simulation parameters is independent of the cloth model, we can, in principle, select a specific model that meets a set of criteria such as accuracy or simulation speed. Our choice of a cloth model was guided by two principles, realism and practicality. We wanted to use a model that was sophisticated enough to capture the detailed dynamic behavior found in real fabrics but still straightforward to implement. Because our intention was to apply the learned cloth model parameters to arbitrary garments with varying triangle resolution, it was also important that the cloth parameters correctly scale to varying resolutions of cloth. We used the model described by Baraff and Witkin as the basis for our cloth simulator
4 . This model has sufficient richness to produce a wide variety of cloth behaviors. The underlying meshing is triangular, making clothing modelling easier. More importantly, its input parameters are independent of meshing, so that parameters recovered on one mesh (the test swatch) can safely be transferred to another (the skirt). While nonlinear models such as the buckling behavior of Choi and Ko 9 could potentially capture more realistic details of cloth, there is no straightforward way to scale the parameters of these models to meshes of varying resolutions. We expect that future application of our parameterestimation framework to other scale-invariant cloth models will provide even more realistic results. The model developed by Baraff and Witkin formulates the energy of a particular triangle in terms of so-called condition functions C(x) such that the total potential energy associated with the system is given by E u = k s C(x)C T (x) ( 1 ) 2 where k s is a stiffness coefficient associated with the particular condition function. Forces are then simply calculated by
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Bhat et al. / Estimating Cloth Simulation Parameters from Video
F = ∇ x E u ( 2 )
Damping forces are similarly fomulated in terms of the
C(x), d = −k C(x)
̇
( 3 )
dC d dx
We thus associate a stiffness coefficient k s and a damping coefficient k d with each of the C(x). In their paper, Baraff and Witkin describe a set of C(x) consisting of an in-plane stretch term, an in-plane shear term, and an out-of-plane bending term, giving a total of six parameters we can use to tune the internal cloth model. We refer the reader to their paper 4
for the full details. We note, however, that (as they allude to in footnote 5) energy should scale linearly with triangle area to ensure scale independence. Therefore, we need to be careful when substituting C(x) for stretch and shear into eq. 1 that the resulting formula is linear in a rather than quadratic. In the course of running our experiments, we discovered that a linear drag model such as that used in previous cloth work
4 , 9
was not able to capture dynamic aspects of cloth. In order to add additional air-drag degrees of freedom to our cloth model without resorting to fully modeling aerodynamics
25 , we developed a simple nonlinear alternative. To calculate the drag force on a triangle, we decompose the average velocity on the face into two components, one normal to the surface (v N ) and one tangential (v T ). Total drag force is then a linear function of tangential velocity and a quadratic function of normal velocity, with an additional term k f that controls the degree of nonlinearity,
f drag = −a 1 + k N k |v f |v N | N 2 | 2 |v v N N | + k T v T where a is the area of the given triangle. The linear term
is merely Stokes’s law 1 ; the quadratic term matches better the experimental behavior of macroscopic bodies in low Reynold’s number flow
14 . The addition of the |v N | 2 term in the denominator which makes the force asymptotic as v N → ∞ was partially motivated by the observed phenomenon of drag crisis 14
, where under certain circumstances the drag can actually drop at the onset of turbulence 1 . The optimizer is free to eliminate this behavior or other terms of this equation by setting the corresponding parameters to zero. Initially, we used a first-order implicit Euler time integration scheme similar to the one described by Baraff and Witkin
4 . Unfortunately, we found that implicit integration introduced damping which could not be eliminated by optimizing cloth parameters. We had more success in matching realistic cloth motions by using higher-order explicit methods. The results in this paper all use an adaptive 4thorder accurate Runge-Kutta methods with embedded error estimation
2 . While this method offers the advantages of familiarity and automatic bounding of error, it is rather slow, and recent work suggests that using 2nd-order backward differences 9 or Newmark schemes 7 may be a better choice. For collision handling, we use a model similar to Bridson and colleagues 6 which combines repulsion forces with impulses to robustly prevent all collisions before they occur. However, separating repulsion forces from the cloth internal dynamics and applying them outside the Runge-Kutta solver affected stability and resulted in visible artifacts. Instead, we apply repulsion forces inside the solver loop, so that the solver’s own internal error estimation can remove these artifacts. The drawback of this technique is speed, because the system must check for collisions every time it evaluates the state derivatives (as opposed to once every collision timestep as in Bridson et al. 6 ).
To achieve acceptable performance, we used a number of collision culling algorithms, including hybrid top-down/bottom-up update
24 , fast triangle reject tests
26 , and a curvature-based criterion for rejecting self-collisions that was first introduced by Volino and Thalmann 38 and later refined by Provot
30 .
<H1>4. A Metric for Matching Simulation to Video</H1>
We use a perceptually motivated metric to compare the motion of cloth in simulation with a video sequence of real fabric motion. Our algorithm compares the two sequences frame by frame and computes an average error across the entire sequence. Real fabrics exhibit a wide variety of motion ranging from soft and flowing (satin) to stiff (linen). Our metric captures the complex dynamics of cloth motion and also helps to distinguish between different fabrics. Researchers in computational neurobiology hypothesize that the human perceptual system is sensitive to moving edges in video
11 , 12 , 36 . Studies have shown that the receptive fields of simple cells in the macaque cortex act as edge or line detectors, responding to oriented edges or lines in natural scenes
19 , 35 , 10 . In cloth, these edges correspond to folds, which are regions of high variation in shape. Hence, our perceptually motivated metric for cloth compares two video sequences, one from simulation and one from the real world, and returns a number that measures the differences in their folds. The metric also penalizes the silhouette mismatch between the two sequences. Fold Detection and Representation: Folds appear as soft edges in video whose appearance is dependent on material properties and lighting. Haddon and Forsyth
15 , 16 describe a learning approach for detecting and grouping folds (and grooves) in images of fabrics. Their technique can handle lighting effects caused by diffuse inter-reflections in cloth. However, most fabrics have very complicated reflectance properties. In our experiments, we normalize the effects of lighting and material reflectance by projecting a structured light pattern of horizontal stripes onto the fabric. From the light-striped video sequence, we compute the dominant orientation for each edge pixel by convolving it with a steerable filter bank
13 . In our implementation, we use the G2/H2 quadrature pair with kernel size 12 as the basis filters. Details of computing the dominant orientation from the coefficients of filter bank response are given in Appendix I of Freeman and Adelson
13 . We convolve the image with the filter bank, compute the filter coefficient responses, blur the coefficients using a gaussian kernel, and compute the dominant orientation from these coefficients. We define the resulting orientation image as an angle map, shown in Fig. 1 . The angle map, which measures the local orientation of the projected pattern, has a constant value when the surface is planar and varies at folds. We threshold the gradient of the angle map to get a gradient mask M k for each frame of video ( Fig. 1 ). M k (i, j) = 1, 0, δ(i, δ(i, j) j) ≥ < τ τ ( 4 ) where τ is a user defined threshold and δ(i, j) is the magnitude of the gradient of the angle map at (i, j). The gradient mask is non-zero at regions of high gradients, corresponding to folds, and zero at planar regions. Fold Comparison: Our metric computes the frame by frame sum of squared differences (SSD) between masked angle maps in simulation with video. We preprocess the input video sequence to compute the angle map at each frame. Similarly, in simulation, we render the cloth shape using the current parameter values and project the same striped pattern, to get a striped simulation sequence. We compute the angle map at every frame in simulation from this sequence. We then compute the SSD of the angle values for all overlapping points in the two angle maps. We pre-multiply this difference with the gradient mask, which helps to emphasize the differences in fold regions over planar regions ( Fig. 2 ). We sum the error across all frames to compute the overall error across the entire sequence. The error at any particular frame k along the sequence is S x S y E k f old = ∑ ∑ M k (i, j) · (θ real k (i, j) − θ sim k (i, j)) 2 ( 5 ) i=0 j=0 where (S x , S y ) is the size of the angle maps and θ real , θ sim are the angle values from real and simulation angle maps respectively. Silhouette Comparison: In addition to the angle map error, we penalize the silhouette mismatch between the simulation and the video of real cloth. This penalty is proportional to the difference between the two silhouettes, i.e., the number of mismatched pixels. S x S y E k silh = ∑ ∑ | A k real (i, j) − A k sim (i, j) | ( 6 ) i=0 j=0 where 1, inside silhouette A k (i, j) = 0, otherwise ( 7 ) The total error in frame k is E k = E k f old + αE k silh ( 8 ) where α is a user-defined weight that controls the relative contribution of the two terms. We used a value of 0.1 for α in our experiments. The error across the entire sequence of length N frames is given by N E = ∑ E k ( 9 ) k=1
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Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 1: Top Row: Input light striped image. Bottom Row (left to right): angle map and gradient mask.
Figure 2: The stages in the metric pipeline. Top row (left to right): Angle map from video, angle map from simulation. Bottom row (left to right): angle map difference, final metric value for this frame (angle map difference multiplied by gradient mask from video).
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 3: This plot shows angle map error as a function of bend and stretch stiffness parameters. Dark areas indicate regions of small error and bright areas correspond to large errors. Note that the space is fairly noisy. The minimum found by the optimizer is contained in the large dark region in the lower portion of the plot.
<H1>5. Parameter Identification</H1>
We use optimization to estimate the parameters of the cloth simulator from video. Before we describe the details of the optimizer, we look at the error space of the angle map metric, which gives us useful insight about the parameters of the system. Fig. 3 shows the variation of error for different values of bend stiffness and stretch stiffness coefficients for satin. To generate this error map, we compared the angle map from one frame in video with several angle maps in simulation. From the figure, it is evident that the error space is fairly noisy, with many local minima, motivating the need for a global optimization technique. In addition to the parameter values, we estimate the relative importance of each parameter for a given experiment by performing a perturbation analysis at the solution point. The importance or sensitivity of a parameter p depends on its local gradient ∂E ∂p ; it relates a small change in parameter value to a change in the error value. Instead of computing the gradient, we robustly compute the variability of the param∂p eters, defined as ∂E . To compute the variability, we perturb each parameter of the simulator individually up to ±0.20% of its value, compute the error and fit a quadratic to the data ( Fig. 4 ). From the quadratic, the variability is computed as the change in parameter values that results in a 1% change in the error. Parameters with low variability have high sensitivity and are estimated reliably for a given experiment.
109 108.5 108 107.5 error 107 106.5 106 105.5 105 104.5 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 % change in parameter value
Figure 4: Perturbation analysis at the solution for bend stiffness parameter.
<H1>6. Optimization Framework</H1>
We use simulated annealing to find the parameters that minimize the error function given in eq. 9. Simulated annealing initially explores the space in a semi-random fashion and eventually takes downhill steps. The likelihood that it will take a step in a direction that is not locally optimal is a function of the temperature ( Fig. 5 ). We chose to use the continuous simulated annealing method presented in Press et al.
28
, which combines the Metropolis algorithm with the downhill simplex method for continuous n-variable optimization. We found it useful to reset the simplex with the current best solution when the temperature reduces by a factor of 3. Prior to optimization, we perform an exhaustive search for each fabric, where we choose four values for each cloth parameter across its entire range. This corresponds to a very coarse sampling of the parameter space. We simulate the fabric for all points in this coarse set and compute the error for each point by comparing against the real fabric. We initialize the optimizer with the point corresponding to the minimum error. We have found that this strategy allows the optimizer to locate a good minimum of the space.
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Bhat et al. / Estimating Cloth Simulation Parameters from Video
1000 Temperature 900 350 800 300 700 250 600 Temperature 500 200 Error 400 150 300 100 200 50 100 0 0 50 100 150 200 250 300 350 400 Iteration Number
Figure 5: Progress of the simulated annealing optimizer as measured by error. The temperature decrease is governed by a geometric cooling schedule.
<H1>7. Experiments</H1>
We designed a few simple experiments to capture the dynamics of the different types of fabrics and the air/cloth interaction. The experiments are easy to perform, capture, and repeat; yet they demonstrate the complex dynamics of cloth motion. The parameters obtained from the simple experiments were used to simulate skirts and other complex fabric motions. In essence, our experiments were designed to be a calibration setup for estimating the static and dynamic parameters of a cloth simulator. We perform two estimation experiments for each fabric, a static test and waving test. We used four types of fabrics: linen, fleece, satin and knit. These fabrics exhibit a wide range of static and dynamic behavior and span a large range of real fabrics. We perform the static and waving tests on a small swatch of each fabric. In the static test, the two top corners of the fabric are held stationary, and the fabric is allowed to sag under gravity. For a fixed separation between the top corners, different fabrics attain different static shapes as shown in Fig. 6 . The static test give a good estimate for the static stiffness and bend parameters. In the waving test, one of the top corners of the fabric is fixed and the other corner is moved back and forth ( Fig. 7 ). The waving motion of fabrics in simulation is affected by their dynamic parameters. We see from the accompanying videos that real fabrics exhibit a wide range of interesting motions. Different fabrics also exhibit different types of motion for the same input excitation. We designed the waving motion to roughly match the types of motion occurring in real garments such as skirts. This gives reasonable estimates for cloth parameters while avoiding the need to optimize directly on complex fabric geometries (e.g. skirts) involving many collisions.
Figure 6: The static test with four real fabrics. Top row (left to right): linen and fleece. Bottom row: satin and knit. Top corner separation is identical across all four fabrics.
Figure 7: Three frames from the waving test for satin.
<H1>8. Results</H1>
In this section, we report the results of simulation parameters obtained using our technique applied to four fabrics: linen, fleece, satin and knit. We measured the mass and dimensions of the fabrics. We also accurately measure the position of the two top corners using a Vicon motion capture system. We compute the projection matrices for the camera and projector using a calibration grid comprising of several motion capture markers. We performed two trials per experiment, each with slightly different initial conditions and optimized on the first 50 frames of video in each trial. Each trial took approximately 50 hours to converge on a 2.8GHz Intel Xeon processor (approximately 600 iterations of simulated annealing). For this reason, we started the optimizations on the two trials (per fabric) with the same initial guess and chose parameters (optimized) that minimized the total error on the two trials. Static test. We perform optimization on two trials for each fabric; the results are shown in Fig. 8 and Fig. 9 . The two trials have different separation distances between the top corners. For each fabric, we optimize for six parameters: stiffness and damping parameters for stretch, shear, and bend. The air drag parameters were fixed for this experiment to the mid point of their range of values. The initial values for the two trials are obtained from a coarse exhaustive search (four values per parameter). The initial values and final values of the estimated parameters are summarized in Table 1 . Figs. 8 and 9 show a very good visual match between the simulations with their counterpart real fabrics. However, there is a significant disparity in the final optimized values from the two trials. In order to understand this disparity, we performed a set of optimizations (on a single fabric) with very similar initial values. Table 2 shows the parameter values for satin from five optimizations where the initial conditions were randomly varied by ±5%. From the table, we see that the final error values are very close. We get consistent estimates for parameters that have lower variability (e.g., bend, stretch). Parameters with high variability are estimated poorly, because their values do not contribute sufficiently to the error. This result is consistent with our intuition that static tests cannot be used to estimate dynamic parameters like stretch and shear damping or air drag and motivates the waving test, which excites both the static and waving parameters. Waving test. We optimize for nine parameters in the waving test: the six cloth stiffness and damping parameters and three air drag parameters ( Fig. 10 ). As with the static test, we initialize the static parameters in this test from a coarse exhaustive search. The dynamic parameters were initialized using a random guess. We optimized on the first 50 frames of the sequence. The initial values and final values of the optimized parameters for two trials are reported in Table 3 . The final values of the parameters from the two trials differ in part because the variability of the parameters is still fairly high ( Fig. 11 ). Different motions or larger sequence might further reduce the variability of the parameters. We choose the parameter set that minimizes the sum of the error from the two trials. For instance, in the following example of fleece waving, we choose the parameters from experiment 2. Error: Exp 1 Error: Exp 2 Total Error Pars: Exp 1 4257.2 10913.5 15170.7 Pars: Exp 2 4566.2 7144.3 11710.5 This approach seems to produce plausible results with skirts and other validation experiments. However, we believe that a more general solution for parameter identification using our framework is to simultaneously optimize across multiple trials of different experiments. Optimization progress. Fig. 12 shows the static shape of the simulation before and after optimization. Fig. 13 shows the corresponding angle map comparison. These two figures show the progress of the optimization and indicate that the minimum corresponds to a visually compelling match. Metric validation. We compare each of the four optimized angle maps from simulation (corresponding to the four fabrics) with the four angle maps computed from video. In Fig. 14 , each curve shows one fabric (e.g., fleece) compared with four simulations, corresponding to each fabric type. We see that each fabric in simulation has a minimum error when compared to its counterpart in reality. Fig. 14 also demonstrates that our approach could be potentially useful for recognizing different types of fabrics in video. Generalization. We evaluated the parameters obtained from optimization on longer sequences (150 frames). Fig. 10 and the accompanying videos show a good visual match between corresponding frames in simulation and video. All videos are available off our web page and/or included in the DVD. The videos also show that the parameters obtained from optimization generalize well on new sequences. We also validated the estimated parameters on a long sequence actuated by a robot ( Fig. 15 ). We used a a Mitsubishi PA-10 robot arm to move the corner point along a simple sinusoidal trajectory, thereby ensuring that we had the same input motion across different fabrics. Finally, we used the optimized parameters to simulate a skipping motion of a human actor wearing a skirt ( Fig. 16 ). Here, the actor repeats the same skipping motion (approximately) for the four different skirts. We used data from a full body optical motion capture of the actor performing the same skipping motion (in another trial) to drive the character for the cloth simulation. The results show that the parameters obtained from our optimization approach approximately capture the static shape and dynamic properties of skirts of different materials.
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Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 8: Results of optimization for the static test, trial 1. Top row: real fabrics (left to right) linen, fleece, satin and knit. Bottom row: Corresponding fabrics in simulation.
Figure 9: Results of optimization for the static test, trial 2. Top row: real fabrics. Bottom row: Corresponding fabrics in simulation.
Linen Fleece Satin Knit
Pars
Start
Exp 1
Exp2
Start
Exp 1
Exp 2
Start
Exp 1
Exp 2
Start
Exp 1
Exp 2
1
1e-3
0.009
0.0045
1e-4
0.0001
0.0001
1e-5
1.106e-5
6.94e-6
1e-6
1.52e-6
1.51e-6
2
4000
404.9
3682.1
50
129.2
200.04
50
19.58
19.38
50
27.97
28.36
3
215.442
175.374
208.15
215.442
103.96
31.391
50
76.81
69.65
50
1226.44
2693.07
4
1e-7
9.92e-7
3.22e-7
2.15e-6
2.13e-7
4.11e-7
1e-7
2.49e-7
3.98e-7
1e-7
1.01e-7
2.27e-7
5
10
12.16
10.17
10
4.78
0.064
10
14.42
3.68
10
10.12
11.83
6
10
2.19
13.17
10
13.86
3.75
10
4.11
4.56
10
0.13
4.04
Linen Fleece Satin Knit Pars Start Exp 1 Exp2 Start Exp 1 Exp 2 Start Exp 1 Exp 2 Start Exp 1 Exp 2 1 1e-3 0.009 0.0045 1e-4 0.0001 0.0001 1e-5 1.106e-5 6.94e-6 1e-6 1.52e-6 1.51e-6 2 4000 404.9 3682.1 50 129.2 200.04 50 19.58 19.38 50 27.97 28.36 3 215.442 175.374 208.15 215.442 103.96 31.391 50 76.81 69.65 50 1226.44 2693.07 4 1e-7 9.92e-7 3.22e-7 2.15e-6 2.13e-7 4.11e-7 1e-7 2.49e-7 3.98e-7 1e-7 1.01e-7 2.27e-7 5 10 12.16 10.17 10 4.78 0.064 10 14.42 3.68 10 10.12 11.83 6 10 2.19 13.17 10 13.86 3.75 10 4.11 4.56 10 0.13 4.04
Table 1: Tabulation of the static parameters from two experiments. Legend: 1=bend, 2=stretch, 3=shear, 4=bend damping, 5=stretch damping, 6=shear damping.
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 10: Waving results for satin. The top picture in each block shows the real fabric and the bottom shows the corresponding frame from simulation.
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Bend Stretch Shear Initial Values 1.0e-05 50 50 Optimization 1 6.93766e-06 19.3832 69.653 Optimization 2 7.77204e-06 20.2884 32.6492 Optimization 3 8.75613e-06 19.8365 50.8304 Optimization 4 9.55647e-06 19.2745 74.7429 Optimization 5 8.47762e-06 20.1119 36.762 Variability (in %) 9.18 8.10 23.01
Table 2: Performance of simulated annealing on several optimizations. All the optimizations start with values which are within ±5% of the initial values given in the first row. Parameters with high variability (e.g., stretch damping) are estimated poorly and vary significantly across the different optimizations. However, parameters with low variability (e.g., bend) are consistent across multiple optimizations.
Linen Fleece Satin Knit
Pars
Start
Exp 1
Exp2
Start
Exp 1
Exp 2
Start
Exp 1
Exp 2
Start
Exp 1
Exp 2
1
1e-3
0.001
0.0008
1e-4
1.13e-5
0.0001
1e-5
6.41e-6
5.64e-6
1e-6
1.12e-6
1.16e-6
2
4000
2016.8
2935.26
50
82.61
89.32
50
26.42
32.37
50
69.75
12.68
3
215.442
167.833
465.73
215.443
255.198
296.861
50
97.77
74.24
50
37.48
59.99
4
1e-7
3.17e-7
4.76e-7
2.15e-6
1.36e-6
1.31e-6
1e-7
1.48e-6
1.24e-7
1e-7
1.04e-7
5.4e-7
5
10
2.71
5.17
10
2.39
5.92
10
0.57
4.48
10
4.52
3.87
6
10
3.89
5.52
10
1.59
9.82
10
6.57
4.73
10
4.93
2.64
7
2
8.73
2.18
2
2.40
1.62
2
4.85
0.85
2
1.54
0.99
8
2
5.56
1.99
2
3.15
0.31
2
1.76
1.48
2
0.52
1.79
9
2
0.44
1.29
2
4.28
1.23
2
0.95
0.79
2
1.22
0.33
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Table 3: Waving parameters from two experiments. Parameters from the experiment shown in bold is selected as the final estimate from this experiment. Legend: 1=bend, 2=stretch, 3=shear, 4=bend damping, 5=stretch damping, 6=shear damping, 7=linear drag, 8=quadratic drag, 9=drag degradation.
50
50
45
45
40
40
35
Variability 35
Variability 30
30
25
25
20
20
15
15
10
10
5
5
0
0
1 2 3
4 5 6 7 8 9
1 2 3
4 5 6 7 8 9
Parameters
Parameters
50 50 45 45 40 40 35 35 Variability 25 30 Variability 25 30 20 20 15 15 10 10 5 5 0 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Parameters Parameters
Figure 11: Bar charts showing the variability analysis results for the waving test. From left to right: linen, fleece, satin and knit. Legend: 1=bend, 2=stretch, 3=shear, 4=bend damping, 5=stretch damping, 6=shear damping, 7=linear drag, 8=quadratic drag, 9=drag degradation.
Bend Damp Stretch Damp Shear Damp Error 2e-07 10 10 179.026 3.98337e-07 3.67932 4.56238 104.747 2.08755e-07 1.95807 10.6535 104.502 2.56854e-07 7.08276 9.25576 103.501 3.14821e-07 5.47909 1.06559 103.243 2.3997e-07 8.38981 11.9167 103.849 21.11 >100 >100
50 50 45 45 40 40 35 35 30 30 Variability 25 Variability 25 20 20 15 15 10 10 5 5 0 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Parameters Parameters
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 12: Showing the improvement in shape match after optimization. The top row compares a video frame of fleece with simulation before optimization. The bottom row shows the corresponding video/simulation pair after optimization.
Figure 13: Comparison of angle maps for the shapes shown in Fig. 12 before and after optimization. Top Row (Before Optimization, from left to right): Angle map from video, angle map from simulation, angle map SSD. Bottom Row: The corresponding anglemaps after optimization.
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Linen Fleece Error 1 Satin Knit 2 Linen Sim Satin Sim Fleece Sim Knit Sim Optimized Simulation Parameters
Figure 14: Comparing the optimized parameters in simulation for each fabric with the four real fabrics. For example, point 1 in the graph shows the error when a simulation with fleece parameters is compared with video of satin. Similarly, point 2 is the error when the satin simulation is compared with real satin. The four curves have a minimum when they are compared to their correct counterparts.
<H1>9. Discussion</H1>
This paper describes an optimization framework for identifying the simulation parameters of cloth from video. We captured the behavior of small swatches of fabric using a set of dynamic and static tests and demonstrated that the optimizer could identify appropriate simulation parameters from those tests. These parameters produced four distinct and recognizable fabrics when applied to a more complex simulation of a skirt as it was driven by motion capture data from a human figure. The cloth model was not the main focus of this research, yet in early versions of the system it was often the bottleneck in achieving appealing results. To match a video sequence accurately, the cloth physics model as well as the collision algorithms must be chosen carefully. Instabilities in the collision handling will cause perceptible quivering in the motion of cloth. Similarly, extra damping introduced by the integration method makes crisp folds impossible to match. The parameters must also be independent of the resolution of the mesh so that they can be identified on low resolution swatches and applied to higher resolution garments. Progress is being made in these areas, however, and cloth models are continually improving. For example, Bridson et al. 7
introduces a scale-independent bend model with encouraging results. Our cloth model does not diverge significantly from previous models discussed in the literature. Our only major addition was a simple nonlinearity we introduced into the drag model. Hence, our approach should generalize to any parametrized cloth model that produces a sufficiently rich set of physically realistic motions. Although the skirt is far more complex than the swatches that were used to determine the parameters, it is not as complex as many garments, for example, a form-fitting pair of pants or a tailored blazer. For more complex garments, choosing the parameters via optimization on small, flat swatches may not be sufficient because the shape of the garment is determined by darts, pleats and by the interplay of different fabrics (wool, lining, and interfacing, for example). More complex garments may require the hand design of additional tests that mimic particular behaviors or elements of the garment in isolation. Moreover, the model might need extra parameters to handle anisotropic effects, hysteresis and coupling effects (stretching along one direction causing shrinking in the other direction), all of which would need specialized tests. En route to the metric used in the experiments described here, we tried a number of other metrics: comparing the overlap of the silhouettes, the distance function between silhouette edges, and using information from internal edges marked on the fabric. The metric that measures folds and silhouettes, in concert with the projector for the light stripes, proved to be a simple and effective metric that far outperformed our earlier attempts. The space of possible metrics is vast, of course, but one class of metrics that we did not experiment with are statistical metrics that compute a function of the shape of the fabric across time rather than evaluating the match on a frame-by-frame basis. The experiments with the swatches were carefully controlled to have initial conditions for the simulation that matched those seen in the video. If instead, we were to optimize on more complicated garments, then such tight control of the initial conditions is unlikely and a statistical metric might be preferable. Such a metric might, for example, compute the average number of folds across a time sequence rather than looking for a fold to appear at a particular location on the swatch. Our hope is that this work will promote a more rigorous evaluation of various cloth models, especially with respect to how accurately they match reality, and perhaps lead to creation of a standardized set of benchmarks for cloth simulation models.
<H1>10. Acknowledgements</H1>
We like to thank Jia-Chi Wu for his help in implementing the cloth simulator and the simplified air drag model. We also thank Roshni Sivasankaran, Bonnie Jang and Priyanka Vaddi for their help with the skirt motion capture experiments, and Mike Stevens for cleaning up the motion capture data. The support of NSF under ITR grant IIS-0113007 and EIA-0196217 is gratefully acknowledged. Finally, we would like to thank the anonymous reviewers for their valuable comments and feedback.
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 15: Validating the estimated parameters using the same input excitation. The top right corner of the fabric is actuated using a Mitsubishi PA-10 robot. Each row shows the match between video (top) and simulation (bottom) at four frames chosen from a 100 frame sequence. The fabrics, from top to bottom, are linen, fleece, satin and knit respectively.
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
Figure 16: Validating the estimated parameters on a more complicated motion and garment. We show (from left to right, top to bottom) several frames of an actor skipping while wearing a fleece skirt. The corresponding frames of the skirt in simulation shows that our technique captures the approximate shape and dynamics of the real skirt. These frames were equally spaced across the entire sequence (0.5 seconds apart). The videos in the webpage show the validation results on all four skirts.
c The Eurographics Association 2003.
Bhat et al. / Estimating Cloth Simulation Parameters from Video
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c The Eurographics Association 2003.
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c The Eurographics Association 2003.
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"text": "['which are of key importance for moving cloth']",
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"text": "['One promising approach for modeling cloth parameters is to automatically search for parameters that match real, observed cloth']",
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"offsets": [
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"text": "['Jojic and Huang fit parameters of a particlebased cloth model to fit a range scan of real cloth in a static rest configuration, draped over a sphere']",
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"text": "['they attacked the problem of measuring the 3D geometry of an object from the resting shape of a piece of cloth draped over it']",
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"text": "['Jojic and Huang did not treat the problem of measuring dynamic parameters']",
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"text": "['demonstrate accurate results across a range of fabric types']",
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"text": "['More distantly related are techniques for computing the geometry of cloth from images']",
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"text": "['Coarse estimates of the time-varying geometry of cloth can be computed using traditional stereo matching techniques by using two or more cameras and treating each time instant independently']",
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"text": "['More accurate results may be obtained by projecting structured light patterns on the cloth']",
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"text": "['Rather than computing shape at every time instant independent from the next, it can be advantageous to integrate images over time to improve accuracy']",
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"text": "['8']",
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"text": "['34']",
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"text": "['Two examples of promising work along these lines are Carceroni and Kutulakos']",
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"text": "['and Torresani et al.']",
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"text": "['both studies demonstrated reconstructions of moving cloth']",
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"text": "['our framework for estimating cloth simulation parameters is independent of the cloth model']",
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"text": "['Our choice of a cloth model was guided by two principles, realism and practicality']",
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"text": "['Because our intention was to apply the learned cloth model parameters to arbitrary garments with varying triangle resolution']",
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"text": "['it was also important that the cloth parameters correctly scale to varying resolutions of cloth']",
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"text": "['We wanted to use a model that was sophisticated enough to capture the detailed dynamic behavior found in real fabrics but still straightforward to implement']",
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"text": "['We used the model described by Baraff and Witkin as the basis for our cloth simulator']",
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"text": "['This model has sufficient richness to produce a wide variety of cloth behaviors']",
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"text": "['making clothing modelling easier']",
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"text": "['The underlying meshing is triangular']",
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"text": "['4']",
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"text": "['parameters recovered on one mesh']",
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"text": "['can safely be transferred to another']",
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"text": "['its input parameters are independent of meshing']",
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"text": "['potentially capture more realistic details of cloth']",
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"text": "['there is no straightforward way to scale the parameters of these models to meshes of varying resolutions']",
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"text": "['9']",
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"text": "['future application of our parameterestimation framework to other scale-invariant cloth models will provide even more realistic results']",
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"text": "['energy should scale linearly with triangle area to ensure scale independence']",
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"text": "['footnote 5']",
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"offsets": [
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"text": "['we need to be careful when substituting C(x) for stretch and shear into eq. 1 that the resulting formula is linear in a rather than quadratic']",
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"text": "['a linear drag model such as that used in previous cloth work']",
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"text": "['was not able to capture dynamic aspects of cloth']",
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"text": "['our experiments']",
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"text": "['9']",
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"text": "['we developed a simple nonlinear alternative']",
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11568,
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"text": "['In order to add additional air-drag degrees of freedom to our cloth model without resorting to fully modeling aerodynamics']",
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"text": "['25']",
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"offsets": [
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12223,
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"text": "['The linear term', 'is merely Stokes’s law 1']",
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"text": "['the quadratic term matches better the experimental behavior of macroscopic bodies in low Reynold’s number flow']",
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"text": "['14']",
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"text": "['The addition of the |v N | 2 term in the denominator which makes the force asymptotic as v N → ∞ was partially motivated by the observed phenomenon of drag crisis']",
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"text": "['14']",
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"text": "['implicit integration introduced damping which could not be eliminated by optimizing cloth parameters']",
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"id": "T121"
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{
"offsets": [
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],
"text": "['We had more success in matching realistic cloth motions by using higher-order explicit methods']",
"type": "own_claim",
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{
"offsets": [
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12777,
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"text": "['we used a first-order implicit Euler time integration scheme similar to the one described by Baraff and Witkin']",
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"text": "['The results in this paper all use an adaptive 4thorder accurate Runge-Kutta methods with embedded error estimation']",
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"text": "['it is rather slow']",
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13351,
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"text": "['recent work suggests that using 2nd-order backward differences']",
"type": "background_claim",
"id": "T129"
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{
"offsets": [
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13419,
13434
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],
"text": "['Newmark schemes']",
"type": "background_claim",
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13437,
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"text": "['may be a better choice']",
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"text": "['9']",
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"text": "['separating repulsion forces from the cloth internal dynamics and applying them outside the Runge-Kutta solver affected stability and resulted in visible artifacts']",
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"text": "['we apply repulsion forces inside the solver loop, so that the solver’s own internal error estimation can remove these artifacts']",
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"text": "['The drawback of this technique is speed']",
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"text": "['the system must check for collisions every time it evaluates the state derivatives']",
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"text": "['We use a perceptually motivated metric to compare the motion of cloth in simulation with a video sequence of real fabric motion']",
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"text": "['Real fabrics exhibit a wide variety of motion ranging from soft and flowing']",
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"text": "['helps to distinguish between different fabrics']",
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"text": "['Researchers in computational neurobiology hypothesize that the human perceptual system is sensitive to moving edges in video']",
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"text": "['Studies have shown that the receptive fields of simple cells in the macaque cortex act as edge or line detectors, responding to oriented edges or lines in natural scenes']",
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"text": "['36']",
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"text": "['In cloth, these edges correspond to folds, which are regions of high variation in shape']",
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"text": "['our perceptually motivated metric for cloth compares two video sequences, one from simulation and one from the real world, and returns a number that measures the differences in their folds']",
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{
"offsets": [
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15964,
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"text": "['Their technique can handle lighting effects caused by diffuse inter-reflections in cloth']",
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"text": "['most fabrics have very complicated reflectance properties']",
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"offsets": [
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15962
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"text": "['describe a learning approach for detecting and grouping folds (and grooves) in images of fabrics']",
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"text": "['15']",
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"text": "['16']",
"type": "data",
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"offsets": [
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16142,
16282
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],
"text": "['we normalize the effects of lighting and material reflectance by projecting a structured light pattern of horizontal stripes onto the fabric']",
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"id": "T161"
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{
"offsets": [
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16869,
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"text": "['We define the resulting orientation image as an angle map']",
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"id": "T162"
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{
"offsets": [
[
16937,
16943
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],
"text": "['Fig. 1']",
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"offsets": [
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17093,
17186
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"text": "['We threshold the gradient of the angle map to get a gradient mask M k for each frame of video']",
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"id": "T164"
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17189,
17195
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"text": "['Fig. 1']",
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17360,
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"text": "['The gradient mask is non-zero at regions of high gradients, corresponding to folds, and zero at planar regions']",
"type": "own_claim",
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"offsets": [
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17489,
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"text": "['Our metric computes the frame by frame sum of squared differences (SSD) between masked angle maps in simulation with video']",
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"id": "T167"
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{
"offsets": [
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18155
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"text": "['We pre-multiply this difference with the gradient mask, which helps to emphasize the differences in fold regions over planar regions']",
"type": "own_claim",
"id": "T168"
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{
"offsets": [
[
18158,
18164
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],
"text": "['Fig. 2']",
"type": "data",
"id": "T169"
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"offsets": [
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18604,
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"text": "['we penalize the silhouette mismatch between the simulation and the video of real cloth']",
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18692,
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"text": "['This penalty is proportional to the difference between the two silhouettes']",
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"offsets": [
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"text": "['The error across the entire sequence of length N frames is given by N E = ∑ E k ( 9 ) k=1']",
"type": "own_claim",
"id": "T172"
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{
"offsets": [
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"text": "['We use optimization to estimate the parameters of the cloth simulator from video']",
"type": "own_claim",
"id": "T173"
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{
"offsets": [
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20493,
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"text": "['we look at the error space of the angle map metric, which gives us useful insight about the parameters of the system']",
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"id": "T174"
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"text": "['it is evident that the error space is fairly noisy, with many local minima, motivating the need for a global optimization technique']",
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"id": "T175"
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"text": "['the figure']",
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"text": "['Fig. 3']",
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"text": "['the variability is computed as the change in parameter values that results in a 1% change in the error']",
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"text": "['The likelihood that it will take a step in a direction that is not locally optimal is a function of the temperature']",
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"text": "['We found it useful to reset the simplex with the current best solution']",
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"text": "['We use simulated annealing to find the parameters that minimize the error function given in eq. 9']",
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"text": "['We chose to use the continuous simulated annealing method presented in Press et a']",
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"text": "['These fabrics exhibit a wide range of static and dynamic behavior and span a large range of real fabrics']",
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"text": "['a fixed separation between the top corners']",
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24890,
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"text": "['Fig. 6']",
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"text": "['The static test give a good estimate for the static stiffness and bend parameters']",
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"text": "['Different fabrics also exhibit different types of motion for the same input excitation']",
"type": "own_claim",
"id": "T206"
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"text": "['This gives reasonable estimates for cloth parameters while avoiding the need to optimize directly on complex fabric geometries']",
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"text": "['involving many collisions']",
"type": "own_claim",
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"text": "['skirts']",
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"text": "['We designed the waving motion to roughly match the types of motion occurring in real garments']",
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"text": "['skirts']",
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"text": "['Each trial took approximately 50 hours to converge on a 2.8GHz Intel Xeon processor']",
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"type": "own_claim",
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"text": "['a very good visual match between the simulations with their counterpart real fabrics']",
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"text": "['Figs. 8']",
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27545,
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"text": "['9']",
"type": "data",
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27647,
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"text": "['there is a significant disparity in the final optimized values from the two trials']",
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"text": "['we see that the final error values are very close']",
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"id": "T220"
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27861,
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"text": "['Table 2']",
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27993,
28002
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"text": "['the table']",
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28055,
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"text": "['We get consistent estimates for parameters that have lower variability']",
"type": "own_claim",
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28133,
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"text": "['bend']",
"type": "data",
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{
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28139,
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"text": "['stretch']",
"type": "data",
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{
"offsets": [
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28149,
28202
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"text": "['Parameters with high variability are estimated poorly']",
"type": "own_claim",
"id": "T226"
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{
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28212,
28268
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"text": "['their values do not contribute sufficiently to the error']",
"type": "own_claim",
"id": "T227"
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{
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"text": "['static tests cannot be used to estimate dynamic parameters like stretch and shear damping or air drag and motivates the waving test, which excites both the static and waving parameters']",
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28988,
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"text": "['The final values of the parameters from the two trials differ in part']",
"type": "own_claim",
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{
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29066,
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"text": "['the variability of the parameters is still fairly high']",
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"id": "T230"
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{
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[
29123,
29130
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"text": "['Fig. 11']",
"type": "data",
"id": "T231"
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{
"offsets": [
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29134,
29225
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"text": "['Different motions or larger sequence might further reduce the variability of the parameters']",
"type": "own_claim",
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{
"offsets": [
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28571,
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"text": "['the six cloth stiffness and damping parameters and three air drag parameters']",
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"text": "['We optimize for nine parameters in the waving test']",
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"id": "T234"
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{
"offsets": [
[
28650,
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"text": "['Fig. 10']",
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{
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29520,
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"text": "['This approach seems to produce plausible results with skirts and other validation experiments']",
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"text": "['a more general solution for parameter identification using our framework is to simultaneously optimize across multiple trials of different experiment']",
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"text": "['the minimum corresponds to a visually compelling match']",
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"id": "T238"
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[
29815,
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"text": "['Fig. 12']",
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{
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29895,
29902
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"text": "['Fig. 13']",
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{
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30380,
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"text": "['each fabric in simulation has a minimum error when compared to its counterpart in reality']",
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{
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30248,
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"text": "['Fig. 14']",
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"text": "['our approach could be potentially useful for recognizing different types of fabrics in video']",
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"id": "T243"
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"text": "['Fig. 14']",
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"text": "['a good visual match between corresponding frames in simulation and video']",
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"text": "['Fig. 10']",
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"text": "['the parameters obtained from optimization generalize well on new sequences']",
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"text": "['the parameters obtained from our optimization approach approximately capture the static shape and dynamic properties of skirts of different materials']",
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"text": "['The results']",
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{
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[
31072,
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],
"text": "['Fig. 15']",
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31369,
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"text": "['Fig. 16']",
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"text": "['the optimizer could identify appropriate simulation parameters from those tests']",
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{
"offsets": [
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"text": "['These parameters produced four distinct and recognizable fabrics when applied to a more complex simulation of a skirt as it was driven by motion capture data from a human figure']",
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{
"offsets": [
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"text": "['The cloth model was not the main focus of this research']",
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{
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"text": "['in early versions of the system it was often the bottleneck in achieving appealing results']",
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{
"offsets": [
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"text": "['To match a video sequence accurately, the cloth physics model as well as the collision algorithms must be chosen carefully']",
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{
"offsets": [
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"text": "['Instabilities in the collision handling will cause perceptible quivering in the motion of cloth']",
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"id": "T259"
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{
"offsets": [
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"text": "['extra damping introduced by the integration method makes crisp folds impossible to match']",
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{
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"text": "['The parameters must also be independent of the resolution of the mesh so that they can be identified on low resolution swatches and applied to higher resolution garments']",
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"text": "['7']",
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"text": "['Our cloth model does not diverge significantly from previous models discussed in the literature']",
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"text": "['Our only major addition was a simple nonlinearity we introduced into the drag model']",
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"text": "['our approach should generalize to any parametrized cloth model that produces a sufficiently rich set of physically realistic motions']",
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{
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"text": "['the skirt is far more complex than the swatches that were used to determine the parameters']",
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"text": "['a form-fitting pair of pants']",
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{
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"text": "['a tailored blazer']",
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{
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47368,
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"text": "['choosing the parameters via optimization on small, flat swatches may not be sufficient']",
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{
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"text": "['more complex garments']",
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{
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"text": "['the shape of the garment is determined by darts, pleats and by the interplay of different fabrics']",
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"text": "['More complex garments may require the hand design of additional tests that mimic particular behaviors or elements of the garment in isolation']",
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{
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"text": "['the model might need extra parameters to handle anisotropic effects, hysteresis and coupling effects']",
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"text": "['all of which would need specialized tests']",
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"text": "['we tried a number of other metrics']",
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"id": "T280"
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{
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47862,
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"text": "['stretching along one direction causing shrinking in the other direction']",
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"text": "['comparing the overlap of the silhouettes']",
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{
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"text": "['the distance function between silhouette edges']",
"type": "data",
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{
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48172,
48230
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],
"text": "['using information from internal edges marked on the fabric']",
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{
"offsets": [
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"text": "['The metric that measures folds and silhouettes, in concert with the projector for the light stripes, proved to be a simple and effective metric that far outperformed our earlier attempts']",
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"text": "['The space of possible metrics is vast']",
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"text": "['a statistical metric might be preferable']",
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"text": "['we were to optimize on more complicated garments']",
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"text": "['Such a metric might']",
"type": "own_claim",
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A29 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A29_F09_Mass-Conserving_Eulerian_Liquid_Simulation_CITATION_PURPOSE_M_v1.xml">
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3x6s
http://dx.doi.org/10.1109/tvcg.2013.19
Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012) P. Kry and J. Lee (Editors)
<Title>Mass-Conserving Eulerian Liquid Simulation</Title>
Nuttapong Chentanez
1
2
Matthias Müller
1
1 NVIDIA Research 2 Chulalongkorn University
<Abstract>We present a GPU friendly, Eulerian, free surface fluid simulation method that conserves mass locally and globally without the use of Lagrangian components. Local mass conservation prevents small scale details of the free surface from disappearing, a problem that plagues many previous approaches, while global mass conservation ensures that the total volume of the liquid does not decrease over time. Our method handles moving solid boundaries as well as cells that are partially filled with solids. Due to its stability, it allows the use of large time steps which makes it suitable for both off-line and real-time applications. We achieve this by using density based surface tracking with a novel, unconditionally stable, conservative advection scheme and a novel interface sharpening method. While our approach conserves mass, volume loss is still possible but only temporarily. With constant mass, local volume loss causes a local increase of the density used for surface tracking which we detect and correct over time. We also propose a density post-processing method to reveal sub-grid details of the liquid surface. We show the effectiveness of the proposed method in several practical examples all running either at interactive rates or in real-time.</Abstract>
NVIDIA Research Chulalongkorn University
<H1>1. Introduction</H1>
Tracking the free surface of a liquid is an important and challenging problem. For an overview of existing methods we recommend the class notes of Wojtan et al. [WMFB11]. The most popular approach is to advect a scalar field with the fluid and define the liquid surface to be one of the isosurfaces. The main advantage of this class of methods is that they handle topological changes implicitly in contrast to mesh-based tracking methods. Until recently, the level set method was the method of choice in graphics. Here, the signed distance field is used as the scalar field with the zeroiso-surface as the liquid surface. A well known drawback of the level set method is that volume is lost both globally and locally. With global volume loss the water level decreases over time while local volume loss causes small detail such as thin sheets and droplets to disappear. A way to alleviate this problem is to introduce Lagrangian components such as particles
[FF01], [EMF02] or triangle meshes [BGOS05]. Even though these methods reduce volume loss, they cannot guarantee complete volume conservation. Moreover, Lagrangian components add significant run-time cost and complicate the implementation significantly, especially for GPUs. As an alternative to the signed distance field, [MMTD07] in- troduced the idea of using a density field as the scalar field for surface tracking with the liquid surface being the 0.5 isosurface. This density field is not to be mistaken for the density field of the liquid. In incompressible fluid simulations, the fluid-density is 1 everywhere and therefore not stored. So in what follows, we use the symbol ρ for the surface density. We chose to use surface density instead of the signed distance field because there are advection methods that strictly conserve quantities like density such as the one proposed by
[LAF11]. Their advection method is unconditionally stable and fully conservative. With this approach, the overall mass defined by the surface density is conserved. Since the surface density can deviate from 1 temporarily, the overall volume may vary over time though. However, in contrast to the level set method where such variations go unnoticed, volume deviations are reflected in the density field. In this paper we propose several methods to preserve volume both globally and locally using the information stored in the density field. Ideally, the surface density has the form of a step function at the liquid-air interface. Over time, however, the initially sharp boundary blurs out due to numerical diffusion. Therefore, [MMTD07] apply a sharpening filter at each time step which conserves mass globally but not locally. We propose a new sharpening method which conserves mass both locally and globally. Our main contributions are: • A GPU friendly, purely Eulerian liquid simulator that conserves mass locally and globally without any need for Lagrangian components. • A new GPU friendly sharpening method which conserves mass locally and globally. • Modifications to reduce the computational cost of the conservative advection method of [LAF11] and to make it more GPU friendly. • Additional novel steps to handle non-axis aligned and moving solid boundaries. • A density post processing technique to bring out sub-grid detail.
c The Eurographics Association 2012.
Figure 1: A liquid jet with large flow rate inside a rectangular tank simulated at a resolution of 256x128x128 cells. The simulation time step is 1/30 seconds (CFL 25) which is much larger than what is typically used in other grid based liquid simulation approach. The liquid moves across many grid cells in a single time step, a situation our method handles without difficulty. Left) Surface rendering. Right) Volumetric rendering showing intricate detail of the density field.
<H1>2. Related Work</H1>
3D Eulerian liquid simulation was introduced to computer graphics by Foster and Metaxas
[FM96]
who used a finite difference approach to solve the governing equations. Later Foster and Fedkiw
[FF01]
employed the semi-Lagrangian method introduced by Stam [Sta99]
to solve the advection term and the level set method and particles to track liquid surface.
Enright et al.
[EMF02]
devised the Particle Level Set (PLS) method which uses particles on both sides of liquid-air interface to reduce volume loss. Since then, many methods have been proposed to further improve the accuracy of Eulerian surface tracking. Various approaches have been proposed to track the liquid domain more faithfully. [BGOS05] used a triangle mesh representation in connection with a level set grid, [HK10]
augmented the level set grid with quadrature points.
Grid-less methods work with Lagrangian elements only such as particles
[ZB05], [APKG07] or
[YT10], triangles meshes
[M 09], ̈
[BB09] and [WTGT10]. In this paper we focus on fluid mass and volume conservation. A popular way to compensate volume gain or loss is to modify the divergence of the velocity field as proposed in
[FOA03].
This technique was extended and used for con∗ serving volume of bubbles
[KLL 07], highly deformable objects
[ISF07] and liquids [MMTD07].
The problem of loss of liquid mass and momentum has also been addressed by proposing elaborate advection methods such as BFECC [KLLR05], modified MacCormack ∗
[SFK 08], derivatives advection
[KSK08] and conservative semi-Lagrangian advection
[LGF11], [LAF11]. As an alternative to level-set, the fluid domain can be tracked with a Volume-of-Fluid (VOF) approach
[HN81]
where the volume fraction of fluid in each cell is evolved over time. With proper care, VOF can be made mass conserving. However, despite several improvements in subsequent works such
[PP04], [AGDJ08], reconstructing surface normal and curvature from VOF is still difficult. Sussman and Puckett
[SP00]
proposes coupled Level Set and Volume-ofFluid (CLVOF) which track the fluid interface with both representations, where VOF is used for re-initializing the Level Set. The surface can then be extracted from the Level Set. CLVOF is extended to handle multiple interfaces in [KPyNS10]. The downside of CLVOF is the need to use two representations which can be quite computationally intensive. An alternative to VOF is to track a smeared-out surface density and keep it relatively sharp with a sharpening operation. This method was introduced to computer graphics by Mullen et al. [MMTD07]. Our fluid domain tracking approach builds upon this work and make it conserve mass both locally and globally.
Apart from the Eulerian formulation we use, there are many alternative models to simulate 3D liquids such as the LatticeBoltzmann method
[TR04] and [TR09], approaches based on the discrete sin-cosine transform
[LR09] or particle based ∗ methods such as
[MCG03], [PTB 03], [APKG07],
[SP09], and
[SG11].
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Nuttapong Chentanez & Matthias Müller / Mass-Conserving Eulerian Liquid Simulation
<H1>3. Methods</H1>
We simulate the liquid by solving the inviscid Euler Equations,
1
∂u f p = −(u · )u + − , ∂t d d
subject to the incompressibility constraint
2
· u = 0,
where u = [u, v, w] T is the fluid velocity field, p is the pressure, t is time, d the fluid density and f is a field of external forces.
The equations are solved in the domain specified by a surface density field ρ [MMTD07], in the region where ρ > 0.5. The surface density itself is advected with the fluid via
3
∂ρ = −u · ρ ∂t
and periodically sharpened to prevent the 0.5 iso-contour from being blurred by numerical damping. The interaction of the liquid with the environment is handled by considering the appropriate Dirichlet and Neumann boundary conditions.
<H2>3.1. Discretization</H2>
We discretize the simulation domain using a regular staggered grid
[HW65]. The x, y and z components of fluid velocity u = (u, v, w) are stored at the center of the faces perpendicular to the x, y and z axis, respectively. The scalar pressure p and the density ρ are stored at cell centers following [MMTD07].
<H2>3.2. Time integration</H2>
Our time integration scheme is summarized in Algorithm 1. The overall structure is the same as the one proposed in [MMTD07] with our novel modifications to the advection, sharpening and pressure incompressibility enforcement steps.
Algorithm 1: 2: 3: 4:
1 Time step Velocity extrapolation Density advection and density sharpening Velocity advection and external force addition Incompressibility enforcement
First, we extrapolate the velocity field into the air region. Then, we advect the surface density field and sharpen it. After this we advect the velocity field and take external forces into account. Finally, we enforce incompressibility by making the velocity field divergence free.
<H2>3.3. Velocity Extrapolation</H2>
To extrapolate the velocities from inside the liquid into the surrounding air we use the scheme described in [CM11b], i.e. we apply the method of
[JRW07] to derive the velocities a few grid cells away from the interface and then extrapolate based on a hierarchy of grids to obtain velocities far away from the surface.
<H2>3.4. Density Advection</H2>
We advect ρ using our unconditionally stable conservative advection method which we derived from the method proposed by Lentine et al. in
[LGF11] and [LAF11]
and improved in terms of computational cost. Lentine et al.
[LGF11]
modified the semi-Lagrangian advection scheme to conserve mass by ensuring that each cell distributes all its mass of the current time step among some cells at the next time step. Let A be the matrix of the discretized advection operator such that ρ n+1 = Aρ n , where ρ n and ρ n+1 are the density in the current and the next time step respectively. Let w ij − (and w + ij ) represent the fraction of value that cell i gives to cell j which is found by backward (and forward) tracing and computing the tri-linear interpolation weights. The entries of A in the standard semi-Lagrangian advection is hence A i j = w − ji . Then, β j = ∑ i A i j is the fraction of mass from cell j that gets advected. To ensure that mass is conserved, A needs to be modified such that all the β j are 1. Lentine et al.
[LGF11]
achieve this by first iterating through all cells j with β j > 1 and re-scaling all A i j to A i j /β j . In a second step they iterate through all cells j with β j < 1 and forward trace the velocity field to adding the weights (1 − β j ) by distributing them among the A k j , where k are the cells reached by forward tracing and tri-linear interpolation. At this point, all the β j are 1, i.e. A is mass conserving. This method works well for compressible flow on fine grids. However, as discussed in
[LAF11]
, the scheme produces artifacts when used for incompressible flow on coarser grids. The problem is due to the clamping of the β j by re-scaling which limits the amount of density that reaches certain cells. An indicator of this amount are the γ i = ∑ j A i j . The traditional semi-Lagrangian method ensures that all the γ i are 1 while the β j are arbitrary. In contrast, the scheme described above ensures that all the β j are 1, while the γ i are arbitrary. Lentine et al.
[LAF11]
propose a method to ensure the β j are all 1 while the γ i stay close to 1. To this end they keep track of the cumulative γ i over time as separate variables. The matrix A is computed by performing multiple forward and backward traces as follows: 1. Advect γ i using the backward semi-Lagrangian method (set to 1 in the first time step). 2. Compute A by performing a backward tracing step as before, i.e. A i j = w − ji . 3. Scale A by the γ i , i.e. A i j A i j /γ i . 4. Compute the β j from A. 5. Forward trace the velocity field to add the weights (1 − β j ) to A for all cells j where β j < 1 by distributing them among the A k j , where k are the cells reached by forward tracing and tri-linear interpolation as before, i.e. A k j += (1 − β j )w + jk . 6. Compute the new γ i from the updated matrix A. 7. Scale A by the γ i , i.e. A i j ← A i j /γ i . 8. Re-compute the β j from the updated matrix A. 9. Clamp the β j to 1 by re-scaling A i j ← A i j /β j . 10. Re-compute the γ i from the updated A. 11. Evaluate ρ n+1 = Aρ n . At this point, all the β j are 1 but the γ i might still deviate from 1. To bring them even closer to 1 Lentine et al. apply a diffusion step on ρ n+1 and the γ i . They iterate through all the cells dimension-by-dimension. If, for two neighboring cells i and j, γ j > γ i , they move ρ j (γ j − γ i )/2γ j from cell j to cell i and set both γ j and γ i to γ j +γ 2 i . If γ j < γ i , the flow happens in the opposite direction. This process is repeated 1 to 7 times per time step. Note that these diffusion iterations do not affect the β j , so they remain 1. Implementing the method described above on a GPU would require 5 scatter passes per iteration in steps 4, 6, 8, 10, and 11. Scattering is expensive on GPU’s because it either requires atomic operations or a prefix-scan. We propose a modification of this method. The basic idea is to reorder the forward tracing and the re-scaling steps to simplify the calculations. The resulting discrete conservative advection operator is not the same as the one computed with the original scheme. However, both are just approximations to the doubly-stochastic matrix (all rowand column sums are one) closest to the original discrete advection operator. While the visual results are of similar quality as shown in Figure 2 and the accompanying video, our simplification reduces the number of scatter passes from 5 to 3. Another advantage of our new scheme is that A does not need to be stored explicitly because the order of the operations allow for updating ρ n+1 , β and γ directly. Not storing A explicitly is possible in the original scheme as well but it would complicate the process considerably and would require even more scatter operations. Here is our modified scheme: 1. Advect γ i using the semi-Lagrangian method (set to 1 in the first time step). 2. Initialize β ← 0. 3. Add the weights γ i to β by distributing them among the β l , where l are the cells reached by backward tracing and tri-linear interpolation, i.e. β l += w − li γ i .
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Nuttapong Chentanez & Matthias Müller / Mass-Conserving Eulerian Liquid Simulation
←
Figure 2: Snapshots from a simulation of a 2D ball of liq-
uid dropping into an empty box at a resolution of 128 2 cells. Top) Using conservative advection method of Lentine et al. 2011. Bottom) Using our conservative advection method. The result are of similar visual quality.
4. Evaluate ρ n+1 from ρ n and γ from γ by backward tracing and tri-linear interpolation from cells l but this time scale the weights by max(1,β γ i l ) , i.e. ρ n+1 i += ∑ l max(1,β γ i l ) w − li ρ n l ) 5. γ ← γ . (This can be done in-place during the previous step). 6. For each cell j whose β j < 1, add ρ n j (1 − β j ) to ρ n+1 by distributing the value among the ρ n+1 k , where k are the cells reached by the forward tracing and tri-linear interpolation, i.e. ρ k n+1 += ρ n j (1 − β j )w + jk . 7. Similarly, for each cell j whose β j < 1, add (1 − β j ) to γ by distributing the value among the γ k , where k are the cells reached by the forward tracing and tri-linear interpolation, i.e. γ k n+1 += γ n j (1 − β j )w + jk . These two steps can be done concurrently. 8. Apply diffusion as in the original approach. This modified method only requires 3 scatter passes in the steps 3, 6, and 7.
As demonstrated in Table 1 , our method keeps γ in a similar range to that of
[LAF11], while
[LGF11]
has a much larger range, resulting in visible compressibility artifacts.
Method
Minimum γ
Maximum γ
Our Method
0.627
2.403
[LAF11]
0.627
2.502
[LGF11]
0.271
9.793
Method Minimum γ Maximum γ Our Method 0.627 2.403 [LAF11] 0.627 2.502 [LGF11] 0.271 9.793
Table 1: Minimum and maximum γ of our method, LAF11 and LGF11 for the situation of Figure 2 . Our method and LAF11 have similar range, while LGF11 has a much larger range which explains the incompressibility artifacts.
<H2>3.5. Density sharpening</H2>
The technique above guarantees that mass is conserved. However, the density field smooths out over time blurring the 0.5 iso-contour with the effect that we can no longer track the movement of the liquid surface accurately. We solve this problem by manipulating ρ to sharpen the interface. Following [MMTD07], we first compute the mass change of each cell due to unit velocity along the x axis as
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Nuttapong Chentanez & Matthias Müller
4
δ x i + = ( · (ρ[1, 0, 0] T )∆T )dV,
5
C i − δ i x = ( · (ρ[−1, 0, 0] T )∆T )dV, C i
where ∆T is the fictitious time step, which we set to 3 times that of the simulation time step in all of our examples. We discretize δ x i + and δ i x − using an upwind scheme to get
6
δ i x + ≈ −(ρ i − ρ i−(1,0,0) )∆x∆T,
7
− δ i x ≈ −(ρ i+(1,0,0) − ρ i )∆x∆T.
y + The mass change due to unit velocity along y and z axes, δ i , δ i y − , δ i z + , and δ i z − are computed similarly. The maximum mass increase and mass decrease due to any unit velocity in each cell is:
8
∆T | ρ| + i = ∆x 1 2 (max(max(δ i x + , 0) 2 , min(δ i x − , 0) 2 ) +
9
y + 2 y − 2 max(max(δ i , 0) , min(δ i , 0) ) +
10
z + 2 z − 2 1 max(max(δ i , 0) , min(δ i , 0) )) 2
and
11
∆T | ρ| − i = ∆x 1 2 (max(min(δ i x + , 0) 2 , max(δ i x − , 0) 2 ) +
12
y + 2 y − 2 max(min(δ i , 0) , max(δ i , 0) ) +
13
z + 2 z − 2 1 max(min(δ i , 0) , max(δ i , 0) )) 2 .
We then compute
14
w i (ρ) = (ρ i − 0.5) 3 (1 − min(1, max j∈א(C i ) (|ρ i − ρ j |) )), τ
where א(C i ) is the set of cells adjacent to C i . The parameter τ controls the maximum difference in density between two adjacent cells, which we set to 0.4 as in [MMTD07]. This yields the following density correction:
15
| + ≥
15
∆ρ i = w i (ρ) ∆T | ρ| i − if if w w i (ρ) (ρ) < 0 0 .
i i ρ can then be sharpened by updating the density of each cell using
16
ρ ← ρ + ∆ρ .
16
i i i
This update sharpens the interface. However, it does not conserve mass. Mullen et al. [MMTD07] modify it to conserve mass by summing up the mass change due to this update across all cells. Then they distribute a fraction of the total mass change back to each cell based on a local area measure. This successfully conserves mass globally. One artifact of this approach is that mass moves far, potentially across the entire simulation domain. This problem can be reduced by re-distributing mass only within connected regions as pro∗ posed by
[KLL 07]. However, even with this technique, local mass loss can still occur due to moving mass away from small features resulting in the disappearance of small surface details. Figure 3 top shows a scene where liquid balls are thrown into a pool of water. The mass conserving sharpening method of [MMTD07] transfers the mass from the liquid balls to the pool causing them to disappear mid-air. The left side of Figure 4 illustrates the situation in greater detail. We propose a novel method to conserve mass during the sharpening phase that conserves mass both locally and globally. After evaluating ∆ρ i using Equation 15, we modify it as follows:
Figure 3: Top) The density sharpening method used by Mullen et al. [2007] conserves mass globally but not locally, causing the mass from a liquid ball (marked with the arrow) to disappear in mid air. Bottom) Our density sharpening method conserves mass both globally and locally preserving the mass of the liquid ball.
17
i i i i ∆ρ i ← 0 if ρ i > 0.5 ∆ρ i otherwise, −5
where we use ε = 10 in all examples. In the first line we make sure that ρ ≥ 0 at the next time step. We also clamp small positive densities to zero so that we do not have to apply the sharpening operator to this cell at the next time step, thus reducing computation cost. In the second line we make sure that cells with ρ > 0.5 are not modified. This way mass only moves from the air side to the liquid side. Then we update ρ i using this modified ∆ρ i in Equation 16. We then add back −∆ρ i by using Algorithm 2. TraceAlongField determines where to put the lost mass. It starts from the cell center and follows the gradient field of the density ρ until it reaches the 0.5 iso-contour. The tracing stops if a predefined distance D∆x is reached or if it crosses a solid boundary. This is done using multiple forward Euler sub-steps. ScatterValue deposits −∆ρ i to nearby grid points using tri-linear weights. If a grid point is in a solid we set the corresponding weight to zero and re-normalize the weights. We use values of D between 1.1 to 3.1 in all of our examples. Figure 5 shows the result of ball dropping into a pool using various values of D. Increasing D visually resembles the effect of surface tension.
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Nuttapong Chentanez & Matthias Müller / Mass-Conserving Eulerian Liquid Simulation
Mullen et al. 2007
1.0 Before sharpening 0.5 0.0 1.0 After adding 0.5 0.0 1.0 After mass correction 0.5 0.0
Figure 4: Comparison between the sharpening scheme of Mullen et al. 07 (Left) and ours (Right). The bigger hump has a large area with 0.5 < ρ < 1.0. In this particular case, − ∑ i ∆ρ i is negative in the scheme of Mullen et al. 2007. This negative mass is distributed to all the cells near interface, causing the smaller hump to become even smaller. Our method does not have this problem because −∆ρ > 0 is only added to the nearby cells around the 0.5 iso-contour. This prevents mass from being transported from one hump to another.
Figure 5: Left most) Initial condition of a ball dropping into a liquid pool. Others) Liquid surface at frame 40 of simulations with various values of parameter D.
2 Local mass conservation for sharpening each cell i do
p = TraceAlongField(Position(i), ρ, ρ, D∆x) ScatterValue(p, −∆ρ i ).
Algorithm 1: for 2: 3: 4: end for
<H2>3.6. Handling Solid Boundaries</H2>
So far, the method does not take solid fraction and solid velocity into account. We use u s = (u s , v s , w s ) for the solid velocity and V i for the fraction of non-solid matter, i.e. fluid and f f f air in cell i. The scalars V i+( 1 ,0,0) , V i+(0, 1 ,0) , and V i+(0,0, 1 ) 2 2 2 represent the fraction of non-solid area of the positive x, y, and z faces respectively.
18
δ x i + ≈−(ρ i V i+( f 1 ,0,0) − ρ i−(1,0,0) V i−( f 1 ,0,0) )∆x∆T,
19
2 2 δ i x − ≈−(ρ i+(1,0,0) V i+( f 1 ,0,0) − ρ i V i−( f 1 ,0,0) )∆x∆T. 2 2
During the simulation, the value of ρ i can become larger than V i in some cells which is a non-valid state. We handle the situation differently depending on whether the cell is partially solid (V i < 1) or completely non-solid (V i = 1). If the cell is partially solid, we first compute the excess density d = ρ i − V i . When then follow the gradient of the solid signed distance function away from the solid for a distance of S∆x and scatter d to nearby grid points. After this we subtract d from ρ i . This method keeps ρ i ≤ V i in most cells near solid boundary and guarantees ρ i = 0 inside the solid. We use S = 1 in all of our examples. With this choice excess density gets removed from solid quickly enough to not cause visual artifacts. The case where V i = 1 is handled in the incompressibility enforcement step described in the next section.
Our method 1.0 0.5 0.0 1.0 0.5 0.0
1.0 0.5 0.0
<H2>3.7. Enforcing Incompressibility</H2>
To enforce incompressibility, we first compute the pressure using a variational framework
[BBB07] and then use the pressure gradient to make velocity field divergence free. The tricky part in our case is to determine the fraction of liquid in each cell. This fraction is used to decide whether a cell is included in the linear pressure solve. It is also needed in the ghost fluid method [ENGF03] to accurately handle the liquid-air boundary. However, we cannot directly use ρ because a cell with V < 0.5 will likely have ρ < 0.5 causing the solver to treat it erroneously as air. To fix this, we define ρ as follows:
20
0 if V i = 0 ρ i = V ρ i i otherwise .
Notice that cells that are completely solid (V = 0) have ρ = 0. We then extrapolate ρ from cells that have V > 0 to adjacent cells with V = 0 so they are included in the linear system. For the ghost fluid method, we also need a signed distance function near the free surface. We approximate this field by defining φ i = −(ρ i − 0.5)∆x and use the method of [CM11a] to compute the coefficients of linear system for pressure projection. To handle the cells with ρ i > 1 (whether or not V = 1 or V < 1), we add min(λ(ρ i −1),η) to the divergence, where we ∆x use λ = 0.5 and η = 1 in all our examples. This artificial divergence pushes the excess density away from the cells whose ρ > 1. Mullen et al. [MMTD07] also added this term to the divergence but with λ = 1 and η = ∞ which can cause stability problems when ρ is much larger than 1. A scenario in which this happens is when liquid flows very fast towards a solid boundary and gets reflected due to our method for removing excess density from the solid.
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Nuttapong Chentanez & Matthias Müller / Mass-Conserving Eulerian Liquid Simulation
Figure 6: A crown splash simulated with our method at resolution of 128 3 cells. The density field is post-processed with
the method proposed in this paper to enhance sub-grid details.
Adding additional divergence is important because in our case, ρ > 1 results in visual volume loss. With the method described above, this problem gets gradually corrected over time. We solve for the pressure p with the multigrid method of [CM11a] which enforces separating solid boundary conditions. Finally, we use the pressure field to make the velocity field divergence free.
<H2>3.8. Density Post Processing</H2>
For rendering, we extract the triangle mesh of the 0.5 isocontour of ρ using the marching cubes method
[LC87]. This approach is typically used in level-set based liquid simulations as well to extract the zero contour of the signed distance field
[EMF02]. The surface density ρ contains small scale details that are not captured by the 0.5 iso-contour. This problem is clearly visible on the right side of Figure 1 and in the bottom row of Figure 12 . Here, the regions where 0 < ρ < 0.5 represent features such as small splashes and thin sheets that are too small to be captured with the grid resolution used. In the level-set approach, these features are destroyed by the redistancing step. To bring out these small scale details in surface rendering, we propose a post processing method. An important observation is that regions in which 0 < ρ < 0.5 do not necessarily represent small scale features. They also exist on the air side of the surface of large liquid regions. In the latter case, we want to leave ρ unchanged but in the former we want to scale up ρ so that the features appear in the 0.5 iso-surface. To this end, we define an additional function γ i = 2 min(ρ i , 0.5) and define the regions in which ρ needs to be scaled up as the regions where γ ≤ 0.5. So far, the two cases above are not distinguished. However, this can be achieved by applying a Gaussian blur filter to γ. Now, since γ > 0.5 inside liquid, those values spread across the interface and cause γ to raise toward 1. In contrast, since γ < 0.5 everywhere inside small scale features, blurring will still result in γ being small. We then define ρ i = min(max(γ ρ i i ,θ),1) and extract the liquid surface as the 0.5 iso-surface of this modified density field. The effect of this post processing method is shown in Figure 6 and in the accompanying video. We used σ = 2∆x for the Gaussian blur filter and θ = 0.01 in this example. A way to improve the results further would be to apply thinning to the parts of the surface that come from region with ρ < 0.5 in order to compensate for the density up-scaling. This is part of our future work.
<H1>4. Results</H1>
We implemented our method using CUDA and ran the simulations on an NVIDIA GTX 680. For all the examples we used a time step size of 1/30s, ∆x = 0.05m, gravity 10m/s 2 and D = 2.1. Density post-processing was turned off unless otherwise stated. Our code run at interactive rates in all examples. The simulation times and CFL numbers are listed in Table 2 . Parameter tuning to get visually appealing results did not take much time. We compared our method with the particle level set (PLS) approach
[EMF02]. The results of this comparison are shown in Figure 7 and in the accompanying video. Our method conserves the liquid’s mass as expected and prevents the water level from decreasing. In contrast, with PLS, most of the liquid disappears in the course of the simulation due to the large time step size used. We used the PLS implementation of [MF] and set the number of particles per cell to 64. Figure 1 shows a simulation of a liquid jet in a rectangular tank. The jet has a very fast flow rate and generates fast moving liquid splashes and sheets. The accompanying video of this example also shows how we fill the tank from a completely dry state by adding liquid balls. These are difficult cases for level set approaches while our method handles them without any problem. With our approach we were able to create, for the first time, a 3d water demo that is both simulated and ray-traced in real time. The scene starting with a dam break initial setup and subsequent additions of water balls is shown in Figure 9 . We achieved a frame rate of over 30fps with two GPUs, one for simulation and one for ray-tracing. Figures 8 and 12 show a dam break and dropping balls in a spherical container. In the accompanying video we shake the container. These examples demonstrate the ability of our method to simulate liquid in a non-axis aligned moving container. One way coupling with fast moving solids is shown in Figure 11 and the accompanying video. Several solid objects move at high speed across the tank sloshing the liquid up to the air. Our method conserves mass and prevents volume loss in this difficult case as well. We computed the mass and the volume enclosed by the 0.5 iso-contour of the liquid over time in various examples. The corresponding plots are shown in Figure 10 . The total mass is computed by integrating ρ over the whole simulation grid. To measure the volume we used marching cubes to extract the 0.5 contour triangle mesh of ρ and determined the enclosed volume. Our method conserves mass in all examples and generally keeps the volume close to the true liquid volume. However, there are several situations where our method loses volume visually. One such case is when a liquid ball hits the ground and spreads out until it becomes thinner than the grid spacing. Even though the density values are nonzero, marching cubes does not generate surface meshes in those regions. Another case is when the ratio of surface area to volume is large. In this case, there are large regions with ρ < 0.5 that do not contribute to the volume because the 0.5 iso-contour is empty. However, in contrast to PLS, when such features join the main body of water again, they correctly contribute to its volume so that the global level remains constant.
c The Eurographics Association 2012.
Figure 7: Liquid ball dropped inside a box simulated on a 128 3 resolution grid. Left) Initial condition. Shape of the surface at frame 40 computed with our method (middle) and with PLS (right). PLS loses most of the mass due to the large time step used.
Figure 8: Snapshots of a dam breaking scene in a spherical container simulated at a resolution of 128 3 cells.
Name
Grid
CFL
Time (ms)
Figure 1
256x128x128
25
113.2
Figure 5
128x128x128
8
54.2
Figure 9
128x128x64
24
26.7
Figure 8
128x128x128
14
53.4
Figure 11
256x128x128
32
118.6
Figure 12
128x128x128
20
53.8
Name Grid CFL Time (ms) Figure 1 256x128x128 25 113.2 Figure 5 128x128x128 8 54.2 Figure 9 128x128x64 24 26.7 Figure 8 128x128x128 14 53.4 Figure 11 256x128x128 32 118.6 Figure 12 128x128x128 20 53.8
Table 2: CFL Number and simulation time per frame for various examples. We use the time step of 1/30s in all examples. All timing are done on GTX680.
<H1>5. Conclusion and Discussion</H1>
We proposed a method for simulating liquids that conserves mass and is effective in keeping the volume defined by the 0.5 iso-contour close to constant. We have demonstrated the strength of our technique in various scenarios. The method has its limitations as well. First, although our sharpening scheme ensures that the ρ = 0.5 interface is sharp, it does not modify regions where ρ > 0.5. It could theoretically be possible that the region with ρ slightly above 0.5 expands so that the volume defined by the 0.5 iso-contour grows by a factor of two while keeping its original mass. This, however,
Figure 12: Simulation of a liquid ball dropping inside a spherical container at a resolution of 128 3 cells. Top) Sur-
face rendering. Bottom) Volume rendering, showing many sub-grid details not visible in the surface rendering.
does not happen in practice because the divergence free velocity field prevents the liquid from expanding significantly. An alternative to our sharpening method is to perform antidiffusion step
[SHA11]
, which is an interesting avenue for future work. Another limitation is the possibility of losing local volume temporarily as discussed in the previous section. The density post processing method we proposed is an effective way to alleviate this effect. Even though our method cannot guarantee complete volume conservation at all times, it reduces this problem significantly in comparison to all the previous methods we have investigated.
<H1>References</H1>
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[ENGF03] ENRIGHT D., NGUYEN D., GIBOU F., FEDKIW R.: Using the particle level set method and a second order accurate pressure boundary condition for free surface flows. In Proc. 4th ASME-JSME Joint Fluids Eng. Conf., number FEDSM2003U ̋ 45144. ASME (2003), pp. 2003–45144. 6
[FF01] FOSTER N., FEDKIW R.: Practical animation of liquids. In Proc. SIGGRAPH (Aug. 2001), pp. 23–30. 1, 2
[FM96] FOSTER N., METAXAS D.: Realistic animation of liquids. Graph. Models Image Process. 58, 5 (1996), 471–483. 2
[FOA03] FELDMAN B. E., O’BRIEN J. F., ARIKAN O.: Animating suspended particle explosions. In the Proceedings of ACM SIGGRAPH 2003 (July 2003), pp. 708–715. 2
[HK10] HEO N., KO H.-S.: Detail-preserving fully-eulerian interface tracking framework. ACM Trans. Graph. 29 (December 2010), 176:1–176:8. 2
[HN81] HIRT C. W., NICHOLS B. D.: Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 39, 1 (Jan. 1981), 201–225. 2
[HW65] HARLOW F., WELCH J.: Numerical calculation of timedependent viscous incompressible flow of fluid with a free surface. The Physics of Fluids 8 (1965), 2182 2189. 3
[ISF07] IRVING G., SCHROEDER C., FEDKIW R.: Volume conserving finite element simulations of deformable models. In ACM SIGGRAPH 2007 papers (New York, NY, USA, 2007), SIGGRAPH ’07, ACM. 2
[JRW07] JEONG W.-K., ROSS, WHITAKER T.: A fast eikonal equation solver for parallel systems. In SIAM conference on Computational Science and Engineering (2007). 3
[KLL 07] KIM B., LIU Y., LLAMAS I., JIAO X., ROSSIGNAC J.: Simulation of bubbles in foam with the volume control method. ACM Trans. Graph. 26 (July 2007). 2, 5
[KLLR05] KIM B., LIU Y., LLAMAS I., ROSSIGNAC J.: Flowfixer: Using bfecc for fluid simulation. In NPH (2005), pp. 51–56. 2
[KPyNS10] KANG N., PARK J., YONG NOH J., SHIN S. Y.: A hybrid approach to multiple fluid simulation using volume fractions. Comput. Graph. Forum 29, 2 (2010), 685–694. 2
[KSK08] KIM D., SONG O.-Y., KO H.-S.: A semi-lagrangian cip fluid solver without dimensional splitting. Computer Graphics Forum 27, 2 (April 2008), 467–475. 2
[LAF11] LENTINE M., AANJANEYA M., FEDKIW R.: Mass and momentum conservation for fluid simulation. In Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on ComputerAnimation (New York, NY, USA, August 2011), SCA ’11, ACM, pp. 91–100. 1, 2, 3, 4
[LC87] LORENSEN W. E., CLINE H. E.: Marching cubes: A high resolution 3d surface construction algorithm. SIGGRAPH Comput. Graph. 21 (August 1987), 163–169. 7
[LGF11] LENTINE M., GRÉTARSSON J. T., FEDKIW R.: An unconditionally stable fully conservative semi-lagrangian method. J. Comput. Phys. 230 (April 2011), 2857–2879. 2, 3, 4
[LR09] LONG B., REINHARD E.: Real-time fluid simulation using discrete sine/cosine transforms. In Proc. ACM SIGGRAPH symposium on Interactive 3D Graphics and Games (2009),pp. 99–106. 2
[M ̈09] MÜLLER M.: Fast and robust tracking of fluid surfaces. In Proc. ACM SIGGRAPH/Eurographics Symposium on Computer Animation (2009). 2
[MCG03] MÜLLER M., CHARYPAR D., GROSS M.: Particle based fluid simulation for interactive applications. In ACM SIGGRAPH/Eurographics Symposium on Computer Animation (2003), pp. 154–159. 2
[MF] MOKBERI E., FALOUTSOS P.: A particle level set library.
[MMTD07] MULLEN P., MCKENZIE A., TONG Y., DESBRUN M.: A variational approach to eulerian geometry processing. In ACM SIGGRAPH 2007 papers (New York, NY, USA, 2007), SIGGRAPH ’07, ACM. 1, 2, 3, 5, 6
[PP04] PILLIOD JR. J. E., PUCKETT E. G.: Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J. Comput. Phys. 199, 2 (Sept. 2004), 465–502. 2
[PTB 03] PREMOZE S., TASDIZEN T., BIGLER J., LEFOHN A. E., WHITAKER R. T.: Particle-based simulation of fluids. Comput. Graph. Forum 22, 3 (2003), 401–410. 2
[SFK 08] SELLE A., FEDKIW R., KIM B., LIU Y., ROSSIGNAC J.: An unconditionally stable MacCormack method. J. Sci. Comput. 35, 2-3 (2008), 350–371. 2
[SG11] SOLENTHALER B., GROSS M.: Two-scale particle simulation. ACM Trans. Graph. 30 (Aug. 2011), 81:1–81:8. 2
[SHA11] SO K. K., HU X. Y., ADAMS N. A.: Anti-diffusion method for interface steepening in two-phase incompressible flow. J. Comput. Phys. 230, 13 (June 2011), 5155–5177. 8
[SP00] SUSSMAN M., PUCKETT E. G.: A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162, 2 (Aug.2000), 301–337. 2
[SP09] SOLENTHALER B., PAJAROLA R.: Predictive-corrective incompressible sph. In Proc. SIGGRAPH (2009), pp. 1–6. 2
[Sta99] STAM J.: Stable fluids. In Proc. SIGGRAPH (Aug. 1999), pp. 121–128. 2
[TR04] THÜREY N., RÜDE U.: Free Surface Lattice-Boltzmann fluid simulations with and without level sets. Proc. of Vision, Modelling, and Visualization VMV (2004), 199–207. 2
[TR09] THÜREY N., RÜDE U.: Stable free surface flows with the lattice Boltzmann method on adaptively coarsened grids. Computing and Visualization in Science 12 (5) (2009). 2
[WMFB11] WOJTAN C., MÜLLER-FISCHER M., BROCHU T.: Liquid simulation with mesh-based surface tracking. In ACM SIGGRAPH 2011 Courses (New York, NY, USA, 2011), SIGGRAPH’11, ACM, pp. 8:1–8:84. 1
[WTGT10] WOJTAN C., THÜREY N., GROSS M., TURK G.: Physics-inspired topology changes for thin fluid features. In Proc. SIGGRAPH (2010), no. 4, pp. 1–8. 2
[YT10] YU J., TURK G.: Enhancing fluid animation with adaptive, controllable and intermittent turbulence. In Proc. ACM SIGGRAPH/Eurographics Symposium on Computer Animation (2010). 2
[ZB05] ZHU Y., BRIDSON R.: Animating sand as a fluid. In Proc.SIGGRAPH (2005), pp.965–972. 2
ÜLLER
ARSCHNER
GUYEN IBOU EDKIW
[AGDJ08] A J. C., G C., D M. A., J K.: Discrete multi-material interface reconstruction for volume fraction data. 3 (2008). 2 [APKG07] A B., P M., K R., G L. J.: Adaptively sampled particle fluids. (July 2007). 2 [BB09] B T., B R.: Robust topological operations for dynamic explicit surfaces. 4 (2009), 2472–2493. 2 [BBB07] B C., B F., B R.: A fast variational framework for accurate solid-fluid coupling. In (2007), p. 100. 6 [BGOS05] B A. W., G T. G., O’B J. F., S J. A.: A semi-lagrangian contouring method for fluid simulation. (2005). 1, 2 [CM11a] C N., M M.: A multigrid fluid pressure solver handling separating solid boundary conditions. In (New York, NY, USA, 2011), SCA ’11, ACM, pp. 83–90. 6, 7 [CM11b] C N., M M.: Real-time eulerian water simulation using a restricted tall cell grid. In (2011), p. 82. 3 [EMF02] E D., M S., F R.: Animation and rendering of complex water surfaces. In (2002), pp. 736–744. 1, 2, 7 [ENGF03] E D., N D., G F., F R.: Using the particle level set method and a second order accurate pressure boundary condition for free surface flows. In
c The Eurographics Association 2012.
Nuttapong Chentanez & Matthias Müller / Mass-Conserving Eulerian Liquid Simulation
Figure 9: Snapshots of a dam break and ball drop scene in a glass box at a resolution of 128x128x64 cells. The simulation and raytracing run in parallel on dual NVIDIA GTX680 GPUs in real-time at 30fps.
Figure 10: Time evolution of mass and volume relative to the ground truth in various examples. In all examples, the time axis is re-scaled to start at 0 and end at 1. The volume of the liquid was measured by evaluating the volume enclosed by the marching cubes 0.5 iso-contour triangle mesh of ρ. Our method conserves mass up to an arithmetic error. The volume generally stays close to the true value, but can decrease significantly in some cases. For example, when the liquid ball spreads out on the floor and becomes thinner than the grid spacing or when the ratio of surface area to volume is large.
In Proc. 4th ASME-JSME Joint Fluids Eng. Conf., number FEDSM2003 U45144. ̋ ASME OSTER EDKIW Proc. SIGGRAPH OSTER ETAXAS Graph. Models Image Process. 58, ELDMAN RIEN RIKAN the Proceedings of ACM SIGGRAPH 2003
EO O
ACM Trans. Graph. 29
IRT ICHOLS
Journal of Compu-
tational Physics 39,
ARLOW ELCH
The Physics of Fluids 8
RVING CHROEDER EDKIW
ACM
SIGGRAPH 2007 papers
EONG OSS HITAKER SIAM conference on Computational Science and Engineering ∗ IM IU LAMAS IAO OSSIGNAC
(2003), pp. 2003–45144. 6 [FF01] F N., F R.: Practical animation of liquids. In (Aug. 2001), pp. 23–30. 1, 2 [FM96] F N., M D.: Realistic animation of liquids. 5 (1996), 471–483. 2 [FOA03] F B. E., O’B J. F., A O.: Animating suspended particle explosions. In (July 2003), pp. 708–715. 2 [HK10] H N., K H.-S.: Detail-preserving fully-eulerian interface tracking framework. (December 2010), 176:1–176:8. 2 [HN81] H C. W., N B. D.: Volume of fluid (VOF) method for the dynamics of free boundaries. 1 (Jan. 1981), 201–225. 2 [HW65] H F., W J.: Numerical calculation of timedependent viscous incompressible flow of fluid with a free surface. (1965), 2182–2189. 3 [ISF07] I G., S C., F R.: Volume conserving finite element simulations of deformable models. In (New York, NY, USA, 2007), SIGGRAPH ’07, ACM. 2 [JRW07] J W.-K., R , W T.: A fast eikonal equation solver for parallel systems. In (2007). 3 [KLL 07] K B., L Y., L I., J X., R J.: Simulation of bubbles in foam with the volume control method. (July 2007). 2, 5
ACM Trans. Graph. 26
IM IU LAMAS
OSSIGNAC NPH
ANG ARK YONG OH HIN
Comput. Graph. Forum 29, IM ONG O Computer Graphics Forum 27, ENTINE ANJANEYA EDKIW Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation
ORENSEN LINE
Comput. Graph. 21
SIGGRAPH
ENTINE RÉTARSSON EDKIW
J. Comput. Phys. 230 ONG EINHARD Proc. ACM SIGGRAPH symposium on Interactive 3D Graphics and Games
ÜLLER Proc. ACM SIGGRAPH/Eurographics Symposium on Computer Animation
ÜLLER HARYPAR ROSS
[KLLR05] K B., L Y., L I., R J.: Flowfixer: Using bfecc for fluid simulation. In (2005), pp. 51– 56. 2 [KPyNS10] K N., P J., N J., S S. Y.: A hybrid approach to multiple fluid simulation using volume fractions. 2 (2010), 685–694. 2 [KSK08] K D., S O.-Y., K H.-S.: A semi-lagrangian cip fluid solver without dimensional splitting. 2 (April 2008), 467–475. 2 [LAF11] L M., A M., F R.: Mass and momentum conservation for fluid simulation. In (New York, NY, USA, August 2011), SCA ’11, ACM, pp. 91–100. 1, 2, 3, 4 [LC87] L W. E., C H. E.: Marching cubes: A high resolution 3d surface construction algorithm. (August 1987), 163–169. 7 [LGF11] L M., G J. T., F R.: An unconditionally stable fully conservative semi-lagrangian method. (April 2011), 2857–2879. 2, 3, 4 [LR09] L B., R E.: Real-time fluid simulation using discrete sine/cosine transforms. In (2009), pp. 99–106. 2 [M 09] ̈ M M.: Fast and robust tracking of fluid surfaces. In (2009). 2 [MCG03] M M., C D., G M.: Particlebased fluid simulation for interactive applications. In
ACM
Figure 11: Simulation of a solid duck moving across a liquid tank at high speed, sloshing the liquid up in the air on a grid of 256x128x128 cells.
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"text": "['they move ρ j (γ j − γ i )/2γ j from cell j to cell i and set both γ j and γ i to γ j +γ 2 i']",
"type": "own_claim",
"id": "T191"
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"text": "['for two neighboring cells i and j, γ j > γ i']",
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15130,
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"text": "['γ j < γ i']",
"type": "data",
"id": "T193"
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15145,
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"text": "['the flow happens in the opposite direction']",
"type": "own_claim",
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"text": "['This process is repeated 1 to 7 times per time step']",
"type": "own_claim",
"id": "T195"
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15252,
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"text": "['these diffusion iterations do not affect the β j']",
"type": "own_claim",
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"text": "['they remain 1']",
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"text": "['Scattering is expensive on GPU’s']",
"type": "own_claim",
"id": "T198"
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{
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15486,
15539
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],
"text": "['it either requires atomic operations or a prefix-scan']",
"type": "data",
"id": "T199"
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{
"offsets": [
[
15541,
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"text": "['We propose a modification of this method']",
"type": "own_claim",
"id": "T200"
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{
"offsets": [
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15583,
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],
"text": "['The basic idea is to reorder the forward tracing and the re-scaling steps to simplify the calculations']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
15687,
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],
"text": "['The resulting discrete conservative advection operator is not the same as the one computed with the original scheme']",
"type": "own_claim",
"id": "T202"
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{
"offsets": [
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"text": "['both are just approximations to the doubly-stochastic matrix']",
"type": "own_claim",
"id": "T203"
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{
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"text": "['closest to the original discrete advection operator']",
"type": "own_claim",
"id": "T204"
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"text": "['our simplification reduces the number of scatter passes from 5 to 3']",
"type": "own_claim",
"id": "T205"
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{
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"text": "['the visual results are of similar quality']",
"type": "own_claim",
"id": "T206"
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{
"offsets": [
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16020,
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],
"text": "['Figure 2']",
"type": "data",
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{
"offsets": [
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16033,
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"text": "['the accompanying video']",
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"offsets": [
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],
"text": "['Another advantage of our new scheme is that A does not need to be stored explicitly']",
"type": "own_claim",
"id": "T209"
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"offsets": [
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16218,
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],
"text": "['the order of the operations allow for updating ρ n+1 , β and γ directly']",
"type": "data",
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"offsets": [
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"text": "['Not storing A explicitly is possible in the original scheme as well']",
"type": "own_claim",
"id": "T211"
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{
"offsets": [
[
18091,
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"text": "['This modified method only requires 3 scatter passes in the steps 3, 6, and 7']",
"type": "own_claim",
"id": "T212"
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{
"offsets": [
[
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],
"text": "['our method keeps γ in a similar range to that of']",
"type": "own_claim",
"id": "T213"
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{
"offsets": [
[
18189,
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],
"text": "['Table 1']",
"type": "data",
"id": "T214"
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{
"offsets": [
[
18250,
18255
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],
"text": "['LAF11']",
"type": "data",
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"offsets": [
[
18274,
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],
"text": "['has a much larger range, resulting in visible compressibility artifacts']",
"type": "own_claim",
"id": "T216"
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{
"offsets": [
[
18266,
18271
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],
"text": "['LGF11']",
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"offsets": [
[
19122,
19161
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"text": "['Our method and LAF11 have similar range']",
"type": "own_claim",
"id": "T218"
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{
"offsets": [
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19175,
19245
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],
"text": "['has a much larger range which explains the incompressibility artifacts']",
"type": "own_claim",
"id": "T219"
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{
"offsets": [
[
19169,
19174
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],
"text": "['LGF11']",
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19355,
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"text": "['mass is conserved']",
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"id": "T221"
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19319,
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],
"text": "['The technique above guarantees']",
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},
{
"offsets": [
[
19383,
19541
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],
"text": "['the density field smooths out over time blurring the 0.5 iso-contour with the effect that we can no longer track the movement of the liquid surface accurately']",
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"id": "T223"
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{
"offsets": [
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19629,
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"text": "['we first compute the mass change of each cell due to unit velocity along the x axis']",
"type": "own_claim",
"id": "T224"
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{
"offsets": [
[
19620,
19626
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],
"text": "['MMTD07']",
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},
{
"offsets": [
[
19842,
19877
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],
"text": "['δ x i + = ( · (ρ[1, 0, 0] T )∆T )dV']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
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19919,
19964
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"text": "['C i − δ i x = ( · (ρ[−1, 0, 0] T )∆T )dV, C i']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
20225,
20260
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],
"text": "['δ i x + ≈ −(ρ i − ρ i−(1,0,0) )∆x∆T']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
20302,
20337
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],
"text": "['− δ i x ≈ −(ρ i+(1,0,0) − ρ i )∆x∆T']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
20101,
20157
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],
"text": "['We discretize δ x i + and δ i x − using an upwind scheme']",
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"offsets": [
[
20360,
20375
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"text": "['The mass change']",
"type": "own_claim",
"id": "T231"
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{
"offsets": [
[
20455,
20477
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"text": "['are computed similarly']",
"type": "own_claim",
"id": "T232"
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{
"offsets": [
[
20383,
20452
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],
"text": "['unit velocity along y and z axes, δ i , δ i y − , δ i z + , and δ i z']",
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{
"offsets": [
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20479,
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"text": "['The maximum mass increase and mass decrease']",
"type": "own_claim",
"id": "T234"
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{
"offsets": [
[
20527,
20560
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],
"text": "['to any unit velocity in each cell']",
"type": "data",
"id": "T235"
},
{
"offsets": [
[
20596,
20665
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],
"text": "['∆T | ρ| + i = ∆x 1 2 (max(max(δ i x + , 0) 2 , min(δ i x − , 0) 2 ) +']",
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"id": "T236"
},
{
"offsets": [
[
20706,
20753
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],
"text": "['y + 2 y − 2 max(max(δ i , 0) , min(δ i , 0) ) +']",
"type": "data",
"id": "T237"
},
{
"offsets": [
[
20795,
20845
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],
"text": "['z + 2 z − 2 1 max(max(δ i , 0) , min(δ i , 0) )) 2']",
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"id": "T238"
},
{
"offsets": [
[
20899,
20968
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],
"text": "['∆T | ρ| − i = ∆x 1 2 (max(min(δ i x + , 0) 2 , max(δ i x − , 0) 2 ) +']",
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"id": "T239"
},
{
"offsets": [
[
21010,
21057
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],
"text": "['y + 2 y − 2 max(min(δ i , 0) , max(δ i , 0) ) +']",
"type": "data",
"id": "T240"
},
{
"offsets": [
[
21099,
21149
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],
"text": "['z + 2 z − 2 1 max(min(δ i , 0) , max(δ i , 0) )) 2']",
"type": "data",
"id": "T241"
},
{
"offsets": [
[
21357,
21442
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],
"text": "['The parameter τ controls the maximum difference in density between two adjacent cells']",
"type": "own_claim",
"id": "T242"
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{
"offsets": [
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"text": "['we set to 0.4 as in']",
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"id": "T243"
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{
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[
21471,
21477
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"text": "['MMTD07']",
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"offsets": [
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21480,
21524
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"text": "['This yields the following density correction']",
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"id": "T245"
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21558,
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],
"text": "['| + ≥']",
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"id": "T246"
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[
21605,
21660
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"text": "['∆ρ i = w i (ρ) ∆T | ρ| i − if if w w i (ρ) (ρ) < 0 0']",
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"id": "T247"
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{
"offsets": [
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21680,
21750
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"text": "['i i ρ can then be sharpened by updating the density of each cell using']",
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"id": "T248"
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21783,
21793
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"text": "['ρ ← ρ + ∆ρ']",
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"id": "T249"
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"offsets": [
[
21860,
21894
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"text": "['This update sharpens the interface']",
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"id": "T250"
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{
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[
21905,
21930
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"text": "['it does not conserve mass']",
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"id": "T251"
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{
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[
21932,
21945
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],
"text": "['Mullen et al.']",
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21955,
22047
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"text": "['modify it to conserve mass by summing up the mass change due to this update across all cells']",
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{
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21947,
21953
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"text": "['MMTD07']",
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22155,
22196
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"text": "['This successfully conserves mass globally']",
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"id": "T255"
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22198,
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"text": "['One artifact of this approach is that mass moves far, potentially across the entire simulation domain']",
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22301,
22389
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"text": "['This problem can be reduced by re-distributing mass only within connected regions as pro']",
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{
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22391,
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"text": "['posed by']",
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"text": "['KLL 07']",
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{
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22420,
22444
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"text": "['even with this technique']",
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22446,
22477
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"text": "['local mass loss can still occur']",
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{
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22485,
22577
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],
"text": "['moving mass away from small features resulting in the disappearance of small surface details']",
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{
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22658,
22698
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"id": "T263"
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{
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22708,
22794
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"text": "['transfers the mass from the liquid balls to the pool causing them to disappear mid-air']",
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"id": "T264"
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{
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22700,
22706
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"text": "['MMTD07']",
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{
"offsets": [
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22867,
22983
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"text": "['We propose a novel method to conserve mass during the sharpening phase that conserves mass both locally and globally']",
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23087,
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"id": "T267"
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{
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23146,
23169
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"text": "['conserves mass globally']",
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23174,
23185
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"text": "['not locally']",
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"id": "T269"
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{
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23195,
23222
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"text": "['the mass from a liquid ball']",
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{
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23247,
23270
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"text": "['to disappear in mid air']",
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23125,
23145
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"text": "['Mullen et al. [2007]']",
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"offsets": [
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23280,
23389
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"text": "['Our density sharpening method conserves mass both globally and locally preserving the mass of the liquid ball']",
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{
"offsets": [
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23582,
23587
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"text": "['ρ ≥ 0']",
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23588,
23609
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"text": "['at the next time step']",
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23755,
23780
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"text": "['reducing computation cost']",
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23666,
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"text": "['we do not have to apply the sharpening operator to this cell at the next time step']",
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23819,
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"text": "['cells with ρ > 0.5 are not modified']",
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"id": "T278"
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24292,
24343
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"text": "['This is done using multiple forward Euler sub-steps']",
"type": "own_claim",
"id": "T279"
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24451,
24519
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"text": "['we set the corresponding weight to zero and re-normalize the weights']",
"type": "own_claim",
"id": "T280"
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{
"offsets": [
[
24424,
24450
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],
"text": "['a grid point is in a solid']",
"type": "data",
"id": "T281"
},
{
"offsets": [
[
24666,
24727
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],
"text": "['Increasing D visually resembles the effect of surface tension']",
"type": "own_claim",
"id": "T282"
},
{
"offsets": [
[
25148,
25205
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],
"text": "['The bigger hump has a large area with 0.5 < ρ < 1.0']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
25232,
25271
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],
"text": "['− ∑ i ∆ρ i is negative in the scheme of']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
25272,
25290
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],
"text": "['Mullen et al. 2007']",
"type": "data",
"id": "T285"
},
{
"offsets": [
[
25207,
25230
]
],
"text": "['In this particular case']",
"type": "data",
"id": "T286"
},
{
"offsets": [
[
25292,
25357
]
],
"text": "['This negative mass is distributed to all the cells near interface']",
"type": "own_claim",
"id": "T287"
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{
"offsets": [
[
25367,
25406
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],
"text": "['the smaller hump to become even smaller']",
"type": "own_claim",
"id": "T288"
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{
"offsets": [
[
25408,
25445
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],
"text": "['Our method does not have this problem']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
25454,
25525
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],
"text": "['−∆ρ > 0 is only added to the nearby cells around the 0.5 iso-contour']",
"type": "data",
"id": "T290"
},
{
"offsets": [
[
25527,
25593
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],
"text": "['This prevents mass from being transported from one hump to another']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
26082,
26153
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],
"text": "['the method does not take solid fraction and solid velocity into account']",
"type": "own_claim",
"id": "T292"
},
{
"offsets": [
[
26292,
26447
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],
"text": "['The scalars V i+( 1 ,0,0) , V i+(0, 1 ,0) , and V i+(0,0, 1 ) 2 2 2 represent the fraction of non-solid area of the positive x, y, and z faces respectively']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
26706,
26749
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],
"text": "['the value of ρ i can become larger than V i']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
26750,
26763
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],
"text": "['in some cells']",
"type": "data",
"id": "T295"
},
{
"offsets": [
[
26770,
26790
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],
"text": "['is a non-valid state']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
26792,
26923
]
],
"text": "['We handle the situation differently depending on whether the cell is partially solid (V i < 1) or completely non-solid (V i = 1)']",
"type": "own_claim",
"id": "T297"
},
{
"offsets": [
[
26957,
27006
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],
"text": "['we first compute the excess density d = ρ i − V i']",
"type": "own_claim",
"id": "T298"
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{
"offsets": [
[
26928,
26955
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],
"text": "['the cell is partially solid']",
"type": "data",
"id": "T299"
},
{
"offsets": [
[
27192,
27293
]
],
"text": "['This method keeps ρ i ≤ V i in most cells near solid boundary and guarantees ρ i = 0 inside the solid']",
"type": "own_claim",
"id": "T300"
},
{
"offsets": [
[
27332,
27432
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],
"text": "['With this choice excess density gets removed from solid quickly enough to not cause visual artifacts']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
27449,
27509
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],
"text": "['V i = 1 is handled in the incompressibility enforcement step']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
27523,
27539
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],
"text": "['the next section']",
"type": "data",
"id": "T303"
},
{
"offsets": [
[
27757,
27816
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],
"text": "['we first compute the pressure using a variational framework']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
27819,
27824
]
],
"text": "['BBB07']",
"type": "data",
"id": "T305"
},
{
"offsets": [
[
27901,
27980
]
],
"text": "['The tricky part in our case is to determine the fraction of liquid in each cell']",
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<Document xmlns:gate="http://www.gate.ac.uk" name="A16_M07_Momentum-based_Parameterization_of_Dynamic_Character_Motion_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1028523.1028546
Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2004) R. Boulic, D. K. Pai (Editors)
<Title>Momentum-based Parameterization of Dynamic Character Motion</Title>
Yeuhi Abe C. Karen Liu Zoran Popović
University of Washington
<Abstract>This paper presents a system for rapid editing of highly dynamic motion capture data. At the heart of this system is an optimization algorithm that can transform the captured motion so that it satisfies high-level user constraints while enforcing that the linear and angular momentum of the motion remain physically plausible. Unlike most previous approaches to motion editing, our algorithm does not require pose specification or model reduction, and the user only need specify high-level changes to the input motion. To preserve the dynamic behavior of the input motion, we introduce a spline-based parameterization that matches the linear and angular momentum patterns of the motion capture data. Because our algorithm enables rapid convergence by presenting a good initial state of the optimization, the user can efficiently generate a large number of realistic motions from a single input motion. The algorithm can then populate the dynamic space of motions by simple interpolation, effectively parameterizing the space of realistic motions. We show how this framework can be used to produce an effective interface for rapid creation of dynamic animations, as well as to drive the dynamic motion of a character in real-time.</Abstract>
Categories and Subject Descriptors (according to ACM CCS) : I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Animation
<H1>1. Introduction</H1>
Despite great advances in recent years, creating effective tools for synthesis of realistic human motion remains an open problem in computer animation. This is particularly true for synthesis of highly dynamic character motion such as running, leaping, jumping and other athletic and acrobatic maneuvers that frequently occur in feature special effects and video games. Synthesizing such motions can be challenging because any physical inaccuracies in these motions are particularly noticeable. Both spacetime optimization and controller synthesis approaches have been proposed for direct synthesis of dynamic character motion. Although these methods do satisfy physical laws, they tend to appear overly smooth and at times robotic. Furthermore, these methods do not provide interactive control, often requiring considerable offline processing time before the animation sequence is generated. In addition, it is difficult to achieve a graceful degradation of realism for the purpose of greater control. In contrast to direct synthesis, methods based on adaptation of motion capture data produce highly realistic motion, especially in the neighborhood of captured motion samples. They also run at interactive speeds, as they employ data interpolation techniques. Unfortunately, these methods require a large number of motion samples. If the animator wants to interactively control a specific parameter of the animation such as the landing foot position in a particular acrobatic stunt, the need for a large dataset is particularly pronounced: the interpolation techniques would require an already existing family of motion sequences where the only difference in motion is the landing foot position. Gathering such a datataset is not only laborious, but it also requires that the captured family of motions is similar in all other respects (e.g. other landing points, initial and final state, overall style) — an aspect that is quite hard to reproduce by real actors. In fact, the process of generating such parameterized motions is the most challenging aspect of data acquisition for video game production [Buc]. In addition, the animators often wish to create non-realistic motions that defy the laws of physics, a space where motion capture simply fails to provide any samples. We take the approach to acquiring similar motions is to adapt a single motion sequence several times to synthesize a family of motions that preserve physics constraints. Motions created in this manner can satisfy an animator’s exact specifications with a minimum of deviation from the initial motion sequence. Ideally, we would like to use a minimal source of motion data, perhaps a single captured movement, to create a wide range of additional motions. Recently a number of dynamic motion adaptation methods have been proposed
[PW99, ZH99, TSK02, SP04, SHP04]
, and the work presented in this paper falls into this category. In this paper, we describe the momentum-based motion editing technique. In contrast to the existing methods, our proposed framework is particularly robust to large-scale motion modifications. For example, we can adapt a forward leaping movement, to a collection of leaping movement in different directions including a backward leap, or a 360 ◦ leaping spin. Using our motion editing framework, we show how a family of dynamic movements can be synthesized based on the animator’s needs for interactive control. Because our family of motions samples the space widely, satisfies exact constraints, and otherwise deviates minimally from the original source sequence, we can use simple interpolation techniques to allow real-time exploration of this synthetic motion space. We describe a number of real-time animation tools that can be constructed using these synthetic motion families, such as interactive displacement of constraints (e.g. varying foot landing position), as well as inverse control examples such as the determination of the natural volleyball spike that would hit the ball arriving at a specific position in space. In addition, we describe how the same synthetic sampling/interpolation approach can be used to develop realtime controllers for leaping character motion, all synthesized from a single motion-captured leap.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
<H1>2. Related work</H1>
Recent research in computer animation focused on techniques for remapping existing data to given specifications of a new scenario. In this paper, we build on the research in both physicsand interpolation-based motion editing methods.
<H2>2.1. Physics-based motion editing</H2>
Optimal trajectory methods introduced by Witkin and Kass
[WK88]
provide a powerful framework for enforcing dynamic constraints while searching for the most favorable motion judged by the objective function. Extending physicsbased optimization to a full human figure, however, has presented a significant challenge mainly due to the nonlinearity of the dynamic constraints, and sensitivity to the starting point of the optimization. The dependency on the initial point has been somewhat alleviated by starting out with the captured motion sequence. Popović and Witkin in 1999 developed a first method that transforms motion capture data while preserving physical properties
[PW99]. They found solutions by performing optimizations on the reduced character model. More recently, editing motion capture data based on spacetime optimization has become a popular strategy for producing realistic character animations [RGBC96, SP04, SHP04]. These methods provide control for modifying data while retaining physically plausible properties of captured motion by restricting the optimization space with additional kinematic constraints (e.g. [RGBC96]), or by solving within the PCA-reduced space of motions
[SHP04]. It has recently been shown that relying on simplifications of dynamic constraints is not necessary if proper scaling and estimation of joint angles, torques, and Lagrange multipliers are provided
[SP04]. Our work uses a similar spacetime optimization framework. In contrast to other approaches, we formulate significantly simpler momentum constraints on a complex character model, without solving for muscle forces explicitly, similar to
[LP02]. Since we do not compute internal torques for joints, scaling and convergence issues are less critical in our optimization framework. Our physics-based motion editing approach is based on the momentum constraints introduced by Liu and Popović
[LP02]. In that work, momentum constraints were used for synthesis of highly dynamic motion from simple animations that did not contain sufficient information to synthesize the full motion. As a result, transition poses had to be introduced to further restrict the optimization space. There are two main advantages of momentum constraints over the full dynamics constraints. First, since dynamic constraints are reduced to only global momentum patterns, we are solving for a much smaller set of unknowns, and over a much “better behaved” set of constraints. This allows us to find solutions quickly. Also, in our experience, these constraints do not suffer from many local minima, thus enabling us to find solutions significantly further away from the original motion. The second advantage of momentum constraints is that they encode more about the natural motion than just physical correctness. For example in natural motion, passive elements such as tendons and ligaments store and release energy during ballistic motion. To model this with a full dynamic system, one would have to include a complex muscle model. Momentum constraints effectively record the aggregate effect of the natural torque usage and energy storage/release in a specific momentum pattern. This additional information embedded within the momentum constraints ensures that adapted motion is not just physically correct, but that it also constrains the motion within the momentum exchange patterns observed in nature. In contrast to the original paper that introduced momentum constraints, our method applies momentum constraints directly on the motion capture data. Our algorithm does not require any additional pose constraints at the transition points between flight and ground phases. Furthermore, we introduce a novel spline-based representation for the momentum patterns that can be used to intrinsically enforce the similarity between the resultant motion and the input motion. Instead of formulating a physics-based optimization, dynamic filtering is an efficient alternative for motion editing of smaller amplitude. Per-frame based frameworks largely reduce the computation time, providing an interactive editing interface to the user
[TSK02, SKG03]. Unfortunately, the per-frame approach means that animators can modify the spatial position of constraints, but not their position in time. Tak et al. applied Kalman filter to estimate an optimal pose for the current frame subject to the given constraints. The result of the estimation is then rectified by least-square-fit to ensure a physically sound motion
[TSK02]. Shin et al. approximated the adjustment made to the original motion capture data by correcting the momentum of the character during flight and using the balance constraints on the ground
[SKG03]. In general, these methods are geared toward the local modification compared to the overall motion, such as improving the balance, whereas our approach is able to handle global changes of the motion such as transforming a forward jump to a 360 ◦ backward spin jump. Another branch of dynamic filtering employs dynamic tracking
[ZH99, PR01]. These methods combine motion capture data and dynamic simulation to retain human-like details from the data while presenting interaction with the environment. These methods produce motions that do not deviate significantly from the input motion, relying on the existence of captured motion that is similar to what the user intends to do.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
<H2>2.2. Interpolation-based motion editing</H2>
Straightforward interpolation of joint angles usually fails to preserve physical realism from the original data. However, many methods have shown that small modification of the motion can be easily done by linear interpolation of joint angles
[BW95, WP95, WH97]. Combining interpolation with kinematics constraints, Gleicher adapted original motion to a new character while maintaining environmental constraints such as foot contacts on the floor
[Gle98].
A more sophisticated interpolation was presented using radial basis functions to blend motion sequences with various inverse-kinematic goals
[RSC01] or different style
[RCB98]. Unfortunately, data acquisition and post-processing for these methods present a significant challenge since motion sequences need to be carefully crafted so that they contain the same content yet different in style. Our approach only requires one single motion capture sequence as the seed. This seed is used to generate a family of motion sequences that parameterize the dynamic space. Lee and Shin presented a multi-level B-spline representation by which they transform existing motion to satisfy desired constraints adaptively through direct manipulation
[LS99]. Using B-spline representation, the motion edits can be limited to user-specified frequency bands, providing a more effective optimization framework. Our work adapts the idea of using spline-based representation to constrain the search of the optimization. We model the momentum curves by a B-spline representation which are fitted to the original motion so that the search space in the optimization is limited to solutions that have similar dynamic behavior of the original motion.
Mocap Data Momentum Curve Pre-fitting Spacetime User specification Optimization Optimized Optimized Optimized motion motion motion Linear Interpolation Interpolated ..... Interpolated ..... Interpolated motion motion motion
Figure 1: System overview
<H1>3. Overview</H1>
Our system is based on an optimization algorithm that can transform the captured motion to satisfy high-level user constraints while preserving physical realism. As input, the system takes a single motion capture sequence and the userspecified modification. We describe the algorithm in three separate components: Motion pre-fitting, optimization, and interpolation (see Figure 1 ). The pre-fitting optimizes a set of coefficients used to model momentum curves so that they are constrained to the similar shapes of the original motion. The system then formulates a spacetime optimization that solves for a new motion, where both high-level physical constraints and the user specification are met. With a family of such optimized motions that parameterize certain dynamic space, we can apply a simple linear interpolation to generate arbitrary new motion within the dynamic space in real-time.
<H1>4. Motion editing with momentum constraints</H1>
Our algorithm adapts the momentum-based constraints
[LP02]
for the task of motion editing. Instead of filling in missing data, motion editing must solve the converse problem of preserving the original data while still satisfying animator-imposed constraints. There is no need for keyframing of any kind because the motion already starts in a good initial state. Any underlying physical model employed by the system must be flexible enough to precisely describe the initial state of the motion and, at the same time, rigid enough to maintain a semblance of the original motion throughout the editing process.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
<H2>4.1. Motion pre-fitting</H2>
At the heart of our algorithm is a set of full-body angular and linear momentum curves. These curves constrain the edited motion to the realm of physical realism without the need to simulate expensive dynamical properties such as joint torques and contact forces. The momentum curves are parameterized by a set of coefficients that are pre-solved to closely match the input motion. The advantage of this approach is twofold. First, a good initial state of the momentum coefficients results in rapid convergence of the optimization. Second, the coefficients that control the shape of the curves can be fixed throughout the editing process, effectively performing a biased search for similar motions in the momentum space. After the motion is captured using an optical system and processed to fit the character’s skeletal structure, we employ the constraint detection technique described in
[LP02]
to partition the motion into ground-contact and flight stages. Since the the animator may at times wish to produce physically impossible jumps that are not constrained to the earth’s gravity, and because the sampling rate varies for each input motion sequence, we also need to determine the time interval between two animation frames. Gravity and time step are directly related because we can equivalently choose to find the right gravitational constant that makes the motion realistic for a given unit time step. During free-fall stages, the linear momentum is only affected by gravity and the angular momentum remains constant. By observing that the center of mass (COM) of the model must follow a parabolic trajectory, p(t), we can compute the gravitational constant, g, by solving a system of equations = 1/2gt 2 + 0 t + 0
p(t) v C p(t n C n p(t n/2 C n/2
) = ) =
where t 0..n are time steps in the free-fall stage, C 0..n are corresponding values of the COM, and v 0 is the unknown initial velocity of the COM. When the body is in contact with external forces, the momentum curves can no longer be represented by a simple set of linear equations. Instead, we represent the momentum curves with a 3rd-order non-uniform B-splines for their flexibility and convenient knot based parameterization. In our spline representation, the first and last knots have duplicity 4 to ensure interpolation of the end points (see [FvDFH92]). A defining characteristic of motion is the shape and magnitude of its momentum curve (see Figure 2 ). In the case of our spline representation, the control points determine the magnitude of the curve and the spacing of the knots influence the shape. We note that this formulation can capture a greater variability of momentum patterns than the previously used hardwired patterns
[LP02]. This is especially important when dealing with motion capture data due to wide range of different maneuvers possible in the real world. To find a set of control points, {c i |i ∈ 1..k}, and knots, {u i |i ∈ 1..k + 4}, that closely match the momentum pattern of the input motion, we solve the following constrained optimization problem for each momentum spline 0..k , 0..k+4 ):
S(t, c u
n
S(0) m S(n) m
= 0 =
min S i=0 ∑ (m i − S(t i )) 2 subject u i − u to i−1 < ε, for S(n) S(0) ̇ ̇ i ∈ = = 1..k v v n 0 + n 4 where m i is the momentum of the input motion at time step i, and v i = gM, where g is the gravitational constant in the adjacent flight stage and M is the body mass of the character. In other words, we perform a least-squares regression over the momentum curve in the ground stage, while maintaining C 1 continuity through the transitions to the flight stages. There are few exceptions to the problem described above. When there is no adjacent flight stage, we remove the constraint corresponding to v i from the statement of the problem. Also, the constraint corresponding to v 0 is entirely removed when pre-fitting the vertical linear momentum curve since the transition from a free-fall stage to a ground stage is typically dominated by impulsive forces, which are not C 1 continuous in the vertical momentum component.
<H2>4.2. Motion editing and optimization</H2>
In this section we discuss the process of editing motions using our system. As in [LP02]
we model motion as an optimal dynamic process with a set of realistic constraints. In general terms, our condition for optimality is that the output motion be both as smooth, and as similar, to the original motion as possible. Constraints on the solution ensure that the character’s limb do not bend unnaturally, that the character’s feet do not pass through the ground, and that the character’s full-body momentum curve follows the path of the pre-fit momentum splines. The degrees of freedom to be optimized are contained in Q G, where Q is the set of joint angles through time describing the motion and G is the set of the control points controlling the momentum splines. In the initial state of the optimization, Q is a good initial guess at the target motion formed by linearly interpolating the original motion between user specified translations and orientations, and G contains the pre-fit momentum coefficients. In addition to the constraints and objectives used in
[LP02]
, we also introduce a similarity objective and a pseudo balance objective as described in the following sections.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
20 8 19 20 8 19 8 24 40 56 72 88 8 24 40 56 72 88
Figure 2: Linear momentum of a jumping motion in vertical direction. The gray area indicates the flight stage. Left: The control points {c i |i ∈ 1..k}, visible as red circles, determine the magnitude of the curve. The spacing of the knots {u i |i ∈ 1..k + 4}, visible as blue triangles, influence the shape. Pre-fitting phase solves for a set of control points and knots that closely match the momentum pattern of the input motion (shown as green squares). Right: During the spacetime optimization, u i is held fixed while c i is part of free variables. In this example, the optimized control points c i result in a more energetic jumping motion.
<H3>4.2.1. Similarity objective</H3>
The similarity objective is intended to keep the optimized motion as similar to the original as possible. We formulate this objective as the squared distance between the original vector of DOFs, Q init , and the solution vector, Q. Each joint DOF is scaled by its natural bound. The energy function we wish to minimize is then, E s (Q) = (Q init − Q) 2
<H3>4.2.2. Pseudo balance objective</H3>
Since we do not model the specific human preference to stay out of extreme leaning movements that in real life can often cause foot slipping on the ground, there are some instances when the resulting motion would leave the character unnaturally leaning without a means of support. To pull the optimized solution away from these unstable regions, we include a pseudo balance objective. The objective we use attempts to minimize the squared distance between the COM, C(t) of model in the first time-step, t 0 , and last timestep, t f , of the initial and final ground stages of the motion. For interior ground stages, we instead minimize the distance between the COM of the model in the middle frame of the stage, C(t m ), and the COM of the linearly interpolated input motion, C orig (t m ), in the same frame. In other words, we introduce an additional objective function term, E b (Q) = (C(t 0 ) − )) 2 , for the initial and final ground
C(t
f
stage, and E b (Q) = (C orig (t m ) − C(t m )) 2 for each interior ground stages. We find that the correct weight of these objectives do not vary much from motion to motion and, in fact, as long as the weight is well scaled w.r.t. other parts of the objective function, one value tends to “fit all”.
<H3>4.2.3. Spacetime optimization</H3>
To summarize, the unknowns of our system, Q and G, are the character DOFs and the control points for the momentum splines. Note that spline knots are omitted to maintain the similar momentum pattern of the original motion. The optimization enforces two types of constraints: environment constraints, K e , such as feet positions on the ground, and momentum constraints, K m . The following spacetime formulation finds the unknowns Q and G that minimize the objective function while satisfying all the constraints: min Q,G E s (Q) + E b (Q) subject to K K e m (Q) (Q, G) = = 0 0
<H3>4.2.4. User interface</H3>
Our system provides several high level motion specification tools so that the animator never has to think of editing in terms of constrained optimization. First, motions are automatically partitioned into alternating flight and ground stages. Alternatively, the user can manually adjust the partitioning to make corrections. Next, the user manipulates ground stages with the mouse to translate their position and turns a dial to change the orientations as desired. The system treats these specifications as offsets from the original state of a ground stage. In other words, given the original translation, q T , and original orientation, θ, of the ground stage, the user specifies offsets ∆q T and ∆θ. The new translation and rotation of the ground stage is then altered to be q T + ∆q T and θ + ∆θ, respectively. To form a good initial guess at the solution for the frames of the flight stages, the system linearly interpolates the offsets of the adjacent ground stages over each time step of the flight stage. The resulting motion is a crude approximation of the final result, but provides a good initial state for the spacetime optimization. The animator can also change the height of the trajectory in a flight stage by interactively shaping a visualization of the trajectory. This is particularly useful when creating non-realistic motion that defies gravity, as will be explained below. Once the user is satisfied with the edits, the optimization process takes between 1 to 5 minutes per motion. Alternatively, several motions can be generated together in a batch mode.
<H2>4.3. Populating the dynamic space of motions</H2>
In this section we describe a technique for generating a continuous ranges of physically plausible motions from a single motion capture sequence. The technique constructs an output motion in real-time by performing a simple weighted average over the DOFs values from a set of sample motions. A family of motions can be populated from the input motion by systematically varying the position and orientation of one or more ground stages and then performing a sequence of similar optimization.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
Figure 3: Left: Line motion family that varies the translation of a ground stage along a line. Middle: Grid motion family that varies the translation of the ground stage along a 2 dimensional grid. Right: Circle motion family that varies both the translation and orientation of the ground stage along a semi-circle such that the orientation of the character is always aligned with the normal vector on the arc.
<H3>4.3.1. Motion families</H3>
We provide a user interface for the three most useful types of motion families(see Figure 3 ). The first type varies the translation of a ground stage along a line, the second type varies the translation of the ground stage along a 2 dimensional grid, and the third type varies both the translation and orientation of the ground stage along a semi-circle such that the orientation of the character is consistently aligned along the normal vector of the arc. The size of the sample space as well as the density at which it is sampled can both be adjusted as necessary. Other types of motion families can be easily added. Once a motion family is populated, we are able to generate arbitrary intermediary motions by blending the nearest 2 n samples, where n is the number of dimensions in the parameterized space. We chose to use a simple linear blending method for several reasons. First and foremost, the algorithm is very fast and well suited to any application where the output motion must be generated “on the fly”. Since motion families are produced offline, they can be as densely populated as necessary to increase the accuracy of the interpolation. Second, since the members of a motion family are produced by the same optimization setup, varying only in specific dimensions (e.g. landing positions, height, orientation, etc), it is often the case that they blend very well and need not be sampled very densely at all. In our results section, 9 samples is the most we ever required to adequately sample the dynamic space of a motion.
<H3>4.3.2. Foot glide</H3>
Although foot glide is among the most troublesome artifacts for most motion blending techniques, we find that it is imperceptible for both the line and grid motion families. However, when the global orientation and the translation of the motion are interpolated simultaneously, as is the case in the circle motion family, a very miniscule amount of foot glide becomes perceptible. A simple fix is to apply a per-frame inverse kinematic (IK) solver to slightly adjust the lower body to satisfy the positional constraints on each foot. Solving IK on the lower body not only has the effect of planting the foot firmly on the ground without changing the overall looks of the motion, but is also light-weight enough to converge in real-time, as the motion is being displayed.
<H3>4.3.3. Inverse control</H3>
So far we have shown how to populate the space of dynamic motion by interpolating between samples. Here we will discuss a more intuitive way of controlling these animations. In many applications the most important aspect to control is the position and time at which the character makes contact with an object in the environment. Consider the example of a soccer header motion, where it is required that the character’s head always makes contact with the soccer ball at the correct moment in time. Starting from a single input motion we can generate an arbitrary header by creating a grid motion family that varies the translation of the landing stage. The joint configuration at each time-step in the output motion is then defined as a vector function q(x, y,t) of the landing position, (x, y), and the time-step, t. If we denote the position of the character’s head by the function h(q), the problem of finding the motion that constrains the characters head to ball position p c at time t c , is reduced to that of finding values (x, y) such that p c = h(q(x, y,t c )). This is, in turn, analogous to minimizing the energy function E(x, y) = (p c − h(q(x, y,t c ))) 2 , which can be solved efficiently by a simple gradient descent method. The gradients are computed using finite differences. One caveat is that q is actually a piecewise function that performs a bi-linear interpolation of the 4 nearest sample motions. When one sample motion is replaced by another in the set of 4, q ceases to be C 1 continuous, causing convergence problems with the gradient descent method. A simple solution is to replace the linear blending functions f (x) = x and g(x) = (x − 1) with smooth in/out functions such as f (x) = sin 2 (x) and g(x) = cos 2 (x), thereby maintaining C 1 continuity through the transitions.
<H2>4.4. Interactive control</H2>
One advantage of our motion generation algorithm is that it provides for a wide range of physically plausible animations in real-time. To demonstrate the full benefit of this approach, we have created a video game interface where the user controls the trajectory of a jumping character with a multi-directional control pad (see Figure 6 ). We start with a motion capture sequence of a character making two consecutive jumps. The interesting aspect of this motion is that the character must exhibit foresight in the motion of the first jump, so that the correct contact forces can be generate in the intermediate ground stage, to create the necessary momentum for the second jump. The spacetime approach is
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
p c p c = h(q(x,y,t c )) (x, y) (x, y)
Figure 4: In the example of the soccer header motion, the user specifies the contact point of the head and the soccer ball p c at timestep t c . Inverse control mechanism is able to immediately determine the four nearest neighbors among the
sampled motions as well as their weights that interpolate the desired motion. An efficient gradient descent method solves for the landing position (x, y) by optimizing E(x, y) = (p c − y,t c ))) 2 .
h(q(x,
ideal for editing such a motion because of the way it intrinsically models the interdependencies between different stages of a motion. Our approach inherits the same key benefit from spacetime, but allow us generate motions in realtime. In this demonstration we wish to control the horizontal translation vectors of the first and second jumps, d 1 and d 2 , respectively. First we generate a motion family by varying both the first and last ground stages along a 3x3 grid. The entire motion family then consists of 81 optimal motions resulting from permuting the 9 possible starting positions with 9 possible ending positions. This is necessary in order to sample the entire range of possible ground stage transitions between the two jumps. We are then able to populate the space between sampled motions by linearly interpolating the nearest neighbor optimal solutions. In this case, we have 4 dimensions in our sample space corresponding to the values of d 1 and d 2 , making for a total of 2 4 (or 16) nearest neighbor motions. Therefore, we can express the output motion as vector function q(d 1 , d 2 ), whenever d 1 and d 2 are within the bounds of the sample space. To make our demonstration even more interesting, we chain our jumping motion end to end, such that it continuously loops upon itself. This is done by blending the second flight stage of the first motion, q a (d a1 , d a2 ), into the first flight stage of the second motion,q b (d b1 , d b2 ). In order to make the blending work, we simply require that d a2 = d b1 . In order words, we require the length and direction of the blended jumps be the same. The end result is an interactive jumping simulation where the user controls the direction that the character jumps and then sees the motion carried out in a physically plausible manner. Due to the foresight discussed earlier, the character must always have prior knowledge of the next two directions it will jump. This causes some lag time between when the user specifies a direction and when that motion will occur, but this is only natural given the deterministic nature of the ballistic motion.
Motion
Sequences
Frames
Time
Forward jumps
1
46
2 min
Two-step hop
1
49
3.5 min
360 degree spin
1
79
3.5 min
Volleyball slam
9
44
17 min
Interactive controller
81
56
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Motion Sequences Frames Time Forward jumps 1 46 2 min Two-step hop 1 49 3.5 min 360 degree spin 1 79 3.5 min Volleyball slam 9 44 17 min Interactive controller 81 56 4.5 h
Table 1: Computation time for optimizations
<H1>5. Results</H1>
The motion sequences in our demonstration were captured at 120 frames per second using an optical motion capture system. The character is composed of 18 rigid links and 43 degrees of freedom. S0(3) rotations are expressed in exponential map representation. The mass distribution of the model is an appropriately scaled version of the population average as obtained from
[dL96]. We used SNOPT
[GSM96]
, a nonlinearly-constrained optimization package, for solving spacetime optimization, as well as for pre-fitting the momentum curves. Most edits shown in the accompanying video clips were done in less than 1 minute. The optimization process for each motion took on the order of 2 to 4 minutes to fully converge on a 2Ghz Pentium 4 machine (see Table 1 ).
<H2>5.1. Motion editing</H2>
Our system provides a set of UI tools to help the user rapidly specify modifications to existing motions. In a hopping example, the animator interactively manipulates the position, height, and orientation of each ground stage. The character must cover a longer distance, reach a greater height and assume a new orientation in the modified hopping motion, so she must lower her center of mass, lean farther to the right, and pivot slightly in preparation for the take-off. Despite these changes, the resultant motion remains stylistically similar to the original. To show that our system is capable of making drastic changes from the original motion, we edited the same hopping motion to exhibit a 360 ◦ spin followed by a 180 ◦ spin in the opposite direction(see Figure 5 ).
<H2>5.2. Real-time interpolation</H2>
In order to demonstrate real-time motion interpolation we modified a motion with two consecutive leaps. We let the user control the landing and take-off positions along an evenly spaced grid to generate a set of parameterized motions. Since the interpolation can be performed in real-time, we are able to generate a jumping motion with arbitrary takeoff and landing positions within the parameterized space in an interactive fashion. Another example shows a soccer header motion observed to miss its target. First, we correct the motion by increasing the height of the jump to meet the ball at the point of contact. Next, we use our editing algorithm to generate a motion family parameterized over the space of the landing position of the motion. By interpolating between the optimal motions, we are able to generate arbitrary intermediary motions where the character contacts the ball at any location within the sampled space, in real-time.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
<H2>5.3. Inverse control</H2>
A more intuitive way to edit motion capture data with arbitrary positional constraints is to use our real-time inverse control mechanism. In the volleyball slam example, the user interactively specifies the position of the character’s hand in mid-flight. Our system immediately determined the correct linear interpolation of 4 nearest neighbor samples to meet the positional constraint on the hand. The brightness of the sample motions on the floor indicates the weights associated with each sample. We used 9 sampled motions which are all edits of the same input sequence. The demonstration shows various slam motions being generated in real-time by using the trajectory of the volleyball to guide the character’s motion. (see Figure 6 ).
<H2>5.4. Non-realistic motion</H2>
Our system can also be used to create a class of nonrealistic motions that allow the character to exhibit superhuman strength and to defy the laws of physics. Consider an example where we wish to edit a jumping motion to reach a higher mid-point in the same time span as the the original motion. The first observation to make is that this is physically impossible without altering the gravitational constant, which dictates the maximum rate at which the character returns to the ground from the height of the jump. In our system it is easy to alter the gravitational constant in one or more ground stages. Still, the character must gain the momentum required to achieve the specified height on takeoff and, subsequently, absorb the same amount of momentum on landing. This requires a super-human muscle strength, but since we do not directly model muscle forces, and we place no limits on their magnitude, our system can easily handle these imaginary circumstances. From the animators perspective, editing non-realistic motion is the same as editing any other motion. To increase the height of a flight stage, the animator simply manipulates a visualization of the trajectory of the motion in the flight stage to the required height, and then specifies whether the system should change gravity or, alternatively, the total time in the flight stage. If the animator chooses to leave the gravity unaltered, the system increases the length of the time-step in each frame of the flight stage and then continues the editing process as normal. In one example, we edited a forward jump into a 2-meter-long backward jump (see Figure 7 ).
<H1>6. Conclusion</H1>
This work builds on the research in both physics-based motion synthesis and interpolation-based motion editing approaches. In this paper we suggest that using physics-based adaptation to create motion samples for the purpose of data interpolation is perhaps a "sweetspot" between these two approaches. Once the dataset is created, this paradigm allows animators to interactively edit the realistic dynamic motion. The primary contribution of this work is a new momentum-based method for adaptation of ballistic character movement. In contrast to previous dynamic-based adaptation methods, our framework can produce an wide range of motions that are significantly different from the original motion. Our method does not require model reduction, or a reduced motion space. Because we do not solve for the generalized forces for each joint angle, our method is also significantly faster than other physics-based transformation methods. This speed allows us to create a large number of motions within a reasonable time. Once the family of parameterized motion samples has been generated, we describe an interactive framework where the animator can explore the space of realistic motions. We also show how the same framework can be adapted for inverse control. Finally, we show how real-time data-driven controllers for realistic human motion can be constructed from a single motion capture sequence. Naturally, our framework does not handle all realistic character motions. It specifically applies to highly-dynamic motions with ballistic stages. We suspect that momentumbased approach would not be well suited for less energetic motions such as walking. Furthermore, the number of samples required is exponentially proportional to the number of dimensions, thus the current framework is hindered by the offline computation of a large dataset. There are several ways to facilitate the computation by taking advantage of the fact that we are solving a sequence of very similar problems. A more intelligent sampling strategy is essential for generalizing our approach to a multi-dimensional dynamic space. Because our model does not account for realistic muscle strength, and friction during ground contact there are some extreme cases which do not produce realistic motion. Adding heuristics such as balance during contact can to a large extent eliminate these problems.
<H1>7. Acknowledgments</H1>
Special thanks go to Mira Doncheva for her assistance with creating videos. We also thank Keith Grochow for his help with processing motion capture data. This work was supported by the UW Animation Research Labs, NSF grants CCR-0092970, ITR grants IIS-0113007, EIA-0121326, NSF REU grant, Alfred P. Sloan Fellowship, Electronic Arts Corporation, Sony and Microsoft Research.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
Figure 5: A forward hopping motion (shown in yellow) is modified to make a 360 degree spin in the clockwise direction followed by a 180 degree spin in the opposite direction (shown in blue).
Figure 6: Left: For a volleyball slam motion, the user interactively specifies the position of the character’s hand in mid-flight. The system then determines the correct linear interpolation of the sampled motions to meet the positional constraint on the hand. Middle: The volleyball motion in profile. Right: The user interactively controls the direction the character jumps with a multi-directional control pad.
Figure 7: The timeline of this animation goes from left to right. To demonstrate a motion that is impossible to achieve in the real world, the animator altered a forward jump to a 2-meter-long backward jump.
c The Eurographics Association 2004.
Yeuhi Abe, C. Karen Liu, & Zoran Popović / Momentum-based Parameterization of Dynamic Character Motion
<H1>References</H1>
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c The Eurographics Association 2004.
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"text": "['Extending physicsbased optimization to a full human figure']",
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"text": "['the nonlinearity of the dynamic constraints']",
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"text": "['sensitivity to the starting point of the optimization']",
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"text": "['The dependency on the initial point has been somewhat alleviated by starting out with the captured motion sequence']",
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"text": "['a first method that transforms motion capture data while preserving physical properties']",
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7539,
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"text": "['PW99']",
"type": "data",
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"text": "['They found solutions by performing optimizations on the reduced character model']",
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"text": "['More recently, editing motion capture data based on spacetime optimization has become a popular strategy for producing realistic character animations']",
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"text": "['RGBC96']",
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7786,
7790
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"text": "['SP04']",
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"text": "['SHP04']",
"type": "data",
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"text": "['These methods provide control for modifying data while retaining physically plausible properties of captured motion by restricting the optimization space with additional kinematic constraints']",
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"text": "['or by solving within the PCA-reduced space of motions']",
"type": "background_claim",
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"text": "['RGBC96']",
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8065,
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],
"text": "['SHP04']",
"type": "data",
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"offsets": [
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8105,
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],
"text": "['relying on simplifications of dynamic constraints is not necessary']",
"type": "background_claim",
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"text": "['proper scaling and estimation of joint angles, torques, and Lagrange multipliers are provided']",
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"text": "['SP04']",
"type": "data",
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"text": "['Our work uses a similar spacetime optimization framework']",
"type": "own_claim",
"id": "T82"
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"text": "['In contrast to other approaches, we formulate significantly simpler momentum constraints on a complex character model, without solving for muscle forces explicitly']",
"type": "own_claim",
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8514,
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"text": "['LP02']",
"type": "data",
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8574,
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"text": "['scaling and convergence issues are less critical in our optimization framework']",
"type": "own_claim",
"id": "T85"
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8527,
8572
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"text": "['we do not compute internal torques for joints']",
"type": "data",
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8654,
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"text": "['Our physics-based motion editing approach is based on the momentum constraints']",
"type": "own_claim",
"id": "T87"
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{
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8765,
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],
"text": "['LP02']",
"type": "data",
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8772,
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],
"text": "['In that work, momentum constraints were used for synthesis of highly dynamic motion from simple animations that did not contain sufficient information to synthesize the full motion']",
"type": "background_claim",
"id": "T89"
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{
"offsets": [
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8954,
9047
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],
"text": "['As a result, transition poses had to be introduced to further restrict the optimization space']",
"type": "background_claim",
"id": "T90"
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{
"offsets": [
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9049,
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],
"text": "['There are two main advantages of momentum constraints over the full dynamics constraints']",
"type": "own_claim",
"id": "T91"
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{
"offsets": [
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9218,
9320
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],
"text": "['we are solving for a much smaller set of unknowns, and over a much “better behaved” set of constraints']",
"type": "own_claim",
"id": "T92"
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{
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9146,
9216
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],
"text": "['since dynamic constraints are reduced to only global momentum patterns']",
"type": "data",
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"offsets": [
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9322,
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"text": "['This allows us to find solutions quickly']",
"type": "own_claim",
"id": "T94"
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9450,
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"text": "['enabling us to find solutions significantly further away from the original motion']",
"type": "own_claim",
"id": "T95"
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9389,
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"text": "['these constraints do not suffer from many local minima']",
"type": "data",
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9586,
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"text": "['they encode more about the natural motion than just physical correctness']",
"type": "background_claim",
"id": "T97"
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{
"offsets": [
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9672,
9786
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],
"text": "['in natural motion, passive elements such as tendons and ligaments store and release energy during ballistic motion']",
"type": "data",
"id": "T98"
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"offsets": [
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9788,
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"text": "['To model this with a full dynamic system, one would have to include a complex muscle model']",
"type": "background_claim",
"id": "T99"
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{
"offsets": [
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9880,
10026
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],
"text": "['Momentum constraints effectively record the aggregate effect of the natural torque usage and energy storage/release in a specific momentum pattern']",
"type": "background_claim",
"id": "T100"
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{
"offsets": [
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10028,
10252
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],
"text": "['This additional information embedded within the momentum constraints ensures that adapted motion is not just physically correct, but that it also constrains the motion within the momentum exchange patterns observed in nature']",
"type": "background_claim",
"id": "T101"
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{
"offsets": [
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10326,
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"text": "['our method applies momentum constraints directly on the motion capture data']",
"type": "own_claim",
"id": "T102"
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{
"offsets": [
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10403,
10523
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],
"text": "['Our algorithm does not require any additional pose constraints at the transition points between flight and ground phases']",
"type": "own_claim",
"id": "T103"
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{
"offsets": [
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10538,
10719
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],
"text": "['we introduce a novel spline-based representation for the momentum patterns that can be used to intrinsically enforce the similarity between the resultant motion and the input motion']",
"type": "own_claim",
"id": "T104"
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{
"offsets": [
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10774,
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"text": "['dynamic filtering is an efficient alternative for motion editing of smaller amplitude']",
"type": "background_claim",
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"text": "['providing an interactive editing interface to the user']",
"type": "background_claim",
"id": "T106"
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10861,
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"text": "['Per-frame based frameworks largely reduce the computation time']",
"type": "data",
"id": "T107"
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{
"offsets": [
[
10982,
10987
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"text": "['TSK02']",
"type": "data",
"id": "T108"
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"offsets": [
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10989,
10994
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"text": "['SKG03']",
"type": "data",
"id": "T109"
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"offsets": [
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11012,
11134
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"text": "['the per-frame approach means that animators can modify the spatial position of constraints, but not their position in time']",
"type": "background_claim",
"id": "T110"
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{
"offsets": [
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11147,
11251
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],
"text": "['applied Kalman filter to estimate an optimal pose for the current frame subject to the given constraints']",
"type": "background_claim",
"id": "T111"
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{
"offsets": [
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11136,
11146
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],
"text": "['Tak et al.']",
"type": "data",
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"offsets": [
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11253,
11355
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"text": "['The result of the estimation is then rectified by least-square-fit to ensure a physically sound motion']",
"type": "background_claim",
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{
"offsets": [
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11358,
11363
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],
"text": "['TSK02']",
"type": "data",
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"offsets": [
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11378,
11552
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],
"text": "['approximated the adjustment made to the original motion capture data by correcting the momentum of the character during flight and using the balance constraints on the ground']",
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"id": "T115"
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11555,
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"text": "['SKG03']",
"type": "data",
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"offsets": [
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11575,
11660
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"text": "['these methods are geared toward the local modification compared to the overall motion']",
"type": "background_claim",
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11670,
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"text": "['improving the balance']",
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11701,
11760
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"text": "['our approach is able to handle global changes of the motion']",
"type": "own_claim",
"id": "T119"
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11769,
11826
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],
"text": "['transforming a forward jump to a 360 ◦ backward spin jump']",
"type": "data",
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"offsets": [
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11828,
11888
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],
"text": "['Another branch of dynamic filtering employs dynamic tracking']",
"type": "background_claim",
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"offsets": [
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11891,
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"text": "['ZH99']",
"type": "data",
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"offsets": [
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11897,
11901
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],
"text": "['PR01']",
"type": "data",
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"offsets": [
[
11904,
12061
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"text": "['These methods combine motion capture data and dynamic simulation to retain human-like details from the data while presenting interaction with the environment']",
"type": "background_claim",
"id": "T124"
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{
"offsets": [
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12063,
12148
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"text": "['These methods produce motions that do not deviate significantly from the input motion']",
"type": "background_claim",
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"offsets": [
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12150,
12240
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"text": "['relying on the existence of captured motion that is similar to what the user intends to do']",
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"id": "T126"
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{
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12481,
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],
"text": "['Straightforward interpolation of joint angles usually fails to preserve physical realism from the original data']",
"type": "background_claim",
"id": "T127"
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"offsets": [
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12603,
12723
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"text": "['many methods have shown that small modification of the motion can be easily done by linear interpolation of joint angles']",
"type": "background_claim",
"id": "T128"
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{
"offsets": [
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12726,
12730
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],
"text": "['BW95']",
"type": "data",
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12732,
12736
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],
"text": "['WP95']",
"type": "data",
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12738,
12742
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"text": "['WH97']",
"type": "data",
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"offsets": [
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12798,
12893
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"text": "['Gleicher adapted original motion to a new character while maintaining environmental constraints']",
"type": "background_claim",
"id": "T132"
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"offsets": [
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12902,
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"text": "['foot contacts on the floor']",
"type": "data",
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"offsets": [
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12745,
12796
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],
"text": "['Combining interpolation with kinematics constraints']",
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"offsets": [
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12931,
12936
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],
"text": "['Gle98']",
"type": "data",
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"offsets": [
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12940,
13080
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"text": "['A more sophisticated interpolation was presented using radial basis functions to blend motion sequences with various inverse-kinematic goals']",
"type": "background_claim",
"id": "T136"
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{
"offsets": [
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13090,
13108
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"text": "['or different style']",
"type": "background_claim",
"id": "T137"
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"offsets": [
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13083,
13088
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],
"text": "['RSC01']",
"type": "data",
"id": "T138"
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"offsets": [
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13111,
13116
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],
"text": "['RCB98']",
"type": "data",
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"offsets": [
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13134,
13220
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"text": "['data acquisition and post-processing for these methods present a significant challenge']",
"type": "background_claim",
"id": "T140"
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{
"offsets": [
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13227,
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],
"text": "['motion sequences need to be carefully crafted so that they contain the same content yet different in style']",
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"offsets": [
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13335,
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"text": "['Our approach only requires one single motion capture sequence as the seed']",
"type": "own_claim",
"id": "T142"
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{
"offsets": [
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13410,
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],
"text": "['This seed is used to generate a family of motion sequences that parameterize the dynamic space']",
"type": "own_claim",
"id": "T143"
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{
"offsets": [
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13529,
13676
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],
"text": "['a multi-level B-spline representation by which they transform existing motion to satisfy desired constraints adaptively through direct manipulation']",
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{
"offsets": [
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13679,
13683
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"text": "['LS99']",
"type": "data",
"id": "T145"
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"offsets": [
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13686,
13782
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"text": "['Using B-spline representation, the motion edits can be limited to user-specified frequency bands']",
"type": "background_claim",
"id": "T146"
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{
"offsets": [
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13784,
13833
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"text": "['providing a more effective optimization framework']",
"type": "background_claim",
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"offsets": [
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13835,
13940
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"text": "['Our work adapts the idea of using spline-based representation to constrain the search of the optimization']",
"type": "own_claim",
"id": "T148"
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{
"offsets": [
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13942,
14166
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],
"text": "['We model the momentum curves by a B-spline representation which are fitted to the original motion so that the search space in the optimization is limited to solutions that have similar dynamic behavior of the original motion']",
"type": "own_claim",
"id": "T150"
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{
"offsets": [
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14505,
14665
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],
"text": "['Our system is based on an optimization algorithm that can transform the captured motion to satisfy high-level user constraints while preserving physical realism']",
"type": "own_claim",
"id": "T151"
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{
"offsets": [
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14667,
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],
"text": "['As input, the system takes a single motion capture sequence and the userspecified modification']",
"type": "own_claim",
"id": "T152"
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{
"offsets": [
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14888,
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],
"text": "['The pre-fitting optimizes a set of coefficients used to model momentum curves so that they are constrained to the similar shapes of the original motion']",
"type": "own_claim",
"id": "T153"
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"offsets": [
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15041,
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"text": "['The system then formulates a spacetime optimization that solves for a new motion, where both high-level physical constraints and the user specification are met']",
"type": "own_claim",
"id": "T149"
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{
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15202,
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"text": "['With a family of such optimized motions that parameterize certain dynamic space, we can apply a simple linear interpolation to generate arbitrary new motion within the dynamic space in real-time']",
"type": "own_claim",
"id": "T154"
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{
"offsets": [
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15542,
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"text": "['for the task of motion editing']",
"type": "own_claim",
"id": "T155"
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{
"offsets": [
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15481,
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"text": "['Our algorithm adapts the momentum-based constraints']",
"type": "own_claim",
"id": "T156"
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{
"offsets": [
[
15535,
15539
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],
"text": "['LP02']",
"type": "data",
"id": "T157"
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{
"offsets": [
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15574,
15740
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],
"text": "['Instead of filling in missing data, motion editing must solve the converse problem of preserving the original data while still satisfying animator-imposed constraints']",
"type": "own_claim",
"id": "T158"
},
{
"offsets": [
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15742,
15785
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],
"text": "['There is no need for keyframing of any kind']",
"type": "own_claim",
"id": "T159"
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{
"offsets": [
[
15794,
15843
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],
"text": "['the motion already starts in a good initial state']",
"type": "data",
"id": "T160"
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"offsets": [
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15845,
16089
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"text": "['Any underlying physical model employed by the system must be flexible enough to precisely describe the initial state of the motion and, at the same time, rigid enough to maintain a semblance of the original motion throughout the editing process']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
16314,
16400
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],
"text": "['At the heart of our algorithm is a set of full-body angular and linear momentum curves']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
16402,
16535
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],
"text": "['These curves constrain the edited motion to the realm of physical realism without the need to simulate expensive dynamical properties']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
16544,
16557
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],
"text": "['joint torques']",
"type": "data",
"id": "T164"
},
{
"offsets": [
[
16562,
16576
]
],
"text": "['contact forces']",
"type": "data",
"id": "T165"
},
{
"offsets": [
[
16696,
16737
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],
"text": "['The advantage of this approach is twofold']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
16746,
16844
]
],
"text": "['a good initial state of the momentum coefficients results in rapid convergence of the optimization']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
16854,
16951
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],
"text": "['the coefficients that control the shape of the curves can be fixed throughout the editing process']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
16953,
17033
]
],
"text": "['effectively performing a biased search for similar motions in the momentum space']",
"type": "own_claim",
"id": "T170"
},
{
"offsets": [
[
17041,
17143
]
],
"text": "['the motion is captured using an optical system and processed to fit the character’s skeletal structure']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
17145,
17199
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],
"text": "['we employ the constraint detection technique described']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
17212,
17273
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],
"text": "['to partition the motion into ground-contact and flight stages']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
17205,
17209
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],
"text": "['LP02']",
"type": "data",
"id": "T173"
},
{
"offsets": [
[
17473,
17545
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],
"text": "['we also need to determine the time interval between two animation frames']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
17285,
17402
]
],
"text": "['the animator may at times wish to produce physically impossible jumps that are not constrained to the earth’s gravity']",
"type": "data",
"id": "T175"
},
{
"offsets": [
[
17416,
17471
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],
"text": "['the sampling rate varies for each input motion sequence']",
"type": "data",
"id": "T176"
},
{
"offsets": [
[
17547,
17589
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],
"text": "['Gravity and time step are directly related']",
"type": "own_claim",
"id": "T177"
},
{
"offsets": [
[
17598,
17724
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],
"text": "['we can equivalently choose to find the right gravitational constant that makes the motion realistic for a given unit time step']",
"type": "data",
"id": "T178"
},
{
"offsets": [
[
17803,
17840
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],
"text": "['the angular momentum remains constant']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
17726,
17798
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],
"text": "['During free-fall stages, the linear momentum is only affected by gravity']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
18680,
18764
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],
"text": "['A defining characteristic of motion is the shape and magnitude of its momentum curve']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
18770,
18778
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],
"text": "['Figure 2']",
"type": "data",
"id": "T183"
},
{
"offsets": [
[
18943,
19058
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],
"text": "['this formulation can capture a greater variability of momentum patterns than the previously used hardwired patterns']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
19061,
19065
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],
"text": "['LP02']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
19068,
19134
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],
"text": "['This is especially important when dealing with motion capture data']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
19142,
19202
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],
"text": "['wide range of different maneuvers possible in the real world']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
20028,
20083
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],
"text": "['There are few exceptions to the problem described above']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
20567,
20641
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],
"text": "['In this section we discuss the process of editing motions using our system']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
20657,
20738
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],
"text": "['we model motion as an optimal dynamic process with a set of realistic constraints']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
20650,
20654
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],
"text": "['LP02']",
"type": "data",
"id": "T190"
},
{
"offsets": [
[
20758,
20882
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],
"text": "['our condition for optimality is that the output motion be both as smooth, and as similar, to the original motion as possible']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
20884,
21126
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],
"text": "['Constraints on the solution ensure that the character’s limb do not bend unnaturally, that the character’s feet do not pass through the ground, and that the character’s full-body momentum curve follows the path of the pre-fit momentum splines']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
21578,
21628
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],
"text": "['In addition to the constraints and objectives used']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
21642,
21713
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],
"text": "['we also introduce a similarity objective and a pseudo balance objective']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
21734,
21752
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],
"text": "['following sections']",
"type": "data",
"id": "T195"
},
{
"offsets": [
[
21634,
21638
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],
"text": "['LP02']",
"type": "data",
"id": "T196"
},
{
"offsets": [
[
22715,
22819
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],
"text": "['The similarity objective is intended to keep the optimized motion as similar to the original as possible']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
[
23295,
23418
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],
"text": "['there are some instances when the resulting motion would leave the character unnaturally leaning without a means of support']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
23145,
23293
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],
"text": "['we do not model the specific human preference to stay out of extreme leaning movements that in real life can often cause foot slipping on the ground']",
"type": "data",
"id": "T199"
},
{
"offsets": [
[
23420,
23522
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],
"text": "['To pull the optimized solution away from these unstable regions, we include a pseudo balance objective']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
24204,
24294
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],
"text": "['We find that the correct weight of these objectives do not vary much from motion to motion']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
24353,
24390
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],
"text": "['other parts of the objective function']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
24392,
24420
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],
"text": "['one value tends to “fit all”']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
24309,
24351
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],
"text": "['as long as the weight is well scaled w.r.t']",
"type": "data",
"id": "T204"
},
{
"offsets": [
[
25130,
25195
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],
"text": "['Our system provides several high level motion specification tools']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
25204,
25283
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],
"text": "['the animator never has to think of editing in terms of constrained optimization']",
"type": "own_claim",
"id": "T206"
},
{
"offsets": [
[
24714,
24764
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],
"text": "['The optimization enforces two types of constraints']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
24766,
24794
]
],
"text": "['environment constraints, K e']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
[
24839,
24864
]
],
"text": "['momentum constraints, K m']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
24805,
24833
]
],
"text": "['feet positions on the ground']",
"type": "data",
"id": "T210"
},
{
"offsets": [
[
24867,
25003
]
],
"text": "['The following spacetime formulation finds the unknowns Q and G that minimize the objective function while satisfying all the constraints']",
"type": "own_claim",
"id": "T211"
},
{
"offsets": [
[
25005,
25068
]
],
"text": "['min Q,G E s (Q) + E b (Q) subject to K K e m (Q) (Q, G) = = 0 0']",
"type": "data",
"id": "T212"
},
{
"offsets": [
[
26411,
26493
]
],
"text": "['This is particularly useful when creating non-realistic motion that defies gravity']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
26632,
26704
]
],
"text": "['Alternatively, several motions can be generated together in a batch mode']",
"type": "own_claim",
"id": "T214"
},
{
"offsets": [
[
26523,
26630
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],
"text": "['Once the user is satisfied with the edits, the optimization process takes between 1 to 5 minutes per motion']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
26213,
26273
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],
"text": "['provides a good initial state for the spacetime optimization']",
"type": "own_claim",
"id": "T217"
},
{
"offsets": [
[
26142,
26207
]
],
"text": "['The resulting motion is a crude approximation of the final result']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
26806,
26934
]
],
"text": "['we describe a technique for generating a continuous ranges of physically plausible motions from a single motion capture sequence']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
26936,
27080
]
],
"text": "['The technique constructs an output motion in real-time by performing a simple weighted average over the DOFs values from a set of sample motions']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
27082,
27279
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],
"text": "['A family of motions can be populated from the input motion by systematically varying the position and orientation of one or more ground stages and then performing a sequence of similar optimization']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
27942,
28020
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],
"text": "['We provide a user interface for the three most useful types of motion families']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
28025,
28033
]
],
"text": "['Figure 3']",
"type": "data",
"id": "T222"
},
{
"offsets": [
[
28510,
28560
]
],
"text": "['Other types of motion families can be easily added']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
28842,
28958
]
],
"text": "['the algorithm is very fast and well suited to any application where the output motion must be generated “on the fly”']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
29004,
29095
]
],
"text": "['they can be as densely populated as necessary to increase the accuracy of the interpolation']",
"type": "own_claim",
"id": "T225"
},
{
"offsets": [
[
28966,
29002
]
],
"text": "['motion families are produced offline']",
"type": "data",
"id": "T226"
},
{
"offsets": [
[
29301,
29365
]
],
"text": "['they blend very well and need not be sampled very densely at all']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
29105,
29222
]
],
"text": "['since the members of a motion family are produced by the same optimization setup, varying only in specific dimensions']",
"type": "data",
"id": "T228"
},
{
"offsets": [
[
29391,
29480
]
],
"text": "['9 samples is the most we ever required to adequately sample the dynamic space of a motion']",
"type": "data",
"id": "T229"
},
{
"offsets": [
[
28753,
28820
]
],
"text": "['We chose to use a simple linear blending method for several reasons']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
28400,
28508
]
],
"text": "['The size of the sample space as well as the density at which it is sampled can both be adjusted as necessary']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
28037,
28105
]
],
"text": "['The first type varies the translation of a ground stage along a line']",
"type": "data",
"id": "T232"
},
{
"offsets": [
[
28107,
28192
]
],
"text": "['the second type varies the translation of the ground stage along a 2 dimensional grid']",
"type": "data",
"id": "T233"
},
{
"offsets": [
[
28198,
28398
]
],
"text": "['the third type varies both the translation and orientation of the ground stage along a semi-circle such that the orientation of the character is consistently aligned along the normal vector of the arc']",
"type": "data",
"id": "T234"
},
{
"offsets": [
[
29548,
29634
]
],
"text": "['foot glide is among the most troublesome artifacts for most motion blending techniques']",
"type": "background_claim",
"id": "T235"
},
{
"offsets": [
[
29636,
29711
]
],
"text": "['we find that it is imperceptible for both the line and grid motion families']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
29861,
29918
]
],
"text": "['a very miniscule amount of foot glide becomes perceptible']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
29727,
29859
]
],
"text": "['the global orientation and the translation of the motion are interpolated simultaneously, as is the case in the circle motion family']",
"type": "data",
"id": "T238"
},
{
"offsets": [
[
29920,
30071
]
],
"text": "['A simple fix is to apply a per-frame inverse kinematic (IK) solver to slightly adjust the lower body to satisfy the positional constraints on each foot']",
"type": "own_claim",
"id": "T239"
},
{
"offsets": [
[
30073,
30216
]
],
"text": "['Solving IK on the lower body not only has the effect of planting the foot firmly on the ground without changing the overall looks of the motion']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
30222,
30308
]
],
"text": "['is also light-weight enough to converge in real-time, as the motion is being displayed']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
30379,
30469
]
],
"text": "['we have shown how to populate the space of dynamic motion by interpolating between samples']",
"type": "own_claim",
"id": "T242"
},
{
"offsets": [
[
30546,
30699
]
],
"text": "['In many applications the most important aspect to control is the position and time at which the character makes contact with an object in the environment']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
30725,
30867
]
],
"text": "['a soccer header motion, where it is required that the character’s head always makes contact with the soccer ball at the correct moment in time']",
"type": "data",
"id": "T244"
},
{
"offsets": [
[
32241,
32374
]
],
"text": "['One advantage of our motion generation algorithm is that it provides for a wide range of physically plausible animations in real-time']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
32376,
32563
]
],
"text": "['To demonstrate the full benefit of this approach, we have created a video game interface where the user controls the trajectory of a jumping character with a multi-directional control pad']",
"type": "own_claim",
"id": "T246"
},
{
"offsets": [
[
32569,
32577
]
],
"text": "['Figure 6']",
"type": "data",
"id": "T247"
},
{
"offsets": [
[
32666,
32780
]
],
"text": "['The interesting aspect of this motion is that the character must exhibit foresight in the motion of the first jump']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
[
32790,
32919
]
],
"text": "['the correct contact forces can be generate in the intermediate ground stage, to create the necessary momentum for the second jump']",
"type": "own_claim",
"id": "T249"
},
{
"offsets": [
[
32921,
32946
]
],
"text": "['The spacetime approach is']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
33667,
33698
]
],
"text": "['ideal for editing such a motion']",
"type": "own_claim",
"id": "T251"
},
{
"offsets": [
[
33710,
33800
]
],
"text": "['the way it intrinsically models the interdependencies between different stages of a motion']",
"type": "data",
"id": "T252"
},
{
"offsets": [
[
33802,
33859
]
],
"text": "['Our approach inherits the same key benefit from spacetime']",
"type": "own_claim",
"id": "T253"
},
{
"offsets": [
[
33865,
33902
]
],
"text": "['allow us generate motions in realtime']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
37849,
37953
]
],
"text": "['Our system provides a set of UI tools to help the user rapidly specify modifications to existing motions']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
38344,
38410
]
],
"text": "['the resultant motion remains stylistically similar to the original']",
"type": "own_claim",
"id": "T256"
},
{
"offsets": [
[
38425,
38497
]
],
"text": "['our system is capable of making drastic changes from the original motion']",
"type": "own_claim",
"id": "T257"
},
{
"offsets": [
[
38499,
38607
]
],
"text": "['we edited the same hopping motion to exhibit a 360 ◦ spin followed by a 180 ◦ spin in the opposite direction']",
"type": "data",
"id": "T258"
},
{
"offsets": [
[
37955,
38074
]
],
"text": "['In a hopping example, the animator interactively manipulates the position, height, and orientation of each ground stage']",
"type": "data",
"id": "T259"
},
{
"offsets": [
[
38207,
38319
]
],
"text": "['she must lower her center of mass, lean farther to the right, and pivot slightly in preparation for the take-off']",
"type": "own_claim",
"id": "T261"
},
{
"offsets": [
[
38076,
38202
]
],
"text": "['The character must cover a longer distance, reach a greater height and assume a new orientation in the modified hopping motion']",
"type": "data",
"id": "T263"
},
{
"offsets": [
[
38612,
38620
]
],
"text": "['Figure 5']",
"type": "data",
"id": "T260"
},
{
"offsets": [
[
38982,
39124
]
],
"text": "['we are able to generate a jumping motion with arbitrary takeoff and landing positions within the parameterized space in an interactive fashion']",
"type": "own_claim",
"id": "T262"
},
{
"offsets": [
[
38933,
38980
]
],
"text": "['the interpolation can be performed in real-time']",
"type": "data",
"id": "T264"
},
{
"offsets": [
[
39485,
39632
]
],
"text": "['we are able to generate arbitrary intermediary motions where the character contacts the ball at any location within the sampled space, in real-time']",
"type": "own_claim",
"id": "T265"
},
{
"offsets": [
[
39442,
39483
]
],
"text": "['interpolating between the optimal motions']",
"type": "data",
"id": "T266"
},
{
"offsets": [
[
39854,
39990
]
],
"text": "['A more intuitive way to edit motion capture data with arbitrary positional constraints is to use our real-time inverse control mechanism']",
"type": "own_claim",
"id": "T267"
},
{
"offsets": [
[
40659,
40816
]
],
"text": "['Our system can also be used to create a class of nonrealistic motions that allow the character to exhibit superhuman strength and to defy the laws of physics']",
"type": "own_claim",
"id": "T268"
},
{
"offsets": [
[
41174,
41263
]
],
"text": "['In our system it is easy to alter the gravitational constant in one or more ground stages']",
"type": "own_claim",
"id": "T269"
},
{
"offsets": [
[
41565,
41623
]
],
"text": "['our system can easily handle these imaginary circumstances']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
41272,
41364
]
],
"text": "['the character must gain the momentum required to achieve the specified height on takeoff and']",
"type": "own_claim",
"id": "T271"
},
{
"offsets": [
[
41380,
41425
]
],
"text": "['absorb the same amount of momentum on landing']",
"type": "own_claim",
"id": "T272"
},
{
"offsets": [
[
41427,
41470
]
],
"text": "['This requires a super-human muscle strength']",
"type": "own_claim",
"id": "T273"
},
{
"offsets": [
[
41526,
41563
]
],
"text": "['we place no limits on their magnitude']",
"type": "data",
"id": "T274"
},
{
"offsets": [
[
41482,
41520
]
],
"text": "['we do not directly model muscle forces']",
"type": "data",
"id": "T275"
},
{
"offsets": [
[
42064,
42195
]
],
"text": "['the system increases the length of the time-step in each frame of the flight stage and then continues the editing process as normal']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
42011,
42062
]
],
"text": "['the animator chooses to leave the gravity unaltered']",
"type": "data",
"id": "T277"
},
{
"offsets": [
[
41657,
41725
]
],
"text": "['editing non-realistic motion is the same as editing any other motion']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
41630,
41655
]
],
"text": "['the animators perspective']",
"type": "data",
"id": "T279"
},
{
"offsets": [
[
42342,
42463
]
],
"text": "['This work builds on the research in both physics-based motion synthesis and interpolation-based motion editing approaches']",
"type": "own_claim",
"id": "T280"
},
{
"offsets": [
[
42479,
42642
]
],
"text": "['we suggest that using physics-based adaptation to create motion samples for the purpose of data interpolation is perhaps a \"sweetspot\" between these two approaches']",
"type": "own_claim",
"id": "T281"
},
{
"offsets": [
[
42644,
42754
]
],
"text": "['Once the dataset is created, this paradigm allows animators to interactively edit the realistic dynamic motion']",
"type": "own_claim",
"id": "T282"
},
{
"offsets": [
[
42756,
42871
]
],
"text": "['The primary contribution of this work is a new momentum-based method for adaptation of ballistic character movement']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
42931,
43039
]
],
"text": "['our framework can produce an wide range of motions that are significantly different from the original motion']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
43041,
43111
]
],
"text": "['Our method does not require model reduction, or a reduced motion space']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
43186,
43273
]
],
"text": "['our method is also significantly faster than other physics-based transformation methods']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
43121,
43184
]
],
"text": "['we do not solve for the generalized forces for each joint angle']",
"type": "data",
"id": "T287"
},
{
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A34 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A34_C04_Collision_Resolutions_in_Cloth_Simulation_CITATION_PURPOSE_M_v1.xml">
0bd8ef45a16f87a35fe8437e9ed149cb9a7cc62f61dae7f711d590388cf978d7
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http://dx.doi.org/10.1109/ca.2001.982385
<Title>Collision Resolutions in Cloth Simulation</Title>
Suejung Huh
Dimitris N. Metaxas
Norman I. Badler Center of Human Modeling
Simulation Lab
Moore Bldg. University of Pennsylvania Philadelphia
PA
USA (215)
fax (215) 573-7453 f suejung
dnm
badler g
@graphics.cis.upenn.edu
<Abstract>We present a new collision resolution scheme for cloth collisions. Our main concern is to find dynamically convincing resolutions, i.e. positions and velocities of cloth elements, for any kinds of collisions occuring in cloth simulation (cloth-cloth, cloth-rigid, and cloth-cloth-rigid). We define our cloth surface as connected faces of mass particles where each particle is controlled by its internal energy functions. Our collision resolution method finds appropriate next positions and velocities of particles by conserving the particles’ momentums as accurately as possible. Cloth-cloth collision resolution is a special case of deformable N-body collision resolution. So to solve deformable N-body collision resolutions, we propose a new collision resolution method, which groups cloth particles into parts and resolves collisions between parts using the law of momentum conservation. To resolve collisions, we solve a system of linear equations derived from the collision relationships. A system of linear equations is built using a scheme adapted from the simultaneous resolution method for rigid N-body collisions [ 1 ]. For the special case where we can find cyclic relationships in collisions, we solve a system of linear inequalities derived from the collision relationships.</Abstract>
<H1>1. Introduction</H1>
Collision handling in Computer Graphics has two phases. One is to detect collisions and the other is to resolve collisions. In cloth collision detection, the computation time to detect collisions is not negligible because the number of geometrical entities (nodes, faces, edges) the collision detection algorithm has to handle is considerable (over 10,000 particles for regular attire). For this reason, several approaches have tried to expedite the collision detection processes [ 16 , 3 ]. Collision resolution is to find the correct next positions and velocities of colliding objects. Cloth resolution methods so far have found non-penetrating positions, velocities and accelerations of cloth surface particles
[ 14 , 3 , 15 ]. This scheme works fine for cloth-rigid collisions and for the special case of cloth-cloth collisions where the dynamic interactions between cloth surfaces in cloth-cloth collisions do not have to be noticeable. Volino et. al. [ 15 ]
applied the conjugate gradient method to find the actual particles’ positions where a group of particles are colliding into each other. By preserving barycentric relationships of collision entities, their method resolves collisions where numerous cloth surfaces are colliding together as a group, which is a novel way to resolve multiple collisions at once. However it does not conserve the momentum of cloth surfaces in cloth-cloth collisions. Another method for cloth-cloth collisions has been proposed by Provot
[ 12 ]
, which resolves collisions by giving an average velocity to all the particles of collisions. Provot’s method is easy to implement but it cannot give proper visual effect of collisions since we cannot get dynamic interactions between particles once the particles collide into each other. Cloth-cloth collision resolution is a special case of deformable N-body collision resolution. To solve deformable N-body collision resolutions, we propose a new collision resolution method which gives a visually reasonable response by ensuring the conservation of N-body momentums. Our cloth system is particle-based, as many systems are in other cloth research groups [ 14 , 6 , 12 , 3 ]. To resolve collisions, we first divide the colliding particles into parts and build a system of linear equations based on the collision relationships between these parts. Then we solve the whole system using the law of N-body momentum conservation. The system of equalities is based on the scheme adapted from the simultaneous resolution method for rigid N-body collisions proposed by Baraff
[ 1 ]. However his original inequality relationships between relative velocities before and after collisions are purely heuristic, which may not be physically correct. Though this physical inaccuracy has been an inherent problem of simultaneous collision resolutions, it appears to give graphically agreeable results. Hence with the help of the law of N-body momentum conservation, we found the results of our resolutions are visually acceptable. In the next section, we discuss the issues involved in collision detection, especially to define a way to store collision information. In Sec. 3, we propose our collision resolution scheme. In Sec. 3.1 and 3.2, we examine some special cases of collisions which need some prior processes before we actually resolve collisions. From Sec. 3.3 to Sec. 3.5, the actual collision resolution method is presented. Finally Sec. 4 discusses results.
<H1>2 Collision Detection</H1>
<H2>2.1 Collision detection using swept-volumes</H2>
A swept volume is a volume made by two sets of positional entities of a face one at time t and one at time t + t . Connecting these old and new positions of all particles in a face gives us a volume. Any collision happening within an integration time step always can be detected by this swept volume method, unless the motions of faces are highly rotational. An interesting case is where the faces are not actually intersecting but two swept volumes report a intersection anyway. Though this case is not an actual collision, it happens only when two faces are very close. Hence we resort to the collision report of this case, since we consider this case as a violation of the proximity law. We use classical edge-polygon detection algorithms to detect collisions among swept volumes. We use this swept volume approach for cloth and the dynamic rigid body alike, but for the non-moving rigid body only the surface faces are used for collision detection. In addition, we add proximity regions to the normal directions of faces of a swept volume to add proximity violation regions.
<H2>2.2 Saving detected collisions</H2>
Figure 1. Impact Zones
Though detected collisions are reported as pairs of face-face, we cannot respond to each collision individually since these individual responses may introduce another new collision or one face may possibly be related to several other collisions. So we save all detected collisions in a data structure, i.e. a set of zones of impact
[ 12 ] during the collision detection phase. All stored detected collisions will be resolved comprehensively by the rule described in the next section. Originally a zone of impact (IZ) is an area where multiple self-collisions occur
[ 12 ]. We extend Provot’s definition of an IZ to an area where collisions happen, including collisions against bodies and self-collisions. An object O is a set of particles, faces, and edges, where faces and edges are defined based on the positions of particles by the rule comprising cloth surface. An area A is a subset of O such that all the particles and edges constituting a face in A are members of A . An area A is called separable when, for all faces 2 A , does not collide with any face in A . An area A is called colliding when, for all faces 2 A , collides with at least one face in A . An IZ is a separable colliding area. Fig. 1 shows a cross section view of a collision situation where three cases of IZs are shown by dotted circles. An area A is called visitable when, for all particles P 2 A , P can be encountered by traversing from any other particle in A using edges in A . Otherwise, the area A is non-visitable. We call a visitable subset area of an IZ a collision cluster (CC). When a face-face collision is detected, the entities of each colliding face (the particle and the edges of , and itself) are inserted into a CC, where the CC can be encountered by traversing from the particles in using only edges in . When there is no such CC, becomes a CC. When two or more such CCs are found, these CCs are merged into one CC connected by . In Fig. 1 , a CC is denoted as a shaded area.
A widely used method for detecting cloth collisions is to put small repellent proximity forces between the cloth surface and the rigid or cloth surface [ 4 , 14 , 3 ] while the actual collisions are tested with pairs of particle-face or face-face of the current positions. When objects are moving fast, however, these preventive proximity forces cannot prevent collisions since a particle can pass through the proximity violation region during one integration time step. This problem can be negligible when the integration time step is very small, so we rarely have those pass-through cases. However since implicit integration methods [ 3 , 5 , 7 ] have been used times for as cloth large simulations, as to meet the the frame time step rate has of 30 become 1 sec/frame.
larger, Consesomequently this penetration problem is no longer negligible, so we use the swept volume approach
[ 13 , 8 ] which will be described in the next section.
<H1>3 Collision Resolutions</H1>
Numerous approaches
[ 14 , 12 , 3 , 15 ] have been introduced for cloth collision resolution: the correct next positions and velocities of colliding cloth particles. So far, however, no cloth collision resolution method which considers cloth-cloth momentum conservation has been introduced, while we cannot achieve realistic cloth interactions in cloth animation without conserving cloth-cloth momentums. Having this characteristic is visually distinctive when cloth surfaces are moving fast and interact with each other.
For rigid N-body collisions by graphics and robotics groups
[ 11 , 1 , 10 , 9 ] and for flexible-rigid collision resolution
[ 2 ], several approaches have been suggested. But they are not directly applicable for deformable N-body collision resolutions, which is the case cloth requires. Cloth resolution methods so far compute non-penetrating positions, velocities or accelerations of particles
[ 14 , 3 , 15 ]
, which work fine for the collisions against fixed bodies. Using these methods, however, we cannot achieve visually satisfying dynamics of cloth-cloth collisions. Adjusting particle orientations after collision resolution as suggested by
[ 14 ]
to sustain the geometrical consistency of colliding faces also does not warrant reasonable dynamic movements of cloth-cloth collisions. Handling collisions in an IZ as a bundle, proposed by Provot
[ 12 ]
, also does not give a proper visual effect. Since all particles are given the same velocities after collision resolution, particles tend to get stuck once they collide into each other in an IZ. We propose our cloth collision resolution method which resolves simultaneous collisions while ensuring conservation of momentum as accurately as possible. Since simultaneous resolution does not blindly resolve a collision without considering neighboring collisions within an IZ, we do not introduce any new collisions while resolving a collision. The collision resolution module receives collision information as a set of IZs as described in Sec. 2.2. Within an IZ, where possibly multiple collisions reside, collisions are resolved simultaneously as described in Sec. 3.4. After we handle each IZ separately, we check whether any new collisions between IZs are introduced by collision resolutions, and handle them if there are any. In an IZ, we first check whether it has CCs from rigid bodies. In case we do not find any CCs from rigid bodies, the collisions in that IZ are categorized as cloth-cloth collisions. If we find CCs from rigid bodies in an IZ, we extract them temporarily from the IZ so that only cloth-cloth collisions remain in the IZ. After resolving these cloth-cloth collisions, we take care of cloth collisions against rigid bodies so that the resolutions against rigid bodies will be done on top of the result of self-collision resolutions. This sequence of resolutions is chosen to avoid the case where self-collisions are ignored while collisions against rigid bodies are handled.
<H2>3.1 Single Collision Cluster in an IZ</H2>
Figure 2. Segmenting a Collision Cluster into parts
For some cases, an IZ has only one CC (for example, in the case of extreme bending). When an IZ has only one CC, we cannot handle the particles in that CC as a bundle as usual. Since the particles in that CC will stick together after resolution, the movements of cloth would not be natural and satisfactory. Hence we divide one CC into parts so that we can find proper collision responses within these parts. Segmenting one CC into parts is performed by identifying border edges. A border edge is an edge where we identify a “significant” bending between two faces adjoining in that edge. Empirically and N 2 is identified a bending as between significant two when faces N 1 with N 2 face 0 normals . EliminatN 1 ing the particles in the border edges segments a CC into several sub-CCs as shown Fig. 2 .
<H2>3.2 Merging Multiple CCs in an IZ</H2>
Figure 3. Merging Collision Clusters
When an IZ has three or more CCs, we reduce the total number of CCs by merging closely located CCs. This merger is performed to prevent undesirable collision resolution. If CCs are closely located, it means the cloth patches represented by these CCs are closely located. We do not want to handle closely located CCs separately since it might instantly introduce instabilities to the system by allowing closely located CCs to have different velocities. However, there is an exception. When we find a significant bending between these closely located CCs, we have to resolve collisions between these CCs by handling them separately. Bending between CCs is considered significant in the same way as in the case of bending between faces. We do not want to handle closely located CCs separately except for the case where the bending is significant (CCs are considered to be closely located heuristically when they can be connected using at most two edges which are not members of both CCs). Hence the candidates of the CC merger are the CCs closely located, where we do not witness any significant bending between the CCs. Fig. 3 shows this merging process. After merging, we still possibly have more than two CCs. Multiple CC collision resolution is discussed at length in Sec. 3.4.
<H2>3.3 Collision resolution for two CCs</H2>
Figure 4. Determining the colliding direction. Showing two different colliding directions where the solid arrow shows the direction computed by CC face normals and the dotted arrow shows the direction computed by connecting centers of masses of CCs.
By definition, an IZ is a set of CCs. Since we pre-processed a single CC IZ previously, we assume an IZ always has two or more CCs. The important part of the collision resolution of these multiple CCs is to find the proper directions of collisions. Collision direction is a direction to which two CCs collide into each other. Since the velocities of CCs after collision are computed based on this collision direction, finding the correct collision direction is important to achieve proper visual effect of collisions. In the case of the two billiard ball collision, the collision direction is computed by connecting the two ball centers of mass. But in cloth-cloth collisions, connecting two centers of CC masses is not a proper way to decide the collision direction. We choose the collision direction to be the average direction of the two face normals of colliding CCs. To have the proper average direction, the CC face normals, N 1 and N 2 , have to be properly signed as N 1 N 2 0 . The face normal of a CC is the average normal of all faces in the CC. Fig. 4 shows two different colliding directions between two collision clusters CC1 and CC2; one by the average face normal and the other by connecting centers of masses. We handle a CC as a sphere mass where the diameter of the sphere reflects the minimum proximity region. This approach serves us well empirically. The velocity of a CC is defined as the average velocity of all particles in that CC.
<H2>3.4 Simultaneous collisions</H2>
When we have collisions of three or more CCs in an IZ, it is not straightforward to resolve the collisions. As has been discussed in multiple collisions of rigid bodies, we can think of two ways to solve this multiple cloth collision problem. One way of resolving these multiple collisions is to handle them as staggered collisions [ 11 , 10 ]; the other way is to handle them as simultaneous collisions
[ 1 ].
The staggered collision approach handles multiple collisions as a series of single collisions
[ 11 ] or desynchronized groups of collisions
[ 10 ]. The simultaneous collision approach treats multiple collisions as simultaneous collisions within a time-step. The staggered collision approach gives us a more physically correct solution than the other. In the synchronized staggered collision method, we have to find the first collision among multiple collisions. After we resolve it, we march the time step until we find the next collision. Then we repeat the same procedure. This whole process is not only computationally expensive but also we have to consider the possibility that the resolution of a collision can create new multiple collisions, which we have to employ another strategy to resolve. In the desynchronized staggered collision method, we identify groups of collisions, and redefine the integration front-end by allowing time desynchronization. In addition to the substantial computational expense and complexity, the visual advantage of those staggered methods is not considered significant compared to that of the simultaneous collision method. The simultaneous collision handling method, proposed by Baraff
[ 1 ]
, resolves multiple rigid body collisions by solving a system of linear inequalities, where the system of linear inequalities is based on the colliding relationships between rigid objects. As we treat CCs as mass balls, we adapt this approach for the cloth after collision body If collision, N i before is resolution the number collision, given problem. v i of , , and colliding we have v A i + CC is to bodies, the is find considered velocity v v i + i , for is the of all as the velocity i a 1 body. same i of body N the . When CC bodies are considered as vertices, an edge exists between two vertices where the bodies represented by those two vertices collide. We call the resultant graph a collision graph. When the collision graph of an IZ has a loop, we call the collisions in the IZ cyclic. When N bodies are colliding without having any cyclic collision, we can build a system of N , 1 linear equations based on the changes of relative velocities of each colliding pair, such as
1
v i;j + = , C e v i;j ,
and pair where CC after CC j collision, before i is the collision, i th and CC, C e v v is i;j i;j , + an is is elastic the the relative relative coefficient. velocity velocity of of the the same CC i In addition, the law of N-body momentum conservation says,
2
m 1 v 1 + + m 2 v 2 + + + m N v N + = m 1 v 1 , + m 2 v 2 , + + m N v N , :
all v Using i + for eq. 1 1 i and N 2, . we can deterministically find the solution of Since we use swept-volumes of faces to detect collisions, we handle fast moving cloth and rigid body objects. However a problem arises when collision resolutions of an IZ create new collisions against objects around the IZ. This happens when objects do not move fast enough to penetrate objects outside an IZ, but just fast enough to make the result of collision resolutions penetrate the proximity region of objects outside the IZ. To our relief, this case appears to be very rare. However we can resolve this case by maintaining the barycentric relationship between cloth surfaces and the newly introduced colliding entities. Apparently, in the worst case, this involves repetitious processes as we may introduce other new collisions when we resolve the current collisions.
<H3>3.4.1 Cyclic Collisions</H3>
For the special case where we observe cyclic collisions in an IZ, we build a system of linear inequalities based on the collision rela- tionships between grouped particle parts. We find the feasible solution of the linear inequality system, while trying to minimize the energy we introduce into the simulation artificially. If an IZ has cyclic colliding relationships between CCs, we have N or more linear inequalities such that
(3)
where the notations are as in eq. 1. This inequality relationship between the relative velocities before and after collision is an artificial relationship set up heuristically, not based on physics. This inequality relationship, first used for rigid body multiple collisions
[ 1 ]
, appears to serve the graphical purpose well. mize When the difference we solve this between system v i;j + of and inequalities, , C e v i;j , , which we want would to minirepresent the extra energy we introduce into the system. Hence, we define an objective function, X
(4)
to be minimized when we solve the system. The system of inequalities with an objective function can be solved using a Linear Programming Method.
<H2>3.5 Moving and Fixed Rigid Bodies</H2>
If an IZ has CCs from rigid bodies (rigid CCs) along with CCs from cloth (cloth CCs), the collision resolutions against rigid bodies are performed after cloth-cloth collisions are resolved. When an IZ has rigid CCs, collision responses are different based on whether rigid CCs are moving or fixed or a mixture of both. If the rigid CCs in an IZ are all fixed, we handle particles in that IZ individually. Where N face is the normal of a rigid face, V is a particle velocity, V normal and V tangential are the normal and tangential components of V with respect to the rigid face, particles are considered separating if V normal N face 0 . Particles are ignored if they are not in the vicinity of a face in fixed rigid CCs, where the size of vicinity is the thickness of cloth. Furthermore, particles separating from the rigid bodies are also ignored. The new particle velocity V new is , C e V normal + C f V tangential , where C e is an elastic coefficient and C f is a frictional coefficient. If the rigid CCs in an IZ are all moving, we handle particles as a bundle as long as particles are in the vicinity of moving rigid CCs. We find the x and the velocity V rigid of a moving rigid CC, where V rigid is defined as the translational velocity of the center of mass of the moving rigid CC. Then the positions of all particles we have to handle will be incremented by x and the velocities of the particles will be updated as V rigid . If an IZ has both moving and fixed rigid CCs along with cloth CCs, collision resolutions against rigid bodies are done based on the proximities of particles to the rigid CCs. Cloth collision resolutions against rigid CCs will be computed based on the closest rigid CC. Collisions between rigid bodies (rigid-rigid) have to be handled independently from cloth collisions.
<H1>4 Results</H1>
Fig. 5 shows cloth-rigid and cloth-cloth collisions, where no friction was assumed between the rod and the cloth. Fig. 6 shows various kinds of collisions; cloth-rigid(fixed), cloth-rigid(moving), and cloth-cloth. All our simulations were done on SGI Octane with R10000 CPU and R10010 FPU. For numerical integration, we used the CG method proposed by Baraff
[ 3 ]. Details of simulations are noted in Tab. 1. Notably t in Fig. 6 (above) is 5 times bigger than that of Fig. 6 (below). This mainly comes from the fact that the fabric in Fig. 6 (below) is stiffer than the one in Fig. 6 (above).
<H1>5 Acknowledgments</H1>
We sincerely appreciate Dr. Turteltaub for the discussion about rigid N-body momentum conservations, and also appreciate Koji Ashida for helping in video production. This work was supported in part by NSF grants IRI-9624604. The work of the second author was supported in part by NSF-Career Award (Interactive Virtual Environment for Modeling Anatomy and Physiology).
<H1>References</H1>
[1] D. Baraff. Analytical methods for dynamic simulation of non-penetrating rigid bodies. Computer Graphics (Proc. SIGGRAPH), 23(3):223–232, 1989.
[2] D. Baraff and A. Witkin. Dynamic simulation of non-penetrating flexible body simulation. Computer Graphics (Proc. SIGGRAPH), 26(2):303–308, 1992.
[3] D. Baraff and A. Witkin. Large steps in cloth simulation. Computer Graphics (Proc. SIGGRAPH), pages 43–53, 1998.
[4] D. Breen, D. House, and M. Wozny. Predicting the drape of woven cloth using interacting particles. Computer Graphics (Proc. SIGGRAPH), pages 365–372, 1994.
[5] M. Desbrun, P. Schroder, and A. Barr. Interactive animation of structured deformable objects. Graphics Interface, Kingston, Canada, June 1999.
[6] B. Eberhardt, A. Weber, and W. Strasser. A fast, flexible particle-system model for cloth draping. IEEE Computer Graphics and Applications, 1996.
[7] Y.-M. Kang, J.-H. Choi, and H.-G. Cho. Fast and stable animation of cloth with an approximated implicit method. Computer Graphics International, Geneva, Switzerland, June 2000.
[8] E.-A. Karabassi, G. Papaioannou, T. Theoharis, and A. Boehm. Interaction test for collision detection in particle systems. ACM Journal of Graphics Tools, AK Peters, Ltd., 4(1):25–37, 1999.
[9] D. Marhefka and D. Orin. Simulation of contact using a nonlinear damping model. Proc. IEEE Int. Conf. On Robotics and Automation, Minneapolis, MN, 1996.
[10] B. Mirtich and J. Canny. Impulse-based real-time dynamic simulation. Proc. Workshop on Algorithmic Foundations of Robotics, K. Goldberg et al. (eds.). AK Peters, 1994.
[11] M. Moore and J. Wilhelms. Collision detection and response for computer animation. Computer Graphics (Proc. SIGGRAPH), 22:289–298, 1988.
[12] X. Provot. Collision and self-collision handling in cloth model dedicated to design garments. Proc. Graphics Interface, pages 177–189, 1997.
[13] S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. Anuual Symposium of Computational Geometry, Miami, FL, pages 163–172, June 1999.
[14] P. Volino, M. Courchesne, and N. Magnenat-Thalmann. Versatile and efficient techniques for simulating cloth and other deformable objects. Computer Graphics (Proc. SIGGRAPH), pages 137–144, 1995.
[15] P. Volino, M. Courchesne, and N. Magnenat-Thalmann. Accurate collision response on polygonal meshes. Proc. of Computer Graphics, pages 179–188, 2000.
[16] P. Volino and N. Magnenat-Thalmann. Efficient self-collision detection on smoothly discretized surface animations using geometrical shape regularity. Eurographics, 13(3):155–166, 1994.
Simulation
ST ST
CD(%) CD(%)
Fig. 5(above)
0.0005 0.0005
0.36 0.36
.20(56) .20(56)
Fig. 5(below)
0.0001 0.0001
0.22 0.22
.099(45) .099(45)
Fig. 6(above)
0.0005 0.0005
1.22 1.22
.98(80) .98(80)
Fig. 6(below)
0.0001 0.0001
1.15 1.15
.90(78) .90(78)
Simulation ST CD(%) Fig. 5 (above) 0.0005 0.36 .20(56) Fig. 5 (below) 0.0001 0.22 .099(45) Fig. 6 (above) 0.0005 1.22 .98(80) Fig. 6 (below) 0.0001 1.15 .90(78)
Table 1. Simulation logs Units are in seconds unless noted otherwise. ST is the average Simulation Time for one time step, CD is the average Collision Detection time, CH is the average Collision Handling time(including re-integration), Int is the average Integration time, Leftover time is used for displaying and saving frames, etc., N p is the number of particles used. l edge is the length of a warp or weft directional edge.
Figure 5. Movements of two different types of fabrics
Figure 6. Dropping balls onto two different types of fabrics
CH(%) Int edge (cm) 0.07(19) 0.07(20) 961 1 0.039(18) 0.07(32) 961 1 0.12(10) 0.1(8) 1681 1 0.10(9) 0.1(9) 1681 1
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"text": "['By preserving barycentric relationships of collision entities, their method resolves collisions where numerous cloth surfaces are colliding together as a group, which is a novel way to resolve multiple collisions at once']",
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"offsets": [
[
8710,
8745
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],
"text": "['the entities of each colliding face']",
"type": "own_claim",
"id": "T63"
},
{
"offsets": [
[
8791,
8902
]
],
"text": "['are inserted into a CC, where the CC can be encountered by traversing from the particles in using only edges in']",
"type": "own_claim",
"id": "T64"
},
{
"offsets": [
[
8910,
8929
]
],
"text": "['there is no such CC']",
"type": "data",
"id": "T65"
},
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"offsets": [
[
8931,
8943
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],
"text": "['becomes a CC']",
"type": "own_claim",
"id": "T66"
},
{
"offsets": [
[
8950,
8980
]
],
"text": "['two or more such CCs are found']",
"type": "data",
"id": "T67"
},
{
"offsets": [
[
8982,
9027
]
],
"text": "['these CCs are merged into one CC connected by']",
"type": "own_claim",
"id": "T68"
},
{
"offsets": [
[
9098,
9249
]
],
"text": "['A widely used method for detecting cloth collisions is to put small repellent proximity forces between the cloth surface and the rigid or cloth surface']",
"type": "background_claim",
"id": "T69"
},
{
"offsets": [
[
9252,
9253
]
],
"text": "['4']",
"type": "data",
"id": "T70"
},
{
"offsets": [
[
9256,
9258
]
],
"text": "['14']",
"type": "data",
"id": "T71"
},
{
"offsets": [
[
9261,
9262
]
],
"text": "['3']",
"type": "data",
"id": "T72"
},
{
"offsets": [
[
9265,
9369
]
],
"text": "['while the actual collisions are tested with pairs of particle-face or face-face of the current positions']",
"type": "background_claim",
"id": "T73"
},
{
"offsets": [
[
9376,
9399
]
],
"text": "['objects are moving fast']",
"type": "data",
"id": "T74"
},
{
"offsets": [
[
9410,
9469
]
],
"text": "['these preventive proximity forces cannot prevent collisions']",
"type": "background_claim",
"id": "T75"
},
{
"offsets": [
[
9476,
9567
]
],
"text": "['a particle can pass through the proximity violation region during one integration time step']",
"type": "background_claim",
"id": "T76"
},
{
"offsets": [
[
9569,
9599
]
],
"text": "['This problem can be negligible']",
"type": "background_claim",
"id": "T77"
},
{
"offsets": [
[
9605,
9644
]
],
"text": "['the integration time step is very small']",
"type": "data",
"id": "T78"
},
{
"offsets": [
[
9649,
9688
]
],
"text": "['we rarely have those pass-through cases']",
"type": "own_claim",
"id": "T79"
},
{
"offsets": [
[
9704,
9732
]
],
"text": "['implicit integration methods']",
"type": "background_claim",
"id": "T80"
},
{
"offsets": [
[
9747,
9761
]
],
"text": "['have been used']",
"type": "background_claim",
"id": "T81"
},
{
"offsets": [
[
9768,
9771
]
],
"text": "['for']",
"type": "background_claim",
"id": "T82"
},
{
"offsets": [
[
9775,
9780
]
],
"text": "['cloth']",
"type": "background_claim",
"id": "T83"
},
{
"offsets": [
[
9787,
9798
]
],
"text": "['simulations']",
"type": "background_claim",
"id": "T84"
},
{
"offsets": [
[
9735,
9736
]
],
"text": "['3']",
"type": "data",
"id": "T85"
},
{
"offsets": [
[
9739,
9740
]
],
"text": "['5']",
"type": "data",
"id": "T86"
},
{
"offsets": [
[
9743,
9744
]
],
"text": "['7']",
"type": "data",
"id": "T87"
},
{
"offsets": [
[
9815,
9818
]
],
"text": "['the']",
"type": "background_claim",
"id": "T88"
},
{
"offsets": [
[
9825,
9834
]
],
"text": "['time step']",
"type": "background_claim",
"id": "T89"
},
{
"offsets": [
[
9840,
9843
]
],
"text": "['has']",
"type": "background_claim",
"id": "T90"
},
{
"offsets": [
[
9850,
9856
]
],
"text": "['become']",
"type": "background_claim",
"id": "T91"
},
{
"offsets": [
[
9871,
9877
]
],
"text": "['larger']",
"type": "background_claim",
"id": "T92"
},
{
"offsets": [
[
9896,
9944
]
],
"text": "['this penetration problem is no longer negligible']",
"type": "own_claim",
"id": "T93"
},
{
"offsets": [
[
9949,
9981
]
],
"text": "['we use the swept volume approach']",
"type": "own_claim",
"id": "T94"
},
{
"offsets": [
[
9985,
9987
]
],
"text": "['13']",
"type": "data",
"id": "T95"
},
{
"offsets": [
[
9990,
9991
]
],
"text": "['8']",
"type": "data",
"id": "T96"
},
{
"offsets": [
[
10095,
10114
]
],
"text": "['Numerous approaches']",
"type": "background_claim",
"id": "T97"
},
{
"offsets": [
[
10137,
10188
]
],
"text": "['have been introduced for cloth collision resolution']",
"type": "background_claim",
"id": "T98"
},
{
"offsets": [
[
10118,
10120
]
],
"text": "['14']",
"type": "data",
"id": "T99"
},
{
"offsets": [
[
10123,
10125
]
],
"text": "['12']",
"type": "data",
"id": "T100"
},
{
"offsets": [
[
10128,
10129
]
],
"text": "['3']",
"type": "data",
"id": "T101"
},
{
"offsets": [
[
10132,
10134
]
],
"text": "['15']",
"type": "data",
"id": "T102"
},
{
"offsets": [
[
10279,
10385
]
],
"text": "['no cloth collision resolution method which considers cloth-cloth momentum conservation has been introduced']",
"type": "background_claim",
"id": "T103"
},
{
"offsets": [
[
10393,
10499
]
],
"text": "['we cannot achieve realistic cloth interactions in cloth animation without conserving cloth-cloth momentums']",
"type": "background_claim",
"id": "T104"
},
{
"offsets": [
[
10501,
10551
]
],
"text": "['Having this characteristic is visually distinctive']",
"type": "background_claim",
"id": "T105"
},
{
"offsets": [
[
10557,
10616
]
],
"text": "['cloth surfaces are moving fast and interact with each other']",
"type": "data",
"id": "T106"
},
{
"offsets": [
[
10619,
10678
]
],
"text": "['For rigid N-body collisions by graphics and robotics groups']",
"type": "background_claim",
"id": "T107"
},
{
"offsets": [
[
10682,
10684
]
],
"text": "['11']",
"type": "data",
"id": "T108"
},
{
"offsets": [
[
10687,
10688
]
],
"text": "['1']",
"type": "data",
"id": "T109"
},
{
"offsets": [
[
10691,
10693
]
],
"text": "['10']",
"type": "data",
"id": "T110"
},
{
"offsets": [
[
10696,
10697
]
],
"text": "['9']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
10700,
10743
]
],
"text": "['and for flexible-rigid collision resolution']",
"type": "background_claim",
"id": "T112"
},
{
"offsets": [
[
10747,
10748
]
],
"text": "['2']",
"type": "data",
"id": "T113"
},
{
"offsets": [
[
10752,
10790
]
],
"text": "['several approaches have been suggested']",
"type": "background_claim",
"id": "T114"
},
{
"offsets": [
[
10796,
10872
]
],
"text": "['they are not directly applicable for deformable N-body collision resolutions']",
"type": "background_claim",
"id": "T115"
},
{
"offsets": [
[
10908,
11015
]
],
"text": "['Cloth resolution methods so far compute non-penetrating positions, velocities or accelerations of particles']",
"type": "background_claim",
"id": "T116"
},
{
"offsets": [
[
11019,
11021
]
],
"text": "['14']",
"type": "data",
"id": "T117"
},
{
"offsets": [
[
11024,
11025
]
],
"text": "['3']",
"type": "data",
"id": "T118"
},
{
"offsets": [
[
11028,
11030
]
],
"text": "['15']",
"type": "data",
"id": "T119"
},
{
"offsets": [
[
11035,
11090
]
],
"text": "['which work fine for the collisions against fixed bodies']",
"type": "background_claim",
"id": "T120"
},
{
"offsets": [
[
11092,
11111
]
],
"text": "['Using these methods']",
"type": "background_claim",
"id": "T121"
},
{
"offsets": [
[
11122,
11194
]
],
"text": "['we cannot achieve visually satisfying dynamics of cloth-cloth collisions']",
"type": "background_claim",
"id": "T122"
},
{
"offsets": [
[
11196,
11270
]
],
"text": "['Adjusting particle orientations after collision resolution as suggested by']",
"type": "background_claim",
"id": "T123"
},
{
"offsets": [
[
11274,
11276
]
],
"text": "['14']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
11280,
11414
]
],
"text": "['to sustain the geometrical consistency of colliding faces also does not warrant reasonable dynamic movements of cloth-cloth collisions']",
"type": "background_claim",
"id": "T125"
},
{
"offsets": [
[
11416,
11476
]
],
"text": "['Handling collisions in an IZ as a bundle, proposed by Provot']",
"type": "background_claim",
"id": "T126"
},
{
"offsets": [
[
11480,
11482
]
],
"text": "['12']",
"type": "data",
"id": "T127"
},
{
"offsets": [
[
11487,
11528
]
],
"text": "['also does not give a proper visual effect']",
"type": "background_claim",
"id": "T128"
},
{
"offsets": [
[
11536,
11606
]
],
"text": "['all particles are given the same velocities after collision resolution']",
"type": "background_claim",
"id": "T129"
},
{
"offsets": [
[
11608,
11678
]
],
"text": "['particles tend to get stuck once they collide into each other in an IZ']",
"type": "background_claim",
"id": "T130"
},
{
"offsets": [
[
11680,
11833
]
],
"text": "['We propose our cloth collision resolution method which resolves simultaneous collisions while ensuring conservation of momentum as accurately as possible']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
11841,
11957
]
],
"text": "['simultaneous resolution does not blindly resolve a collision without considering neighboring collisions within an IZ']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
[
11959,
12025
]
],
"text": "['we do not introduce any new collisions while resolving a collision']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
12027,
12105
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],
"text": "['The collision resolution module receives collision information as a set of IZs']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
12122,
12130
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],
"text": "['Sec. 2.2']",
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[
12132,
12227
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],
"text": "['Within an IZ, where possibly multiple collisions reside, collisions are resolved simultaneously']",
"type": "own_claim",
"id": "T136"
},
{
"offsets": [
[
12244,
12252
]
],
"text": "['Sec. 3.4']",
"type": "data",
"id": "T137"
},
{
"offsets": [
[
12254,
12411
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],
"text": "['After we handle each IZ separately, we check whether any new collisions between IZs are introduced by collision resolutions, and handle them if there are any']",
"type": "own_claim",
"id": "T138"
},
{
"offsets": [
[
12526,
12593
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],
"text": "['the collisions in that IZ are categorized as cloth-cloth collisions']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
12484,
12524
]
],
"text": "['we do not find any CCs from rigid bodies']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
12598,
12636
]
],
"text": "['we find CCs from rigid bodies in an IZ']",
"type": "data",
"id": "T141"
},
{
"offsets": [
[
12638,
12677
]
],
"text": "['we extract them temporarily from the IZ']",
"type": "own_claim",
"id": "T142"
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{
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[
12686,
12730
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],
"text": "['only cloth-cloth collisions remain in the IZ']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
12732,
12831
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],
"text": "['After resolving these cloth-cloth collisions, we take care of cloth collisions against rigid bodies']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
12840,
12940
]
],
"text": "['the resolutions against rigid bodies will be done on top of the result of self-collision resolutions']",
"type": "own_claim",
"id": "T145"
},
{
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12942,
13082
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],
"text": "['This sequence of resolutions is chosen to avoid the case where self-collisions are ignored while collisions against rigid bodies are handled']",
"type": "own_claim",
"id": "T146"
},
{
"offsets": [
[
13259,
13296
]
],
"text": "['For some cases, an IZ has only one CC']",
"type": "own_claim",
"id": "T147"
},
{
"offsets": [
[
13372,
13434
]
],
"text": "['we cannot handle the particles in that CC as a bundle as usual']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
13349,
13370
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],
"text": "['an IZ has only one CC']",
"type": "data",
"id": "T149"
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{
"offsets": [
[
13442,
13503
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],
"text": "['the particles in that CC will stick together after resolution']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
13505,
13565
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],
"text": "['the movements of cloth would not be natural and satisfactory']",
"type": "own_claim",
"id": "T151"
},
{
"offsets": [
[
13573,
13600
]
],
"text": "['we divide one CC into parts']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
13609,
13666
]
],
"text": "['we can find proper collision responses within these parts']",
"type": "own_claim",
"id": "T153"
},
{
"offsets": [
[
13668,
13737
]
],
"text": "['Segmenting one CC into parts is performed by identifying border edges']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
13972,
14044
]
],
"text": "['ing the particles in the border edges segments a CC into several sub-CCs']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
13960,
13968
]
],
"text": "['Eliminat']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
14054,
14060
]
],
"text": "['Fig. 2']",
"type": "data",
"id": "T157"
},
{
"offsets": [
[
14233,
14260
]
],
"text": "['an IZ has three or more CCs']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
14262,
14326
]
],
"text": "['we reduce the total number of CCs by merging closely located CCs']",
"type": "own_claim",
"id": "T159"
},
{
"offsets": [
[
14328,
14396
]
],
"text": "['This merger is performed to prevent undesirable collision resolution']",
"type": "own_claim",
"id": "T160"
},
{
"offsets": [
[
14401,
14424
]
],
"text": "['CCs are closely located']",
"type": "data",
"id": "T161"
},
{
"offsets": [
[
14426,
14497
]
],
"text": "['it means the cloth patches represented by these CCs are closely located']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
14499,
14554
]
],
"text": "['We do not want to handle closely located CCs separately']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
14561,
14678
]
],
"text": "['it might instantly introduce instabilities to the system by allowing closely located CCs to have different velocities']",
"type": "own_claim",
"id": "T164"
},
{
"offsets": [
[
14689,
14710
]
],
"text": "['there is an exception']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
14717,
14780
]
],
"text": "['we find a significant bending between these closely located CCs']",
"type": "data",
"id": "T166"
},
{
"offsets": [
[
14782,
14857
]
],
"text": "['we have to resolve collisions between these CCs by handling them separately']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
14859,
14960
]
],
"text": "['Bending between CCs is considered significant in the same way as in the case of bending between faces']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
14962,
15070
]
],
"text": "['We do not want to handle closely located CCs separately except for the case where the bending is significant']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
15220,
15344
]
],
"text": "['the candidates of the CC merger are the CCs closely located, where we do not witness any significant bending between the CCs']",
"type": "own_claim",
"id": "T170"
},
{
"offsets": [
[
15381,
15436
]
],
"text": "['After merging, we still possibly have more than two CCs']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
15932,
15974
]
],
"text": "['we pre-processed a single CC IZ previously']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
15976,
16018
]
],
"text": "['we assume an IZ always has two or more CCs']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
16020,
16135
]
],
"text": "['The important part of the collision resolution of these multiple CCs is to find the proper directions of collisions']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
16220,
16304
]
],
"text": "['the velocities of CCs after collision are computed based on this collision direction']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
16306,
16404
]
],
"text": "['finding the correct collision direction is important to achieve proper visual effect of collisions']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
16454,
16532
]
],
"text": "['the collision direction is computed by connecting the two ball centers of mass']",
"type": "own_claim",
"id": "T177"
},
{
"offsets": [
[
16421,
16452
]
],
"text": "['the two billiard ball collision']",
"type": "data",
"id": "T178"
},
{
"offsets": [
[
16538,
16654
]
],
"text": "['in cloth-cloth collisions, connecting two centers of CC masses is not a proper way to decide the collision direction']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
16760,
16872
]
],
"text": "['To have the proper average direction, the CC face normals, N 1 and N 2 , have to be properly signed as N 1 N 2 0']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
17219,
17259
]
],
"text": "['This approach serves us well empirically']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
17426,
17474
]
],
"text": "['we have collisions of three or more CCs in an IZ']",
"type": "data",
"id": "T181"
},
{
"offsets": [
[
17476,
17527
]
],
"text": "['it is not straightforward to resolve the collisions']",
"type": "own_claim",
"id": "T183"
},
{
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A25 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
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29dd9660c10228b001729ce8e0bc4153ee4445b609ab646c168d467337a763fb
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http://dx.doi.org/10.1145/1141911.1141961
Computer Graphics Proceedings, Annual Conference Series, 2006
<Title>Fluid Animation with Dynamic Meshes</Title>
Bryan M. Klingner Bryan E. Feldman Nuttapong Chentanez James F. O’Brien University of California
Berkeley
Figure 1: Top: A paddle mixes smoke in a tank. Bottom: A cross-section of the simulation meshes used for each frame.
<Abstract>This paper presents a method for animating fluid using unstructured tetrahedral meshes that change at each time step. We show that meshes that conform well to changing boundaries and that focus computation in the visually important parts of the domain can be generated quickly and reliably using existing techniques. We also describe a new approach to two-way coupling of fluid and rigid bodies that, while general, benefits from remeshing. Overall, the method provides a flexible environment for creating complex scenes involving fluid animation.</Abstract>
Keywords: Natural phenomena, physically based animation, computational fluid dynamics. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation; I.6.8 [Simulation and Modeling]: Types of Simulation—Animation.
<H1>1 Introduction</H1>
Although systems for physically based fluid animation have developed rapidly in recent years and can now reliably generate production-quality results, they still have some limitations. Simulation domains can change substantially from step to step because of deforming boundaries, moving obstacles, and evolving fluid motion, yet current systems based on fixed grids are not ideally suited to handle these situations.
E-mail: {klingner|feldman|nchentan|job}@eecs.berkeley.edu From the ACM SIGGRAPH 2006 conference proceedings. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ACM SIGGRAPH 2006, Boston, MA c Copyright ACM 2006
We propose a method to simulate fluids with such rapidly changing domains by generating a new tetrahedral simulation mesh at each time step. When generating the mesh, we use the position and shape of boundaries as well as criteria based on the visually important parts of the fluid and velocity field to construct a sizing field that dictates the desired edge length for tetrahedra throughout the domain. We then use an efficient and reliable meshing algorithm adapted from [ Alliez et al., 2005 ]
to produce a mesh that is refined according to this field. We use unstructured tetrahedral meshes because they conform to curved and irregular boundaries better than axis-aligned grids with the same number of grid elements and allow for precise control of refinement throughout the domain. We transfer the physical properties of the simulation from the old mesh to the new mesh using a generalization of the semi-Lagrangian velocity advection technique that introduces no additional smoothing. We then perform a mass conservation step that has been extended to allow a new, single-step solution of two-way coupling between fluid and rigid bodies. Overall, this approach provides a flexible framework for fluid simulation that opens the door to many features. We have implemented the system and tested it in a variety of scenarios such as the one shown in Figure 1 . We have found that the combination of unstructured tetrahedral domains and dynamic remeshing creates a versatile environment for the creation of complex and visually interesting fluid animations.
<H1>2 Background</H1>
The animation of fluids through physical simulation has become an important tool in the visual effects industry. One approach that has been popular in recent years makes use of a spatial discretization based on regular, fixed, hexahedral grids.
Some examples of this approach can be found in
[Foster and Metaxas, 1996], [Foster and Metaxas, 1997], [Stam, 1999], [Yngve et al., 2000], [Fedkiw et al., 2001], [Foster and Fedkiw, 2001], [Enright et al., 2002], [Carlson et al., 2002], [Feldman et al., 2003], and
[Goktekin et al., 2004]. The most commonly used storage scheme for these approaches is the “staggered grid” scheme. This method offsets storage of different quantities on the grid, and was first described by
[Harlow and Welch, 1965]. Efforts have been made to enhance these methods to allow for better conformance to irregular boundaries such as the free surface of liquids, complex obstacles, or irregularly shaped domains. [Losasso et al., 2004]
described an octree-based method that retains many of the advantages of regular grids while allowing computational effort to be focused in particular parts of the simulation domain; this enables detailed tracking of moving boundaries such as liquid surfaces. Both [Carlson et al., 2004] and [Guendelman et al., 2005]
have demonstrated methods for two-way coupling of obstacles to fluid. Unstructured tetrahedra have also been used for fluid simulation within the graphics community. Two examples of this are
[Feldman et al., 2005a] and [Elcott et al., 2005]. The first method uses a velocity-based approach while the second uses a vorticity-based formulation. It is a blend of ideas from these two papers, along with a generalization of the semi-Lagrangian velocity advection technique for moving meshes described in [Feldman et al., 2005b]
that forms the heart of our method. The idea of moving meshes independent of a fixed or particle-centric coordinate system is not a new one; arbitrary Lagrangian-Eulerian (ALE) methods were designed for just this purpose. They have proven useful in the simulation of highly deformable elastic materials. ALE was first described in
[Hirt et al., 1974]
, where it was used with finite differences to solve compressible fluid problems. [Donea et al., 1977]
went on to apply ALE in a finite element setting. An excellent survey of the development of ALE methods appears in
[Donea et al., 2004]. Examples within the graphics literature that feature moving meshes without remeshing include
[Shah et al., 2004] and [Rasmussen et al., 2004]
, both of which translate the grid to follow the visually important portion of the fluid. Another approach to handling changing domains is to dispense with the mesh altogether, instead using Lagrangian particles for simulation of fluids.
A few examples of this approach are
[Terzopoulos et al., 1989], [Desbrun and Cani, 1996], [Cani and Desbrun, 1997], [Stora et al., 1999], [Müller et al., 2003], [Premo ze et al., 2003], and
[Müller et al., 2004]. These meshless methods are particularly well suited to changing domains because points can move freely without concerns about mesh quality. Because we regenerate a new simulation mesh at each time step, the viability of our method hinges on fast, high-quality, reliable tetrahedral mesh generation. While a history of unstructured mesh generation is outside the scope of this paper,
[Owen, 1998] and [Teng and Wong, 2000]
provide good surveys of the field. For our mesh generator we selected the approach described in
[ Alliez et al., 2005 ]. This innovative method produces meshes which conform to domains of arbitrary topology quickly and reliably. Also, it allows for the local edge length of the tetrahedra to be specified arbitrarily throughout space, which allows us to easily perform adaptive mesh refinement from step to step. The meshes produced by this technique are Delaunay, which provides improved gradient estimation and allows us to significantly simplify some of the expressions that arise when interpolating velocity values stored on the mesh.
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<H1>3 Methods</H1>
The key contribution of our method is to demonstrate the freedom granted by remeshing at each simulation time step. The core of our system is based on the simple, efficient methods for discretizing the inviscid Euler equations on tetrahe- dral meshes described in
[Elcott et al., 2005] and [Feldman et al., 2005a]. We have made a few modifications in order to combine the best aspects of both approaches that are described below. Once we have a good discretization, we need a way to propagate information from one mesh to the next. [Feldman et al., 2005b]
details a generalization of the standard semiLagrangian velocity advection technique that allows simulation state to be transferred between deforming domains without incurring additional smoothing. We demonstrate that their approach can easily be applied to transfer information between two arbitrary, topologically unrelated meshes, which is required to achieve more general evolution of the simulation domain from step to step. Finally, we need to quickly and reliably generate a new tetrahedral mesh for each time step that suits the current simulation conditions, such as conformance to boundaries and obstacles as well as any desired refinement. Although methods have long existed to mesh arbitrary domains, most are relatively slow in comparison to simulation running times or don’t reliably terminate under realistic conditions. The availability of efficient, versatile meshing algorithms such as
[ Alliez et al., 2005 ]
has made the generation of a new mesh at each time step practical. Any changes that were required to make these pieces work together harmoniously are discussed below. Also, we describe a new, single-step method to achieve two-way coupling between obstacle and fluid motion.
<H2>3.1 Discretization</H2>
We use a staggered fluid state storage scheme that stores pressures at tetrahedron circumcenters and “face-normal velocities,” the component of velocity in the direction of the face normal, at the face circumcenters. Similar schemes have been used in [Botta and Hempel, 1996], [Elcott et al., 2005] and [Feldman et al., 2005a]. These methods are a generalization of the staggered grid scheme originally proposed by
[Harlow and Welch, 1965]. This staggered method is used to discretize the inviscid Euler equations:
1
∂u p f = − (u · ) u − + ∂t ρ ρ
subject to the mass conservation constraint for incompress-
2
ible fluids: · u =0 .
In these equations, u is the fluid velocity, t time, p pressure, ρ density, and f any external forces. The symbol denotes T the vector of differential operators = [∂/∂x, ∂/∂y, ∂/∂z] . We account for the changes in the mesh over a time step directly during semi-Lagrangian advection (see Section 3.2).
<H3>3.1.1 Discrete Derivative Operators</H3>
Divergence and gradient operators are needed as part of the mass conservation step. We make discrete estimates of these derivatives following the formulation presented in [Losasso et al., 2004] and [Elcott et al., 2005]. The divergence of a tetrahedron is computed as an area weighted sum of the tetrahedron’s face normal velocities. The gradient at a face circumcenter in the direction of the face’s normal is computed using finite differences. The difference in circumcenter pressures adjacent to a face is divided by the distance between these circumcenters. In Delaunay meshes, the line connecting adjacent tetrahedra circumcenters passes through the circumcenter of the face between them and is in the direction of that face’s normal. This property of Delaunay meshes motivates our storage scheme at circumcenters because the gradient estimate is equivalent to the gradient of a piecewise linear function that interpolates the circumcenter values.
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Current step
<H3>3.1.2 Velocity Interpolation</H3>
The staggered scheme stores only the component of velocity in the face normal direction. For both the semi-Lagrangian step and to advect smoke particles for rendering, a full velocity vector must be found at arbitrary positions in the mesh. We interpolate velocity vectors from face normal velocities using the two-step method developed in
[Elcott et al., 2005]. First, a velocity vector, u t , is computed at each tetrahedron circumcenter, then we interpolate within Voronoi cells using u t values at the cell vertices. Velocity u t for tetrahedron t is found by solving the small linear system N t u t = z t where N t is a matrix containing 4 rows of the face normals of t and z t is a vector of the 4 face normal velocities associated with t. For a divergence-free field, this solution has the remarkable property that interpolating back to the face circumcenters exactly recovers the original face-normal velocities. Thus interpolating the u t velocities also exactly interpolates the face-normal velocity components, and does not incur the error one would otherwise expect from a twostep interpolation method. To find a velocity at an arbitrary point we interpolate within the Voronoi cell using the tetrahedra velocities associated with the cell. This interpolation is based on the method of
[Warren et al., 2004]
, which presents a way to interpolate within a general convex polytope. They interpolate the value at the point x as a weighted sum of the polytope’s node values where node t’s unnormalized weight is computed as
3
|N t | w t (x) = . f ∈σ t n f · x + d f
Here, σ t is the set of polytope faces that intersect at node t. The denominator is the product of distances from x to the faces in σ t computed using the face normals,n f , and plane offsets, d f . |N t | is the determinant of a matrix of face normals in σ. Weights from all nodes are normalized to sum to 1 before use in the weighted sum. To simplify this computation we take advantage of two properties: 1) in a Delaunay mesh, edges are in the direction of the Voronoi cell’s face normals and 2) the volume of tetrahedron t is 1/6|E t | where E t is a matrix formed from the three vectors of edges emanating from a common node of t. After some manipulation, which is omitted for brevity, Equation (3) applied to node weights within a Voronoi cell can be simplified to
4
6Vol(t) w t (x) = 3 i=1 (p i − p v ) · (c t − x)
where w t (x) is the weight associated with the node at tetrahedra t’s circumcenter, Vol(t) is the volume of tetrahedron t, p v is the position of the node associated with the Voronoi cell, p i are positions of the other nodes of t, c t the circumcenter of t, and x the interpolation position. A similar observation appears in [Ju et al., 2005]
, and we find that with it the velocity interpolation is quite efficient. All quantities appearing in Equation (4) are already stored for use in other parts of the timestep, saving the need to compute the terms in Equation (3). When advecting large numbers of particles, velocities at nodes of tetrahedra can be first be found using Equation (4) and then quickly interpolated in a linear fasion over the tetrahedra to advect the particles.
<H2>3.2 Generalized Semi-Lagrangian Step</H2>
The simple and stable semi-Lagrangian method has become the standard tool for advection of the velocity field for graphical applications
[Stam, 1999]. The basic idea of the method x A two-dimensional representation of the generalized semi-Lagrangian advection step. We trace back from the position where a velocity is stored in the new mesh, x i = (x, y), interpolate the velocity using the old mesh and velocity field, and update the velocity in the new mesh. is that we can find a velocity that will advect to a point by tracing back from that point and interpolating the old velocity field. This method does not rely on velocities being stored at any particular place, as long as the velocity can be interpolated throughout space. We can extend this technique naturally to meshes which change arbitrarily at each time step as in
[Feldman et al., 2005b]. This extension does not incur any additional smoothing compared to using semi-Lagrangian advection with static meshes. Suppose at time t velocities are stored at locations x (t) (in our case, the face circumcenters), and we want to find (t) the velocity at a particular face location x i . We trace back (t) from x i through the velocity field of the previous time step to a point x i , which has no necessary correspondence to any feature of the old mesh. Then, we update the velocity at (t) x i to the value interpolated from the old velocity field at x i . Because the velocities from the previous step are stored on a different mesh, we have to trace back and interpolate using this previous mesh (see Figure 2 ).
y x i (t) x Previous step x i y
Figure 2:
<H2>3.3 Remeshing</H2>
The domain boundaries, obstacles, and smoke are free to move and change from step to step of the simulation. By regenerating the mesh at each time step we can ensure that our domain conforms well to boundaries and is refined in visually important areas. We accomplish this by using the variational tetrahedral meshing algorithm presented in
[ Alliez et al., 2005 ]. This method allows for generation of tetrahedral meshes that conform well to an arbitrary input surface mesh, have no restrictions on topology (i.e., allow nested voids), and allow for sizing of tetrahedra throughout the domain based on arbitrary criteria. Our implementation differs from the original algorithm in a couple of details. As in the original method, refinement of the mesh is controlled by a sizing function μ(x) that, for any point x in the simulation domain, returns the desired local edge length of the tetrahedra. While the original algorithm builds this sizing function by finding the minimum combination of local feature size and distance to a boundary point Left: a visualization of the sizing field for a rectangular domain with an irregular obstacle at the top and a plume of smoke at the bottom. Right: the resulting simulation mesh. Obstacle faces are colored green.
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Computer Graphics Proceedings, Annual Conference Series, 2006
Figure 3:
from x, we instead formulate it as follows:
5
μ(x) = k 0 + min (k d d(x), k s (1 − s(x)) , k ω (1 − ω(x)))
In this equation, k 0 is an offset value that controls the minimum value of the sizing field, and hence the minimum local edge length of tetrahedra. d(x) is the distance to the closest obstacle or boundary which demands refinement, s(x) is a function of the density of smoke particles, and ω(x) is a function of the vorticity of the velocity field. The parameters k d , k s , and k ω respectively control the weight each of these functions has on the sizing field. These three factors are the same as those used for octree refinement in
[Losasso et al., 2004]. The overall goal of the sizing field is to focus computational effort in the most visually important parts of the scene, that is, near closed boundaries, where the velocity field varies most, and where smoke is visible. Figure 3 shows an example of a sizing field and the resulting mesh. Figure 4 demonstrates the benefits of refinement near areas of high vorticity and smoke density. This meshing method is iterative, so the mesh from the previous simulation time step can be used as an initial guess for the node placement in the mesh at the next simulation time step. Because there is, in general, strong temporal coherence between steps of the simulation, the sizing field does not change too much and so the nodes from the previous step are often a good initialization. Before the algorithm proceeds, the initial node placement is corrected to match the sizing field of the current step. One other modification we made to the algorithm is that, when optimizing the node positions, we move nodes to the average of the barycenters of the surrounding tetrahedra instead of the circumcenters. We have found that while this tends to slightly decrease the average quality of tetrahedra in the mesh, it often leads to substantial improvements in the quality of the worst elements of the mesh, which are of more concern for numerical simulation. Of course, remeshing takes time, so it is important to consider the impact it has on overall simulation performance. The time spent generating meshes for each simulation step varies, but generally accounts for less than a quarter of the overall simulation time. In Section 4 we show timing information for several examples.
<H2>3.4 Two-way Coupling and Mass Conservation</H2>
The motion of fluid and rigid bodies that mutually effect each other can be complex and visually appealing. The interaction occurs as a consequence of the conditions that:
1. The velocities in the normal direction are the same at the interface of the fluid and the rigid body surface.
2. The fluid velocity is divergence free and the rigid body velocity is rigid. 3. The linear and angular momentum of the combined system is conserved.
In [Carlson et al., 2004]
these conditions are enforced sequentially. While for many cases this produces results that look very good, under some situations artifacts can be created because enforcing one of the conditions in general will break a previously enforced one. Examples of such artifacts might be fluid leaking through solid boundaries or poor performance in piston-like situations. Our implementation differs from
[Carlson et al., 2004]
in a couple of ways, but most significantly we enforce these conditions simultaneously within the mass conservation step. In general, the mass conservation step solves for pressures that accelerate the velocity field to be divergence free. In previous works, including those with two-way coupling, the mass conservation step treats faces to behave as fluid or explicitly prescribes their velocities. For fluid faces, the pressure accelerates the velocity proportional to the gradient of the pressure while for prescribed faces, the pressure does not effect the fluid. For a more complete discussion of fluid/prescribed-velocity mass conservation see
[Fedkiw et al., 2001]. We extend mass conservation to include a dynamic, rigid body. To do so, we solve for acceleration of the fluid and the rigid body, ignoring pressure for both. We then solve for a pressure term that satisfies boundary and incompressibility constraints to find the final accelerations. The rigid body accelerations can be computed by creating a matrix R that is multiplied by a vector of the pressures that surround a rigid body. R can be formed by a series of matrix multiplications:
6
b 1 R = . . . M 0 −1 I −1 0 A 1 b T 1 ··· A k b T k b k
where b i = n T i | (r i × n i ) T , n i is the normal of the ith face, r i is the vector from the rigid objects center of mass to position of the ith face, and A i is the area of that face. The rightmost matrix finds the net force-torque couple acting on a rigid body by summing up the contribution due to pressure forces acting on rigid body mesh faces. The force-torque couple is converted to a linear and angular acceleration of A comparison between uniform and selectively refined simulation meshes. Left: a frame from a simulation using approximately 43000 uniformly sized tetrahedra. Right: the same frame using approximately 32000 tetrahedra refined near areas of high vorticity and smoke density. The refined mesh preserves the fine detail in the velocity field and near the visible smoke, enhancing vortex action and natural movement. The runtimes of the two are equivalent. the body by the middle (6×6) block matrix. M is a diagonal matrix with the mass of the rigid body on the diagonals and I is the inertia matrix. The leftmost matrix in the multiplication returns the acceleration of the fluid-rigid faces in the direction of the face normal due to the linear and angular acceleration of the rigid body. By construction, accelerations generated by this matrix behave rigidly. Computing pressure accelerations of both the fluid and fluid-rigid faces can be expressed as a matrix A multiplied by a vector of all the pressures. A row of A that corresponds to a face with fluid on both sides contains the same entries as the standard gradient matrix multiplied by −1/ρ. A row of A that belongs to a face at the fluid-rigid interface has element values obtained from the corresponding row of R. The elements of this row are placed at columns corresponding to the pressures that surround the rigid-body. With A built, mass conservation including two way coupling proceeds much in the same way as in the all-fluid case, with A replacing the role of the discrete gradient matrix. For a given vector of pressures, p, the intermediate velocity field, z ∗ , is accelerated to the end-of-step velocity, z, by z = z ∗ + ∆tAp. For the fluid faces, z ∗ is found by applying all terms of Equation (1) except the pressure term. For the fluid-rigid faces, z ∗ is found using a rigid body simulator without pressure forces applied. To find a particular pressure that accelerates z ∗ such that z is divergence free we solve the linear system
Figure 4:
823
ACM SIGGRAPH 2006, Boston, MA, July 30–August 3, 2006
Figure 5:
Red particles are transfered from the left tank to the right by squeezing and releasing the central bulb. The blue valves are coupled to the fluid simulation and prevent backflow. Remeshing time Total time Percent per frame (mean) per frame (mean) remeshing Figure 1 13.2 sec 64.8 sec 20.3% Figure 5 8.33 sec 44.5 sec 18.7% 5.76 sec 35.8 sec 16.1% 313 sec 796 sec 39.3% A comparison of remeshing and simulation time for selected examples.
7
∆tDAp = −Dz ∗ .
This linear system can be solved efficiently using PCG since the the matrix DA, which replaces the discrete Laplacian from the all fluid case, is also a positive-definite symmetric matrix. Using the same machinery, we can also interact with constrained rigid bodies. This simply requires finding an R matrix that correctly computes face accelerations due to pressure. For example, one could easily alter R such that the body was constrained to just rotate about the origin by replacing b i in Equation (6) with b i = (r i × n i ) T and using only the I −1 block for the center matrix. This idea could be extended further to include even articulated bodies.
<H1>4 Results and Discussion</H1>
We implemented the method described above in matlab 1 and C, making use of Pyramid [Jonathan Shewchuck, personal communication] for Delaunay triangulation and pixie 2 for all renderings. Typical simulation times for meshes with 100,000 tetrahedra were about 1 minute per frame. Table 1 compares remeshing and simulation times for several of the examples presented in this paper. The images in Figure 1 show smoke in a tank mixed by the scripted motion of a paddle. Refinement of the simulation mesh near the paddle ensures good conformance to its curved surfaces that produce interesting vortex effects in the smoke. 1 http://www.mathworks.com 2 http://sourceforge.net/projects/pixie In Figure 5 , a pump transfers particles from the left tank to the right tank as the bulb in the middle is squeezed and released. The blue valves on either side of the bulb prevent backflow. The motion of these valves is not scripted. Instead, they are modeled as rigid bodies constrained to rotate about an axis and their motion is caused by two-way interaction with the fluid. Figure 6 demonstrates the two-way interaction of the Stanford bunny with smoke cannons. On the left is a lighter bunny which is tossed about by the force of the cannons and also affects the motion of the smoke. On the right is a heavier bunny that drops quickly to the ground. In Figure 7 , smoke moves through an array of obstacles in a higher resolution mesh of over 500,000 tetradra. Although quality of the mesh elements does not suffer at this level of refinement, the proportion of time spent meshing increases to 39.3%. The motion of the smoke at the higher resolution is more lively and exhibits more fine-scale detail. A vorticity enhancement method, such as those in
[Fedkiw et al., 2001] and [Selle et al., 2005]
could be used to further enhance the fluid motion but we do not find such enhancement necessary and so have not implemented it. We have presented a system for performing fluid animation using unstructured tetrahedral domains that can change arbitrarily at each time step. Although our current implementation models completely fluid-filled domains, we believe it would be well-suited for use with surface tracking techniques for liquid simulation.
Table 1:
<H1>Acknowledgments</H1>
We thank the other members of the Berkeley Graphics Group for their helpful criticism and comments. This work was supported in part by California MICRO 04-066 and 05-044, and by generous support from Apple Computer, Pixar Animation Studios, Autodesk, Intel Corporation, Sony Computer Entertainment America, and the Alfred P. Sloan Foundation. Klingner and Feldman were supported by NSF Graduate Fellowships.
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"text": "['Müller et al., 2003']",
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"text": "['Premo ze et al., 2003']",
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"text": "['These meshless methods are particularly well suited to changing domains']",
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"text": "['any desired refinement']",
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"text": "['Similar schemes have been used in']",
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"text": "['Botta and Hempel, 1996']",
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"text": "['These methods are a generalization of the staggered grid scheme originally proposed by']",
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"text": "['Harlow and Welch, 1965']",
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"text": "['This staggered method is used to discretize the inviscid Euler equations']",
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"text": "['∂u p f = − (u · ) u − + ∂t ρ ρ']",
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"text": "['This property of Delaunay meshes motivates our storage scheme at circumcenters']",
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"text": "['the gradient estimate is equivalent to the gradient of a piecewise linear function that interpolates the circumcenter values']",
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"text": "['this solution has the remarkable property']",
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"text": "['After some manipulation']",
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"text": "['refinement of the mesh is controlled by a sizing function μ(x)']",
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18510
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"text": "['k 0 is an offset value that controls the minimum value of the sizing field']",
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19505,
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19892
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"text": "['These three factors are the same as those used for octree refinement in']",
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19915
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"text": "['Losasso et al., 2004']",
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19918,
20037
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"text": "['The overall goal of the sizing field is to focus computational effort in the most visually important parts of the scene']",
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"text": "['Figure 3']",
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"text": "['Figure 4']",
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{
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20303,
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20340,
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"text": "['the mesh from the previous simulation time step can be used as an initial guess']",
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"text": "['the nodes from the previous step are often a good initialization']",
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"text": "['Before the algorithm proceeds']",
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"text": "['this tends to slightly decrease the average quality of tetrahedra in the mesh']",
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"text": "['it often leads to substantial improvements in the quality of the worst elements of the mesh']",
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21215,
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"text": "['are of more concern for numerical simulation']",
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"text": "['it is important to consider the impact it has on overall simulation performance']",
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21378,
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21448,
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"text": "['generally accounts for less than a quarter of the overall simulation time']",
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"text": "['can be complex and visually appealing']",
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"text": "['for many cases']",
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"text": "['enforcing one of the conditions in general will break a previously enforced on']",
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"text": "['Examples of such artifacts might be fluid leaking through solid boundaries or poor performance in piston-like situations']",
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"text": "['Our implementation differs from']",
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22568,
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"text": "['Carlson et al., 2004']",
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"text": "['the mass conservation step solves for pressures that accelerate the velocity field to be divergence free']",
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"text": "['the mass conservation step treats faces to behave as fluid or explicitly prescribes their velocities']",
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"text": "['Fedkiw et al., 2001']",
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"text": "['For a more complete discussion of fluid/prescribed-velocity mass conservation see']",
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"text": "['\\uf8ee b 1 \\uf8f9 R = \\uf8f0']",
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"text": "['\\uf8fb M 0 −1 I −1 0 A 1 b T 1 ··· A k b T k b k']",
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"text": "['construction']",
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10.1002/cav.132
COMPUTER ANIMATION AND VIRTUAL WORLDS Comp. Anim. Virtual Worlds 2006; 17: 281–292 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cav.132 *******************************************************************************************************************
<Title>Curve skeleton skinning for human and creature characters</Title>
By Xiaosong Yang
Arun Somasekharan
Jian J. Zhang
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*Correspondence to: Jian J. Zhang, National Centre for Computer Animation, Bournemouth Media School, Bournemouth University, Poole, Dorset, BH12 5BB, UK. E-mail: [email protected] Contract/grant sponsor: British Arts and Humanities Research Council; contract/grant number: B/RG/AN5263/APN12727.
<Abstract>The skeleton driven skinning technique is still the most popular method for animating deformable human and creature characters. Albeit an industry de facto due to its computational performance and intuitiveness, it suffers from problems like collapsing elbow and candy wrapper joint. To remedy these problems, one needs to formulate the non-linear relationship between the skeleton and the skin shape of a character properly, which however proves mathematically very challenging. Placing additional joints where the skin bends increases the sampling rate and is an ad hoc way of approximating this non-linear relationship. In this paper, we propose a method that is able to accommodate the inherent non-linear relationships between the movement of the skeleton and the skin shape. We use the so-called curve skeletons along with the joint-based skeletons to animate the skin shape. Since the deformation follows the tangent of the curve skeleton and also due to higher sampling rates received from the curve points, collapsing skin and other undesirable skin deformation problems are avoided. The curve skeleton retains the advantages of the current skeleton driven skinning. It is easy to use and allows full control over the animation process. As a further enhancement, it is also fairly simple to build realistic muscle and fat bulge effect. A practical implementation in the form of a Maya plug-in is created to demonstrate the viability of the technique. Copyright # 2006 John Wiley & Sons, Ltd. Received: 10 April 2006; Revised: 2 May 2006; Accepted: 10 May 2006 </Abstract>
Keywords: skeleton; curve-skeleton; deformation
<H1>Introduction</H1>
A realistic and visually accurate character animation necessitates proper skin deformation of the character models. Skin deformation owes a large part to proper rigging of the characters. The virtual skeleton forms the interface by which the animator can pose or animate the characters. The joint-based skeleton has been very popular in the animation industry for many years and has nearly become a de facto standard. Other technologies like inverse kinematics, forward kinematics, motion capture etc. are built on this hierarchical system of joints. It is plain to see why the joint-based skeleton system is thoroughly integrated into the current production pipeline in animation. Where visual fidelity is of the utmost importance, with respect to film quality animation, a combination of techniques including muscle simulation is used to achieve the realistic best in mesh deformation. The attachment of mesh geometry to the underlying skeleton rig is called ‘skinning’ and this can be understood as a function mapping of the skeleton parameters to a deformation field. 1 One of the common skinning methods in interactive systems is known by the following nomenclatures: sub-space deformation (SSD), smooth skinning, linear blend skinning and enveloping. The process followed by this technique is to assign influence joints and blend weights to each vertex of the character. Transforming the vertex by a weighted combination of the joints local coordinate frames completes skin computation. But in spite of computational performance and ease of use, the joint skeleton skinning is not without its share of problems, particularly where skin deformation is concerned. Unusual deformation artifacts appear in the skin while deforming. Some of the commonly seen problems in joint-based skinning during deformation are: candy wrapper effect during twist deformation and collapsing joints, which would create a rubber-tube like effect. There are certain solutions to circumvent these problems (which we will examine later in the paper) but they have their own drawbacks. Nevertheless, the joint-based system is popular owing to its interactivity and use of minimal animation data. More importantly, it is almost an integral part of the current animation workflow and animators are reluctant to abandon their familiar production practice. The relationship between a skeleton and the skin shape is highly non-linear. The problems of joint-based skeleton skinning mentioned above, in essence arises from under-sampling. The transformations of the two related joints are too far from each other. And with that low-rate sampling they fail to give a good approximation of the deformed skin surface. In this paper, we introduce a novel method called curve skeleton skinning, to overcome the persisting drawbacks of joint skeleton skinning. The basic idea is to represent the relationship between the skeletal movements and the skin deformation in a non-linear continuous fashion. Since a lot of contemporary animation technology is built upon the hierarchical joint-skeleton based system, it is not wise to entirely replace the current practice. What we propose to do is to enhance the current joint skinning system using the curve skeleton skinning and retain the current animation production pattern that the animators are familiar with. While the joint skeleton is a discrete centre line representation of an object, the curve skeleton offers a continuous skeletal representation. Thus a character will have two skeletons: the ordinary joint skeleton and a curve skeleton. The curve skeleton being continuous gives the maximum sampling rate and provides skin deformation transformation without any artifacts. In addition, we will demonstrate how the curve skeleton technique can drive muscle-based systems to achieve realistic muscle deformation during animation. Using our technique, the animator is able to work without digressing from the familiarity of the current joint-based system, but at the same time achieves maximum visual realism in terms of skin deformation. What needs pointing out is that the term curve skeleton has been used for other applications, such as virtual navigation, reduced-model formulation, visualization improvement, surface reconstruction and it was defined as ‘a 1D subset of the medial surface of a 3D object.’ 2 Despite some similarity, it should not be confused with what we are presenting in this paper. One should neither confuse this with the inverse kinematics (IK) spline handle tool provided by the animation package Maya. Despite their seeming similarity, they are in essence very different techniques.
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<H1>Related Work</H1>
In this Section we briefly review some relevant skinning techniques.
<H2>Character Skinning</H2>
Mesh deformations due to skeletal joint influence have undergone significant improvements in the recent years. Some of the normal deformation techniques like free form deformations (FFDs) or lattices can be used in skin deformation techniques. Singh and Kokkevis 3 demonstrate this in their paper. They use surface-oriented FFDs for skinning. An interactive deformation technique for complex geometric objects using curves or wires is detailed in Reference
[ 4 ]. There are basically two main approaches to modeling skin deformations, namely, anatomy-based approach and skin-shape based approach (e.g., example-based skinning). The anatomical approach derives its name from its implementation using anatomical models of muscles and skeletons and other relevant interior structures. These modules undergo deformation when the body moves and a skin simulation and collision detection algorithm is run which would realistically deform the skin where and whenever it is required. Reference [ 5 ]
details a technique of efficient muscle shape deformation using the anatomical skin deformation technique. Reference [ 5 ]
resorts to the creation of a muscle model, which is categorized into two layers: an action line and a surface mesh.
Basically, the action line is the mechanism that drives the deformation. They also implement attractive and repulsive force fields in the form of ellipsoid metaballs to stabilize the action line. Simulation of complex dynamics and performing complex collisions and also providing a visually realistic output form the main strength of the anatomical approach. Incorporating physical properties of anatomy structures can potentially improve realism. Physics can be used either at the muscle level 6 or used to help character rigging. 7 Reference [ 8 ] presents another approach to deformation using an elastic surface layer model. It uses a layered structure of anatomical parts from the inside out, skeleton->bone->fat->skin. The surface is discretized and finite differencing techniques are used to evolve the deformation through time. The drawback comes in the form of computation expense. The anatomy-based approach is therefore used mainly in high-quality film visual effects where anatomical accuracy is a must for believable computer generated characters. The example-based approach forms a suitable alternative where computational expenses are to be minimized. This method takes an interpolative approach to deformation. An artist models certain key poses of the characters where a correlation is maintained for the degrees of freedom, in this case, it would be the joint positions or rotational angles. New poses are interpolated from these key poses. A modified least square fitting technique is used to compute the weights of the deformation and the subsequent generalization of skin movement to other animated poses. In Reference [ 9 ] the algorithm is trained in a statistical manner so that deformation computation for an arbitrary animated pose can be done. They use a technique called multiweight enveloping in place of single-weight enveloping for better deformation. Reference [ 10 ] also implements an example-based approach to deforming meshes by using radial basis functions to supply the interpolation weights and also for shape interpolation. A variation of example-based approach where key example poses are derived from arbitrary unrelated examples is detailed in Reference
[ 11 ] where a range scan is used. Thus example-based approaches have the advantage over anatomical approaches by being computationally faster and also due to the fact that creating example poses are much easier compared to creating detailed anatomically correct models. Most of the described techniques are built upon the existing hierarchical skeletal joint system and modify 10 or even create 9,12 new weight calculations to rectify any sort of physical artifacts in the skin deformation. The example-based approach relies on key sample poses to derive a generalization of deformation, and this becomes a major disadvantage, as this in itself is an expensive and time-consuming process. It is not desirable to create many examples and train the system. Our approach builds upon the existing system using the curve skeleton for a continuous sampling of the skin surface thereby facilitating skin deformations devoid of geometry artifacts. Since our technique falls in between the two approaches, seamless integration with the two is also possible and becomes its strong advantages. A relevant technique to ours is the sweep-based skinning. 13 The body of a character is segmented with a large number of sweep planes which will be transformed by the joint skeleton. These planes are used to guide the transformation of every skin point during animation. With our method, the skin surface does not need to be approximated by sweep surfaces. It will be deformed directly by the underlying curve skeleton, leading to a simpler process.
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<H2>Our Contribution</H2>
Skeleton and skin relationship in the present production pipeline is strictly linear, whereas observation of the various geometry artifacts like candy wrapper and collapsing joints intuitively point to the fact that linear blending or skeletal space deformation falls short in accurately depicting skin deformations because of their non-linear nature. This non-linear nature is explored in Reference
[ 12 ] where a spherical blending is proposed. Only the translation factor is most commonly used for the skin vertices and the rotation factor is not considered. It is our knowledge that the problem reduces after weight painting only when the joint influence fall-off follows a curve pattern. Wang and Philips 9 introduce a multiweight technique to eliminate this problem in a normal joint-based skeleton skinning. However, this requires the generation of a large number of pre-modeled examples in the first place. The solution to the collapsing joints problem, which is to place additional joints 1,14 (placing additional joints is basically bringing a curve nature to the joint chain) near the main joint, has the added problems of: (1) creating a new joint in the hierarchy; (2) joint connections have to be done again to connect the new joint in the existing chain; and (3) painting of weights have to be adjusted to accommodate the new joint. With our curve skeleton technique, the curve serves as a duplicated skeleton to the actual underlying joint skeleton. Effectively any point on the curve can be considered as a joint. In other words, the skeleton is equipped with an infinite number of joints, which will influence the skin deformation. The curve nature of the skeleton makes it easier to manipulate it with a great order of flexibility. The idea to use a curve skeleton side by side with the traditional joint skeleton is conceptually simple and functionally efficient giving realistic skin deformations even under extreme mesh duress. Our curve skeleton technique takes full advantage of the nonlinearity of the skeleton-skin relationship.
<H1>Curve Skeleton</H1>
The curve skeleton can be generated in two ways depending on what the animator supplied in the first place. If the animator supplies a skin model and a skeleton model in the traditional manner, the curve skeleton generation is easy. If on the other hand, the animator supplies only a 3D surface model (not a voxelized representation), the generation of the skeleton becomes slightly more complex in that an additional step is required. A temporary copy of the surface model can be created (during runtime) and voxelized. Once voxelized, a curve skeleton is created using the repulsive force field function. 15 Then the temporary mesh can be deleted and the skeleton can be used with the original surface model. The whole structure of a curve skeleton may involve several curves, which depend on the topology of the original joint skeleton. The three types of topology for skeleton segments used here are:
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<H2>Linear Linkage (Joint WithT wo Links)</H2>
In a linear linkage the centre of the joint gives the first control point (CP) of the curve. Then one Bone_CP each is inserted on the opposite sides of the Joint_CP ( Figure 1a ). Both Bone_CPs have floating positions along the two neighbouring bones, its position being constrained by the angle between the two bones. The reason for the floating position is to eliminate the selfintersection of the skin mesh ( Figure 1 -a1). Before we can predict the exact movement of the Bone_CP, first we should estimate the approximate distance d from the skin surface to the relevant link of the skeleton. The condition for non-self-intersection is to check if the local radius of curvature r at the joint is not less than d. If we analyse the curve function, we can extract the exact expression from the position of Bone_CP, but because the distance d is only an approximate result, it may not fit exactly in the animation. So here the floating position of the Bone_CP is left to the animator to define interactively ( Figure 1 -a2). By providing the animator with more parameters, which he/she can tweak, we grant flexibility and freedom to adjust the animation. One curve is generated.
Figure 1. Three types of skeleton topology: (a) linear linkage structure, it shows a schematic diagram of the curve skeleton with the Bone_CPs inserted on the two sides of the Joint_CP; (a1) shows the curve skeleton within the actual skin mesh, the white crosshairs denote the Bone_CP which is placed to avoid mesh self-intersection; (a2) shows the Floating Bone_CP, the pink line denotes the curve skeleton, the joint skeleton is shown in green, the Bone_CPs on either side of the Joint_CP changes position as the joint angle varies. (b) Bifurcation linkage structure, it shows the schematic blow-up of the abdomen area of the skin model. Here we can see that there are three curves for the linkage. (c) Cross linkage structure, it shows the schematic blow-up of the neck area of the skin model. Here we can see that there are two curves for the linkage.
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<H2>Bifurcation Linkage (Joint With Three Links)</H2>
In anatomical areas like the hip ( Figure 1b ), a fork exists in the joint chain. Hence, three curves are generated, two curves starting from the central link to the two limbs linkage (in the example) and one curve linking the two links.
<H2>Cross Linkage (Joint With Four Links)</H2>
In the neck area, a cross exists in the joint chain. Although it appears to have four links, we only need to generate two curves for the curve skeleton, as seen in Figure 1(c) . As can be seen from the above classification, for a human character, we will use a maximum of three curves for each joint. In most cases, one curve is sufficient.
<H1>Curve Skeleton Motion Synthesis</H1>
<H2>Representation</H2>
In our method, the parameter t on the curve plays an important role in the deformation. We use B-splines to represent the curve skeleton.
<H2>Local Frame Def|nition</H2>
Similar to the joint-based skeleton, each point on the curve in a curve skeleton has a local frame (similar to a Frenet frame) defining the space transformation sampled at that point. This local frame is a function of the parameter t of the curve point. The frame on the curve can be defined in two phases: (i) For the local frame of the point associated with original joints in the joint skeleton. These points normally form the curve segment endings. They can be easily found from the curve definition. At these points, the local frame is defined from the related joint’s local coordinate axis: the x-axis x 0 is the tangent direction at that point on the curve ( Figure 2a ) Then the y 0 and z 0 can be determined by ( Figure 2 ) y 0 1⁄4 z  x 0 z 0 1⁄4 x 0  y 0 (1Þ (ii) For the local frame at any point inside the curve segment: This can be calculated by using the frames at the two endings to perform a linear interpolation.
Figure 2. Local frame definition. (a) Shows the local coordinated axis for both the curve skeleton and the joint. (b) Shows the local coordinate axis at each curve point on the curve skeleton transposed onto the joint skeleton.
h xðtÞ; yðtÞ; zðtÞ i ! ð1 À tÞ h x 1 ; y 1 ; z 1 i þ t h x 2 ; y 2 ; z 2 i (2)
An important feature of the local frame is the centre of the local coordinate system. If the centre is lying on the curve, the deformed skin will move out from underneath the skeleton. The underlying structures like muscles or bones will be exposed. So here the centre of the local frame is translated on to the original skeleton shown in Figure 2b .
<H2>T wist</H2>
When the bone twists around its local x-axis, it will not have any effect on the associated curve skeleton. This is not acceptable. In order to remedy this problem, here on the curve skeleton we define two extra attributes, twist angle and twist distribution. The twist angle can be easily queried from the associated joint. The rotation angle is for the curve ending. For each point on the curve, we still need a twist distribution to define how the curve twists along its path. Normally it is not evenly distributed as can be seen from the twist of a forearm. In order to perform even distribution of twisting, we provide the animator with the freedom to control how the curve twists by manipulating the distribution curve. The distribution curve ( Figure 3a ) is very much like the animation curves in Maya. Here the twist angle is distributed along the distribution curve so that the twisting is smooth and natural.
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Figure 3. Twist operation on curve skeleton: (a) The twist distribution curve is shown in green in the bottom. It helps to evenly distribute the twist along the curve skeleton; (b) Arm twist action on muscles using curve skeleton. Left: The muscle deformation Right: The corresponding skin layer deformation.
<H1>Skinning</H1>
<H2>Binding Skin to the Curve Skeleton</H2>
The process of skin binding is to transform each skin surface point hx,y,zi at the binding pose to the local frame coordinate system hi, t, u, di, where i is the index to the specified curve segment, t is the parameter along that curve segment, u is the rotation angle around the x-axis from the y-axis, d is the distance from the local frame centre. Actually the triple parameter ht, u, di may be considered as being expressed in a cylindrical coordinate system. The values ht, u, di can be easily computed if we can settle the associated curve segment. Thus the challenging part of the work is to find the associated curve segment, and assign the weighting parameter for each curve segment—skin binding. There is a lot of work 16 associated with the traditional joint-based method, like the containment-binding algorithm, point-to-line mapping, Delaunay tetrahedralization. The relevant default weight factors w i of a skin point for the ith curve segment is determined by the distance between the skin point concerned with the relevant curve segments. If a skin point is related with only one curve, which represents the majority of cases, the weight factor is always 1. For those skin points associated with two curve segments, the default weights are proportional to the distances to the relevant curve segments, that is, the further away a skin point is from the curve segment, the smaller the weight is. This is also the case for any skin points associated with three curve segments. In all cases, the summation of the weights are constrained to one, P w i 1⁄4 1. The animator will have freedom to edit the weighting factors in the same way as the smooth skinning. Given that we have a maximum of only three curve segments for each skin point, weight assignment for a skin point is simpler than the traditional smooth skinning method and the computation for skin deformation is computationally cheaper. Smooth skinning usually involves three weights for each skin point and in many cases there could be as many as five weights. This is worsened if additional joints are placed in order to remedy the unpleasant artefacts. The more the joints, the trickier it is to determine the weight distribution. With our curved skeleton, this problem will almost certainly not arise.
<H2>Deforming the Skin With the Curve Skeleton</H2>
Once the skin is bound with the curve skeleton, deforming the skin is pretty straightforward. The local coordinates of each skin surface point are transformed with the associated local frame to obtain the new position in the world coordinate system. So the new point P, is defined by X P 1⁄4 w i M ði;tÞ P Lðu;dÞ (3) i where w i is the weight for the specific curve segment i, M (i,t) is the new transformation matrix at the parameter t position along the curve segment i. As discussed earlier, we use on average a smaller number of weights. This leads to a smaller number of summation terms needed for the calculation of the deformed skin points (see Equation (3)). As a result, our computation speed is at the same order, but is slightly faster than that of the traditional smooth skinning technique.
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Figure 4. Whole body deformation: (a) Rendered result for the character deformation in different stretched poses; (b) Internal curve skeleton layout. The curve skeleton is shown as lines.
<H1>Muscle Deformation</H1>
So far, we have discussed how to realistically skin a character without taking into account the anatomical structures. But muscles will give an added layer of realism to the deformation, especially in regions where the skin is visibly influenced by the underneath muscles. With our curve skeleton technique, muscle deformation can be fully integrated where the muscles are driven and animated by our curve skeletons. One of the best third party muscle simulation systems available called muscleTK 17 deforms the muscle using the so-called action lines. The action line is basically a curve, which defines the direction of deformation. But the disadvantage is that the action line has to be manually animated each frame during animation. Using the proposed curve skeleton, we can realistically deform not only the skin directly (as explained earlier), but also the muscles, in a unified manner. Effectively, each action line is deformed by a curve skeleton, and the action line in turn deforms the muscle. Therefore, we can achieve sophisticated muscle deformations without the tediousness of animating the action lines manually every frame. When the effect of a muscle bending around the joint or the bone is required, we can first transform the control points (CP) of the action line from world space to the associated curve skeleton local frame. These CPs will then be transformed with the curve skeleton, resulting in the muscle bending around the joint or bone being automatically created ( Figure 3b ).
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Figure 5. Muscle deformation: (a) Muscle deformation at the elbow during twist and bend. The action line deformed under the curve skeleton movement; (b) skin deformation for the actions in (a); (c) muscle deformation around the neck; (d) Shows the corresponding skin deformation.
<H1>Implementation of the Curve Skeleton in Maya as a Plug-In</H1>
Maya is the most widely used 3D animation package in the industry. In order for scalability and increasing the feature base of Maya, Autodesk has provided Maya APIs for developers to expand the functionality of Maya. Maya 6.5 and Visual Studio. NET were used in the implementation of the curve skeleton. From an interface point of view, the artist basically works with normal edit point (EP) curve tools to generate the curves according to his/her wish. Once the curve is selected and the plug-in activated, the curves become the skeleton for the skin mesh. Internally, the curve cluster is bound to the joint skeleton so that any movement of the joints affects the curvature of the curve. The local transformations of the curve points are applied to the skin mesh vertices thereby generating deformations on the skin.
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<H1>Conclusion and Future Work</H1>
Skin deformation is closely linked with the movement of the skeleton of a character. It is understandable that the relationship between both is highly non-linear, which poses a challenge if the relationship is to be modeled mathematically correctly. Existing skeletondriven techniques regard it as a much-simplified linear problem, which however, has resulted in unrealistic skin deformation in certain regions of the character body. In this paper we have presented a technique, known as the curve skeleton based skinning, by considering it as a proper non-linear problem. The main advantage of this technique is its consistency with the current animation production practice and the ability to overcome the undesirable drawbacks of skeleton-driven skinning. From the algorithmic point of view, the technique reduces a level of complexity in the skinning and deformation. By layering the curve skeleton on top of the existing joint skeleton, we allow the animator to work conventionally (as in a joint-based system) and yet receive good results. Through a combination of existing practices and newly designed ones, we have successfully created a fusion, which maximizes the efficiency of surface deformation during animation. For an articulated character, we use no more than three weights for any skin point. In fact, for the majority of cases, there is only one weight, which is 1, to be is used. In comparison with the traditional smooth skinning technique that usually requires on average 3–5 weights, our computation speed is faster. One should not confuse the curve skeleton technique with the inverse kinematics (IK) spline handle tool provided by the animation package Maya. Despite their seeming similarity, the objective of the Maya IK spline handle tool is to control the joint positions using a spline. Skin deformation is achieved using the traditional smooth skinning technique. Our curve skeleton is controlled by the joints of a character. The skin is directly deformed by the curve skeleton. The Maya plug-in implementation of the curve skeleton technique has given satisfactory results. One of the main advantages of the curve skeleton skin deformation technique is that, the curve skeleton needs not necessarily be placed on the underlying joint skeleton. With a slight modification utilizing a linear mapping of curve points to the joint skeleton, the plug-in can make use of a displaced curve skeleton which would be useful for subtle deformation on anatomical areas like the armpits. The results of the Maya plug-in can be seen in Figures 4–7. Our current implementation allows both skin and muscle deformation to be modeled within a unified framework. As future work, we will further improve the skinning realism by adding the fat effect. Fat usually deposits between the skin and the muscles. Effective realism occurs when the skin actually slides over the fat. This is especially true in the areas near joints where acute deformation happens. Turner and Thalmann 8 defines the fat layer as a thickness specified at each point on the skin surface and make use of reaction constraints to push the skin the required distance out from the underlying layers. Yang and Zhang 18 devises a fast method for simulating fat in which a fat bulge distribution function is described. They have used a geometric method instead of resorting to a physical simulation method, and gives convincing results without the computational expense of physical simulation. With a small modification, the fat bulge effect can be made even in a curve skeleton-based skinning. The function can be defined under the local frame of the curve skeleton. In the present context, since the skeleton is a curve, distribution and deformation can be linked with the tangent angle at a given number of curve points around the joints. As fat is largely incompressible, when a joint bends, flesh between the adjacent bones will be squeezed, producing bulges immediately near the joint and at the sides. Using curve tangents will provide for an accurate distribution in any given time frame because of the integrated results from the sample multiple curve points.
Figure 6. Limb stretch: (a) shoulder deformation; (b) frontal view of the hip and lower abdomen during limb stretching. The curve skeleton is visible as a curving line. (c) Side view hip and lower abdomen while limb stretching.
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Figure 7. Compare joint skeleton with curve skeleton: (a) collapsing elbow while using joint skeleton; (b) natural effect using the curve skeleton; (c) the candy wrapper problem in joint-based twist. (d) It is absent in the curve skeleton based twist.
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Copyright # 2006 John Wiley & Sons, Ltd.
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CURVE SKELETON SKINNING FOR HUMAN AND CREATURE CHARACTERS
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<H1>ACKNOWLEDGEMENTS</H1>
We are grateful to Autodesk for its donation of the Maya software licenses. We are also grateful to 500 3D-Objects ( http://www.taschen.com ) without whose high-detailed models practical results would have been impossible.
<H1>References</H1>
1. Mohr A, Gleicher M. Building efficient, accurate character skins from examples. ACM Transactions on Graphics 2003; 22(3): 562–568.
2. Cornea ND, Silver D, Min P. Curve skeleton applications. In Proceedings IEEE Visualization, 2005; 95–102.
3. Singh K, Kokkevis E. Skinning character using surfaceoriented free-form deformations. In Proceedings of Graphics Interface, 2000.
4. Singh K, Fiume E. Wires: a geometric deformation technique. Computer Graphics 1998; 405–414.
5. Aubel A, Thalmann D. Interactive modeling of the human musculature. In Proceedings of Computer Animation, 2001.
6. Chadwick J, Haumann D, Parent R. Layered construction for deformable animated characters, computers, SIGGRAPH’89 proceedings, 243–252.
7. Capell S, Burkhart M, Curless B, Duchamp T, Popovi ́ Z. Physically based rigging for deformable characters. In Proceedings of the 2005 ACM Siggraph/Eurographics Symposium on Computer Animation (Los Angeles California, July 29–31, 2005). SCA’05; 2005.
8. Turner R, Thalmann D. The Elastic surface layer model for animated character construction. In Proceedings of CG International, Lausanne, Switzerland, 1993; 399–412.
9. Wang XC, Phillips C. Multi-weight enveloping: leastsquares approximation techniques for skin animation. In SCA’02: Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation, 2002; 129–138.
10. Lewis JP, Cordner M, Fong N. Pose Space Deformation: a unified approach to shape interpolation and skeleton driven deformation. In SIGGRAPH’00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, 2000; 165–172.
11. Allen B, Curless B, Popovic Z. Articulated body deformation from range scan data. In Proceedings of the 29th annual conference on Computer graphics and interactive techniques, 2002; 612–619.
12. Kavan L, Zara J. Spherical blend skinning: a real-time deformation of articulated models. In Proceedings of the 2005 Symposium on Interactive 3D Graphics and Games,2005; 9–16.
13. Hyun D, et al. Sweep-based human deformation. The Visual Computer 2005; 21(8–10): 542–550.
14. Weber J. Run-Time Skin Deformation. In Proceedings of Game Developers Conference, 2000.
15. Cornea N, Silver D, Yuan X, Balasubramanian R. Computing hierarchical curve-skeletons of 3D objects. The Visual Computer 2005; 945–955.
16. http://graphics.ucsd.edu/courses/cse169_w05/3- Skin.htm
17. http://www.cgtoolkit.com/muscletk.htm
18. Yang X, Zhang JJ. Realistic skeleton driven skin deformation. In Third Technical Session on Computer Graphics and Geometric Modeling, TSCG’2005 Suntec Singapore, Springer-Verlag: Singapore, 2005.
Authors’ biographies:
Xiaosong Yang is a research fellow in the National Centre for Computer Animation, Bournemouth Media School, Bournemouth University, United Kingdom. He received his Bachelors (1993) and Masters degree (1996) in computer science from Zhejiang University (China), Ph.D. (2000) in computing mechanics from Dalian University of Technology (China). He worked as postdoc (2000–2002) in the Department of Computer Science and Technology of Tsinghua University for 2 years, and research assistant (2001–2002) at the ‘Virtual Reality, Visualization and Imaging Research Centre’ of Chinese University of Hong Kong. His research interests include 3D modeling, animation, real-time rendering, virtual reality, virtual surgery simulation and computer-aided design. Arun Somasekharan After completing his B. Tech. in computer systems (2001), Arun Somasekharan worked as a software engineer for 3 years during which he helped in the development of an architectural visualisation program. He also worked on client-server based technologies and database systems. During 2004–2005, he took a break from the rigors of the software industry to pursue his MSc in computer animation from Bourne- mouth University. Upon graduation, to further his academic interests, he is currently pursuing his PhD in computer graphics and animation from Bournemouth University. His research interests include game-related technologies, character rigging and animation and programmable visual art. Jian J. Zhang is professor of computer graphics at the National Centre for Computer Animation, Bournemouth Media School, Bournemouth University and head of research at Bournemouth Media School. His research interests include computer graphics, computer animation, physically based simulation, geometric modeling, medical simulation and visualisation.
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X. YANG, A. SOMASEKHARAN AND J. J. ZHANG
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"text": "['presents another approach to deformation using an elastic surface layer model']",
"type": "background_claim",
"id": "T84"
},
{
"offsets": [
[
10029,
10096
]
],
"text": "['It uses a layered structure of anatomical parts from the inside out']",
"type": "background_claim",
"id": "T85"
},
{
"offsets": [
[
10098,
10132
]
],
"text": "['skeleton->bone->fat->skin']",
"type": "data",
"id": "T86"
},
{
"offsets": [
[
10245,
10298
]
],
"text": "['The drawback comes in the form of computation expense']",
"type": "background_claim",
"id": "T87"
},
{
"offsets": [
[
10134,
10243
]
],
"text": "['The surface is discretized and finite differencing techniques are used to evolve the deformation through time']",
"type": "background_claim",
"id": "T88"
},
{
"offsets": [
[
10300,
10329
]
],
"text": "['The anatomy-based approach is']",
"type": "background_claim",
"id": "T89"
},
{
"offsets": [
[
10340,
10468
]
],
"text": "['used mainly in high-quality film visual effects where anatomical accuracy is a must for believable computer generated characters']",
"type": "background_claim",
"id": "T90"
},
{
"offsets": [
[
10470,
10574
]
],
"text": "['The example-based approach forms a suitable alternative where computational expenses are to be minimized']",
"type": "background_claim",
"id": "T91"
},
{
"offsets": [
[
10576,
10634
]
],
"text": "['This method takes an interpolative approach to deformation']",
"type": "background_claim",
"id": "T92"
},
{
"offsets": [
[
10636,
10749
]
],
"text": "['An artist models certain key poses of the characters where a correlation is maintained for the degrees of freedom']",
"type": "background_claim",
"id": "T93"
},
{
"offsets": [
[
10819,
10866
]
],
"text": "['New poses are interpolated from these key poses']",
"type": "background_claim",
"id": "T94"
},
{
"offsets": [
[
10868,
11034
]
],
"text": "['A modified least square fitting technique is used to compute the weights of the deformation and the subsequent generalization of skin movement to other animated poses']",
"type": "background_claim",
"id": "T95"
},
{
"offsets": [
[
11051,
11052
]
],
"text": "['9']",
"type": "data",
"id": "T96"
},
{
"offsets": [
[
11055,
11103
]
],
"text": "['the algorithm is trained in a statistical manner']",
"type": "background_claim",
"id": "T97"
},
{
"offsets": [
[
11112,
11178
]
],
"text": "['deformation computation for an arbitrary animated pose can be done']",
"type": "background_claim",
"id": "T98"
},
{
"offsets": [
[
11180,
11290
]
],
"text": "['They use a technique called multiweight enveloping in place of single-weight enveloping for better deformation']",
"type": "background_claim",
"id": "T99"
},
{
"offsets": [
[
11304,
11306
]
],
"text": "['10']",
"type": "data",
"id": "T100"
},
{
"offsets": [
[
11314,
11471
]
],
"text": "['implements an example-based approach to deforming meshes by using radial basis functions to supply the interpolation weights and also for shape interpolation']",
"type": "background_claim",
"id": "T101"
},
{
"offsets": [
[
11473,
11592
]
],
"text": "['A variation of example-based approach where key example poses are derived from arbitrary unrelated examples is detailed']",
"type": "background_claim",
"id": "T102"
},
{
"offsets": [
[
11614,
11640
]
],
"text": "['where a range scan is used']",
"type": "background_claim",
"id": "T103"
},
{
"offsets": [
[
11609,
11611
]
],
"text": "['11']",
"type": "data",
"id": "T104"
},
{
"offsets": [
[
11647,
11749
]
],
"text": "['example-based approaches have the advantage over anatomical approaches by being computationally faster']",
"type": "background_claim",
"id": "T105"
},
{
"offsets": [
[
11780,
11876
]
],
"text": "['creating example poses are much easier compared to creating detailed anatomically correct models']",
"type": "background_claim",
"id": "T106"
},
{
"offsets": [
[
11878,
12097
]
],
"text": "['Most of the described techniques are built upon the existing hierarchical skeletal joint system and modify 10 or even create 9,12 new weight calculations to rectify any sort of physical artifacts in the skin deformation']",
"type": "background_claim",
"id": "T107"
},
{
"offsets": [
[
12099,
12194
]
],
"text": "['The example-based approach relies on key sample poses to derive a generalization of deformation']",
"type": "background_claim",
"id": "T108"
},
{
"offsets": [
[
12200,
12233
]
],
"text": "['this becomes a major disadvantage']",
"type": "background_claim",
"id": "T109"
},
{
"offsets": [
[
12238,
12295
]
],
"text": "['this in itself is an expensive and time-consuming process']",
"type": "background_claim",
"id": "T110"
},
{
"offsets": [
[
12297,
12361
]
],
"text": "['It is not desirable to create many examples and train the system']",
"type": "background_claim",
"id": "T111"
},
{
"offsets": [
[
12363,
12546
]
],
"text": "['Our approach builds upon the existing system using the curve skeleton for a continuous sampling of the skin surface thereby facilitating skin deformations devoid of geometry artifacts']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
12554,
12603
]
],
"text": "['our technique falls in between the two approaches']",
"type": "own_claim",
"id": "T113"
},
{
"offsets": [
[
12605,
12689
]
],
"text": "['seamless integration with the two is also possible and becomes its strong advantages']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
12691,
12747
]
],
"text": "['A relevant technique to ours is the sweep-based skinning']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
12749,
12751
]
],
"text": "['13']",
"type": "data",
"id": "T116"
},
{
"offsets": [
[
12752,
12872
]
],
"text": "['The body of a character is segmented with a large number of sweep planes which will be transformed by the joint skeleton']",
"type": "background_claim",
"id": "T117"
},
{
"offsets": [
[
12874,
12960
]
],
"text": "['These planes are used to guide the transformation of every skin point during animation']",
"type": "background_claim",
"id": "T118"
},
{
"offsets": [
[
12962,
13046
]
],
"text": "['With our method, the skin surface does not need to be approximated by sweep surfaces']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
13048,
13139
]
],
"text": "['It will be deformed directly by the underlying curve skeleton, leading to a simpler process']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
13684,
13768
]
],
"text": "['Skeleton and skin relationship in the present production pipeline is strictly linear']",
"type": "background_claim",
"id": "T121"
},
{
"offsets": [
[
13900,
13999
]
],
"text": "['linear blending or skeletal space deformation falls short in accurately depicting skin deformations']",
"type": "background_claim",
"id": "T122"
},
{
"offsets": [
[
14011,
14034
]
],
"text": "['their non-linear nature']",
"type": "background_claim",
"id": "T123"
},
{
"offsets": [
[
14036,
14070
]
],
"text": "['This non-linear nature is explored']",
"type": "background_claim",
"id": "T124"
},
{
"offsets": [
[
14087,
14089
]
],
"text": "['12']",
"type": "data",
"id": "T125"
},
{
"offsets": [
[
14098,
14130
]
],
"text": "['a spherical blending is proposed']",
"type": "background_claim",
"id": "T126"
},
{
"offsets": [
[
14132,
14245
]
],
"text": "['Only the translation factor is most commonly used for the skin vertices and the rotation factor is not considered']",
"type": "background_claim",
"id": "T127"
},
{
"offsets": [
[
14272,
14313
]
],
"text": "['the problem reduces after weight painting']",
"type": "background_claim",
"id": "T128"
},
{
"offsets": [
[
14324,
14376
]
],
"text": "['the joint influence fall-off follows a curve pattern']",
"type": "data",
"id": "T129"
},
{
"offsets": [
[
14395,
14396
]
],
"text": "['9']",
"type": "data",
"id": "T130"
},
{
"offsets": [
[
14378,
14394
]
],
"text": "['Wang and Philips']",
"type": "background_claim",
"id": "T131"
},
{
"offsets": [
[
14397,
14498
]
],
"text": "['introduce a multiweight technique to eliminate this problem in a normal joint-based skeleton skinning']",
"type": "background_claim",
"id": "T132"
},
{
"offsets": [
[
14509,
14598
]
],
"text": "['this requires the generation of a large number of pre-modeled examples in the first place']",
"type": "background_claim",
"id": "T133"
},
{
"offsets": [
[
14683,
14687
]
],
"text": "['1,14']",
"type": "data",
"id": "T134"
},
{
"offsets": [
[
14600,
14645
]
],
"text": "['The solution to the collapsing joints problem']",
"type": "background_claim",
"id": "T135"
},
{
"offsets": [
[
14793,
14818
]
],
"text": "['has the added problems of']",
"type": "background_claim",
"id": "T136"
},
{
"offsets": [
[
14824,
14861
]
],
"text": "['creating a new joint in the hierarchy']",
"type": "background_claim",
"id": "T137"
},
{
"offsets": [
[
14867,
14953
]
],
"text": "['joint connections have to be done again to connect the new joint in the existing chain']",
"type": "background_claim",
"id": "T138"
},
{
"offsets": [
[
14963,
15031
]
],
"text": "['painting of weights have to be adjusted to accommodate the new joint']",
"type": "background_claim",
"id": "T139"
},
{
"offsets": [
[
15033,
15149
]
],
"text": "['With our curve skeleton technique, the curve serves as a duplicated skeleton to the actual underlying joint skeleton']",
"type": "own_claim",
"id": "T140"
},
{
"offsets": [
[
15151,
15214
]
],
"text": "['Effectively any point on the curve can be considered as a joint']",
"type": "own_claim",
"id": "T141"
},
{
"offsets": [
[
15232,
15333
]
],
"text": "['the skeleton is equipped with an infinite number of joints, which will influence the skin deformation']",
"type": "own_claim",
"id": "T142"
},
{
"offsets": [
[
15335,
15434
]
],
"text": "['The curve nature of the skeleton makes it easier to manipulate it with a great order of flexibility']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
15436,
15633
]
],
"text": "['The idea to use a curve skeleton side by side with the traditional joint skeleton is conceptually simple and functionally efficient giving realistic skin deformations even under extreme mesh duress']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
15635,
15738
]
],
"text": "['Our curve skeleton technique takes full advantage of the nonlinearity of the skeleton-skin relationship']",
"type": "own_claim",
"id": "T145"
},
{
"offsets": [
[
15803,
15909
]
],
"text": "['The curve skeleton can be generated in two ways depending on what the animator supplied in the first place']",
"type": "own_claim",
"id": "T146"
},
{
"offsets": [
[
15914,
15995
]
],
"text": "['the animator supplies a skin model and a skeleton model in the traditional manner']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
15997,
16034
]
],
"text": "['the curve skeleton generation is easy']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
16058,
16103
]
],
"text": "['the animator supplies only a 3D surface model']",
"type": "data",
"id": "T149"
},
{
"offsets": [
[
16138,
16237
]
],
"text": "['the generation of the skeleton becomes slightly more complex in that an additional step is required']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
16239,
16322
]
],
"text": "['A temporary copy of the surface model can be created (during runtime) and voxelized']",
"type": "own_claim",
"id": "T151"
},
{
"offsets": [
[
16410,
16412
]
],
"text": "['15']",
"type": "data",
"id": "T152"
},
{
"offsets": [
[
16324,
16408
]
],
"text": "['Once voxelized, a curve skeleton is created using the repulsive force field function']",
"type": "own_claim",
"id": "T153"
},
{
"offsets": [
[
16413,
16512
]
],
"text": "['Then the temporary mesh can be deleted and the skeleton can be used with the original surface model']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
17547,
17637
]
],
"text": "['The reason for the floating position is to eliminate the selfintersection of the skin mesh']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
17640,
17652
]
],
"text": "['Figure 1 -a1']",
"type": "data",
"id": "T156"
},
{
"offsets": [
[
17655,
17822
]
],
"text": "['Before we can predict the exact movement of the Bone_CP, first we should estimate the approximate distance d from the skin surface to the relevant link of the skeleton']",
"type": "own_claim",
"id": "T157"
},
{
"offsets": [
[
17824,
17942
]
],
"text": "['The condition for non-self-intersection is to check if the local radius of curvature r at the joint is not less than d']",
"type": "own_claim",
"id": "T158"
},
{
"offsets": [
[
17978,
18042
]
],
"text": "['we can extract the exact expression from the position of Bone_CP']",
"type": "own_claim",
"id": "T159"
},
{
"offsets": [
[
17947,
17976
]
],
"text": "['we analyse the curve function']",
"type": "data",
"id": "T160"
},
{
"offsets": [
[
18056,
18100
]
],
"text": "['the distance d is only an approximate result']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
18102,
18141
]
],
"text": "['it may not fit exactly in the animation']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
18146,
18235
]
],
"text": "['here the floating position of the Bone_CP is left to the animator to define interactively']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
18238,
18250
]
],
"text": "['Figure 1 -a2']",
"type": "data",
"id": "T164"
},
{
"offsets": [
[
18253,
18381
]
],
"text": "['By providing the animator with more parameters, which he/she can tweak, we grant flexibility and freedom to adjust the animation']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
19929,
19961
]
],
"text": "['a fork exists in the joint chain']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
19881,
19913
]
],
"text": "['In anatomical areas like the hip']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
19916,
19925
]
],
"text": "['Figure 1b']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
19970,
19996
]
],
"text": "['three curves are generated']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
19998,
20064
]
],
"text": "['two curves starting from the central link to the two limbs linkage']",
"type": "own_claim",
"id": "T170"
},
{
"offsets": [
[
20086,
20117
]
],
"text": "['one curve linking the two links']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
20204,
20255
]
],
"text": "['In the neck area, a cross exists in the joint chain']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
20266,
20295
]
],
"text": "['it appears to have four links']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
20297,
20355
]
],
"text": "['we only need to generate two curves for the curve skeleton']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
20368,
20379
]
],
"text": "['Figure 1(c)']",
"type": "data",
"id": "T175"
},
{
"offsets": [
[
20451,
20503
]
],
"text": "['we will use a maximum of three curves for each joint']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
20432,
20449
]
],
"text": "['a human character']",
"type": "data",
"id": "T177"
},
{
"offsets": [
[
20505,
20543
]
],
"text": "['In most cases, one curve is sufficient']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
20670,
20756
]
],
"text": "['In our method, the parameter t on the curve plays an important role in the deformation']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
22184,
22268
]
],
"text": "['An important feature of the local frame is the centre of the local coordinate system']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
22273,
22305
]
],
"text": "['the centre is lying on the curve']",
"type": "data",
"id": "T181"
},
{
"offsets": [
[
22307,
22367
]
],
"text": "['the deformed skin will move out from underneath the skeleton']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
22369,
22432
]
],
"text": "['The underlying structures like muscles or bones will be exposed']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
22594,
22633
]
],
"text": "['the bone twists around its local x-axis']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
22635,
22695
]
],
"text": "['it will not have any effect on the associated curve skeleton']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
22697,
22719
]
],
"text": "['This is not acceptable']",
"type": "own_claim",
"id": "T186"
},
{
"offsets": [
[
22958,
23067
]
],
"text": "['For each point on the curve, we still need a twist distribution to define how the curve twists along its path']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
23069,
23106
]
],
"text": "['Normally it is not evenly distributed']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
23127,
23149
]
],
"text": "['the twist of a forearm']",
"type": "data",
"id": "T189"
},
{
"offsets": [
[
23202,
23313
]
],
"text": "['we provide the animator with the freedom to control how the curve twists by manipulating the distribution curve']",
"type": "own_claim",
"id": "T190"
},
{
"offsets": [
[
23315,
23337
]
],
"text": "['The distribution curve']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
23340,
23349
]
],
"text": "['Figure 3a']",
"type": "data",
"id": "T192"
},
{
"offsets": [
[
23352,
23398
]
],
"text": "['is very much like the animation curves in Maya']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
23400,
23507
]
],
"text": "['Here the twist angle is distributed along the distribution curve so that the twisting is smooth and natural']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
24960,
25002
]
],
"text": "['we can settle the associated curve segment']",
"type": "data",
"id": "T195"
},
{
"offsets": [
[
24913,
24956
]
],
"text": "['The values ht, u, di can be easily computed']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
25009,
25140
]
],
"text": "['the challenging part of the work is to find the associated curve segment, and assign the weighting parameter for each curve segment']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
25178,
25180
]
],
"text": "['16']",
"type": "data",
"id": "T199"
},
{
"offsets": [
[
25181,
25231
]
],
"text": "['associated with the traditional joint-based method']",
"type": "background_claim",
"id": "T197"
},
{
"offsets": [
[
25155,
25177
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A11 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A11_M01_Composable_Controllers_for_Physics-Based_Character_Animation_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/383259.383287
<Title>Composable Controllers for Physics-Based Character Animation</Title>
Petros Faloutsos 1⁄2 Michiel van de Panne 3⁄4 1⁄2 Demetri Terzopoulos
1
University of Toronto, Department of Computer Science 3⁄4 Motion Playground, Inc. New York University, Courant Institute, Computer Science Department
Figure 1: A dynamic “virtual stuntman” falls to the ground, rolls over, and rises to an erect position, balancing in gravity.
<Abstract>An ambitious goal in the area of physics-based computer animation is the creation of virtual actors that autonomously synthesize realistic human motions and possess a broad repertoire of lifelike motor skills. To this end, the control of dynamic, anthropomorphic figures subject to gravity and contact forces remains a difficult open problem. We propose a framework for composing controllers in order to enhance the motor abilities of such figures. A key contribution of our composition framework is an explicit model of the “pre-conditions” under which motor controllers are expected to function properly. We demonstrate controller composition with pre-conditions determined not only manually, but also automatically based on Support Vector Machine (SVM) learning theory. We evaluate our composition framework using a family of controllers capable of synthesizing basic actions such as balance, protective stepping when balance is disturbed, protective arm reactions when falling, and multiple ways of standing up after a fall. We furthermore demonstrate these basic controllers working in conjunction with more dynamic motor skills within a prototype virtual stuntperson. Our composition framework promises to enable the community of physics-based animation practitioners to easily exchange motor controllers and integrate them into dynamic characters.</Abstract>
Keywords: Computer Animation, Character Animation, PhysicsBased Animation Control, Physics-Based Modeling CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation; I.6.8 [Simulation and Modeling]: Types of Simulation—Animation
0 To appear in the Proceedings of SIGGRAPH 2001 (Los Angeles, CA, August 12–17, 2001). In Computer Graphics Proceedings, Annual Conference Series, 2001, ACM SIGGRAPH, in press.
<H1>1 Introduction</H1>
Despite the considerable history of progress in animating virtual humans [ 3 , 7 ], physics-based animated characters with a large repertoire of motor skills have so far been elusive. This may seem surprising in view of the recent successes in implementing a slew of specialist controllers capable of realistically synthesizing the complex dynamics of running, diving, and various gymnastic maneuvers
[ 16 ]. While a divide-and-conquer strategy is clearly prudent in coping with the enormous variety of controlled motions that humans and other animals may perform, little effort has been directed at how the resulting control solutions may be integrated to yield composite controllers with significantly broader functionalities. For example, if researcher A creates a walking controller for a dynamic character while researcher B creates a running controller for the same articulated model, it would be beneficial if they could share their controllers (perhaps through an e-mail exchange) and easily create a composite controller enabling the character to both walk and run. This is a difficult problem, but its resolution would help pave the way towards controller libraries for dynamic animation which communities of practitioners could utilize and to which they could contribute. In this paper, we propose a simple yet effective framework for composing specialist controllers into more general and capable control systems for dynamic characters. In our framework, individual controllers are black boxes encapsulating control knowledge that is possibly gleaned from the biomechanics literature, derived from the robotics control literature, or developed specifically for animation control. Individual controllers must be able to determine two things: (1) a controller should be able to determine whether or not it can take the dynamic character from its current state to some desired goal state, and (2) an active controller should be able to determine whether it is operating nominally, whether it has succeeded, or whether it has failed. Any controller that can answer these queries may be added to a pool of controllers managed by a supervisor controller whose goal is to resolve more complex control tasks. An important technical contribution within our controller composition framework is an explicit model of pre-conditions. Preconditions characterize those regions of the dynamic figure’s state space within which an individual controller is able to successfully carry out its mission. Initially, we demonstrate the successful composition of controllers based on manually determined pre-conditions. We then proceed to investigate the question of whether pre-conditions can be determined automatically. We devise a promising solution which employs Support Vector Machine (SVM) learning theory. Our novel application of this technique learns appropriate pre-conditions through the repeated sampling of individual controller behavior in operation. As a testbed of our techniques, we are developing a physicallysimulated animated character capable of a large repertoire of motor skills. An obvious application of such a character is the creation of a virtual stuntperson: the dynamic nature of typical stunts makes them dangerous to perform, but also makes them an attractive candidate for the use of physics-based animation. The open challenge here lies in developing appropriate control strategies for specific actions and ways of integrating them into a coherent whole. In this paper, we demonstrate families of composable controllers for articulated skeletons whose physical parameters reflect anthropometric data consistent with a fully-fleshed adult male. One family of controllers is for a 37 degree-of-freedom (DOF) 3D articulated skeleton, while a second family of controllers has been developed for a comparable 16 DOF 2D articulated skeleton. While the 3D skeleton illustrates the ultimate promise of the technique, the easier control associated with the 2D skeleton allows for more rapid prototyping of larger families of controllers and more careful analysis of their operation. As has been recognized in the robotics literature, the control of broad skilled repertoires of motion remains very much an open problem even for 2D articulated figures. Fig. 1 illustrates the 3D dynamic character autonomously performing a complex control sequence composed of individual controllers responsible for falling reactions, rolling-over, getting up, and balancing in gravity. The upright balancing dynamic figure is pushed backwards by an external force; its arms react protectively to cushion the impact with the ground; the figure comes to rest in a supine position; it rolls over to a prone position, pushes itself up on all fours, and rises to its feet; finally it balances upright once again. A subsequent disturbance will elicit similar though by no means identical autonomous behavior, because the initial conditions and external forces will usually not be exactly the same. Control sequences of such intricacy for fully dynamic articulated figures are unprecedented in the physics-based animation literature. After reviewing related prior work in the next section, we present the details of our control framework in Section 3. We then investigate the question of determining pre-conditions in Section 4. Section 5 describes the articulated figure models and the software system we use to implement the control framework. Section 6 presents the details of the example in Fig. 1 along with several other examples that demonstrate the effectiveness of our framework. Section 7 concludes the paper and discusses avenues for future research opened up by our work.
<H1>2 Previous Work</H1>
The simulation and animation of human characters is a challenging problem in many respects. Comprehensive solutions must aspire to distill and integrate knowledge from biomechanics, robotics, control, and animation. Models for human motion must also meet a particularly high standard, given our familiarity with what the results should look like. Not surprisingly, a divide-and-conquer strategy is evident in most approaches, focusing efforts on reproducing particular motions in order to yield a tractable problem and to allow for comparative analysis. The biomechanics literature is a useful source of predictive models for specific motions, typically based on experimental data supplemented by careful analysis. These models target applications such as medical diagnosis, the understanding and treatment of motor control problems, the analysis of accidents and disabilities, and high-performance athletics. Computer simulation is becoming an increasingly useful tool in this domain as the motion models evolve to become more complex and comprehensive
[ 26 , 27 , 29 ]. Given the challenge of achieving high-fidelity motion models for individual motions, there have been fewer efforts towards integrated solutions applicable to multiple motions. Reference [ 26 ]
is one such example.
Robotics research has made remarkable progress in the successful design of a variety of legged robots
[ 28 ] and, more recently, bipedal robots with anthropomorphic aspirations
[ 23 ]. Despite their limited motion repertoires and rather deliberate movements, these robotic systems are truly engineering marvels. The work in [ 1 ] provides a good summary of behavioral architectures explored in the context of robotics. A 3 DOF ball-juggling robot is described in
[ 6 ]
which uses a theory of behavior composition, although the practicality of extending the method to high-DOF dynamic models of human motions is unclear. Computer animation is to a large extent unencumbered by the exacting fidelity requirements of biomechanical models and the mechanical limitations of robotic systems. This has spawned a great variety of kinematic and dynamic models for character motion
[ 3 , 4 , 7 ]. While motion capture solutions based on blending and warping techniques may give satisfactory results for such tasks in the short term, controller based approaches reveal more about the physics, planning, and control of such motions and they therefore serve as a basis for more general solutions. Dynamically simulated characters were first proposed over 15 years ago
[ 2 , 34 ]
and since then have progressed in sophistication in a variety of directions. Controllers have been successfully designed for specific human motions such as walking, running, vaulting, cycling, etc.
[ 16 , 22 , 35 ]. Dynamically simulated articulated characters equipped with an integrated, wide-ranging repertoire of motor skills currently remain an unachieved goal. Some positive steps in this direction are evident, however.
Examples include an integrated repertoire of motor controllers for biomechanically animated fish
[ 30 ], a methodology for controller design and integration applicable to simple figures
[ 32 ], a demonstration of successful integration for selected diving and gymnastic motions
[ 35 ], and adapting a controller designed for one character to work on another character
[ 17 ]. The work of Wooten
[ 35 ] is the most relevant as an example of a sequence of successive transitions between several controllers for human motions such as leaping, tumbling, landing, and balancing. Transitions are realized by including the end state of some controllers in the starting states of other controllers. A digital biomechanics laboratory is proposed by Boston Dynamics, Inc. [ 20 ]
as a tool for simulating a wide range of human motion. This currently remains ambitious work in progress. Our work is aimed at creating dynamic human characters with broadly integrated action repertoires. Unlike previous work focusing on specific athletic movements, our methodology is to begin with a core set of simple actions, including balancing, small steps, falling reactions, recovery from falls, standing up from a chair, and others. In the present paper, we do not cover in any appreciable detail the design of individual controllers to effect such basic actions. 1 Rather, our contribution here is a framework for composing individual controllers, however they may be designed, into more capable control systems for dynamic characters. An interesting tech-
1 Full details about the individual controllers that we have designed are presented elsewhere
[ 10 ].
nical contribution within our controller composition framework is the introduction of a learning approach for automatically determining controller pre-conditions. Our pre-condition learning algorithm adds to the growing body of learning algorithms that have been successfully applied in the context of computer animation in recent years [ 14 , 15 ].
<H1>3 Controller Composition Framework</H1>
In our controller composition framework, we consider individual controllers as black boxes which are managed by a simple supervisor controller. When no controller is active, the supervisor polls the pool of controllers, querying each whether it can handle the transition of the dynamic character from its current state to the desired goal state. Individual controllers return an integer confidence/suitability score when queried in order to bid on becoming the active controller. In our implementation, controllers that can perform a sensible action given the current state of the character return an integer in the range 1⁄2 1⁄21⁄4 , while those that can handle the current state as well as guarantee a transition to the desired state, return an integer in the range 1⁄21⁄4 3⁄41⁄4 . Lastly, a value of 1⁄4 means that a controller is unsuited for the current state. The controller that returns the highest score becomes active. While this scoring scheme potentially allows for a nuanced evaluation of the controller suitability in terms of criteria such as probability of success or energy used, our current controllers resort to a simpler scheme. This consists of a binary success/failure evaluation multiplied by a weighting factor assigned to each controller that serves to establish a relative preference ordering. The power of this scheme stems from the following attributes:
Simplicity: The composition method is straightforward and easy to implement. It does not appreciably burden the controller design task. Generality: The composition method does not restrict the design of individual controllers. Each controller can be as primitive or as sophisticated as its designer wishes.
<H2>3.1 Controller Abstraction</H2>
A controller within the pool of available controllers can be as simple as a constant force, or as complex as a structured hierarchy of multiple levels of control abstraction. For example, as more controllers are added to the system, we may wish to group all the walking and running controllers together into a cluster that can be treated as one encapsulated controller. Regardless of the encapsulation, our composition method requires controllers to define pre-conditions, post-conditions and expected performance. Pre-conditions are a set of conditions over the state of the character and the environment. If these conditions are met then the controller can operate and possibly enable the character to satisfy the post-conditions. Assuming that the pre-conditions were met, the post-conditions define a range of states for the final state of the character after the execution of the controller. In other words the controller realizes a mapping between a domain of input states to a range of output states for the character. Because of unexpected changes in the environment, this mapping may not always succeed, which motivates the notion of expected performance. The controller should be able to evaluate its performance in order to detect failure at any point during its operation. To do this, the controller must at all times have knowledge of the current and expected state of the character or the environment. Defining the pre-conditions, post-conditions, and expected performance for complex characters, motions, and environments is not a straightforward task. However, we believe that the effort required to generate these specifications is a fair and necessary price to pay to achieve the benefits of composability. Controllers that adhere to these specifications can form a pool of available controllers managed by the supervising controller. Fig. 2 presents an overview of the supervising controller’s function and its interaction with the individual controllers at every time step of the simulation. Before we elaborate on pre-conditions, post-conditions, and expected ing quantities performance and symbols: in subsequent The state sections, Õ let Ü Ü us 1⁄4 define of a figure the followis the vector of generalized positions Ü and velocities Ü , where the dot indicates a time derivative. The position and velocity of the center of mass are denoted as and respectively. The base of support of a figure (often called the support polygon) is denoted as Ë . It is represented by a polygon that surrounds the foot or feet that are in contact with the ground at any given time.
Supervising controller At every time step: if( no active_controller ) Controller for all controllers i =1: N if( controller[i].can_handle() == true) { Preconditions PostConditions put controller[i] into candidates end if Expected Performance end for active_controller = arbitrate(candidates) else status = active_controller.getStatus() endif
Figure 2: Controller selection and arbitration during simulation.
<H2>3.2 Pre-Conditions</H2>
In general, pre-conditions are relationships and constraints involving several different parameters. We have used the following parameters in our work:
The initial state Õ of the figure. Most of our controllers can operate within a small region of the state space which we denote Ê ́ Õ μ . Environmental parameters. These include the contact points between the character and the ground, as well as the normal of the ground and the amount of friction at the contact points. In the following we denote conditions (generally indicated by the letter ) on the environment parameters as . The balance of the figure. Usually, this is indicated by the relative position and velocity between the figure’s center of mass and the base of support. Typically, if the projection of along the gravity vector does not intersect the base of support Ë , the figure is considered to be unbalanced. We denote the balance conditions as ́ Ë μ . A Ê ́ target Õ Ø μ , which state Õ can Ø , or be in provided general a by target the region user. of the state space
Pre-conditions consist of unions of instances of the above conditions and are denoted
(1)
The determination of pre-conditions is crucial to the success of our composition framework and will be examined in detail in Section 4.
<H2>3.3 Post-Conditions</H2>
Successful operation of a controller brings the character from an initial state, as defined by the pre-conditions, to a desired state or a desired region Ê ́ Õ Ó μ in the state space. This region along with balance and possibly environmental constraints form the postconditions of a controller:
(2)
Note that the pre-conditions may reference a subset of the postconditions that is sufficient to characterize what the controller can achieve. In general, however, the post-conditions are different from the pre-conditions. For example, while a pre-condition for a falling controller requires that the center of mass be moving, the postconditions require that the center of mass be at rest.
<H2>3.4 Expected Performance</H2>
Our framework permits the automatic selection of the appropriate controller based on the information provided by the controllers themselves. Only the individual controllers can detect whether they are operating normally or whether failure is imminent. Failure in our case means that the controller cannot meet its post-conditions Ç . The controller may fail because of a sudden change in the environment or because of badly designed pre-conditions. The sooner a controller can detect failure the sooner another more appropriate controller can take over. This is important for making a character behave naturally. For example, the character should not attempt to continue a walking gait if it has lost its balance and it is falling. In our implementation, the expected performance consists of expressions similar to those of the pre-conditions È . In particular if the controller successfully completes its task in the time interval Ø 1⁄2 , Ø 3⁄4 , then ́ Ø 1⁄2 μ 3⁄4È and ́ Ø 3⁄4 μ 3⁄4Ç .
<H2>3.5 Transitions</H2>
Transitions between controllers are not explicitly modeled as they would be in a finite state machine. They occur implicitly in response to the evolution of the motion over time, as the system state traverses the “regions-of-competency” of the various controllers. Nevertheless, given that most controllers are designed for specific situations, typical patterns of controller activation occur. Fig. 3 shows the family of controllers designed for the 3D dynamic character and their typical transition patterns. For example, the controllers and transitions used in achieving the motion shown in Fig. 1 is given by balance fall default rollover prone-tostanding balance. Fig. 4 similarly shows the family of controllers designed for the 2D dynamic character and their typical transition patterns. Note that not all possible transitions are shown in either of Figs. 3 and 4. For example, the prone-to-standing fall transition can occur if the figure is given a sufficiently strong push while rising. Most of the transitions which are not shown but are still practically feasible are of this nature, dealing with falling behaviors. Note that the fall controller always responds to the specific direction of the current fall. Any transition involves one controller being deactivated and another being activated. A controller can become deactivated (and thereby elicit a transition) for one of three reasons. First, it may relinquish control by declaring success upon reaching its postcondition, as is the case for a standup controller which has successfully returned the character to a standing position. Second, user intervention may elicit a transition. The controllers designed for sitting or balanced standing will retain control until intervention by a user (or by a higher level planner) forces a desired transition. Thus, when the 2D character is balanced a user-driven process must choose among the next plausible actions, namely one of sit, walk, or dive (see Fig. 4 ). Third, a controller may detect failure, as will be the case for unpredictable events such as a push or an unforeseen obstacle causing the character to trip. The transitions in Figs. 3 and 4 are labelled according to the type of controller deactivations which typically elicit the given transition patterns. We note that our framework is designed to work in interactive settings. As such, controllers typically start with slightly different initial conditions each time they are invoked, the user can interact with the character at any time, and generally there are no guarantees that the controller will reach the same end state each time it operates. As a result, the transition graph is dynamic in structure.
Figure 3: Controllers and typical transitions for 3D figure
Figure 4: Controllers and typical transitions for 2D figure
<H1>4 Determining Pre-Conditions</H1>
For controllers associated with complex dynamic characters, determining the exact region of the state space and the general conditions that determine success or failure of the controller is in general a non-trivial matter. In this section, we address this problem via manual and automatic approaches. The manual approach allows designers to incorporate their knowledge within controllers, whereas the automatic approach is based on machine learning techniques.
<H2>4.1 Manual Approach</H2>
For certain cases, suitable pre-conditions for specific controllers may be found in the biomechanics literature
[ 8 , 25 ]. For example Pai and Patton [ 25 ]
present a comprehensive study of balance in the sagittal plane and identify the conditions under which a human can compensate for postural disturbances and maintain balance without stepping. For certain other cases, the pre-conditions are trivially defined by the desired motion itself. Certain controllers function as intermediate stages between other controllers. If controller B is the intermediate step between A and C then the postconditions of A dictate the pre-conditions of B and similarly the pre-conditions of C define the post-conditions of B. Finally, in some cases the pre-conditions are computed by manual experimentation. For example a simple balance controller based on an inverted pendulum model
[ 12 ]
has intrinsic stability that can tolerate small disturbances. After the controller has been designed, repeated testing under disturbances of increasing magnitude can yield an approximation of the pre-conditions and the post-conditions. In any case, the designer of a controller presumably understands the way the controller operates, and thus is able to provide high level conditions on its success or failure. For example, the designer of a walking controller knows if the controller can operate when the walking surface has minimal friction properties. Also, human motion is shaped by notions such as comfort, and only the designer can take this into account. For example, if a person is pushed while standing he/she might take a protective step because it may be more comfortable to do so instead of maintaining an inverted pendulum balancing strategy. Similarly, the way people react to slipping and imbalance and the protective behaviors they employ are largely age dependent.
<H2>4.2 Automatic, Learning Approach</H2>
In this section, we introduce an automatic, machine learning approach to determining pre-conditions, which is based on systematically sampling the performance of controllers. Our method uses a machine learning algorithm attributed to Vapnik
[ 33 ] known as Support Vector Machines (SVMs), which has recently attracted much attention, since in most cases the performance of SVMs matches or exceeds that of competing methods.
<H3>4.2.1 Support vector machines (SVMs)</H3>
SVMs are a method for fitting functions to sets of labeled training data. The functions can be general regression functions or they can be classification functions. In our application, we use simple classification functions with binary outputs which encode the success or failure of a controller. Burges
[ 5 ]
provides an excellent tutorial on SVMs. Mathematically, we are given Ð observations, each consisting of an dimensional vector Ü 3⁄4 1⁄2 Ð and the associated “truth” Ý 3⁄4 1⁄2 1⁄2 provided by a trusted source. Here, Ý 1⁄2 labels a positive example—in our application, the observed success of a controller applied when the dynamic figure is in state Ü — while Ý 1⁄2 labels a negative example—the failure of the controller applied to state Ü . The set of observations Ü Ý is called the training set. The SVM is a machine whose task is to learn the mapping Ü Ý from a training set. The SVM is defined by functional mappings of the form Ü ́ Ü « μ , where « are parameters. A particular choice of « generates a “trained” SVM. In a trained SVM, the sign of the decision function ́ Ü μ represents the class assigned to a test data point Ü . In our application, a properly trained SVM predicts if a controller will succeed ( ́ Ü μ 1⁄4 ) or fail ( ́ Ü μ 1⁄4 ) on a given state Ü of the dynamic character. How does one train an SVM? In the simplest case of a linear SVM with separable training data, there exists a decision boundary separating positive from negative examples which takes the form of a “separating hyperplane” in . The SVM training algorithm computes the separating hyperplane with the largest margin · · , where · ( ) is the shortest distance from the separating hyperplane to the closest positive (negative) example. SVM training requires the solution of a quadratic programming optimization problem involving a Lagrange multiplier « for every datapoint in the training set. Those datapoints in the solution with corresponding « 1⁄4 are called support vectors. The support vectors are critical elements of the training set. They lie closest to the separating hyperplane. If other observations in the training set are moved (subject to certain restrictions) or removed and SVM training is repeated, the same separating hyperplane will result. To use a trained SVM, we simply determine on which side of the decision boundary a given test data point Ü lies and assign the corresponding class label to that point. The linear SVM is easily generalized to nonseparable training data. Furthermore, it is straightforward to generalize the theory to encompass nonlinear SVMs for which the decision boundaries are no longer hyperplanes (i.e., the decision function are no longer linear functions of the data). The trick, in principle, is to map the data to some higher (possibly infinite) dimensional space in which the linear theory can be applied. This is easily done by introducing kernel functions à ́ Ü Ü μ , such as the polynomial kernel à (RBF) ́ Ü Ý kernel μ ́ Ü Ã ¡ ́ Ü Ý · Ý μ 1⁄2μ Ô , ÜÔ ́ or the Ü Gaussian Ý 3⁄4 3⁄4 or 3⁄4 μ radial . For the basis mathematfunction ical details, we refer the reader to
[ 5 ].
<H3>4.2.2 Applying SVMs</H3>
To apply the SVM technique to the problem of determining controller pre-conditions, we train a nonlinear SVM classifier to predict the success or failure of a controller for an arbitrary starting state. Thus, the trained SVM demarcates the boundary of regions in the figure’s state space wherein the controller can successfully do its job. Training sets comprising examples Ü Ý are generated by repeatedly starting the dynamic figure at a stochasticallygenerated initial state Ü , numerically simulating the dynamics of the figure under the influence of the controller in question, and setting Ý ·1⁄2 if the controller succeeds or Ý 1⁄2 if it fails. The distribution of the stochastically-generated initial states is of some importance. The sample points should ideally be located close to the boundaries which demarcate the acceptable precondition region of state-space. However, these boundaries are in fact the unknowns we wish to determine and thus we must resort to a more uniform sampling strategy. Unfortunately, the high dimensionality of the state-space precludes regular sampling. We thus adopt the following stochastic process to generate a suitable distribution of initial states: First, a nominal initial state is chosen, based upon the designer’s knowledge of the controller. A shortduration simulation (typically 0.3s) is then carried out from this initial state while a randomized perturbation process is executed. This currently consists of applying an external force of random (but bounded) magnitude and random direction to the center-of-mass of the pelvis. Simultaneously, the character’s joints are perturbed in a stochastic fashion by setting randomized offset target angles for the joints and using the character’s PD joint controllers to drive the joints towards these perturbed positions. While the perturbation strategy is admittedly ad-hoc, we have found it to be effective in sampling the pre-condition space, as is validated by the online use of the learned pre-condition models. We employ T. Joachims’ SVM Ð Ø software which is available on the WWW
[ 21 ]. The software can accommodate large training sets comprising tens of thousands of observations and it efficiently handles many thousands of support vectors. It includes standard kernel functions and permits the definition of new ones. It incorporates a fast training algorithm which proceeds by solving a sequence of optimization problems lower-bounding the solution using a form of local search. It includes two efficient estimation methods for error rate and precision/recall. The SVM training phase can take hours in our application, but this is done off-line. For example, on a 733 MHz PIII computer, the SVM training time for a training set of 8,013 observations is 2,789 seconds using the polynomial kernel, 2,109 seconds using the linear kernel, and 211 seconds using the radial kernel. For a training set of 11,020 observations, the training time is 8,676 seconds using the polynomial kernel, 3,593 seconds using the linear kernel, and 486 seconds using the radial kernel. Once trained, the SVM classifier can provide answers on-line in milliseconds.
<H3>4.2.3 Pre-condition learning results</H3>
Through systematic experimentation, we have evaluated the performance of our automatic, SVM-based algorithm for learning con-
Controller Training set size Test set size NN SVM StepToStand 8,999 9,110 80.97% 87.29% LyingOnBellyToKneel 4,200 4,223 93.27% 94.46% LyingOnBackToKneel 2,234 1,879 100.0% 100.0% BendToStand 6,926 14,272 98.05% 99.77% StandInPlace 17,317 20,393 83.63% 87.67% Walk 11,020 8,658 92.78% 97.73% StandToSit 1,100 1,286 64.15% 69.60% StandToStep 16,999 17,870 72.12% 79.18% KneelToStand 6,000 11,998 79.45% 85.06% Table 1 : Comparison between learned SVM and NN preconditions.
troller pre-conditions. We compared the performance of the SVM algorithm to that of a nearest neighbor (NN) classifier
[ 9 ]. Given a training set, the nearest neighbor classifier returns for an arbitrary state Ü the same succeed/fail label as the label for that observation in the training set that is closest to Ü . NN classifiers should perform particularly well in cases where the feasible area in the state space is highly fragmented and localized. Note that the NN method requires zero training time, but that it provides an answer in Ç ́ Ò μ time where Ò is size of the training set. Table 1 summarizes the percentage success rates (rightmost columns) of learned pre-conditions for a variety of controllers that we use later in our demonstrations. To compute accuracy rates, we trained the SVM and NN pre-condition learning algorithms using randomly sampled observations collected from each of the controllers. Then we generated test sets of novel observations and compared their true success/fail status against that predicted by the trained NN and SVM pre-conditions to obtain the accuracy percentages listed in the rightmost two columns of the table. The results show that the SVM algorithm consistently outperforms the NN classifier. For the results shown in the table, the SVM algorithm employed polynomial kernel functions. We ran a similar set of experiments using Gaussian RBF kernel functions, but the accuracies were consistently lower than those obtained with polynomial kernel functions.
<H1>5 Implementation</H1>
Our control composition framework is implemented within DANCE , a portable, extensible object-oriented modeling and animation system
[ 24 ]. 2 DANCE provides a platform that researchers can use to implement animation and control techniques with minimal design and implementation overhead. The core of the system supports four base classes, Systems, Simulators, Actuators and Geometries which are loadable as plug-ins in accordance with simple APIs. Articulated objects are a System subclass that support skeleton hierarchies. They have kinematic properties and, usually, fully dynamic physical properties as well. Our virtual actors, which will be described shortly, are dynamic articulated objects implemented as Systems within DANCE . An actuator is a generic concept that includes anything that can exert forces or, in general, interact in any way with systems or other actuators. For example, gravity, the ground, the collision mechanism, the supervisor controller and individual controllers are implemented as actuators. DANCE places no restrictions on the complexity of the controllers. Simulators compute the equations of motion of all the dynamic characters and other systems in DANCE . DANCE offers built in support for SD/FAST, a commercial system which produces optimized simulation code
[ 18 ]. However, any simulator that follows a simple
2 DANCE is freely available for non-commercial use via the URL: www.dgp.toronto.edu/software/dance.htm
Joint Rotational DOFs Rotational DOFs 3D skeleton model 2D terminator model Head 1 1 Neck 3 1 Shoulder 2 1 Elbow 2 1 Wrist 2 Waist 3 1 Hip 3 1 Knee 1 1 Ankle 2 1
Figure 5: Dynamic models and their degrees of freedom (DOFs).
API can be dynamically loaded into the system. Our simulators are automatically produced by SD/FAST from description files. They use Kane’s method for computing articulated dynamics and a fourth order explicit Runge-Kutta time integrator for numerically simulating the motions. Actuators and simulators are implemented as DANCE plug-ins. This allows the user to dynamically load controllers and simulators at runtime. In addition, researchers can exchange, simulators, and controllers in the form of dynamically linked pieces of code. Object collisions (including self collisions) are handled by the Collision actuator. This actuator works on pairs of objects. The DANCE API allows it to work with objects that have different simulators. Collision detection is based on a library that uses oriented bounding boxes
[ 13 ]. Collision resolution uses a penalty method that corrects geometry interpenetration using spring-and-damper forces. As with all penalty methods, it can make the system stiff, but it has performed well in our experiments to date.
<H2>5.1 Virtual Stuntman</H2>
<H3>5.1.1 Dynamic model</H3>
Fig. 5 depicts our 2D and 3D articulated character models. The red arrows indicate the joint positions and axes of rotational degrees of freedom (DOFs) which are also presented in the table. The 3D skeleton model has 37 DOFs, six of which correspond to the global translation and rotation parameters. The table in Fig. 5 lists the DOFs for the skeleton and a 2D “terminator” model. The dynamic properties of both models, such as mass and moments of inertia, are taken from the biomechanics literature and correspond to a fullyfleshed adult male. The models are equipped with natural limits both on the motion of the joints and the strength of their muscles. However, DANCE has no built in muscle model and does not enforce the limits automatically. Users can implement the model they prefer and include code to enforce the limits of the model. Our plug-in control scheme uses rotational spring-and-damper forces for control and enforces the limits on the joints with exponential springs.
<H3>5.1.2 Pose and continuous control</H3>
Most of the controllers for our virtual stuntperson are based on pose control, which has often been used both for articulated objects
[ 31 ] and soft objects
[ 11 ]. Pose control is based on cyclic or acyclic finite state machines with time transitions between the states. Each state of the controller can be static or depend on feedback parameters. For some of our controllers, we use continuous control, in the sense that the control parameters are tightly coupled with some of the feedback sensors. The balance controllers are an example of this.
We designed several controllers based in part on experimental studies of how humans detect loss of balance
[ 25 ] and analysis of protective and falling behaviors
[ 8 ]. The resulting parameterized controllers have been enhanced with appropriate pre-conditions, post-conditions, and expected performance and have been integrated using an arbitration-based supervising controller.
<H3>5.1.3 Sensors</H3>
Each controller has full access to the internal data structures of DANCE including all the information associated with any character or object in the system. This allows the controllers to define arbitrary sensors that keep track of necessary information such as state parameters for feedback loops and the state of the environment. For efficiency, the supervisor controller calculates a number of common sensor values that are available to all the controllers.
<H3>5.1.4 Command interface</H3>
Many controller transitions in the control framework happen autonomously, such as taking a protective step in response to losing balance. However, other actions are initiated in a voluntary fashion. For example, a standing character can do any of (1) remain standing using the balance controller, (2) sit-down, (3) walk, and (4) dive. Currently, the user directs these voluntary motions by interactively entering command strings to the supervisor controller which, in turn, directly increases the suitability score of the designated controller and forces the arbitration process to be invoked to select a new active controller. The control of voluntary motions could equivalently be delegated to a high-level planner, although this kind of planning is beyond the scope of our work at present.
<H1>6 Results</H1>
At the heart of our prototype system is a composite controller that is capable of handling a large number of everyday tasks, such as walking, balancing, bending, falling, and sitting. In addition, we present brief descriptions of the controllers involved in producing several stunt actions. While the given controller descriptions are for the 3D character, the equivalent 2D controllers are very similar. Finally, we discuss motion sequences generated using these families of controllers 3 .
3 See www.dgp.toronto.edu/ pfal/animations.html for the associated animations.
<H2>6.1 Everyday Actions</H2>
We began our implementation with the simple tasks of standing, recovering balance when pushed, and falling. An autonomous human agent should be able to balance, standing naturally in place. Should loss of balance occur, the character ought to react naturally either with a restoring motion or with a protective falling behavior depending on which action is appropriate in each case. Affording a dynamic articulated figure with natural reactions to loss of balance or impending falls is an essential step towards believable autonomous characters.
<H3>6.1.1 Balancing</H3>
A balance controller is responsible for maintaining a natural standing posture. This controller is based on an inverted pendulum model
[ 12 ]
, using the ankles to control the body sway. Despite the fact that the body of the character is not as rigid as the inverted pendulum hypothesis suggests, the approximation works well in practice. As an example of the type of manually defined pre-conditions and post-conditions used for this controller and others, these details are given in Appendix A for the balance controller. An animated character should attempt to maintain balance in response to external disturbances by shifting its weight, taking a step or bending at the waist. If the character cannot maintain balance, it must then resort to a falling behavior.
<H3>6.1.2 Falling</H3>
The manner in which people fall depends on a number of factors such as their physique, their age and their training. For example, the work in [ 19 ]
shows that, during a fall, the elderly are more likely to impact their hip first as compared to younger adults falling under the same conditions. Our fall controller is designed with the average adult in mind. Its main action is thus to absorb the shock of the impact using mostly the hands. The pre-conditions of the fall controller are defined in accordance with those of the balance controller. Situations that are beyond the capabilities of the latter should be handled by the fall controller. Our implementation of the fall controller can handle falls in any direction, responding in different ways to falls in different directions. Fig. 6 depicts frames from falls in a variety of directions. The second frame in Fig. 1 also demonstrates the action of the fall controller within a fall-and-recover sequence.
<H3>6.1.3 Sitting</H3>
Sitting down in a chair and rising from a chair are common everyday tasks. We have implemented a controller that can do both depending on the instructions of the animator. Apart from the command string supplied by the user, the pre-conditions are either a balanced upright posture or a balanced sitting posture. The postconditions are similarly defined. The resulting action is illustrated in Fig. 7 .
<H3>6.1.4 Rising from a supine position</H3>
Getting up off the ground is a surprisingly difficult motion to simulate. It involves rapid changes of the contact points and significant shifting of the figure’s weight. In addition, the frictional properties of the ground model can influence the motion. The pre-conditions for this controller are straightforward. The character must be lying with its back flat on the ground, within some tolerance. The post-conditions are that the character should be on its feet with its center of mass within the support polygon. Then it would be up to another controller to take over and bring the character from a crouching position to a standing one. A snapshot of a resulting motion is shown in Fig. 8 .
Figure 6: Falling in different directions
<H3>6.1.5 Rolling over</H3>
When lying on their back, some people may choose to roll-over to a prone position before attempting to stand. We have implemented a roll-over controller that can emulate this action. The fourth frame in Fig. 1 demonstrates the action of the roll-over controller. The pre-conditions of the roll-over controller require a supine posture, and no movement of the center of mass. The postconditions of the roll controller are fairly simple and they include any prone position for which the character is extended and fairly straight; i.e., no crossing of legs or arms, etc.
<H3>6.1.6 Rising from a prone position</H3>
Frames 5–9 in Fig. 1 demonstrate the action of a controller that enables the virtual stuntperson to rise from the prone position. When lying face-down, the pre-conditions can be fairly relaxed. Our controller assumes that is has the time to change the state of the character to one from which it knows how to rise. As long as the figure is not lying on its arms and the ground is relatively flat it will attempt to get up. The post-conditions are chosen such that they satisfy the pre-conditions of the balance controller.
<H2>6.2 Stunts</H2>
Apart from everyday actions, we want our dynamic character to be able to do a variety of other voluntary actions dictated by the animator. Such actions can potentially include vigorous and/or physically dangerous actions. It is our hope that if a large number of researchers contribute controllers the character can eventually be used as a virtual stuntperson.
<H3>6.2.1 Kip move</H3>
The kip is an athletic motion often seen in martial arts films and is depicted in Fig. 9 . The controller is based on a pose controller whose pre-conditions include a variation of supine positions. As before, the first part of the controller makes sure that the character assumes a position suitable for performing the kip. The larger part of the motion is ballistic, which focuses the control mainly at the kick off and landing phases. The last part of the controller applies continuous control to bring the stuntman to an erect position from which the balance controller can take over.
<H3>6.2.2 Plunge and roll</H3>
Fig. 10 shows the stuntman performing a suicidal dive down stairs. The character can be instructed to lunge forward and upward at a takeoff angle controlled by the user. When the hands contact the ground a front-roll is attempted. The pre-conditions of this controller are defined be an upright position and little movement of the center of mass. We have also experimented with a multiple character scenario, with one character tackling another, Fig. 11 . While the timing of the tackle is scripted, it illustrates the capability of the system to cope with a pair of interacting characters, each equipped with its own supervisory controller.
Figure 11: Two interacting virtual characters.
<H2>6.3 Animation Sequences</H2>
We have produced two relatively long animation sequences that demonstrate the potential of the our framework. The sequence for the 3D skeleton model presented in Fig. 1 involves controllers whose pre-conditions are provided analytically by the designer. Such conditions tend to define square regions within the space defined by the parameters involved. Despite their simple form, such pre-conditions can generally work well as is demonstrated by the intricacy of the animation produced. We expect to investigate the application of SVM-learned pre-conditions to the 3D model in the future. A second animation sequence with the 2D terminator model (see Fig. 12 ) makes use of a set of controllers having a mix of analytic and learned pre-conditions. The sequence of controllers that generated the animation was: balance sit lean-forward rise balance walk step-to-stand balance dive default kneel kneel to stand balance step-forward step-tostand balance step-back step-to-stand balance fall default. The analytical pre-conditions prune large parts of the state space and the svm-classifier provides a more accurate success/failure prediction within the remaining region. During the animation sequence, the svm-classifier correctly refined the analytical answer in several cases.
<H2>6.4 Performance Issues</H2>
Most of the computational burden in our approach lies in the numerical simulation of the equations of motion. The computations associated with the controllers and our composition framework are negligible in comparison. In general, the 2D model simulates in real time, while the 3D model runs between 5 and 9 times slower than real time on a 733 MHz Pentium III system.
Figure 7: Sitting and rising from a chair
Figure 8: Rising from a supine position on the ground and balancing erect in gravity.
Figure 9: Kip move: A more vigorous way of getting up from the supine position as in the first frame of Fig. 8 .
Figure 10: Ouch!
<H1>7 Conclusion</H1>
The challenges of physics-based controller design plus the technical obstacles that researchers face when attempting to share their algorithms has hindered progress in the important area of physicsbased character animation. This paper has presented a methodology for ameliorating the problem with a framework which facilitates the exchange and composition of controllers. Our framework has been implemented within a freely available system for modeling and animating articulated characters. To our knowledge, our system is the first to demonstrate a dynamic anthoropomorphic character with controlled reactions to disturbances or falls in any direction, as well as the ability to pick itself up off the ground in several ways, among other controlled motions. We hope that our system will foster collective efforts among numerous practitioners that will eventually result in complex composite controllers capable of synthesizing a full spectrum of human-like motor behaviors. Given the enormous challenge of building controllers capable of large repertoires of dynamic human-like motion, it is inevitable that the work presented in this paper is incomplete in many ways. Published control methods for 3D walking, running, and stair climbing make obvious candidates for integration into our system. Coping with variable terrain and dynamic environments are dimensions of added complexity that should provide work for years to come. Automatic parameterization of controllers to variations in character dimensions and mass is a necessary step for having solutions adaptable to a variety of characters. Deriving controllers from motion-capture data is an exciting but difficult prospect, although some progress is already being made in this area. Other methods of “teaching” skills to a dynamic character also warrant investigation. Finally, intelligently integrating controllers which affect only subsets of DOFs needs to be addressed in order to allow for the parallel execution of controllers.
<H1>Acknowledgements</H1>
We wish to thank Joe Laszlo for his help with the video editing equipment and for useful discussions. We would also like to thank Symbolic Dynamics Inc. for allowings us to distribute the equations of motion of the 3D human model. This work was supported by grants from NSERC and CITO.
<H1>A Balance controller</H1>
The articulated body must be in a balanced upright position, the velocity and acceleration of the center of mass should not exceed certain threshold values as explained in
[ 25 ], and both feet must maintain contact with the ground at all times. The controller can tolerate small perturbations of the posture and the velocity/acceleration of the center of mass by stiffening the ankle joints. For larger accelerations of the center of mass, the controller actively actuates the ankle joint to reduce the acceleration of the center of mass. The post-conditions are similar to the pre-conditions. In mathematical form using the notation defined in Section 3:
Figure 12: A still image from the terminator sequence.
The dynamic terminator model has been knocked backward by the force of a collision to the head by the red ball. The terminator maintains balance by taking a protective step.
: Acceleration: 1⁄4 1⁄2 m sec 3⁄4 . Velocity: Balance: Posture: ́1⁄2 projection Ò 1⁄4 μ ¿ È m ́ Õ sec. μ 3⁄4Ë . Õ 1⁄4 1⁄4 1⁄2 rad, where ́ thigh knee waist μ and Ò is a normalization parameter. Acceleration: : 1⁄4 1⁄41⁄2 m sec 3⁄4 . Balance: Posture: Velocity: ́1⁄2 projection Ò 1⁄4 μ 1⁄4 È m ́ Õ μ sec. 3⁄4Ë . Õ 1⁄4 1⁄4 1⁄2 rad, where ́ thigh knee waist μ and Ò is a normalization parameter.
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"text": "['This may seem surprising in view of the recent successes in implementing a slew of specialist controllers capable of realistically synthesizing the complex dynamics of running, diving, and various gymnastic maneuvers']",
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"text": "['a divide-and-conquer strategy is clearly prudent in coping with the enormous variety of controlled motions that humans and other animals may perform']",
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"text": "['little effort has been directed at how the resulting control solutions may be integrated to yield composite controllers with significantly broader functionalities']",
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"text": "['This is a difficult problem']",
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"text": "['its resolution would help pave the way towards controller libraries for dynamic animation which communities of practitioners could utilize and to which they could contribute']",
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"text": "['In our framework, individual controllers are black boxes encapsulating control knowledge that is possibly gleaned from the biomechanics literature, derived from the robotics control literature, or developed specifically for animation control']",
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"text": "['a controller should be able to determine whether or not it can take the dynamic character from its current state to some desired goal state']",
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"text": "['an active controller should be able to determine whether it is operating nominally, whether it has succeeded, or whether it has failed']",
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"text": "['Any controller that can answer these queries may be added to a pool of controllers managed by a supervisor controller whose goal is to resolve more complex control tasks']",
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"text": "['An important technical contribution within our controller composition framework is an explicit model of pre-conditions']",
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"text": "['Preconditions characterize those regions of the dynamic figure’s state space within which an individual controller is able to successfully carry out its mission']",
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"text": "['We devise a promising solution which employs Support Vector Machine']",
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"text": "['Our novel application of this technique learns appropriate pre-conditions through the repeated sampling of individual controller behavior in operation']",
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"text": "['As a testbed of our techniques, we are developing a physicallysimulated animated character capable of a large repertoire of motor skills']",
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"text": "['the dynamic nature of typical stunts makes them dangerous to perform']",
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"text": "['The open challenge here lies in developing appropriate control strategies for specific actions and ways of integrating them into a coherent whole']",
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"text": "['we demonstrate families of composable controllers for articulated skeletons whose physical parameters reflect anthropometric data consistent with a fully-fleshed adult male']",
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"text": "['the 3D skeleton illustrates the ultimate promise of the technique']",
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"text": "['the easier control associated with the 2D skeleton allows for more rapid prototyping of larger families of controllers and more careful analysis of their operation']",
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"text": "['the control of broad skilled repertoires of motion remains very much an open problem even for 2D articulated figures']",
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"text": "['A subsequent disturbance will elicit similar though by no means identical autonomous behavior']",
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"text": "['the initial conditions and external forces will usually not be exactly the same']",
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"text": "['Control sequences of such intricacy for fully dynamic articulated figures are unprecedented in the physics-based animation literature']",
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"text": "['The simulation and animation of human characters is a challenging problem in many respects']",
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"text": "['Comprehensive solutions must aspire to distill and integrate knowledge from biomechanics, robotics, control, and animation']",
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"text": "['Models for human motion must also meet a particularly high standard']",
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"text": "['our familiarity with what the results should look like']",
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"text": "['a divide-and-conquer strategy is evident in most approaches, focusing efforts on reproducing particular motions in order to yield a tractable problem and to allow for comparative analysis']",
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"text": "['These models target applications such as medical diagnosis, the understanding and treatment of motor control problems, the analysis of accidents and disabilities, and high-performance athletics']",
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"text": "['Computer simulation is becoming an increasingly useful tool in this domain']",
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"text": "['the motion models evolve to become more complex and comprehensive']",
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"text": "['27']",
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"text": "['Robotics research has made remarkable progress in the successful design of a variety of legged robots']",
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"text": "['the practicality of extending the method to high-DOF dynamic models of human motions is unclear']",
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"text": "['motion capture solutions based on blending and warping techniques may give satisfactory results for such tasks in the short term']",
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10816,
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11217,
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"text": "['The work of Wooten']",
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"text": "['leaping']",
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"text": "['tumbling']",
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"text": "['balancing']",
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"text": "['Transitions are realized by including the end state of some controllers in the starting states of other controllers']",
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"text": "['A digital biomechanics laboratory is proposed by Boston Dynamics, Inc.']",
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"text": "['20']",
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"text": "['This currently remains ambitious work in progress']",
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"text": "['Our work is aimed at creating dynamic human characters with broadly integrated action repertoires']",
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"text": "['our methodology is to begin with a core set of simple actions']",
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"text": "['small steps']",
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"text": "['falling reactions']",
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"text": "['recovery from falls']",
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"text": "['standing up from a chair']",
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12923
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"text": "['our contribution here is a framework for composing individual controllers']",
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"text": "['they may be designed, into more capable control systems for dynamic characters']",
"type": "own_claim",
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"text": "['An interesting tech']",
"type": "own_claim",
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"text": "['nical contribution within our controller composition framework is the introduction of a learning approach for automatically determining controller pre-conditions']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
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13316,
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],
"text": "['Our pre-condition learning algorithm adds to the growing body of learning algorithms that have been successfully applied in the context of computer animation in recent years']",
"type": "own_claim",
"id": "T121"
},
{
"offsets": [
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],
"text": "['14']",
"type": "data",
"id": "T122"
},
{
"offsets": [
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13497,
13499
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],
"text": "['15']",
"type": "data",
"id": "T123"
},
{
"offsets": [
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13751,
13921
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],
"text": "['the supervisor polls the pool of controllers, querying each whether it can handle the transition of the dynamic character from its current state to the desired goal state']",
"type": "own_claim",
"id": "T124"
},
{
"offsets": [
[
13726,
13749
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],
"text": "['no controller is active']",
"type": "data",
"id": "T125"
},
{
"offsets": [
[
13923,
14055
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],
"text": "['Individual controllers return an integer confidence/suitability score when queried in order to bid on becoming the active controller']",
"type": "own_claim",
"id": "T126"
},
{
"offsets": [
[
14080,
14209
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],
"text": "['controllers that can perform a sensible action given the current state of the character return an integer in the range 1⁄2 1⁄21⁄4']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
14218,
14358
]
],
"text": "['those that can handle the current state as well as guarantee a transition to the desired state, return an integer in the range 1⁄21⁄4 3⁄41⁄4']",
"type": "own_claim",
"id": "T128"
},
{
"offsets": [
[
14369,
14441
]
],
"text": "['a value of 1⁄4 means that a controller is unsuited for the current state']",
"type": "own_claim",
"id": "T129"
},
{
"offsets": [
[
14511,
14671
]
],
"text": "['this scoring scheme potentially allows for a nuanced evaluation of the controller suitability in terms of criteria such as probability of success or energy used']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
14443,
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],
"text": "['The controller that returns the highest score becomes active']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
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14723
]
],
"text": "['our current controllers resort to a simpler scheme']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
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14725,
14894
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],
"text": "['This consists of a binary success/failure evaluation multiplied by a weighting factor assigned to each controller that serves to establish a relative preference ordering']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
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],
"text": "['The power of this scheme stems from the following attributes']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
14978,
15041
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],
"text": "['The composition method is straightforward and easy to implement']",
"type": "own_claim",
"id": "T135"
},
{
"offsets": [
[
15043,
15100
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],
"text": "['It does not appreciably burden the controller design task']",
"type": "own_claim",
"id": "T136"
},
{
"offsets": [
[
15114,
15191
]
],
"text": "['The composition method does not restrict the design of individual controllers']",
"type": "own_claim",
"id": "T137"
},
{
"offsets": [
[
15193,
15271
]
],
"text": "['Each controller can be as primitive or as sophisticated as its designer wishes']",
"type": "own_claim",
"id": "T138"
},
{
"offsets": [
[
15338,
15511
]
],
"text": "['A controller within the pool of available controllers can be as simple as a constant force, or as complex as a structured hierarchy of multiple levels of control abstraction']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
15529,
15569
]
],
"text": "['more controllers are added to the system']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
15571,
15706
]
],
"text": "['we may wish to group all the walking and running controllers together into a cluster that can be treated as one encapsulated controller']",
"type": "own_claim",
"id": "T141"
},
{
"offsets": [
[
15741,
15851
]
],
"text": "['our composition method requires controllers to define pre-conditions, post-conditions and expected performance']",
"type": "own_claim",
"id": "T142"
},
{
"offsets": [
[
15948,
15972
]
],
"text": "['these conditions are met']",
"type": "data",
"id": "T143"
},
{
"offsets": [
[
15978,
16069
]
],
"text": "['the controller can operate and possibly enable the character to satisfy the post-conditions']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
16414,
16501
]
],
"text": "['this mapping may not always succeed, which motivates the notion of expected performance']",
"type": "own_claim",
"id": "T145"
},
{
"offsets": [
[
16375,
16412
]
],
"text": "['unexpected changes in the environment']",
"type": "data",
"id": "T146"
},
{
"offsets": [
[
16503,
16621
]
],
"text": "['The controller should be able to evaluate its performance in order to detect failure at any point during its operation']",
"type": "own_claim",
"id": "T147"
},
{
"offsets": [
[
16623,
16752
]
],
"text": "['To do this, the controller must at all times have knowledge of the current and expected state of the character or the environment']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
16754,
16904
]
],
"text": "['Defining the pre-conditions, post-conditions, and expected performance for complex characters, motions, and environments is not a straightforward task']",
"type": "own_claim",
"id": "T149"
},
{
"offsets": [
[
16931,
17061
]
],
"text": "['the effort required to generate these specifications is a fair and necessary price to pay to achieve the benefits of composability']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
17063,
17189
]
],
"text": "['Controllers that adhere to these specifications can form a pool of available controllers managed by the supervising controller']",
"type": "own_claim",
"id": "T151"
},
{
"offsets": [
[
18641,
18742
]
],
"text": "['Most of our controllers can operate within a small region of the state space which we denote Ê ́ Õ μ']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
19205,
19288
]
],
"text": "['the projection of along the gravity vector does not intersect the base of support Ë']",
"type": "data",
"id": "T153"
},
{
"offsets": [
[
19291,
19332
]
],
"text": "['the figure is considered to be unbalanced']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
19617,
19707
]
],
"text": "['The determination of pre-conditions is crucial to the success of our composition framework']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
20150,
20280
]
],
"text": "['the pre-conditions may reference a subset of the postconditions that is sufficient to characterize what the controller can achieve']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
20303,
20360
]
],
"text": "['the post-conditions are different from the pre-conditions']",
"type": "own_claim",
"id": "T157"
},
{
"offsets": [
[
20381,
20464
]
],
"text": "['a pre-condition for a falling controller requires that the center of mass be moving']",
"type": "own_claim",
"id": "T158"
},
{
"offsets": [
[
20466,
20527
]
],
"text": "['the postconditions require that the center of mass be at rest']",
"type": "own_claim",
"id": "T159"
},
{
"offsets": [
[
20601,
20740
]
],
"text": "['Our framework permits the automatic selection of the appropriate controller based on the information provided by the controllers themselves']",
"type": "own_claim",
"id": "T160"
},
{
"offsets": [
[
20742,
20851
]
],
"text": "['Only the individual controllers can detect whether they are operating normally or whether failure is imminent']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
20935,
20958
]
],
"text": "['The controller may fail']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
20970,
21004
]
],
"text": "['a sudden change in the environment']",
"type": "data",
"id": "T163"
},
{
"offsets": [
[
21019,
21048
]
],
"text": "['badly designed pre-conditions']",
"type": "data",
"id": "T164"
},
{
"offsets": [
[
21050,
21153
]
],
"text": "['The sooner a controller can detect failure the sooner another more appropriate controller can take over']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
21155,
21212
]
],
"text": "['This is important for making a character behave naturally']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
21227,
21286
]
],
"text": "['the character should not attempt to continue a walking gait']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
21290,
21331
]
],
"text": "['it has lost its balance and it is falling']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
21356,
21445
]
],
"text": "['the expected performance consists of expressions similar to those of the pre-conditions È']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
21465,
21546
]
],
"text": "['the controller successfully completes its task in the time interval Ø 1⁄2 , Ø 3⁄4']",
"type": "data",
"id": "T170"
},
{
"offsets": [
[
21554,
21589
]
],
"text": "[' ́ Ø 1⁄2 μ 3⁄4È and ́ Ø 3⁄4 μ 3⁄4Ç']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
21655,
21713
]
],
"text": "['Transitions between controllers are not explicitly modeled']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
21717,
21756
]
],
"text": "['they would be in a finite state machine']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
21758,
21918
]
],
"text": "['They occur implicitly in response to the evolution of the motion over time, as the system state traverses the “regions-of-competency” of the various controllers']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
22000,
22047
]
],
"text": "['typical patterns of controller activation occur']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
21945,
21998
]
],
"text": "['most controllers are designed for specific situations']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
22539,
22586
]
],
"text": "['the prone-to-standing fall transition can occur']",
"type": "own_claim",
"id": "T177"
},
{
"offsets": [
[
22590,
22649
]
],
"text": "['the figure is given a sufficiently strong push while rising']",
"type": "data",
"id": "T178"
},
{
"offsets": [
[
22459,
22524
]
],
"text": "['not all possible transitions are shown in either of Figs. 3 and 4']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
22651,
22780
]
],
"text": "['Most of the transitions which are not shown but are still practically feasible are of this nature, dealing with falling behaviors']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
22792,
22873
]
],
"text": "['the fall controller always responds to the specific direction of the current fall']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
22875,
22959
]
],
"text": "['Any transition involves one controller being deactivated and another being activated']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
22961,
22996
]
],
"text": "['A controller can become deactivated']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
23031,
23055
]
],
"text": "['for one of three reasons']",
"type": "own_claim",
"id": "T184"
},
{
"offsets": [
[
23064,
23142
]
],
"text": "['it may relinquish control by declaring success upon reaching its postcondition']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
23262,
23303
]
],
"text": "['user intervention may elicit a transition']",
"type": "own_claim",
"id": "T186"
},
{
"offsets": [
[
23305,
23411
]
],
"text": "['The controllers designed for sitting or balanced standing will retain control until intervention by a user']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
23443,
23470
]
],
"text": "['forces a desired transition']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
23512,
23578
]
],
"text": "['a user-driven process must choose among the next plausible actions']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
23483,
23511
]
],
"text": "['the 2D character is balanced']",
"type": "data",
"id": "T190"
},
{
"offsets": [
[
23618,
23624
]
],
"text": "['Fig. 4']",
"type": "data",
"id": "T191"
},
{
"offsets": [
[
23635,
23666
]
],
"text": "['a controller may detect failure']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
23948,
24005
]
],
"text": "['our framework is designed to work in interactive settings']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
24016,
24113
]
],
"text": "['controllers typically start with slightly different initial conditions each time they are invoked']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
24115,
24167
]
],
"text": "['the user can interact with the character at any time']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
24183,
24278
]
],
"text": "['there are no guarantees that the controller will reach the same end state each time it operates']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
24293,
24337
]
],
"text": "['the transition graph is dynamic in structure']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
[
24664,
24825
]
],
"text": "['determining the exact region of the state space and the general conditions that determine success or failure of the controller is in general a non-trivial matter']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
24608,
24662
]
],
"text": "['controllers associated with complex dynamic characters']",
"type": "data",
"id": "T199"
},
{
"offsets": [
[
24844,
24903
]
],
"text": "['we address this problem via manual and automatic approaches']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
24905,
24991
]
],
"text": "['The manual approach allows designers to incorporate their knowledge within controllers']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
25001,
25063
]
],
"text": "['the automatic approach is based on machine learning techniques']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
25123,
25234
]
],
"text": "['For certain cases, suitable pre-conditions for specific controllers may be found in the biomechanics literature']",
"type": "background_claim",
"id": "T203"
},
{
"offsets": [
[
25238,
25239
]
],
"text": "['8']",
"type": "data",
"id": "T204"
},
{
"offsets": [
[
25242,
25244
]
],
"text": "['25']",
"type": "data",
"id": "T205"
},
{
"offsets": [
[
25260,
25274
]
],
"text": "['Pai and Patton']",
"type": "background_claim",
"id": "T206"
},
{
"offsets": [
[
25277,
25279
]
],
"text": "['25']",
"type": "data",
"id": "T207"
},
{
"offsets": [
[
25283,
25472
]
],
"text": "['present a comprehensive study of balance in the sagittal plane and identify the conditions under which a human can compensate for postural disturbances and maintain balance without stepping']",
"type": "background_claim",
"id": "T208"
},
{
"offsets": [
[
25474,
25568
]
],
"text": "['For certain other cases, the pre-conditions are trivially defined by the desired motion itself']",
"type": "background_claim",
"id": "T209"
},
{
"offsets": [
[
25570,
25647
]
],
"text": "['Certain controllers function as intermediate stages between other controllers']",
"type": "background_claim",
"id": "T210"
},
{
"offsets": [
[
25652,
25705
]
],
"text": "['controller B is the intermediate step between A and C']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
25711,
25836
]
],
"text": "['the postconditions of A dictate the pre-conditions of B and similarly the pre-conditions of C define the post-conditions of B']",
"type": "background_claim",
"id": "T212"
},
{
"offsets": [
[
25847,
25918
]
],
"text": "['in some cases the pre-conditions are computed by manual experimentation']",
"type": "background_claim",
"id": "T213"
},
{
"offsets": [
[
25932,
25995
]
],
"text": "['a simple balance controller based on an inverted pendulum model']",
"type": "background_claim",
"id": "T214"
},
{
"offsets": [
[
25999,
26001
]
],
"text": "['12']",
"type": "data",
"id": "T215"
},
{
"offsets": [
[
26005,
26065
]
],
"text": "['has intrinsic stability that can tolerate small disturbances']",
"type": "background_claim",
"id": "T216"
},
{
"offsets": [
[
26067,
26239
]
],
"text": "['After the controller has been designed, repeated testing under disturbances of increasing magnitude can yield an approximation of the pre-conditions and the post-conditions']",
"type": "background_claim",
"id": "T217"
},
{
"offsets": [
[
26254,
26337
]
],
"text": "['the designer of a controller presumably understands the way the controller operates']",
"type": "background_claim",
"id": "T218"
},
{
"offsets": [
[
26348,
26414
]
],
"text": "['is able to provide high level conditions on its success or failure']",
"type": "background_claim",
"id": "T219"
},
{
"offsets": [
[
26429,
26558
]
],
"text": "['the designer of a walking controller knows if the controller can operate when the walking surface has minimal friction properties']",
"type": "background_claim",
"id": "T220"
},
{
"offsets": [
[
26566,
26615
]
],
"text": "['human motion is shaped by notions such as comfort']",
"type": "background_claim",
"id": "T221"
},
{
"offsets": [
[
26621,
26665
]
],
"text": "['only the designer can take this into account']",
"type": "background_claim",
"id": "T222"
},
{
"offsets": [
[
26683,
26716
]
],
"text": "['a person is pushed while standing']",
"type": "data",
"id": "T223"
},
{
"offsets": [
[
26717,
26752
]
],
"text": "['he/she might take a protective step']",
"type": "background_claim",
"id": "T224"
},
{
"offsets": [
[
26761,
26859
]
],
"text": "['it may be more comfortable to do so instead of maintaining an inverted pendulum balancing strategy']",
"type": "background_claim",
"id": "T225"
},
{
"offsets": [
[
26872,
26985
]
],
"text": "['the way people react to slipping and imbalance and the protective behaviors they employ are largely age dependent']",
"type": "background_claim",
"id": "T226"
},
{
"offsets": [
[
27084,
27240
]
],
"text": "['we introduce an automatic, machine learning approach to determining pre-conditions, which is based on systematically sampling the performance of controllers']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
27242,
27307
]
],
"text": "['Our method uses a machine learning algorithm attributed to Vapnik']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
27311,
27313
]
],
"text": "['33']",
"type": "data",
"id": "T229"
},
{
"offsets": [
[
27316,
27348
]
],
"text": "['known as Support Vector Machines']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
27357,
27400
]
],
"text": "['which has recently attracted much attention']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
27408,
27490
]
],
"text": "['in most cases the performance of SVMs matches or exceeds that of competing methods']",
"type": "background_claim",
"id": "T232"
},
{
"offsets": [
[
27573,
27645
]
],
"text": "['SVMs are a method for fitting functions to sets of labeled training data']",
"type": "background_claim",
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"text": "['The functions can be general regression functions or they can be classification functions']",
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"text": "['we use simple classification functions with binary outputs which encode the success or failure of a controller']",
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"text": "['Burges']",
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"text": "['5']",
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"text": "['provides an excellent tutorial on SVMs']",
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"text": "['The SVM is a machine whose task is to learn the mapping Ü Ý from a training set']",
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"text": "['A particular choice of « generates a “trained” SVM']",
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"text": "['a properly trained SVM predicts if a controller will succeed ( ́ Ü μ 1⁄4 ) or fail ( ́ Ü μ 1⁄4 ) on a given state Ü of the dynamic character']",
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"text": "['there exists a decision boundary separating positive from negative examples which takes the form of a “separating hyperplane” in']",
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"text": "['SVM training requires the solution of a quadratic programming optimization problem involving a Lagrange multiplier « for every datapoint in the training set']",
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"text": "['other observations in the training set are moved']",
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"text": "['or removed and SVM training is repeated']",
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"text": "['the same separating hyperplane will result']",
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"text": "['The linear SVM is easily generalized to nonseparable training data']",
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"text": "['it is straightforward to generalize the theory to encompass nonlinear SVMs for which the decision boundaries are no longer hyperplanes']",
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"text": "['The trick']",
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"text": "['is to map the data to some higher (possibly infinite) dimensional space in which the linear theory can be applied']",
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"text": "['This is easily done by introducing kernel functions à ́ Ü Ü μ']",
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"id": "T253"
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"text": "['the polynomial kernel à (RBF) ́ Ü Ý kernel μ ́ Ü Ã ¡ ́ Ü Ý · Ý μ 1⁄2μ Ô , ÜÔ']",
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"text": "['the Ü Gaussian Ý 3⁄4 3⁄4 or 3⁄4 μ radial']",
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"text": "['ical details, we refer the reader to']",
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"text": "['5']",
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"text": "['For the basis mathemat']",
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"text": "['the trained SVM demarcates the boundary of regions in the figure’s state space wherein the controller can successfully do its job']",
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"text": "['To apply the SVM technique to the problem of determining controller pre-conditions, we train a nonlinear SVM classifier to predict the success or failure of a controller for an arbitrary starting state']",
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"text": "['The distribution of the stochastically-generated initial states is of some importance']",
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"text": "['The sample points should ideally be located close to the boundaries which demarcate the acceptable precondition region of state-space']",
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"text": "['these boundaries are']",
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"text": "['the unknowns we wish to determine']",
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"text": "['we must resort to a more uniform sampling strategy']",
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"text": "['the high dimensionality of the state-space precludes regular sampling']",
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"text": "['We']",
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"text": "['adopt the following stochastic process to generate a suitable distribution of initial states']",
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"text": "['First, a nominal initial state is chosen, based upon the designer’s knowledge of the controller']",
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"text": "['A shortduration simulation']",
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"text": "['typically 0.3s']",
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"text": "['is then carried out from this initial state while a randomized perturbation process is executed']",
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"text": "['Simultaneously, the character’s joints are perturbed in a stochastic fashion by setting randomized offset target angles for the joints and using the character’s PD joint controllers to drive the joints towards these perturbed positions']",
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"text": "['we have found it to be effective in sampling the pre-condition space']",
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"text": "['the online use of the learned pre-condition models']",
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"text": "['We employ T. Joachims’ SVM Ð Ø software which is available on the WWW']",
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"text": "['21']",
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"text": "['The software can accommodate large training sets comprising tens of thousands of observations']",
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"text": "['It includes two efficient estimation methods for error rate and precision/recall']",
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"text": "['The SVM training phase can take hours in our application']",
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"text": "['this is done off-line']",
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"text": "['the training time is 8,676 seconds using the polynomial kernel, 3,593 seconds using the linear kernel, and 486 seconds using the radial kernel']",
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"text": "['a training set of 11,020 observations']",
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"text": "['for a training set of 8,013 observations']",
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"text": "['is 2,789 seconds using the polynomial kernel, 2,109 seconds using the linear kernel, and 211 seconds using the radial kernel']",
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"text": "['Once trained, the SVM classifier can provide answers on-line in milliseconds']",
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"text": "['9']",
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"text": "['Given a training set, the nearest neighbor classifier returns for an arbitrary state Ü the same succeed/fail label as the label for that observation in the training set that is closest to Ü']",
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"text": "['NN classifiers should perform particularly well']",
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"text": "['the feasible area in the state space is highly fragmented and localized']",
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"text": "['it provides an answer in Ç ́ Ò μ time']",
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"text": "['The results show that the SVM algorithm consistently outperforms the NN classifier']",
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"text": "['DANCE offers built in support for SD/FAST, a commercial system which produces optimized simulation code']",
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"text": "['any simulator that follows a simple']",
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"text": "['They use Kane’s method for computing articulated dynamics and a fourth order explicit Runge-Kutta time integrator for numerically simulating the motions']",
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"text": "['Actuators and simulators are implemented as DANCE plug-ins']",
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"text": "['researchers can exchange, simulators, and controllers in the form of dynamically linked pieces of code']",
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"text": "['Object collisions']",
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"text": "['are handled by the Collision actuator']",
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"text": "['This actuator works on pairs of objects']",
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"text": "['The DANCE API allows it to work with objects that have different simulators']",
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"text": "['Collision detection is based on a library that uses oriented bounding boxes']",
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"text": "['13']",
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"text": "['it can make the system stiff']",
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"text": "['Collision resolution uses a penalty method that corrects geometry interpenetration using spring-and-damper forces']",
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"text": "['it has performed well in our experiments to date']",
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"text": "['The dynamic properties of both models']",
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{
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39631,
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"text": "['are taken from the biomechanics literature and correspond to a fullyfleshed adult male']",
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"text": "['mass']",
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"text": "['moments of inertia']",
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"text": "['The models are equipped with natural limits both on the motion of the joints and the strength of their muscles']",
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"id": "T344"
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{
"offsets": [
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"text": "['DANCE has no built in muscle model and does not enforce the limits automatically']",
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{
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"text": "['Users can implement the model they prefer and include code to enforce the limits of the model']",
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{
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"text": "['Our plug-in control scheme uses rotational spring-and-damper forces for control and enforces the limits on the joints with exponential springs']",
"type": "own_claim",
"id": "T347"
},
{
"offsets": [
[
40251,
40384
]
],
"text": "['Most of the controllers for our virtual stuntperson are based on pose control, which has often been used both for articulated objects']",
"type": "own_claim",
"id": "T348"
},
{
"offsets": [
[
40388,
40390
]
],
"text": "['31']",
"type": "data",
"id": "T349"
},
{
"offsets": [
[
40393,
40409
]
],
"text": "['and soft objects']",
"type": "own_claim",
"id": "T350"
},
{
"offsets": [
[
40413,
40415
]
],
"text": "['11']",
"type": "data",
"id": "T351"
},
{
"offsets": [
[
40419,
40524
]
],
"text": "['Pose control is based on cyclic or acyclic finite state machines with time transitions between the states']",
"type": "background_claim",
"id": "T352"
},
{
"offsets": [
[
40526,
40601
]
],
"text": "['Each state of the controller can be static or depend on feedback parameters']",
"type": "background_claim",
"id": "T353"
},
{
"offsets": [
[
40603,
40657
]
],
"text": "['For some of our controllers, we use continuous control']",
"type": "own_claim",
"id": "T355"
},
{
"offsets": [
[
40804,
40910
]
],
"text": "['We designed several controllers based in part on experimental studies of how humans detect loss of balance']",
"type": "own_claim",
"id": "T354"
},
{
"offsets": [
[
40914,
40916
]
],
"text": "['25']",
"type": "data",
"id": "T356"
},
{
"offsets": [
[
40919,
40967
]
],
"text": "['and analysis of protective and falling behaviors']",
"type": "own_claim",
"id": "T357"
},
{
"offsets": [
[
40971,
40972
]
],
"text": "['8']",
"type": "data",
"id": "T358"
},
{
"offsets": [
[
40976,
41184
]
],
"text": "['The resulting parameterized controllers have been enhanced with appropriate pre-conditions, post-conditions, and expected performance and have been integrated using an arbitration-based supervising controller']",
"type": "own_claim",
"id": "T359"
},
{
"offsets": [
[
41255,
41411
]
],
"text": "['Each controller has full access to the internal data structures of DANCE including all the information associated with any character or object in the system']",
"type": "own_claim",
"id": "T360"
},
{
"offsets": [
[
41413,
41509
]
],
"text": "['This allows the controllers to define arbitrary sensors that keep track of necessary information']",
"type": "own_claim",
"id": "T361"
},
{
"offsets": [
[
41518,
41553
]
],
"text": "['state parameters for feedback loops']",
"type": "data",
"id": "T362"
},
{
"offsets": [
[
41558,
41586
]
],
"text": "['the state of the environment']",
"type": "data",
"id": "T363"
},
{
"offsets": [
[
41588,
41715
]
],
"text": "['For efficiency, the supervisor controller calculates a number of common sensor values that are available to all the controllers']",
"type": "own_claim",
"id": "T364"
},
{
"offsets": [
[
41796,
41868
]
],
"text": "['Many controller transitions in the control framework happen autonomously']",
"type": "own_claim",
"id": "T365"
},
{
"offsets": [
[
41878,
41932
]
],
"text": "['taking a protective step in response to losing balance']",
"type": "data",
"id": "T366"
},
{
"offsets": [
[
41943,
41993
]
],
"text": "['other actions are initiated in a voluntary fashion']",
"type": "own_claim",
"id": "T367"
},
{
"offsets": [
[
42008,
42129
]
],
"text": "['a standing character can do any of (1) remain standing using the balance controller, (2) sit-down, (3) walk, and (4) dive']",
"type": "own_claim",
"id": "T368"
},
{
"offsets": [
[
42131,
42422
]
],
"text": "['Currently, the user directs these voluntary motions by interactively entering command strings to the supervisor controller which, in turn, directly increases the suitability score of the designated controller and forces the arbitration process to be invoked to select a new active controller']",
"type": "own_claim",
"id": "T369"
},
{
"offsets": [
[
42424,
42512
]
],
"text": "['The control of voluntary motions could equivalently be delegated to a high-level planner']",
"type": "background_claim",
"id": "T370"
},
{
"offsets": [
[
42523,
42587
]
],
"text": "['this kind of planning is beyond the scope of our work at present']",
"type": "background_claim",
"id": "T371"
},
{
"offsets": [
[
42658,
42781
]
],
"text": "['At the heart of our prototype system is a composite controller that is capable of handling a large number of everyday tasks']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
42791,
42798
]
],
"text": "['walking']",
"type": "data",
"id": "T373"
},
{
"offsets": [
[
42800,
42809
]
],
"text": "['balancing']",
"type": "data",
"id": "T374"
},
{
"offsets": [
[
42811,
42818
]
],
"text": "['bending']",
"type": "data",
"id": "T375"
},
{
"offsets": [
[
42820,
42827
]
],
"text": "['falling']",
"type": "data",
"id": "T376"
},
{
"offsets": [
[
42833,
42840
]
],
"text": "['sitting']",
"type": "data",
"id": "T377"
},
{
"offsets": [
[
42955,
43013
]
],
"text": "['the given controller descriptions are for the 3D character']",
"type": "own_claim",
"id": "T378"
},
{
"offsets": [
[
43015,
43061
]
],
"text": "['the equivalent 2D controllers are very similar']",
"type": "own_claim",
"id": "T379"
},
{
"offsets": [
[
43404,
43484
]
],
"text": "['An autonomous human agent should be able to balance, standing naturally in place']",
"type": "own_claim",
"id": "T380"
},
{
"offsets": [
[
43516,
43677
]
],
"text": "['the character ought to react naturally either with a restoring motion or with a protective falling behavior depending on which action is appropriate in each case']",
"type": "own_claim",
"id": "T381"
},
{
"offsets": [
[
43493,
43508
]
],
"text": "['loss of balance']",
"type": "data",
"id": "T382"
},
{
"offsets": [
[
43679,
43840
]
],
"text": "['Affording a dynamic articulated figure with natural reactions to loss of balance or impending falls is an essential step towards believable autonomous characters']",
"type": "own_claim",
"id": "T383"
},
{
"offsets": [
[
43982,
44036
]
],
"text": "['This controller is based on an inverted pendulum model']",
"type": "own_claim",
"id": "T384"
},
{
"offsets": [
[
44040,
44042
]
],
"text": "['12']",
"type": "data",
"id": "T385"
},
{
"offsets": [
[
44047,
44088
]
],
"text": "['using the ankles to control the body sway']",
"type": "own_claim",
"id": "T386"
},
{
"offsets": [
[
44200,
44240
]
],
"text": "['the approximation works well in practice']",
"type": "own_claim",
"id": "T387"
},
{
"offsets": [
[
44112,
44198
]
],
"text": "['the body of the character is not as rigid as the inverted pendulum hypothesis suggests']",
"type": "own_claim",
"id": "T388"
},
{
"offsets": [
[
44426,
44581
]
],
"text": "['An animated character should attempt to maintain balance in response to external disturbances by shifting its weight, taking a step or bending at the waist']",
"type": "own_claim",
"id": "T389"
},
{
"offsets": [
[
44625,
44632
]
],
"text": "['it must']",
"type": "own_claim",
"id": "T390"
},
{
"offsets": [
[
44586,
44623
]
],
"text": "['the character cannot maintain balance']",
"type": "data",
"id": "T391"
},
{
"offsets": [
[
44638,
44666
]
],
"text": "['resort to a falling behavior']",
"type": "own_claim",
"id": "T392"
},
{
"offsets": [
[
44737,
44799
]
],
"text": "['The manner in which people fall depends on a number of factors']",
"type": "background_claim",
"id": "T393"
},
{
"offsets": [
[
44808,
44822
]
],
"text": "['their physique']",
"type": "data",
"id": "T394"
},
{
"offsets": [
[
44824,
44833
]
],
"text": "['their age']",
"type": "data",
"id": "T395"
},
{
"offsets": [
[
44838,
44852
]
],
"text": "['their training']",
"type": "data",
"id": "T396"
},
{
"offsets": [
[
44881,
44883
]
],
"text": "['19']",
"type": "data",
"id": "T397"
},
{
"offsets": [
[
44867,
44875
]
],
"text": "['the work']",
"type": "background_claim",
"id": "T398"
},
{
"offsets": [
[
44887,
45031
]
],
"text": "['shows that, during a fall, the elderly are more likely to impact their hip first as compared to younger adults falling under the same conditions']",
"type": "background_claim",
"id": "T399"
},
{
"offsets": [
[
45033,
45095
]
],
"text": "['Our fall controller is designed with the average adult in mind']",
"type": "own_claim",
"id": "T400"
},
{
"offsets": [
[
45097,
45115
]
],
"text": "['Its main action is']",
"type": "own_claim",
"id": "T401"
},
{
"offsets": [
[
45121,
45177
]
],
"text": "['to absorb the shock of the impact using mostly the hands']",
"type": "own_claim",
"id": "T402"
},
{
"offsets": [
[
45179,
45283
]
],
"text": "['The pre-conditions of the fall controller are defined in accordance with those of the balance controller']",
"type": "own_claim",
"id": "T403"
},
{
"offsets": [
[
45285,
45383
]
],
"text": "['Situations that are beyond the capabilities of the latter should be handled by the fall controller']",
"type": "own_claim",
"id": "T404"
},
{
"offsets": [
[
45385,
45523
]
],
"text": "['Our implementation of the fall controller can handle falls in any direction, responding in different ways to falls in different directions']",
"type": "own_claim",
"id": "T405"
},
{
"offsets": [
[
45525,
45531
]
],
"text": "['Fig. 6']",
"type": "data",
"id": "T406"
},
{
"offsets": [
[
45770,
45843
]
],
"text": "['Sitting down in a chair and rising from a chair are common everyday tasks']",
"type": "background_claim",
"id": "T407"
},
{
"offsets": [
[
45845,
45940
]
],
"text": "['We have implemented a controller that can do both depending on the instructions of the animator']",
"type": "own_claim",
"id": "T408"
},
{
"offsets": [
[
45942,
46080
]
],
"text": "['Apart from the command string supplied by the user, the pre-conditions are either a balanced upright posture or a balanced sitting posture']",
"type": "own_claim",
"id": "T409"
},
{
"offsets": [
[
46082,
46122
]
],
"text": "['The postconditions are similarly defined']",
"type": "own_claim",
"id": "T410"
},
{
"offsets": [
[
46263,
46335
]
],
"text": "['Getting up off the ground is a surprisingly difficult motion to simulate']",
"type": "own_claim",
"id": "T411"
},
{
"offsets": [
[
46337,
46432
]
],
"text": "['It involves rapid changes of the contact points and significant shifting of the figure’s weight']",
"type": "own_claim",
"id": "T412"
},
{
"offsets": [
[
46447,
46517
]
],
"text": "['the frictional properties of the ground model can influence the motion']",
"type": "own_claim",
"id": "T413"
},
{
"offsets": [
[
46519,
46577
]
],
"text": "['The pre-conditions for this controller are straightforward']",
"type": "own_claim",
"id": "T414"
},
{
"offsets": [
[
46579,
46662
]
],
"text": "['The character must be lying with its back flat on the ground, within some tolerance']",
"type": "own_claim",
"id": "T415"
},
{
"offsets": [
[
46664,
46779
]
],
"text": "['The post-conditions are that the character should be on its feet with its center of mass within the support polygon']",
"type": "own_claim",
"id": "T416"
},
{
"offsets": [
[
46781,
46903
]
],
"text": "['Then it would be up to another controller to take over and bring the character from a crouching position to a standing one']",
"type": "own_claim",
"id": "T417"
},
{
"offsets": [
[
47156,
47238
]
],
"text": "['some people may choose to roll-over to a prone position before attempting to stand']",
"type": "background_claim",
"id": "T418"
},
{
"offsets": [
[
47135,
47154
]
],
"text": "['lying on their back']",
"type": "data",
"id": "T419"
},
{
"offsets": [
[
47240,
47311
]
],
"text": "['We have implemented a roll-over controller that can emulate this action']",
"type": "own_claim",
"id": "T420"
},
{
"offsets": [
[
47393,
47503
]
],
"text": "['The pre-conditions of the roll-over controller require a supine posture, and no movement of the center of mass']",
"type": "own_claim",
"id": "T421"
},
{
"offsets": [
[
47505,
47564
]
],
"text": "['The postconditions of the roll controller are fairly simple']",
"type": "own_claim",
"id": "T422"
},
{
"offsets": [
[
47569,
47656
]
],
"text": "['they include any prone position for which the character is extended and fairly straight']",
"type": "own_claim",
"id": "T423"
},
{
"offsets": [
[
47835,
47916
]
],
"text": "['a controller that enables the virtual stuntperson to rise from the prone position']",
"type": "own_claim",
"id": "T424"
},
{
"offsets": [
[
47788,
47808
]
],
"text": "['Frames 5–9 in Fig. 1']",
"type": "data",
"id": "T425"
},
{
"offsets": [
[
47317,
47339
]
],
"text": "['fourth frame in Fig. 1']",
"type": "data",
"id": "T426"
},
{
"offsets": [
[
46163,
46169
]
],
"text": "['Fig. 7']",
"type": "data",
"id": "T427"
},
{
"offsets": [
[
47940,
47980
]
],
"text": "['the pre-conditions can be fairly relaxed']",
"type": "own_claim",
"id": "T428"
},
{
"offsets": [
[
47923,
47938
]
],
"text": "['lying face-down']",
"type": "data",
"id": "T429"
},
{
"offsets": [
[
47982,
48101
]
],
"text": "['Our controller assumes that is has the time to change the state of the character to one from which it knows how to rise']",
"type": "own_claim",
"id": "T430"
},
{
"offsets": [
[
48184,
48209
]
],
"text": "['it will attempt to get up']",
"type": "own_claim",
"id": "T431"
},
{
"offsets": [
[
48114,
48149
]
],
"text": "['the figure is not lying on its arms']",
"type": "data",
"id": "T432"
},
{
"offsets": [
[
48154,
48183
]
],
"text": "['the ground is relatively flat']",
"type": "data",
"id": "T433"
},
{
"offsets": [
[
48211,
48309
]
],
"text": "['The post-conditions are chosen such that they satisfy the pre-conditions of the balance controller']",
"type": "own_claim",
"id": "T434"
},
{
"offsets": [
[
48409,
48517
]
],
"text": "['we want our dynamic character to be able to do a variety of other voluntary actions dictated by the animator']",
"type": "own_claim",
"id": "T435"
},
{
"offsets": [
[
48519,
48600
]
],
"text": "['Such actions can potentially include vigorous and/or physically dangerous actions']",
"type": "own_claim",
"id": "T436"
},
{
"offsets": [
[
48625,
48677
]
],
"text": "['a large number of researchers contribute controllers']",
"type": "data",
"id": "T437"
},
{
"offsets": [
[
48678,
48739
]
],
"text": "['the character can eventually be used as a virtual stuntperson']",
"type": "own_claim",
"id": "T438"
},
{
"offsets": [
[
48602,
48621
]
],
"text": "['It is our hope that']",
"type": "own_claim",
"id": "T439"
},
{
"offsets": [
[
48800,
48862
]
],
"text": "['The kip is an athletic motion often seen in martial arts films']",
"type": "background_claim",
"id": "T440"
},
{
"offsets": [
[
48882,
48888
]
],
"text": "['Fig. 9']",
"type": "data",
"id": "T441"
},
{
"offsets": [
[
48891,
48996
]
],
"text": "['The controller is based on a pose controller whose pre-conditions include a variation of supine positions']",
"type": "own_claim",
"id": "T442"
},
{
"offsets": [
[
49009,
49122
]
],
"text": "['the first part of the controller makes sure that the character assumes a position suitable for performing the kip']",
"type": "own_claim",
"id": "T443"
},
{
"offsets": [
[
49124,
49235
]
],
"text": "['The larger part of the motion is ballistic, which focuses the control mainly at the kick off and landing phases']",
"type": "own_claim",
"id": "T444"
},
{
"offsets": [
[
49237,
49386
]
],
"text": "['The last part of the controller applies continuous control to bring the stuntman to an erect position from which the balance controller can take over']",
"type": "own_claim",
"id": "T445"
},
{
"offsets": [
[
49533,
49634
]
],
"text": "['The character can be instructed to lunge forward and upward at a takeoff angle controlled by the user']",
"type": "own_claim",
"id": "T446"
},
{
"offsets": [
[
49636,
49695
]
],
"text": "['When the hands contact the ground a front-roll is attempted']",
"type": "own_claim",
"id": "T447"
},
{
"offsets": [
[
49697,
49811
]
],
"text": "['The pre-conditions of this controller are defined be an upright position and little movement of the center of mass']",
"type": "own_claim",
"id": "T448"
},
{
"offsets": [
[
49813,
49910
]
],
"text": "['We have also experimented with a multiple character scenario, with one character tackling another']",
"type": "own_claim",
"id": "T449"
},
{
"offsets": [
[
49912,
49919
]
],
"text": "['Fig. 11']",
"type": "data",
"id": "T450"
},
{
"offsets": [
[
49928,
49964
]
],
"text": "['the timing of the tackle is scripted']",
"type": "own_claim",
"id": "T451"
},
{
"offsets": [
[
49966,
50106
]
],
"text": "['it illustrates the capability of the system to cope with a pair of interacting characters, each equipped with its own supervisory controller']",
"type": "own_claim",
"id": "T452"
},
{
"offsets": [
[
50292,
50400
]
],
"text": "['We have produced two relatively long animation sequences that demonstrate the potential of the our framework']",
"type": "own_claim",
"id": "T453"
},
{
"offsets": [
[
50402,
50439
]
],
"text": "['The sequence for the 3D skeleton mode']",
"type": "own_claim",
"id": "T454"
},
{
"offsets": [
[
50461,
50544
]
],
"text": "['involves controllers whose pre-conditions are provided analytically by the designer']",
"type": "own_claim",
"id": "T455"
},
{
"offsets": [
[
50454,
50460
]
],
"text": "['Fig. 1']",
"type": "data",
"id": "T456"
},
{
"offsets": [
[
50546,
50643
]
],
"text": "['Such conditions tend to define square regions within the space defined by the parameters involved']",
"type": "own_claim",
"id": "T457"
},
{
"offsets": [
[
50672,
50715
]
],
"text": "['such pre-conditions can generally work well']",
"type": "own_claim",
"id": "T458"
},
{
"offsets": [
[
50738,
50777
]
],
"text": "['the intricacy of the animation produced']",
"type": "own_claim",
"id": "T459"
},
{
"offsets": [
[
50653,
50670
]
],
"text": "['their simple form']",
"type": "own_claim",
"id": "T460"
},
{
"offsets": [
[
50779,
50879
]
],
"text": "['We expect to investigate the application of SVM-learned pre-conditions to the 3D model in the future']",
"type": "background_claim",
"id": "T461"
},
{
"offsets": [
[
50881,
50937
]
],
"text": "['A second animation sequence with the 2D terminator model']",
"type": "own_claim",
"id": "T462"
},
{
"offsets": [
[
50953,
51038
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A20 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A20_M13_Physically-Valid_Statistical_Models_for_Human_Motion_Generation_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1966394.1966398
<Title>Physically-Valid Statistical Models for Human Motion Generation</Title>
Xiaolin Wei Jianyuan Min Texas A&M University
Figure 1: Combining statistical motion priors and physical constraints for human motion generation: (a) walking with a heavy shoe; (b) resistance running; (c) stylized walking; (d) running→walking→jumping.
<Abstract>This paper shows how statistical motion priors can be combined seamlessly with physical constraints for human motion modeling and generation. The key idea of the approach is to learn a nonlinear probabilistic force field function from prerecorded motion data with Gaussian processes and combine it with physical constraints in a probabilistic framework. In addition, we show how to effectively utilize the new model to generate a wide range of natural looking motions that achieve the goals specified by the users. Unlike previous statistical motion models, our model can generate physically realistic animations that react to external forces or changes in physical quantities of human bodies and interaction environments. We have evaluated the performance of our system by comparing against ground truth motion data and alternative methods.</Abstract>
CR Categories: I.3.6 [Computer Graphics]: Methodology and Techniques—interaction techniques; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—animation Keywords: Human motion analysis and generation, data-driven animation, physics-based animation, animation from constraints, statistical motion modeling, optimization
Jinxiang Chai
<H1>1 Introduction</H1>
A central goal in human motion modeling and generation is to construct a generative motion model to predict how humans move. The problem has attracted the attention of a large number of researchers because of both its theoretical and applied consequences. A generative motion model, for instance, can be used to generate realistic movement for animated human characters or constrain the solution space for modeling 3D human motion in monocular video streams. Decades of research in computer animation have explored two distinctive approaches for human motion modeling: statistical motion
modeling and physics-based motion modeling. Despite the efforts, accurate modeling of human motion remains a challenging task. Statistical motion models are often represented as a set of mathematical equations or functions that describe human motion using a finite number of parameters and their associated probability distributions. Statistical models are desirable for human motion representation because they can model any human movement as long as relevant motion data are available. A fundamental limitation is that they do not consider the dynamics that cause the motion. Therefore, they fail to predict human motion that reacts to external forces or changes in the physical quantities of human bodies and in the interaction environments. Moreover, when motion data are generalized to achieve new goals, the results are often physically implausible and thereby display noticeable visual artifacts such as unbalanced motions, foot sliding, and motion jerkiness. Physics-based motion models could overcome the aforementioned limitations by applying physics to modeling human movements. However, physical laws alone are often insufficient to generate natural human movement because a motion can be physically correct without appearing natural. One way to address the problem is to define a global performance criterion based on either the smoothness of the movement or the minimization of needed controls or control rates (e.g., minimal muscle usage). These heuristics show promise for highly dynamic motions, but it remains challenging to model low-energy motion or highly stylized human actions. In addition, it is unclear if a single global performance objective such as minimal torque is appropriate to model heterogeneous human actions such as running→walking→jumping. In this paper, we show how statistical modeling techniques can be combined with physics-based modeling techniques to address the limitations of both techniques. Physical motion models and statistical motion models are complementary to each other as they capture different aspects of human movements. On the one hand, physical models can utilize statistical priors to constrain the motion to lie in the space of natural appearance and more significantly, learn an appropriate performance criterion to model natural-looking human actions. On the other hand, statistical motion models can rely on physical constraints to generate physically correct human motion that reacts to external forces, satisfies friction limit constraints, and respects physical quantities of human bodies or interaction environments. By accounting for physical constraints and statistical priors simultaneously, we not only instill physical realism into statistical motion models but also extend physics-based modeling to a wide variety of human actions such as stylized walking. The key idea of our motion modeling process is to learn nonlinear probabilistic force field functions from prerecorded motion data with Gaussian Process (GP) models and combine them with physical constraints in a probabilistic framework. In our formulation, a force field function u = g(q, q) ̇ maps kinematic states (joint poses q and joint velocities q) ̇ to generalized forces (u). We demonstrate the power and effectiveness of our motion model in constraint-based motion generation. We show that we can create a natural-looking animation that reacts to changes in physical parameters such as masses or inertias of human bodies and friction properties of environments ( Figure 1(a) ) or external forces such as resistance forces ( Figure 1(b) ). In addition, we show that a single physically valid statistical model is sufficient to create physically realistic animation for a wide range of style variations within a particular human action such as “sneaky” walking (Figure 1(c)) or transitions between heterogeneous human actions such as running→walking→jumping ( Figure 1(d) ). We evaluate the performance of our model by comparing with ground truth data as well as alternative techniques.
<H1>2 Background</H1>
We introduce a physically valid statistical motion model that combines physical laws and statistical motion priors and use it to create physically realistic animation that achieves the goals specified by the user. Therefore, we will focus our discussion on statistical motion modeling and physics-based motion modeling as well as their applications in constraint-based motion synthesis. Statistical models are desirable for human motion modeling and synthesis because they are often compact and can be used to generate human motions that are not in prerecorded motion data.
Thus far, a wide variety of statistical motion models have been developed; their applications include inverse kinematics
[Grochow et al. 2004; Chai and Hodgins 2005], human motion synthesis and editing
[Li et al. 2002; Chai and Hodgins 2007; Lau et al. 2009; Min et al. 2009], human motion style interpolation and transfer
[Brand and Hertzmann 2000; Ikemoto et al. 2009; Min et al. 2010], and so forth. Nonetheless, the motions generated by statistical motion models are often physically invalid because existing statistical motion models do not consider the forces that cause the motion. Another limitation is that they do not react to perturbations (e.g., external forces) or changes in physical quantities such as masses and inertias of human bodies. Physics-based motion models could overcome the limitations of statistical motion models by applying physics to modeling human movement. However, physics-based motion modeling is a mathematically ill-posed problem because there are many ways to adjust a motion so that physical laws are satisfied, and yet only a subset of motions are natural-looking. One way to address this limitation is by adopting the “minimal principle” strategy, which was first introduced to the graphics community by
Witkin and Kass [1988]. They postulated that an individual would determine a movement in such a way as to reduce the total muscular effort to a minimum, subject to certain constraints. Therefore, a major challenge in physics-based motion modeling is how to define an appropriate performance criterion for the “minimal principle.” Decades of research in computer animation (e.g.,
[Witkin and Kass 1988; Cohen 1992; Liu et al. 1994; Fang and Pollard 2003]
) introduced numerous performance criteria for human motion modeling, e.g., minimal energy, minimal torque, minimal jerk, minimal joint momentum, minimal joint acceleration, or minimal torque change. These heuristics show promise for highly dynamic motions, but it remains very difficult to model low-energy motions and highly stylized human movements.
A number of researchers have recently explored the potential of using prerecorded motion data to improve physics-based optimization methods, including editing motion data with the help of simplified physical models
[Popović and Witkin 1999], initializing optimization with reference motion data
[Sulejmanpasic and Popović 2005], learning parameters of motion styles from prerecorded motion data
[Liu et al. 2005], and reducing the search space for physicsbased optimization
[Safonova et al. 2004; Ye and Liu 2008]. Similar to these methods, our system utilizes both motion data and physics for human motion analysis and generation, but there are two important distinctions. First, we rely on statistical motion models rather than a predefined global performance objective (e.g., minimal muscle usage) to reduce the ambiguity of physics-based modeling. This enables us to extend physics-based modeling to stylistic human motions such as “sneaky walking”. Another attraction of our model is that it learns the mapping from the kinematic states to generalized forces using Gaussian process models. Unlike reference trajectories or linear subspace models adopted in previous work, GP models are capable of modeling both stylistic variations within a particular human action and heterogeneous human behaviors. Our research draws inspiration from the large body of literature on developing control strategies for physics-based simulation. In particular, our nonlinear probabilistic force field functions are conceptually similar to control strategies used for physics-based simulation because both representations aim to map kinematic states to driving forces.
Thus far, researchers in physics-based simulation have explored two approaches for control design, including manually designed control strategies (e.g.
[Hodgins et al. 1995]) and tracking a reference trajectory while maintaining balance
[Zordan and Hodgins 2002; Sok et al. 2007; Yin et al. 2007; da Silva et al. 2008; Muico et al. 2009]. However, our approach is different in that we automatically learn nonlinear probabilistic mapping functions from large sets of motion data. In addition, our goal is different because we aim to generate a desired animation that matches user constraints. Physics-based simulation approaches are not appropriate for our task because forward simulation techniques often do not provide accurate control over simulated motions. Our approach uses Gaussian process to model a nonlinear probabilistic function that maps from kinematic states to generalized forces. GP and its invariants (e.g., GPLVM) have recently been applied to modeling kinematic motion for many problems in computer animation, including nonlinear dimensionality reduction for human poses [Grochow et al. 2004], motion interpolation [Mukai and Kuriyama 2005], motion editing [Ikemoto et al. 2009], and motion synthesis [Ye and Liu 2010]. In particular, Ikemoto and her colleagues [2009]
learned the kinematic mapping from pose information of the source motion to pose and acceleration information of the target motion and applied them to transferring a new source motion into a target motion. Ye and Liu [2010]
used GPLVM to construct a second-order dynamic model for human kinematic data and used them to synthesize kinematic walking motion after a perturbation. Our approach is different in that we focus on modeling the relationship between kinematic data and generalized forces rather than kinematic motion data itself.
<H1>3 Overview</H1>
We construct a physically valid statistical model that leverages both physical constraints and statistical motion priors and utilize it to generate physically realistic human motion that achieves the goals specified by the user. Physics-based dynamics modeling. Our motion model considers both Newtonian dynamics and contact mechanics for a full-body human figure. Therefore, we describe the Newtonian dynamics equations for full-body movement and Coulomb’s friction model for computing the forces caused by the friction between the character and the interaction environment (Section 4). Force field function modeling. We automatically extract force field priors from prerecorded motion data (Section 5). Our force field priors are represented by a nonlinear probabilistic function u = g(q, q) ̇ that maps the kinematic states (q, q) ̇ to the generalized forces u. To achieve this goal, we precompute the generalized forces u from prerecorded kinematic motion data and apply Gaussian process to modeling the force field priors embedded in training data. Motion modeling and synthesis. We show how to combine force field priors with physics-based dynamics models seamlessly in a probabilistic framework and how to use the new motion model to generate physically realistic animation that matches user-defined constraints (Section 6). We formulate the constraint-based motion synthesis problem in a Maximum A Posteriori (MAP) framework and introduce an efficient gradient-based optimization algorithm to find an optimal solution.
Figure 2: Motion data preprocessing for joint pose data (q), joint velocity data ( q) ̇ and generalized force data (u). (top) before the preprocessing; (bottom) after the preprocessing.
<H1>4 Physics-based Dynamics Models</H1>
Our dynamics models approximate human motion with a set of rigid body segments. We describe a full-body character pose with a set of independent joint coordinates q ∈ R 48 , including absolute root position and orientation, and the relative joint angles of 18 joints. These joints are the head, thorax, upper neck, lower neck, upper back, lower back, left and right humerus, radius, wrist, femur, tibia, and metatarsal. Newtonian dynamics. The Newtonian dynamics equations for full-body movement can be described using the following equation
[Jazar 2007]:
1
M (q)q + C(q, q) ̇ + h(q) = τ + f c + f e ≡ u
where q, q, ̇ and q represent the joint angle poses, joint velocities, and joint accelerations, respectively. The quantities M (q), C(q, q) ̇ and h(q) are the joint space inertia matrix, centrifugal/Coriolis and gravitational forces, respectively. The vectors τ , f c , and f e represent joint torques, contact forces, and external forces, respectively. The vector u represent the generalized forces, which can be either calculated from kinematic data or resultant forces of join torques, contact forces, and external forces. Human muscles generate torques about each joint, leaving global position and orientation of the body as unactuated joint coordinates. As a result, the movement of the global position and orientation is completely determined by contact forces f c and external forces f e . Contact mechanics. During ground contact, the feet can only push but not pull on the ground. To keep the body balanced, contact forces should not require an unreasonable amount of friction and the center of pressure must fall within the support polygon of the feet. We use Coulomb’s friction model to compute the forces caused by the friction between the character and the environment. A friction cone is defined to be the range of possible forces satisfying Coulomb’s function model for an object at rest. We ensure the contact forces stay within a basis that approximates the cones with nonnegative basis coefficients. We model the contact between two surfaces with multiple contact points m = 1, ..., M . As a result, we can represent the contact forces f c as a function of the joint angle poses and nonnegative basis coefficients
[Pollard and Reitsma 2001; Liu et al. 2005]: M
2
f c (q, λ) = J m (q) T B m e λ m m=1
where the matrix B m is a 3 × 4 matrix consisting of 4 basis vectors that approximately span the friction cone for the m-th contact force. The 4 × 1 vector e λ m represents nonnegative basis weights for the m-th contact force. The contact force Jacobian J m (q) maps the instantaneous generalized joint velocities to the instantaneous world space cartesian velocities at the m-th contact point under the joint pose q. Note that we remove the nonnegative coefficients constraints by representing the basis weights with exponential functions. Enforcing Newtonian dynamics equations and friction limit constraints would allow us to generate physically correct motion that satisfies friction limit constraints. However, physical constraints alone are insufficient to model natural-looking human movement because a motion can be physically correct without appearing natural. In the next section, we discuss how to learn force field functions from prerecorded motion data to constrain the human motion to lie in the space of natural appearance.
<H1>5 Force Field Function Modeling</H1>
Our system automatically extracts force field priors embedded in prerecorded motion data. Our idea of force field modeling is moti-
(a)
(b)
Figure 3: Modeling human motion with force fields: (a) training data: red dots and red lines represent kinematic states [q, q] ̇ and generalized forces u in the two-dimensional eigenspace, respectively; (b) motion generalization: black dots and black lines represent a motion instance generated by the learned force field model; (c) the generated 3D animation.
vated by recent findings in neuroscience [D’Avella et al. 2006; Bizzi et al. 2008]
, which reveal that the complex spatiotemporal characteristics of the muscle patterns for particular actions can be modeled by a weighted combination of a small number of force fields. We generalize this concept by learning a nonlinear probabilistic force field u = g(q, q), ̇ which maps kinematic states (q, q) ̇ to generalized forces u. Given an initial kinematic state (q 1 , q ̇ 1 ) of a human figure, a force field can predict how humans move by sequentially advancing a Newtonian dynamics model over time.
<H2>5.1 Motion Data Preprocessing</H2>
Constructing force field priors from motion capture data, however, is difficult because current motion capture technologies cannot directly measure generalized forces. Our solution is to compute generalized forces from prerecorded kinematic poses using the following Newtonian dynamics equation:
3
u = M (q)q + C(q, q) ̇ + h(q)
where the vector q represents prerecorded joint poses. The joint velocities q ̇ are computed as a backward difference between current and previous frames. The joint accelerations q are computed as a central difference between previous frames, current frames, and next frames. We have observed that the generalized forces computed from kinematic motion data are often very noisy because they are related to second derivatives of kinematic poses (see Figure 2 ). We thus preprocess generalized force data as well as joint poses and velocities using physics-based trajectory optimization techniques. Our approach follows the spacetime formulation in computer graphics literature
[Witkin and Kass 1988; Cohen 1992]. Briefly, we minimize the deviation from prerecorded kinematic motion data as well as the sum of the squared torques. This optimization is subject to foot-ground contact constraints, friction limit constraints, and the discretization of physics constraints determined by a finite difference scheme. Figure 2 shows the joint poses, joint velocities, and generalized forces before and after the preprocessing step. After motion data preprocessing, we have training data sets consisting of kinematic motion data [q n , q ̇ n ], n = 1, ..., N and their corresponding generalized force data u n , n = 1, ..., N . Our next task is to learn force field priors from the training data sets.
(c)
<H2>5.2 GP Modeling of Force Fields ̇</H2>
A force field is a nonlinear probabilistic function u = g(q, q) that maps the kinematic state (q, q) ̇ to the generalized forces u. We propose to use Gaussian process model to construct a force field from the training data sets. We choose GP model because it can efficiently model nonlinear property of the force fields and its learning process involves very few manual tuning parameters. More specifically, our GP model learns a nonlinear probabilistic function that predicts the generalized forces based on the joint pose and joint velocity (for details, see Appendix):
4
pr(u|q, q) ̇ = N (μ(q, q), ̇ Σ(q, q)) ̇
where both means and covariance matrices are functions of kinematic states [q, q]. ̇ In our implementation, we represent the root translations in the ground plane and the rotations about the up axis at the current frame with respect to the root coordinate system at the previous frame in order to eliminate the effect of absolute positions in the ground plane and the rotations about the up axis. In practice, human motion is highly coordinated, the number of dimensions of joint poses, joint velocities, or generalized forces is often much lower than the number of dimensions of the character’ poses. We, therefore, apply Principal Component Analysis techniques to reducing the dimensionality of both kinematic data [q n , q ̇ n ] and generalized force data u n and employ Gaussian process to model the force fields in reduced subspaces. We automatically determine the dimensions of subspaces by keeping 95% of the original energy. Subspace learning not only reduces the memory space for GP modeling but also significantly speeds up the learning and evaluation process of GP models. Figure 3(a) visualizes the force fields computed from a prerecorded walking database, which includes a wide variety of walking variations such as step sizes, turning angles, walking speeds, and walking slopes. To simplify the visualization, we only show the top two eigen-vectors for the kinematic states (q, q) ̇ as well as the generalized forces u. Given an initial state (q 1 , q ̇ 1 ), the learned force field priors pr(u|q, q) ̇ can produce a physically realistic motion sequence by sequentially advancing a Newtonian dynamics model over time ( Figure 3(b) and Figure 3(c) ).
<H1>6 Human Motion Modeling and Synthesis</H1>
We now discuss how to combine force field priors with physicsbased dynamics models in a probabilistic framework and how to apply the proposed framework to generating physically realistic human motion that achieves the goals specified by the user.
<H2>6.1 Combining Physics with Statistical Priors pr ( c | x ) pr ( x ) arg max x pr(x|c) = arg max x pr ( c ) (8) ∝ arg max x pr(c|x)pr(x)</H2>
We introduce a probabilistic motion model to model how humans move. Let pr(x) represent a probabilistic model of human motion x = {(q t , q ̇ t , u t )|t = 1, ..., T }, where q t , q ̇ t , and u t are joint poses, joint velocities, and generalized forces at frame t, respectively. According to Bayes’ rule, we can decompose the probabilistic motion model pr(x) into the following three terms: pr(x) = pr(q 1 , q ̇ 1 ) · pr(u t |q t , q ̇ t ) · pr(q t+1 , q ̇ t+1 |q t , q ̇ t , u t ) t pr init pr f orcef ield pr physics (5) where the first term pr init represents the probabilistic density function of the initial kinematic pose and velocity. In our experiment, we model the initial kinematic priors pr init with Gaussian mixture models. The second term pr f orcef ield represents the force field priors described in Equation (4). The third term pr physics measures how well the generated motion satisfies the physical constraints. In order to evaluate the third term pr physics , we first use backward difference to compute joint velocities and use central difference to compute joint accelerations. Based on the dynamics equation defined in Equation (1), the joint pose, joint velocities and generalized forces in the current step should completely determine the joint accelerations in the current step. Therefore, the joint pose and velocity in the next frame are also fully determined due to finite difference approximation.
Mathematically, we have
6
pr physics = pr(q t+1 , q ̇ t+1 |q t , q ̇ t , u t ) ∝ pr(q t |q t , q ̇ t , u t )
In practice, as noted by other researchers [Sok et al. 2007; Muico et al. 2009], dynamics models adopted in physics-based modeling are often inconsistent with observed data because of simplified dynamics/contact models, discretization of physics constraints, and approximate modeling of physical quantities of human bodies such as masses and inertias. Accordingly, dynamics equations are often not satisfied precisely. In our formulation, we assume Newtonian dynamics equations are disturbed by Gaussian noise of a standard deviation of σ physics : pr physics ∝ pr(q t |q t , q ̇ t , u t ) ∝ exp − M ( q t ) q ̈ t +C( q t , q ̇ 2σ t )+h( 2 q t )−τ t − f c ( q t ,λ t )− f e 2 physics (7) where the standard deviation σ physics shows our confidence of physics-based dynamics models. If the standard deviation is small, then the Gaussian probability distribution has a narrow peak, indicating high confidence in the physical constraints; similarly, a large standard deviation indicates low confidence. Such a motion model would allow us to generate an infinite number of physically realistic motion instances. In particular, we can sample the initial prior distribution pr init to obtain an initial state for joint poses and velocities and sequentially predict joint torques using the force field priors pr f orcef ield to advance the Newtonian dynamics model pr physics over time. More importantly, we can employ the motion model pr(x) to generate physically realistic animation x that best matches the user’s input c.
<H2>6.2 Constraint-based Motion Synthesis</H2>
We formulate the constraint-based motion synthesis problem in a maximum a posteriori (MAP) framework by estimating the most likely motion x from the user’s input c:
(8)
In our implementation, we minimize the negative logarithm of the posteriori probability density function pr(x|c), yielding the following energy minimization problem:
9
arg min x − ln pr(c|x) + − ln pr(x) ,
9
E c E prior
where the first term E c is the likelihood function measuring how well the generated motion x matches the input constraints c. Similar to [Chai and Hodgins 2007]
, the system allows the user to specify various forms of kinematic constraints throughout the motion or at isolated points in the motion. Typically, the user can define a sparse set of key frames as well as contact constraints to generate a desired animation. The user could also specify a small number of key trajectories to control fine details of a particular human action such as stylized walking. The second term E prior is the prior distribution function defined by our physically valid statistical model in Equation (5). The motion synthesis problem can now be solved by nonlinear optimization methods. Given a sparse set of constraints c, the optimization computes joint poses, joint torques, and contact forces by minimizing the following objective function:
10
argmin { q t ,τ t ,λ t } ω 1 E c + ω 2 E init + ω 3 E f orcef ield +ω 4 E physics
where E init , E f orcef ield , and E physics are the negative log of pr init , pr f orcef ield , and pr physics , respectively. In our experiment, we set the weights for E c , E init , E f orcef ield and E physics to 1000, 1, 1 and 100, respectively 1 . We choose a very large weight for the constraint term because we want to ensure the generated motion can match user constraints accurately. The weight for the physical term is much larger than the statistical prior term because physical correctness has a higher priority than statistical consistency in our system. Thus far, we have not discussed how to incorporate the learned force field priors into the motion optimization framework. Note that in the force field modeling step, we performed dimensionality reduction analysis on both kinematic data and generalized joint torques and learned the force field priors in reduced subspaces. One possible solution to incorporating the force field priors is to perform the optimization in the reduced subspaces. We have implemented this idea and found that performing the optimization in the subspaces can hurt the generalization ability of our model and often cannot match user-specified constraints accurately. To avoid this issue, we choose to perform the optimization in the original configuration space while imposing “soft” subspace constraints on both kinematic states and generalized forces. Let B u and B s denote the subspace matrices for generalized forces u and kinematic states s = [q T , q ̇ T ] T , respectively. We reformulate
1 Note that the weight for the physics term (ω 4 ) corresponds to 2σ 2 1 in Equation 7 physics
Motion examples
Total frames
Durations
Total key frames
Total key trajectories
Initialization times
Synthesis times
Normal walking
270
9s
2
0
9 sec
17 min
Big-step walking
272
9s
2
0
7 sec
16 min
Walking and turning
392
13s
2
0
10 sec
20 min
Running
130
4.3s
2
0
5 sec
10 min
Jumping
168
5.6s
3
0
3 sec
7 min
Heavy foot
235
7.8s
2
0
10 sec
22 min
Resistance running
148
4.9s
2
0
5 sec
13 min
Slippery surfaces
193
6.4s
2
0
8 sec
20 min
Moon walking
193
6.4s
2
0
8 sec
21 min
Sneaky walking
674
22.5s
2
3
20 sec
23 min
Proud walking
302
10.1s
2
2
12 sec
16 min
Long walking sequence
1357
45.2s
8
0
27 sec
51 min
Run→walk→jump
510
17s
6
2
13 sec
21 min
Motion examples Total frames Durations Total key frames Total key trajectories Initialization times Synthesis times Normal walking 270 9s 2 0 9 sec 17 min Big-step walking 272 9s 2 0 7 sec 16 min Walking and turning 392 13s 2 0 10 sec 20 min Running 130 4.3s 2 0 5 sec 10 min Jumping 168 5.6s 3 0 3 sec 7 min Heavy foot 235 7.8s 2 0 10 sec 22 min Resistance running 148 4.9s 2 0 5 sec 13 min Slippery surfaces 193 6.4s 2 0 8 sec 20 min Moon walking 193 6.4s 2 0 8 sec 21 min Sneaky walking 674 22.5s 2 3 20 sec 23 min Proud walking 302 10.1s 2 2 12 sec 16 min Long walking sequence 1357 45.2s 8 0 27 sec 51 min Run→walk→jump 510 17s 6 2 13 sec 21 min
Table 1: Details of all the animations generated by our synthesis algorithm.
Databases
size
Durations
Prep.
GP learning
walking
5227
2.9 min
65 min
40 min
stylized walking
7840
4.4 min
138 min
46 min
locomotion
4571
2.5 min
55 min
47 min
Databases size Durations Prep. GP learning walking 5227 2.9 min 65 min 40 min stylized walking 7840 4.4 min 138 min 46 min locomotion 4571 2.5 min 55 min 47 min
Table 2: Details of three training data sets and the computational times spent on data preprocessing (Section 5.1) and GP learning (Section 5.2).
the force field priors as follows:
− ln pr(B u T u|B s T s) + α 1 u − B u B u T u 2 + α 2 s − B s B s T s 2 E f orcef ield (11) where the first term represents the force field priors in reduced subspaces. The second and third terms impose the “soft” subspace constraints for kinematic states and generalized forces, penalizing them as they deviate from the subspace representations. In our experiment, we set the weights α 1 and α 2 to 10 and 10, respectively. The combined motion models are desirable for human motion generation because they measure both statistical consistency and physical correctness of the motion. With the physical term, our model can react to changes in physical parameters. For example, when a character is pushed by an external force, e.g., elastic forces in resistance running, the external force in the physics term E physics (see Equation 7) will force the system to modify kinematic motion and joint torques as well as contact forces in order to satisfy Newtonian dynamics and contact mechanics. However, without force field priors, the modified motion could be unnatural because there are many ways to adjust a motion so that physical laws are satisfied, and yet only a subset of motions are natural-looking. With force field priors, our system pushes the modified motions towards regions of high probability density in order to be consistent with force field priors.
<H1>7 Implementation Details</H1>
Here we briefly discuss implementation details of our system: Data preprocessing. We used three different motion databases in our experiments, including walking (5227 frames), stylized walking (7840 frames), and locomotion databases (4571 frames). We preprocessed the prerecorded motion data using spacetime optimization (Section 5.1). The computational time for each data set was reported in Table 2 . GP learning. To speed up the learning and evaluation process of GP models, we applied PCA to reduce the dimensionality of training data and learned the GP model in a reduced subspace. We automatically determined the dimension of the subspace by preserving 95% of the original energy. The dimensions of the kinematic states ([q t , q ̇ t ]) in three databases were 19, 22, and 19 respectively. The dimensions of the generalized forces (u) were 8, 10, and 7 respectively. We adopted sparse approximation strategies for Gaussian process modeling
[Quinonero-Candela and Rasmussen 2005]. The GP learning times spent on the three training databases were 65 minutes, 138 minutes, and 55 minutes, respectively. Motion optimization. We follow a standard approach of representing q t and τ t using cubic B-splines. We solved the optimization problem using sequential quadratic programming (SQP)
[Bazaraa et al. 1993]
, where each iteration solves a quadratic programming subproblem. We implemented the system with C++/Matlab and conducted the optimization with the Matlab optimization toolbox. Each optimization often took from ten to thirty minutes to converge without code optimization (for details, see Table 1 ). All the experiments were run on a 2.5GHz dual core computer with 3GB of RAM. Initialization. The performance of our optimization algorithm highly depends on the initialization of the optimization. To obtain a good initial guess for joint poses q t , t = 1, ..., T , we dropped off the physical term E physics in the objective function and used the remaining objective functions to optimize the joint poses across the entire motion sequence. We evaluated the force field term with respect to joint poses because we can calculate current generalized forces using current joint poses, velocities and accelerations as shown in Equation (3). With the initialized joint poses q t 0 , t = 1, ..., T , we dropped off the constraint term as well as the initial prior term, and optimized the joint torques τ as well as contact forces λ using the force field prior term and the physics term. In this step, we evaluated the force field priors in terms of joint torques and contact forces: E f orcef ield (τ, λ) = − ln pr(τ + f c (q 0 , λ) + f e )|q 0 , q ̇ 0 ). Each initialization step often took from less than thirty seconds to converge (for details, see Table 1 ).
<H1>8 Experiments</H1>
This section demonstrated the benefits of combining physical constraints and statistical motion priors for human motion generation. In addition, we evaluated the performance of our algorithm by comparing with ground truth data and results obtained by alternative methods. The details for our experiments are summarized in Table 1. For each example in our experiments, we reported the total number of animation frames, the types and number of animation constraints, and the computational times spend on the initialization and motion synthesis step.
Figure 4: Generating physically realistic motion that reacts to changes in physical quantities of human bodies: walking with a heavy left foot.
<H2>8.1 The Benefits of Physical Constraints</H2>
The incorporation of physics into probabilistic motion models significantly improves the generalizability of statistical motion models. This experiment shows that the system can generate physically realistic motion that reacts to changes in physical quantities of human bodies and interaction environments, a capability that has not been demonstrated in previous statistical motion models. Heavy foot. Our system can react to changes in physical quantities such as masses and inertias of human bodies. For example, we changed the mass of the character by simulating a character wearing a 2.5 kilogram shoe. The accompanying video shows that the simulated character maintained balance by adapting the gait and leaning the body to the right side in order to offset the additional weigh caused by the left shoe. Figure 4 shows sample frames for walking with a heavy foot. Resistance running. In this example, the user specified the start and end poses as well as foot contacts to create an animation for resistance running ( Figure 1(b) ). The resistance forces were determined by Hooke’s law of elasticity, ranging from zero to 450N. We observed that the character moved the upper body forward in order to offset the effect of resistance force. Walking on slippery surfaces. We can generate an animation that reacts to changes in friction properties of environments. In the accompanying video, we show a simulated character walking on a slippery surface by reducing the friction coefficient to 0.05. Moon walking. We can edit an animation by changing the gravity of interaction environments. For example, we generated “moon” walking by setting gravity at 1.62 m/s 2 .
<H2>8.2 The Benefits of Statistical Motion Priors</H2>
This experiment shows that we can extend physics-based modeling techniques to stylized walking, detailed walking variations, and heterogeneous human actions with the help of statistical motion priors. Such actions are often difficult or even impossible to generate with previous physics-based modeling techniques. Stylized walking. Our approach can generate physically-realistic animation for highly stylized human actions. The training data sets for stylized walking included normal walking and ten distinct walking styles. The system constructed a single motion model from the training data sets and used it to generate various forms of stylized walking such as “sneaky” walking and “proud” walking (Figure 1(c)). In addition to keyframes and foot contact constraints, the user specified a sparse number of key trajectories in order to control the fine details of stylized walking. Walking variations. We tested the effectiveness of our algorithm for modeling a wide range of walking variations. We learned a single generative model from a “walking” database and used it to generate a long walking sequence. The synthesized motion displayed a wide variety of walking variations such as walking along a straight line, walking with a sharp turn, walking with a big step, walking on a slope, climbing over an obstacle, and transitionings between different walking examples ( Figure 5 ). Because of memory restrictions, we synthesized the whole motion sequence by sequentially computing each example from sparse constraints and stitching them into a long motion sequence. For each example, the user specified the start and end poses of the generated motion as well as foot contact constraints throughout the whole motion sequence. Heterogeneous actions. We tested the effectiveness of the physically valid statistical model on heterogeneous human actions. We learned a single generative model from a locomotion database and used it to create a long animation sequence consisting of walking, running, jumping, and stopping, as well as their transitions (Figure 1(d)).
<H2>8.3 Evaluation and Comparisons</H2>
We assessed the quality of the generated motions by comparing with ground truth data. We also evaluated the importance of force field priors and physical constraints for human motion generation. Comparison against ground truth data. We evaluated the performance of our algorithm via cross validation techniques. More specifically, we pulled out a testing sequence in the training data, used it to extract the start and end poses and foot contact constraints, and applied the synthesis algorithm to generate motion that matches the “simulated” constraints. The accompanying video shows a side-by-side comparison between the ground truth motion and the synthesized motion. We have observed that the generated motions achieve similar quality to the ground truth motion data. The importance of force field priors. This comparison shows the importance of force field priors for human motion generation. We compared our system with standard physics-based optimization techniques
[Witkin and Kass 1988] by dropping off both force field priors term E f orcef ield and initialization term E init in the objective function defined in Equation (10). For a fair comparison, we added the minimal sum of squared joint torques into the objective function because optimizing the motion with the remaining terms (E c and E physics ) is ambiguous–there are an infinite number of physically correct motions that satisfy user constraints. We also included joint torque limits into the optimization. Without the force field priors, the “sneaky” walking appears ballistic because the
(a)
(b)
Figure 5: Generating a wide variety of physically realistic walking motions: (a) normal walking; (b) walking with a big step; (c) climbing over an obstacle; (d) walking on a slope. All the motions are generated by a single statistical walking model constructed from a prerecorded walking database
“minimal torque principle” is not suitable for stylized low-energy motion. With the force field priors, our system can successfully generate physically realistic stylized walking motion.
(a)
(b)
Figure 6: The importance of the physics term. (a) with the physics term; (b) without the physics term. Note that with the physics term, the simulated character reacts to external elastic forces by leaning the body forward to compensate the resistance forces. Note that “yellow” characters are the starting and ending keyframes used for motion generation; foot contact constraints as shown in “green”.
The importance of the physics term. This experiment demonstrated the importance of physical constraints to our motion model. We dropped off the physics term in the objective function and used the remaining terms to optimize the joint poses across the entire motion sequence. The accompanying video shows a side-by-side comparison for animating the “resistance running”. With the physics term, the character reacted appropriately to external elastic forces by leaning the body forward to compensate for the resistance forces ( Figure 6(a) ). As expected, the character did not respond to external forces without the physics term ( Figure 6(b) ). Comparison against subspace optimization. We computed the eigen-poses using the same set of training data and performed physics-based optimization in a reduced eigen-space similar to
Safonova and her colleagues [2004]. The testing example was running→walking→jumping. Unlike Safonova and her colleagues [2004]
, we did not manually select training data to construct a reduced subspace for human poses. Instead, we used the entire locomotion database (4571 frames), which includes normal walking, running and jumping. We automatically determined the dimension of the subspace (11 dimensions) by preserving 95% of energy of the training data. To implement the subspace optimization algorithm, we formulated the problem in the spacetime framework and optimized the motion in the reduced subspace. Briefly, we minimized the sum of squared torques and smoothness of the root and joint angle trajectories over time. We also added a regularization term to penalize the deviation of eigen coefficients from zero. This optimization was also subject
(c)
(d)
to foot-ground contact constraints, friction limit constraints, and the discretization of physics constraints determined by a finite difference scheme. Unlike Safonova and her colleagues [2004]
, we did not incorporate inverse kinematics as part of optimization in our implementation. We evaluated the performance of the subspace optimization technique using the same set of animation constraints, including the start and end poses as well as trajectories of the head and two feet. The accompanying video shows that subspace optimization produces uncoordinated human movements. For example, the walking character did not swing the right arm properly and the walking gait appeared very stiff. This indicates that a global subspace model for kinematic poses is not sufficient to model heterogeneous human actions. We have also observed that the motions generated by subspace methods often cannot accurately match the trajectory and contact constraints specified by the user; this might be due to compression errors caused by reduced subspace representation. In contrast, the GP-based statistical motion priors can accurately model spatial-temporal patterns in heterogeneous human actions and allow for generating physically realistic animation that matches userdefined constraints.
<H1>9 Discussion and Future Work</H1>
We introduce a statistical motion model for human motion analysis and generation. Our model combines the powers of physics-based motion modeling and statistical motion modeling. We have demonstrated the effectiveness of the new model by generating a wide variety of physically realistic motions that achieve the goals specified by the users. The incorporation of physical constraints into statistical motion models ensures generalized motions are physically plausible, thereby removing noticeable visual artifacts (e.g., unbalanced motions and motion jerkiness) in an output animation. Moreover, it enables us to create motions that react to changes in physical parameters. In our experiments, we have shown that the system can generate new motions such as “resistance running”, “moon walking”, “walking on slippery surfaces”, and “walking with a heavy foot”, a capability that has never been demonstrated in any previous statistical motion synthesis methods. Meanwhile, the use of force field priors for human motion modeling not only ensures that generated motions are natural looking but also extends physically-based modeling techniques to stylized and heterogeneous human actions. For example, we have constructed a single generative model for modeling a wide variety of physically realistic walking variations such as normal walking, walking with a sharp turn, walking on a slope, walking with a big step, and climbing over an obstacle. We have also shown that the system can generate physically realistic motion for stylized walking such as sneaky walking and for heterogeneous human actions such as running→walking→jumping. Such actions are often difficult or even impossible to be synthesized by previous physics-based motion models. We model the force field priors using Gaussian process models because GP can efficiently capture nonlinear properties of the force fields and its learning process involves very few manual tuning parameters. However, Gaussian process needs to retain all of the training data to make predictions and therefore its computational demands grow as the square and cube respectively of the number of training examples. The sparse approximation strategy works well for the current size of training data sets (less than 8,000 frames) but might not scale up for use in very large data sets.
One possibility is to learn a probabilistic regression function for force fields using parametric statistical analysis techniques such as the mixture of experts model
[Jacobs et al. 1991] or its variants
[Jordan 1994]. Another limitation of our system is that it cannot generate a motion that is very different from motion examples because our approach is data-driven. In addition, the system is still unable to handle arbitrary external forces because the force field priors prevent the generated motion from moving away from prerecorded motion data. We choose to model the force field priors based on generalized forces rather than joint torques because we can conveniently compute the generalized forces from current kinematic motion capture databases (e.g., the CMU online mocap database 2 ). However, the learned force field priors can only predict resultant forces of join torques and contact forces. If both joint torque data and contact force data are available, we could construct more accurate force field priors that explicitly predict joint torques or contact forces. In the future, we plan to measure ground-reaction forces with force plates and use them along with the captured kinematic motion data to compute joint torques via inverse dynamics techniques. We formulate the constraint-based motion synthesis problem in a spacetime optimization framework. However, the optimization problem is high-dimensional and highly nonlinear; it might be subject to local minima. We found that the initialization process is critical to the success of our optimization. It not only speeds up the optimization process but also alleviates the local-minimum problem. For a long sequence of animation (e.g., Figure 5 ), we need to decompose the entire optimization into a number of spacetime windows, over which subproblems can be formulated and solved using efficient nonlinear optimization techniques. In the future, we plan to explore alternative techniques to address the local minimum problem. One possibility is the employment of a Markov chain Monte Carlo (MCMC), which comes to its solutions by efficiently drawing samples from the posterior distribution, using a Markov chain based on the Metropolis-Hastings algorithm. Similar to other constraint-based animation systems, our system requires the user to specify a sparse number of constraints, e.g., key frames and contact constraints, to generate a desired animation. However, specifying such constraints, particularly trajectory constraints and contact constraints, is not trivial for a novice user. In our experiment, we created the 3D key frames by using our homegrown data-driven inverse kinematic system
[Wei and Chai 2010a]. Trajectory and contact constraints were either directly modified from reference motion data or rotoscoped from video streams similar to the technique described by
Wei and Chai [2010b]. In the future, we are interested in extending our system to searching the positions and timings of contact events as part of the optimization variables, thereby avoiding the necessity of contact constraints required for constraint-based motion synthesis.
2 http://mocap.cs.cmu.edu/
<H1>APPENDIX A Gaussian Processes</H1>
Gaussian processes (GP) are a powerful, non-parametric tool for regression in high-dimensional space. A GP can be thought of as a “Gaussian over functions”. Here, we briefly discuss the basic concept of Gaussian processes. Let D = {(y n , z n )|n = 1, ..., N } be the training set. For our application, we have y = [q, q] ̇ and z = u. The goal of Gaussian processes is to learn a regression function f (·) that finds the predictive output z ∗ using a testing input y ∗ . We assume both training and testing data points are drawn from the following noisy process:
12
z n = f (y n ) + d
where y n is an input vector in R and z n is a scalar output in R. The noise term is drawn from N (0, σ 2 ). For convenience, the inputs are stacked into a d×N matrix Y = [y 1 , y 2 , ..., y N ] and the outputs are stacked into a N -dimensional vector z = [z 1 , z 2 , ..., z N ]. The joint distribution over the noisy output z given inputs Y is a zero-mean Gaussian, and has the form
13
2 pr(z|Y ) = N (0, K(Y, Y ) + σ n I),
where K(Y, Y ) is the kernel matrix with elements K ij = k(y i , y j ). The kernel function, k(y, y ), is a measure of the “closeness” between inputs. The term σ n 2 I introduces Gaussian noise and plays a similar role to that of in Equation (12). Given a set of test inputs Y ∗ , one would like to find the predictive output z ∗ . The noisy training outputs z and the test output z ∗ are jointly Gaussian:
K(Y ∗ , Y ∗ ) K(Y ∗ , Y ) pr(z ∗ , z|Y ∗ , Y ) = N (0, K(Y, Y ∗ ) K(Y, Y ) + σ n 2 I ) (14) Since z is known, this Gaussian can be conditioned on z to obtain the predictive distribution for z ∗ :
15
pr(z ∗ |z, Y ∗ , Y ) = N (μ, Σ),
where μ = K(Y ∗ , Y )[K(Y, Y ) + σ n 2 I] −1 z, Σ = K(Y ∗ , Y ∗ ) − K(Y ∗ , Y )[K(Y, Y ) + σ n 2 I] −1 K(Y, Y ∗ ). (16) A Gaussian process is fully described by its mean and covariance functions. These equations show that the mean function for the testing output is a linear combination of the training output z, and the weight of each input is directly related to the correlation between the testing input Y ∗ and the training input Y . Meanwhile, the uncertainty for every predictive output (i.e. covariance function) is also estimated. In this paper, we choose the squared exponential function as our kernel function:
17
k(y, y ) = σ f 2 exp −( 1 (y − y ) T W (y − y )) 2 2
where σ f is the signal variance. The diagonal matrix W contains the length scales for each input dimension. The value of W ii is inversely proportional to the importance of the i-th input dimension. The parameters of the kernel function θ = [W, σ f , σ n ] can be automatically learned by maximizing the log likelihood of the training outputs given the inputs: θ max = arg max θ log pr(z|Y, θ).
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"offsets": [
[
19676,
19802
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],
"text": "['Our solution is to compute generalized forces from prerecorded kinematic poses using the following Newtonian dynamics equation']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
19841,
19871
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],
"text": "['u = M (q)q + C(q, q) ̇ + h(q)']",
"type": "own_claim",
"id": "T184"
},
{
"offsets": [
[
20192,
20271
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],
"text": "['the generalized forces computed from kinematic motion data are often very noisy']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
20280,
20337
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],
"text": "['they are related to second derivatives of kinematic poses']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
20343,
20351
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],
"text": "['Figure 2']",
"type": "data",
"id": "T187"
},
{
"offsets": [
[
20491,
20569
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],
"text": "['Our approach follows the spacetime formulation in computer graphics literature']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
20572,
20592
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],
"text": "['Witkin and Kass 1988']",
"type": "data",
"id": "T189"
},
{
"offsets": [
[
20594,
20604
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],
"text": "['Cohen 1992']",
"type": "data",
"id": "T190"
},
{
"offsets": [
[
21386,
21517
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],
"text": "['A force field is a nonlinear probabilistic function u = g(q, q) that maps the kinematic state (q, q) ̇ to the generalized forces u']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
21519,
21614
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],
"text": "['We propose to use Gaussian process model to construct a force field from the training data sets']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
21643,
21706
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],
"text": "['it can efficiently model nonlinear property of the force fields']",
"type": "data",
"id": "T193"
},
{
"offsets": [
[
21711,
21774
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],
"text": "['its learning process involves very few manual tuning parameters']",
"type": "data",
"id": "T194"
},
{
"offsets": [
[
22458,
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],
"text": "['In practice, human motion is highly coordinated']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
22507,
22661
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],
"text": "['the number of dimensions of joint poses, joint velocities, or generalized forces is often much lower than the number of dimensions of the character’ poses']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
23072,
23144
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"text": "['significantly speeds up the learning and evaluation process of GP models']",
"type": "own_claim",
"id": "T197"
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{
"offsets": [
[
22995,
23062
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],
"text": "['Subspace learning not only reduces the memory space for GP modeling']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
23538,
23696
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],
"text": "['the learned force field priors pr(u|q, q) ̇ can produce a physically realistic motion sequence by sequentially advancing a Newtonian dynamics model over time']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
[
23498,
23536
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],
"text": "['Given an initial state (q 1 , q ̇ 1 )']",
"type": "data",
"id": "T200"
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{
"offsets": [
[
23699,
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],
"text": "['Figure 3(b)']",
"type": "data",
"id": "T201"
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{
"offsets": [
[
23715,
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],
"text": "['Figure 3(c)']",
"type": "data",
"id": "T202"
},
{
"offsets": [
[
25932,
26023
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],
"text": "['dynamics models adopted in physics-based modeling are often inconsistent with observed data']",
"type": "background_claim",
"id": "T203"
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{
"offsets": [
[
26035,
26069
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],
"text": "['simplified dynamics/contact models']",
"type": "data",
"id": "T204"
},
{
"offsets": [
[
25912,
25929
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],
"text": "['Muico et al. 2009']",
"type": "data",
"id": "T205"
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{
"offsets": [
[
25851,
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],
"text": "['In practice']",
"type": "background_claim",
"id": "T206"
},
{
"offsets": [
[
25895,
25910
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],
"text": "['Sok et al. 2007']",
"type": "data",
"id": "T207"
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{
"offsets": [
[
26071,
26108
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],
"text": "['discretization of physics constraints']",
"type": "data",
"id": "T208"
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{
"offsets": [
[
26114,
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],
"text": "['approximate modeling of physical quantities of human bodies']",
"type": "data",
"id": "T209"
},
{
"offsets": [
[
26182,
26188
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],
"text": "['masses']",
"type": "data",
"id": "T210"
},
{
"offsets": [
[
26193,
26201
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],
"text": "['inertias']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
26216,
26268
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],
"text": "['dynamics equations are often not satisfied precisely']",
"type": "background_claim",
"id": "T212"
},
{
"offsets": [
[
26270,
26397
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],
"text": "['In our formulation, we assume Newtonian dynamics equations are disturbed by Gaussian noise of a standard deviation of σ physics']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
26400,
26537
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],
"text": "['pr physics ∝ pr(q t |q t , q ̇ t , u t ) ∝ exp − M ( q t ) q ̈ t +C( q t , q ̇ 2σ t )+h( 2 q t )−τ t − f c ( q t ,λ t )− f e 2 physics']",
"type": "data",
"id": "T214"
},
{
"offsets": [
[
26548,
26634
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],
"text": "['the standard deviation σ physics shows our confidence of physics-based dynamics models']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
26677,
26788
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],
"text": "['the Gaussian probability distribution has a narrow peak, indicating high confidence in the physical constraints']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
26639,
26670
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],
"text": "['the standard deviation is small']",
"type": "data",
"id": "T217"
},
{
"offsets": [
[
26801,
26852
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],
"text": "['a large standard deviation indicates low confidence']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
26854,
26960
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],
"text": "['Such a motion model would allow us to generate an infinite number of physically realistic motion instances']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
27252,
27370
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],
"text": "['we can employ the motion model pr(x) to generate physically realistic animation x that best matches the user’s input c']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
26977,
27087
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],
"text": "['we can sample the initial prior distribution pr init to obtain an initial state for joint poses and velocities']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
27092,
27232
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],
"text": "['sequentially predict joint torques using the force field priors pr f orcef ield to advance the Newtonian dynamics model pr physics over time']",
"type": "own_claim",
"id": "T222"
},
{
"offsets": [
[
28132,
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],
"text": "['the system allows the user to specify various forms of kinematic constraints throughout the motion or at isolated points in the motion']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
28095,
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],
"text": "['Similar to']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
28107,
28128
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],
"text": "['Chai and Hodgins 2007']",
"type": "data",
"id": "T225"
},
{
"offsets": [
[
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],
"text": "['the user can define a sparse set of key frames as well as contact constraints to generate a desired animation']",
"type": "background_claim",
"id": "T226"
},
{
"offsets": [
[
28390,
28505
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],
"text": "['The user could also specify a small number of key trajectories to control fine details of a particular human action']",
"type": "background_claim",
"id": "T227"
},
{
"offsets": [
[
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],
"text": "['stylized walking']",
"type": "data",
"id": "T228"
},
{
"offsets": [
[
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28738
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],
"text": "['The motion synthesis problem can now be solved by nonlinear optimization methods']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
29932,
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],
"text": "['One possible solution to incorporating the force field priors is to perform the optimization in the reduced subspaces']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
30096,
30250
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],
"text": "['performing the optimization in the subspaces can hurt the generalization ability of our model and often cannot match user-specified constraints accurately']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
30252,
30437
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],
"text": "['To avoid this issue, we choose to perform the optimization in the original configuration space while imposing “soft” subspace constraints on both kinematic states and generalized forces']",
"type": "own_claim",
"id": "T232"
},
{
"offsets": [
[
36518,
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],
"text": "['The combined motion models are desirable for human motion generation']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
36595,
36675
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],
"text": "['they measure both statistical consistency and physical correctness of the motion']",
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"id": "T234"
},
{
"offsets": [
[
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],
"text": "['With the physical term, our model can react to changes in physical parameters']",
"type": "own_claim",
"id": "T235"
},
{
"offsets": [
[
36862,
36910
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],
"text": "['the external force in the physics term E physics']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
36928,
37081
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],
"text": "['will force the system to modify kinematic motion and joint torques as well as contact forces in order to satisfy Newtonian dynamics and contact mechanics']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
36774,
36816
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],
"text": "['a character is pushed by an external force']",
"type": "data",
"id": "T238"
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{
"offsets": [
[
36824,
36860
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],
"text": "['elastic forces in resistance running']",
"type": "data",
"id": "T239"
},
{
"offsets": [
[
36916,
36926
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],
"text": "['Equation 7']",
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"id": "T240"
},
{
"offsets": [
[
37092,
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],
"text": "['without force field priors, the modified motion could be unnatural']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
37167,
37295
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],
"text": "['there are many ways to adjust a motion so that physical laws are satisfied, and yet only a subset of motions are natural-looking']",
"type": "data",
"id": "T242"
},
{
"offsets": [
[
37297,
37454
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],
"text": "['With force field priors, our system pushes the modified motions towards regions of high probability density in order to be consistent with force field priors']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
38739,
38812
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],
"text": "['We solved the optimization problem using sequential quadratic programming']",
"type": "own_claim",
"id": "T244"
},
{
"offsets": [
[
38844,
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],
"text": "['where each iteration solves a quadratic programming subproblem']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
38821,
38840
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],
"text": "['Bazaraa et al. 1993']",
"type": "data",
"id": "T246"
},
{
"offsets": [
[
39235,
39337
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],
"text": "['The performance of our optimization algorithm highly depends on the initialization of the optimization']",
"type": "own_claim",
"id": "T247"
},
{
"offsets": [
[
41163,
41297
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],
"text": "['The incorporation of physics into probabilistic motion models significantly improves the generalizability of statistical motion models']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
[
41326,
41551
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],
"text": "['the system can generate physically realistic motion that reacts to changes in physical quantities of human bodies and interaction environments, a capability that has not been demonstrated in previous statistical motion models']",
"type": "own_claim",
"id": "T249"
},
{
"offsets": [
[
41565,
41619
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],
"text": "['Our system can react to changes in physical quantities']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
41628,
41634
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],
"text": "['masses']",
"type": "data",
"id": "T251"
},
{
"offsets": [
[
41639,
41663
]
],
"text": "['inertias of human bodies']",
"type": "data",
"id": "T252"
},
{
"offsets": [
[
41804,
41970
]
],
"text": "['the simulated character maintained balance by adapting the gait and leaning the body to the right side in order to offset the additional weigh caused by the left shoe']",
"type": "data",
"id": "T253"
},
{
"offsets": [
[
42312,
42404
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],
"text": "['the character moved the upper body forward in order to offset the effect of resistance force']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
42436,
42526
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],
"text": "['We can generate an animation that reacts to changes in friction properties of environments']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
42555,
42659
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],
"text": "['we show a simulated character walking on a slippery surface by reducing the friction coefficient to 0.05']",
"type": "data",
"id": "T256"
},
{
"offsets": [
[
42675,
42751
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],
"text": "['We can edit an animation by changing the gravity of interaction environments']",
"type": "own_claim",
"id": "T257"
},
{
"offsets": [
[
42766,
42826
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],
"text": "['we generated “moon” walking by setting gravity at 1.62 m/s 2']",
"type": "data",
"id": "T258"
},
{
"offsets": [
[
42949,
43121
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],
"text": "['we can extend physics-based modeling techniques to stylized walking, detailed walking variations, and heterogeneous human actions with the help of statistical motion priors']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
[
43123,
43234
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],
"text": "['Such actions are often difficult or even impossible to generate with previous physics-based modeling techniques']",
"type": "own_claim",
"id": "T260"
},
{
"offsets": [
[
43254,
43344
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],
"text": "['Our approach can generate physically-realistic animation for highly stylized human actions']",
"type": "own_claim",
"id": "T261"
},
{
"offsets": [
[
43447,
43577
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],
"text": "['The system constructed a single motion model from the training data sets and used it to generate various forms of stylized walking']",
"type": "own_claim",
"id": "T262"
},
{
"offsets": [
[
43586,
43602
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],
"text": "['“sneaky” walking']",
"type": "data",
"id": "T263"
},
{
"offsets": [
[
43607,
43622
]
],
"text": "['“proud” walking']",
"type": "data",
"id": "T264"
},
{
"offsets": [
[
43624,
43635
]
],
"text": "['Figure 1(c)']",
"type": "data",
"id": "T265"
},
{
"offsets": [
[
44032,
44101
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],
"text": "['The synthesized motion displayed a wide variety of walking variations']",
"type": "own_claim",
"id": "T266"
},
{
"offsets": [
[
44110,
44139
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],
"text": "['walking along a straight line']",
"type": "data",
"id": "T267"
},
{
"offsets": [
[
44141,
44166
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],
"text": "['walking with a sharp turn']",
"type": "data",
"id": "T268"
},
{
"offsets": [
[
44168,
44191
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],
"text": "['walking with a big step']",
"type": "data",
"id": "T269"
},
{
"offsets": [
[
44193,
44211
]
],
"text": "['walking on a slope']",
"type": "data",
"id": "T270"
},
{
"offsets": [
[
44213,
44238
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],
"text": "['climbing over an obstacle']",
"type": "data",
"id": "T271"
},
{
"offsets": [
[
44244,
44293
]
],
"text": "['transitionings between different walking examples']",
"type": "data",
"id": "T272"
},
{
"offsets": [
[
44296,
44304
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],
"text": "['Figure 5']",
"type": "data",
"id": "T273"
},
{
"offsets": [
[
45736,
45835
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],
"text": "['We have observed that the generated motions achieve similar quality to the ground truth motion data']",
"type": "own_claim",
"id": "T274"
},
{
"offsets": [
[
45875,
45961
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],
"text": "['This comparison shows the importance of force field priors for human motion generation']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
46314,
46360
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],
"text": "['optimizing the motion with the remaining terms']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
46382,
46394
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],
"text": "['is ambiguous']",
"type": "own_claim",
"id": "T277"
},
{
"offsets": [
[
46362,
46379
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],
"text": "['E c and E physics']",
"type": "data",
"id": "T278"
},
{
"offsets": [
[
46395,
46483
]
],
"text": "['there are an infinite number of physically correct motions that satisfy user constraints']",
"type": "data",
"id": "T279"
},
{
"offsets": [
[
46545,
46615
]
],
"text": "['Without the force field priors, the “sneaky” walking appears ballistic']",
"type": "own_claim",
"id": "T280"
},
{
"offsets": [
[
46624,
46627
]
],
"text": "['the']",
"type": "data",
"id": "T281"
},
{
"offsets": [
[
47032,
47105
]
],
"text": "['“minimal torque principle” is not suitable for stylized low-energy motion']",
"type": "data",
"id": "T282"
},
{
"offsets": [
[
47107,
47217
]
],
"text": "['With the force field priors, our system can successfully generate physically realistic stylized walking motion']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
47763,
47850
]
],
"text": "['This experiment demonstrated the importance of physical constraints to our motion model']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
48097,
48250
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],
"text": "['With the physics term, the character reacted appropriately to external elastic forces by leaning the body forward to compensate for the resistance forces']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
48281,
48354
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],
"text": "['the character did not respond to external forces without the physics term']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
48357,
48368
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],
"text": "['Figure 6(b)']",
"type": "data",
"id": "T287"
},
{
"offsets": [
[
48253,
48264
]
],
"text": "['Figure 6(a)']",
"type": "data",
"id": "T288"
},
{
"offsets": [
[
49967,
50027
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],
"text": "['subspace optimization produces uncoordinated human movements']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
49937,
49955
]
],
"text": "['accompanying video']",
"type": "data",
"id": "T290"
},
{
"offsets": [
[
50105,
50141
]
],
"text": "['the walking gait appeared very stiff']",
"type": "data",
"id": "T291"
},
{
"offsets": [
[
50042,
50100
]
],
"text": "['the walking character did not swing the right arm properly']",
"type": "data",
"id": "T292"
},
{
"offsets": [
[
50143,
50261
]
],
"text": "['This indicates that a global subspace model for kinematic poses is not sufficient to model heterogeneous human actions']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
50290,
50422
]
],
"text": "['the motions generated by subspace methods often cannot accurately match the trajectory and contact constraints specified by the user']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
50424,
50505
]
],
"text": "['this might be due to compression errors caused by reduced subspace representation']",
"type": "own_claim",
"id": "T295"
},
{
"offsets": [
[
50520,
50729
]
],
"text": "['the GP-based statistical motion priors can accurately model spatial-temporal patterns in heterogeneous human actions and allow for generating physically realistic animation that matches userdefined constraints']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
50890,
50984
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"text": "['it enables us to create motions that react to changes in physical parameters']",
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"text": "['resistance running']",
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"text": "['the system can generate physically realistic motion for stylized walking']",
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"text": "['sneaky walking']",
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"text": "['and for heterogeneous human actions']",
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"text": "['running→walking→jumping']",
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"text": "['Such actions are often difficult or even impossible to be synthesized by previous physics-based motion models']",
"type": "own_claim",
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"text": "['We model the force field priors using Gaussian process models because GP can efficiently capture nonlinear properties of the force fields and its learning process involves very few manual tuning parameters']",
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"text": "['might not scale up for use in very large data sets']",
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"text": "['The sparse approximation strategy works well for the current size of training data sets']",
"type": "own_claim",
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"text": "['less than 8,000 frames']",
"type": "data",
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"text": "['One possibility is to learn a probabilistic regression function for force fields using parametric statistical analysis techniques']",
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"text": "['the mixture of experts mode']",
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"text": "['its variants']",
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"text": "['Jacobs et al. 1991']",
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"text": "['Another limitation of our system is that it cannot generate a motion that is very different from motion examples']",
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53516,
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"text": "['the system is still unable to handle arbitrary external forces']",
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"text": "['the force field priors prevent the generated motion from moving away from prerecorded motion data']",
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"text": "['We choose to model the force field priors based on generalized forces rather than joint torques because we can conveniently compute the generalized forces from current kinematic motion capture databases']",
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"text": "['the CMU online mocap database 2']",
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"text": "['the learned force field priors can only predict resultant forces of join torques and contact forces']",
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"text": "['we could construct more accurate force field priors that explicitly predict joint torques or contact forces']",
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54044,
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"text": "['both joint torque data and contact force data are available']",
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54214,
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"text": "['In the future, we plan to measure ground-reaction forces with force plates and use them along with the captured kinematic motion data to compute joint torques via inverse dynamics techniques']",
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54513,
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"text": "['the optimization problem is high-dimensional and highly nonlinear']",
"type": "own_claim",
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"text": "['it might be subject to local minima']",
"type": "own_claim",
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"text": "['the initialization process is critical to the success of our optimization']",
"type": "own_claim",
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"text": "['It not only speeds up the optimization process but also alleviates the local-minimum problem']",
"type": "own_claim",
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"text": "['In the future, we plan to explore alternative techniques to address the local minimum problem']",
"type": "own_claim",
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"text": "['One possibility is the employment of a Markov chain Monte Carlo']",
"type": "own_claim",
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55203,
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"text": "['which comes to its solutions by efficiently drawing samples from the posterior distribution, using a Markov chain based on the Metropolis-Hastings algorithm']",
"type": "own_claim",
"id": "T345"
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{
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"text": "['imilar to other constraint-based animation systems, our system requires the user to specify a sparse number of constraints']",
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"text": "['to generate a desired animation']",
"type": "own_claim",
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"text": "['key frames']",
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"text": "['contact constraints']",
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"text": "['specifying such constraints']",
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"text": "['is not trivial for a novice user']",
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"text": "['Trajectory and contact constraints were either directly modified from reference motion data or rotoscoped from video streams similar to the technique described']",
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"text": "['Wei and Chai']",
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"text": "['2010b']",
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A02 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
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http://dx.doi.org/10.1145/1053427.1053429
Page 7 was found to have more than 100 embedded images.Image rendering was limited to the first 20, in reverse order of size.
<Title>Spherical Blend Skinning: A Real-time Deformation of Articulated Models</Title>
Ladislav Kavan ∗ Ji rı Zára ˇ Czech Technical University in Prague
∗
∗ e-mail: [email protected]
<Abstract>Skin deformation based on an underlying skeleton is a common method to animate believable organic models. The most widely used skeletal animation algorithm, linear blend skinning, is also known as skeleton subspace deformation, vertex blending, or enveloping. It runs in real-time even on a low-end hardware but it is also notorious for its failures, such as the collapsing-joints artifacts. We present a new algorithm which removes these shortcomings while maintaining almost the same time and memory complexity as the linear blend skinning. Unlike other approaches, our method works with exactly the same input data as the popular linear version. This minimizes the cost of upgrade from linear to spherical blend skinning in many existing applications: the data structures and models need no change at all. The paper discusses also theoretical properties of rotation interpolation, essential to spherical blend skinning.</Abstract>
<H2>CR Categories: </H2>CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation <H2>Keywords: </H2>skinning, deformation, skeletal animation
<H1>1 Introduction</H1>
Real-time animation of deformable objects is always a compromise between visual fidelity and computation complexity. Other aspects are quite important as well, for example the amount of artists work necessary to design the model. Therefore, there exist many algorithms for modeling deformable objects in the literature. They differ by the intended area of application and generality of allowed models. We focus on the real-time animation systems in this paper. Its most popular representative, known generally as the skeletal animation, is based on simple but versatile structure. It consists of joints, given by their position and orientation. The segments connecting the joints are conveniently interpreted as bones. The skeleton is, formally speaking, a tree whose nodes are identified with the joints and edges with the bones. The only displayed element is a skin, a 3D polygonal mesh, usually equipped with normal and texture data. Although the terminology is adopted from the virtual humanoid modeling, the skeletal animation is not limited to character animation – it can be applied to a wide range of soft objects, including imaginary (cartoon) creatures, plants, furniture, etc. This is an apparent advantage over complex systems which rely on explicit anatomy. The skeleton simplifies the animation task considerably: instead of animating each vertex individually, it is sufficient to manipulate the skeleton, and the skin deforms automatically. The skeletal animation in general does not specify how exactly the skeleton posture should be propagated to the skin. However, there is an established standard used in majority of real-time 3D applications. It comes by many names, all relating to the same algorithm: linear blend skinning (LBS), skeleton subspace deformation, vertex blending, enveloping, or simply skinning. Basically, this algorithm blends between rigidly transformed vertices using vertex weights, which denote the amount of influence of individual joints. Although LBS is very fast and advantageous to graphics hardware, it suffers from inherent artifacts, known as ”collapsing joints”, ”twisting elbow problem” or a ”candy-wrapper artifact”. In general, the mesh deformed by LBS loses volume as the joint rotation increases. The cause of this phenomena is explained in section 3, together with the LBS algorithm itself. The structure of the paper is as follows: in the next section, we summarize the previous work concerning real-time skin deformation and sketch our solution. In section 3, we analyze the problems of the LBS algorithm. Our approach to resolve these problems is presented in section 4. In section 5, we compare the results and discuss possible enhancements.
<H1>2 Related Work</H1>
An early contribution concerning the animation of deformable objects is [Magnenat-Thalmann et al. 1988], which considers the movement of a human hand. First 3D characters used in numerous computer games were animated by simple, often unpublished algorithms. Later on, the basic principles of LBS were described by the game development community [Lander 1998; Lander 1999]. The artifacts of LBS were discovered soon
[Weber 2000]. An improvement based on addition of auxiliary joints has been also proposed in
[Weber 2000]. Although this reduces the artifacts, the skin to joints relationship must be re-designed after joint addition. The number and location of the additional joints remains questionable. Another problem is how the movement of the original skeleton should be propagated into the augmented one. More formal articles consider skin deformation as an interpolation problem, such as
[Lewis et al. 2000]. They use radial basis functions to interpolate between example skins with different shapes. Similar method is presented in
[Sloan et al. 2001] and [Kry et al. 2002]. The latter de-correlates the deformation displacements using principal component analysis, which reduces the memory requirements considerably. The advantage of example based methods is that they capture the designed shape, including effects like muscle bulging. The drawback is the necessity of acquiring the example skins. An interesting generalization of LBS is called multi-weight enveloping
[Wang and Phillips 2002]. It introduces more parameters and therefore greater flexibility to the deformation algorithm. Instead of one weight per influence (joint) as in LBS, the multiweight enveloping uses twelve. These numerous parameters are derived from examples using the least squares optimization. The disadvantage is obvious: while the LBS models can be weighted manually by artists
[Steed 2002]
, this is questionable with multiweight enveloping. Tools that help animators to design the vertex weights are described in
[Mohr et al. 2003]. This article is interesting also from the theoretical point of view, because it describes how to explore the space of all possible LBS deformations. Another deformation algorithm
[Bloomenthal 2002] uses a complex auxiliary structure – a medial. An idea similar to spherical blend skinning (SBS) is bones blending proposed by [Kavan and Zára ˇ 2003]. However, bones blending is limited to vertices attached to only two joints. In addition, it requires hand-tuning of special parameters. Another algorithm removes the LBS artifacts by adding additional joints, and computes the vertex weights automatically using examples [Mohr and Gleicher 2003].
A recent skin deformation algorithm presented in
[Magnenat-Thalmann et al. 2004] seems to give results competitive to SBS, although it is based on a different mathematical fundament
[Alexa 2002].
However, this method is considerably slower than LBS and therefore
[Magnenat-Thalmann et al. 2004] recommends to use rather the standard LBS if the joint rotations are small. To conclude, there are many methods correcting the problems of LBS, but none of them is superior to LBS in all aspects. As a result, the linear blend skinning is still widely used in many applications, in spite of the artifacts.
<H2>2.1 Our Contribution</H2>
We observed that the artifacts of LBS are caused by the straightforward, linear interpolation of vertex positions. Intuitively, a linear blending is not suitable to capture deformations induced by skeleton, because their nature is rather spherical. Our basic idea is to change the interpolation domain: we interpolate transformations itself instead of transformed vertex positions. Because we consider transformations consisting of a translation and rotation, we suggest to use a quaternion representation. The transition to non-linear interpolation domain is not elementary. In order to achieve our goal, we cope with two main problems: determination of the center of rotation, and interpolation of multiple quaternions. The first problem follows from the fact that the choice of the center of rotation influences the result of interpolation considerably. We show how to compute a convenient center of rotation in real-time.
The second problem is simple in the case of two quaternions
[Shoemake 1985], but gets considerably harder for more than two rotations
[Buss and Fillmore 2001; Park et al. 2002; Alexa 2002]. Because the previous methods are not efficient enough for our purpose, we use a simple linear quaternion averaging. We justify both theoretically and experimentally that this solution is appropriate for our task (and probably for many others). Resolving those problems, we obtain a skin animation algorithm that deforms the mesh in much more plausible way then LBS. Because we change only the interpolation domain and not the input data, our program works with exactly the same models as LBS. The proposed algorithm improves a deformed shape even of models that have been designed and carefully tuned for LBS. Considering the high speed and low memory demands of SBS, it provides an attractive alternative to classic LBS.
<H2>2.2 Conventions</H2>
Let us denote matrices by capital letters, while vectors and quaternions by bold. Vectors are considered column vectors, therefore a multiplication of vector v by matrix M is written as Mv. We do not introduce a different notation for the R 3 vectors and their homogeneous R 4 counterparts with last coordinate equal to 1. The same convention is used for matrices. We denote the dot product of two vectors v 1 , v 2 as (v 1 , v 2 ) and the norm v 1 as a shortcut for (v 1 , v 1 ).
<H1>3 Linear Blend Skinning</H1>
The input to LBS consists of a polygonal mesh representing the digital skin, a skeleton, and vertex weights for every vertex of the skin. The polygonal mesh and the skeleton are designed in a reference position, e.g. virtual characters are often posed in the da Vinci posture
[Steed 2002]. Let us label the joints by integer numbers, assigning zero to the root. Each joint in the reference posture is associated with a homogeneous matrix, describing its position and orientation in the world coordinate system. For j-th joint, we denote this matrix by A j , like ”absolute” (or reference) position. This matrix is computed by multiplying all the transformations of individual joints in the chain from root to joint j. To compute the shape of the deformed skin, we need yet another set of matrices, describing the position and orientation of joints in the actual, animated posture. We call them F j , standing for the ”final” placement of joint j. Matrices F j are computed in a similar way as the absolute matrices, but including the actual rotation of each joint in the chain (we do not consider translating and scaling joints). The most simple skin deformation algorithm computes v = F j A −1 j v where v is a vertex in the reference skin associated with joint j and v is its position in the deformed mesh. The interpretation is following: the first matrix A −1 j transforms v to the position with joint j’s coordinate system aligned to the world coordinate system. The following transformation F j returns the vertex to its current position induced by the animated skeleton. Because these transformations usually occur together, we define the ”complete” matrix C j = F j A −1 j . Some older computer games animated characters in this way, even though it does not produce nice, smooth deformations. The linear blend skinning allows assignment of one vertex to multiple bones. Assume that vertex v is attached to joints j 1 , . . . , j n with weights w 1 , . . . , w n . The weights are coefficients of a convex combination, i.e. non-negative and ∑ n i=1 w i = 1. The weight w i represents the amount of influence of joint j i . The vertex position in the mesh deformed by LBS is then computed as
1
n v = ∑ w i C j i v i=1
that is to say, making a convex combination of individual vertex transformations. For example if n = 2 then vertex v lies on the line segment connecting C j 1 v and C j 2 v. The actual position on the segment is given by weight w 1 (or w 2 , because w 1 +w 2 = 1). As explained in the next section, the SBS works on a circular arc instead of segment, see Figure 1 . If the joint rotations are large, the LBS produces non-natural deformations. In the extremal case of rotation by 180 degrees, the skin can collapse to a single point. It is the notorious ”candy-wrapper” artifact, which is demonstrated in Figure 2 . The right shoulder of the model is twisted by 180 degrees, while the left shoulder is left in the reference pose. To understand why this undesirable effect occurs, it is sufficient to re-arrange the equation (1)
mesh bone joint j 1 j 2 vertex v C j 2 v C j 1 v LBS workspace SBS workspace
Figure 1: The set of possible results of LBS is a line segment, while SBS gives a circular arc.
2
n v = ∑ w i C j i v i=1
This formula is less efficient, because it blends matrices instead of vectors, but gives us a valuable insight. It is well known that the component-wise interpolation of matrices produces odd results: it does not preserve the orthogonality of the rotational part of the matrix. In some situations, it does not preserve even the rank of the interpolated matrices. This is exactly what happens in the ”candywrapper” problem: the single point the skin collapses to is a result of transformation by a singular matrix. A similar defect is visible also in the proximity of the singular configuration. Although the matrix is regular, it involves a non-uniform scaling and skewing, which is responsible for the loss of volume of the deformed skin even for small rotations.
<H1>4 Spherical Blend Skinning</H1>
Instead of trying to correct the bad results of LBS, we propose to change the interpolation method in (2). We focus on the interpolation of rotations – the linear interpolation of the translation part of C j i matrices is all right. An established interpolation of two rotations is spherical linear interpolation (SLERP)
[Shoemake 1985]. Its key of success is the use of quaternions to represent rotations. Unfortunately, it is not possible to simply replace matrices C j i in (2) with corresponding pairs quaternion-translation. One of the problems is that the linear interpolation of quaternions is not equivalent to SLERP. However, this is not the most serious difficulty, and we address it in section 4.1. The more important problem is to compute a convenient center of the interpolated rotations. We show that this is really an important problem on an example of human arm. Consider that the arm geometry is influenced by two joints j 1 and j 2 , such that j 1 is a parent of j 2 , as in Figure 1 . The transformation of the whole mesh by C j 1 is illustrated in the top row of Figure 3 and the transformation of the same geometry by C j 2 in the bottom row (note that the results are identical in both columns of these rows). The rows in the middle show the progress of interpolation between C j 1 to C j 2 . The only difference between the two columns in Figure 3 is in the choice of the center of rotation. In the left column, the rotation center r c is set to the translation part of matrix A j 2 (the position of joint j 2 in the reference posture). Note that C j 1 r c = C j 2 r c , therefore also the transformed rotation center is constant during the interpolation. In the right column of the figure, the rotation center r c is set to the translation part of A j 1 . Because C j 1 r c = C j 2 r c , the transformed rotation center is linearly interpolated from C j 1 r c to C j 2 r c . By comparison with the starting mesh (drawn gray in each frame), it is obvious that the center of rotation choice in the left column is much more advantageous. In this case, the interpolation of every single point is a circular arc (as in Figure 1), whereas a disturbing drift is inherent to any other choice of rotation center (such as r c ). Unfortunately, the condition of zero translation cannot be always satisfied, typically for more than two influencing joints. But even if the vertex is attached to only two joints k and l that are not neighbours of each other, some translation may be inevitable. For example consider that there is no relative rotation between C k and C l , but there is a relative translation induced by the joints in the chain between k and l. Clearly no choice of the center of rotation can avoid this translation, because the rotation is identity. Anyway, it is possible to define the rotation center as the point whose transformations by associated matrices are as close as possible. This minimizes the drift and works even if the vertex is assigned to n joints j 1 , . . . , j n . We find the center of rotation r c as the
Figure 2: Up – an extreme shoulder twist deformed by LBS, down – the same posture deformed by SBS
C j 1 C j 1 C j 1 r c C j 1 r c
C j 2 C j 2 C j 2 r c C j 2 r c
Figure 3: The correct center of rotation is chosen in the left column, while the sub-optimal in the right column. In the middle rows, notice the difference of the elbow position with respect to the original skin.
least-squares solution of the system of n 2 linear vector equations C a r c = C b r c , a < b, a, b ∈ { j 1 , . . . , j n } Each homogeneous matrix C i has structure C i = C 0 i rot T C 1 tr i
where C i rot is a 3 × 3 orthogonal matrix and C tr i is a translation vector. This enables us to re-write the linear system to C a rot r c + C tr a = C b rot r c + C tr b (C a rot −C b rot )r c = C tr b − C tr a If we stack all these equations to one matrix D and the right-hand sides to vector e, we can write the whole system as Dr c = e where D is a 3 n 2 × 3 matrix, r c is a 3-dimensional unknown vector and e is 3 2 n -dimensional vector. In general, we cannot make any assumptions about the rank of matrix D, which can vary from 0 to 3 (consider for example n = 2 and C j 1 = C j 2 ). We search the optimal solution r c in the least-squares sense. If there are multiple solutions giving the minimal Dr c − e , the r c with the minimal norm is chosen. This can be done in a robust way using the singular value decomposition (SVD), followed by computation of pseudo-inverse matrix. To perform these computations, we use the LAPACK software
[Anderson et al. 1999]. Even though LAPACK routines are efficient, computation of the center of rotation per each vertex would not result in a real-time algorithm. Fortunately, the center of rotation depends only on the transformations of the joints j 1 , . . . , j n and not the vertex itself. Therefore, if we encounter another vertex assigned to the same set of joints j 1 , . . . , j n , we can re-use the center of rotation computed formerly (cached). Moreover, if there is only one, or two neighboring joints that influence the vertex, we can determine the center of rotation precisely (as indicated in the beginning of this section) and omit the SVD computation at all. It turns out that the number of different non-trivial joint sets, and therefore the number of running the SVD, is surprisingly small for common models – about several tens. This enables the real-time performance.
<H2>4.1 Interpolation of Multiple Rotations</H2>
As mentioned in the introduction, the interpolation of multiple rotations has already received some attention [Buss and Fillmore 2001; Park et al. 2002] as well as interpolation of multiple general transformations [Alexa 2002]. Unfortunately, all these methods are substantially slower then the simple linear interpolation used in LBS. Since our goal is an algorithm with comparable time complexity as LBS, we propose an approximate but fast linear quaternion blending. For the case of two rotations, we compare our method with the established SLERP. Recall that a rotation around axis a (unit length vector) with angle 2 α corresponds to quaternion q = cos α + a sin α . However, this correspondence is not unique, because both quaternions q and −q represent the same rotation. The SLERP of two unit quaternions p, q assumes that their dot product (p, q) ≥ 0. If the dot product (p, q) < 0, we use −p instead of p, which is possible because both p and −p represent the same rotation. The SLERP of p, q with interpolation parameter t ∈ 0, 1 is given by the following formula, see for example
[Eberly 2001].
3
sin((1 − t) θ )p + sin(t θ )q s(t; p, q) = sin θ
where θ is the angle inclined by quaternions p, q, i.e. cos θ = (p, q). The linear interpolation of quaternions (QLERP) is computed as
4
(1 − t)p + tq l(t; p, q) = (1 − t)p + tq
The difference to SLERP is obvious: QLERP interpolates along the shortest segment, and then projects to arc, which does not result in the uniform interpolation of the arc. In spite of this, we claim that QLERP is sufficient for our task. In order to justify this statement, we face an interesting question by itself: how big can be the difference between QLERP and SLERP for the same input rotations? For t = 0, both QLERP and SLERP return of course p. For t > 0, we can imagine that both QLERP and SLERP work by concatenating p with some rotation (multiplying p with some quaternion). For SLERP, we denote this quaternion as r s (t). It can be expressed as p ∗ s(t; p, q), because pr s (t) = pp ∗ s(t; p, q) = s(t; p, q) The rotation r s (t) can be written out as
5
r s (t) = p ∗ s(t; p, q) = sin((1 − t) θ sin )1 + θ sin(t θ )p q
The quaternion 1 represents the identity (zero angle rotation). From the definition of quaternion multiplication it can be seen that the real part of p ∗ q equals (p, q) = cos θ . Since p ∗ q is a unit quaternion, we can express it as p ∗ q = cos θ + u sin θ for some axis of rotation u. If we substitute this into equation (5), we obtain sin((1 − t) θ ) + sin(t θ ) cos θ r s (t) = sin θ + u sin(t θ ) which means that the direction of the axis u is independent on t. Let us examine the rotation r l (t) following p in QLERP: r l (t) = p ∗ l(t; p, q) = (1 (1 − − t)1 t)p + + tp tq ∗ q = (1 − t + t cos θ ) t sin θ = + u (1 − t)p + tq (1 − t)p + tq which shows that the axis of rotation has the same direction. We can conclude with an important property: the SLERP can be written as pr s (t) and QLERP as pr l (t), where the rotations r s (t) and r l (t) have the same axis. Moreover, this axis is constant, i.e. independent on the interpolation parameter t. It follows that the only difference between QLERP and SLERP is in the angle of rotations r s (t) and r l (t). Note that both r s (t) and r l (t) have a form of linear combination of quaternions 1 and p ∗ q. It means that the results of both r s (t) and r l (t) always end up in certain 2D subspace of R 4 . We can restrict our attention to this subspace (the linear hull of 1 and p ∗ q). Since SLERP assumes cos θ = (p, q) ≥ 0, the angle θ cannot exceed π /2. To obtain an upper bound of the maximal difference in the angle, we consider the extremal case with θ = π /2, depicted in Figure 4 . The angle α (t) on the picture can be computed by atan, and β (t) by simple linear interpolation of the right angle, which yields the difference function t π d(t) = α (t) − β (t) = atan − t 1 − t 2 It remains to find the extremes of d(t) on the interval 0, 1 . The elementary mathematical analysis discovers the global extremes in points 1/2 ± (1/ π − 1/4). The absolute value of d(t) in these points is approximately 0.071 radians (4.07 degrees). As mentioned in the introduction of this section the angle of rotation is twice the angle inclined by quaternions. To conclude: both SLERP and QLERP interpolate by multiplying the first quaternion with a rotation with the same, fixed axis. The difference between SLERP and QLERP is only in the angle of this rotation, and is strictly less then 0.143 radians (8.15 degrees) for any interpolation parameter t ∈ 0, 1 . This is an upper bound; practical results are much smaller and could hardly cause an observable defect in the deformed skin. The big advantage of QLERP is that it can be easily generalized to interpolate multiple rotations – it suffices to make a convex combination and re-normalization of multiple quaternions.
1 SLERP QLERP t a (t) b (t) 1-t 1
Figure 4: The difference between QLERP angle α (t) and SLERP β (t)
<H2>4.2 Algorithm Overview</H2>
Now we have prepared all the ingredients to describe how the SBS algorithm works. The task is to transform a vertex v influenced by joints j 1 , . . . , j n with convex weights W = (w 1 , . . . , w n ) to its position v in the animated skin. In order to obtain an appealing deformation, it is necessary to respect the computed center of rotation r c . To achieve this, we extend the QLERP scheme to homogeneous matrices C j i . We denote the interpolation of matrices C j i with weights W as
6
Q m q(W ;C j 1 , . . . ,C j n ) = 0 T 1
and compute Q and m as follows. First, the rotation submatrices C rot j i are converted to quaternions q j i . One of them, for example q j 1 , is chosen as pivot. If (q j 1 , q j i ) < 0 for any i = 2, . . . , n, we replace q j i with −q j i (by analogy to SLERP). Then the QLERP computes s = w 1 q j 1 + . . . + w n q j n , which is subsequently normalized to s n = s/ s . Finally, s n is converted to the rotation matrix Q. The translation part is just linearly interpolated, m = ∑ n i=1 w i C tr j i . In order to change the center of rotation from the origin to r c , we define a homogeneous matrix
T = 0 I T r 1 c (7)
where I is a 3 × 3 identity matrix. Then the interpolation of homogeneous matrices with respect to the center of rotation r c can be written as
8
T q(W ; T −1 C j 1 T, . . . , T −1 C j n T )T −1
Note that the shift of the center of rotation does not influence the interpolated rotation – it manifests only in the translation part. The desired transformation of vertex v is v = T q(W ; T −1 C j 1 T, . . . , T −1 C j n T )T −1 v
9
n = Q(v − r c ) + ∑ w i C j i r c i=1
A detailed derivation of this formula can be found in appendix A. The latter addend represents the translation induced by the new center of rotation. The equation (9) has to be evaluated once per each vertex, and therefore should be as efficient as possible. The basic optimization is to pre-compute the quaternions q j i , because they do not depend on the actual vertex – only on the joint’s transformation, similarly as the rotation centers r c . Nonetheless, QLERP has to be executed for each vertex, since weights w 1 , . . . , w n can vary. In order to challenge the speed of LBS, we apply a following trick. The vertex v can be represented by a quaternion with zero real part. In this representation, its rotation by quaternion q can be expressed as q vq ∗ , which is a quaternion with zero real part as well
[Eberly 2001]. Although this expression is not efficient for computation (because of slow quaternion multiplication), it enables us to write out the rotation of v by quaternion s n as s n vs n ∗ = s 1 2 svs ∗ = (s, 1 s) svs ∗ This suggests to convert already the quaternion s to matrix Q and normalize subsequently by dividing (s, s). Therefore, we can compute the Q matrix from (9) as Q = (s,s) Q and save the sqrt operation. Some attention must be paid because standard routines for quaternion to matrix conversion assume a unit-length quaternion. The conversion of an arbitrary length q = w + xi + y j + zk leads to the following matrix: ⎛ ⎞ x 2 + w 2 − y 2 − z 2 2xy − 2wz 2xz + 2wy ⎝ 2xy + 2wz y 2 + w 2 − x 2 − z 2 2yz − 2wx ⎠ 2xz − 2wy 2yz + 2wx z 2 + w 2 − x 2 − y 2 Vertex normal v n is transformed in a similar way as vertex position, but ignoring the translation v n = Qv n Using the formula (9) we can verify our previous intuitive thinking. First, if we substitute r c in place of v, no rotation occurs, which means that r c is indeed a center of rotation. Second, if n = 2 and C j 1 r c = C j 2 r c (as in the beginning of section 4), the translation part becomes w 1 C j 1 r c + w 2 C j 2 r c = (w 1 + w 2 )C j 1 r c = C j 1 r c which is independent of interpolation parameters (weights), i.e. the translation during interpolation is constant indeed. Third, the equation (9) is nothing but a generalization of LBS to an arbitrary method of rotation interpolation. The choice of QLERP is not important for (9), the matrix Q can be replaced by matrix resulting from any other interpolation scheme, such as
[Buss and Fillmore 2001]. If we substitute Q = ∑ w i C rot j i , i.e. a simple linear combination of rotation matrices, we obtain v = Q(v − r c ) + ∑ w i C j i r c = ∑ w i C rot j i v − ∑ w i C rot j i r c + ∑ w i C rot j i r c + ∑ w i C tr j i = ∑ w i C rot j i v + ∑ w i C tr j i = ∑ w i C j i v which is exactly the LBS equation (1). This also shows that LBS is a special case, which is independent of the center of rotation. The whole algorithm can be summarized in the following steps:
Figure 5: 3D models used for testing
Hand
Woman
Creature
vertices
2402
3356
6802
triangles
4800
5205
13590
joints
23
78
56
Hand Woman Creature vertices 2402 3356 6802 triangles 4800 5205 13590 joints 23 78 56
Table 1: Complexities of example models
• compute matrices C i for all joints and convert their rotation parts to quaternions q i • for each vertex v influenced by joints j 1 , . . . , j n – compute (or re-use a cached) center of rotation r c according to section 4 – blend quaternions q j 1 , . . . q j n using QLERP and convert the result to matrix Q – compute the position of vertex v in the deformed skin using the equation (9)
<H1>5 Results and Comparison</H1>
We tested the SBS algorithm on three models, see Figure 5 and Table 1. We compare the shape of the deformed skin on the model of woman, because human eye is most sensitive to the deformations of human body. Figure 6 presents results of LBS and SBS executed on the same posture of the model. Another example has been presented already in Figure 2 . For small deformations, both algorithms produce similar results, as in the second row of Figure 6 (although a small loss of volume is noticeable even there). It is remarkable that the results of SBS are better even though the models have been optimised to work with the LBS algorithm. The performance of both algorithms is compared in Table 2 . The measured value is an average time in milliseconds necessary to deform one model on a 2.5GHz Athlon PC (rendering time not included). In the last row of the table the number of different nontrivial joint sets is reported (trivial joint set consists of only one joint
Figure 6:
Comparison of deformations by LBS (left) and SBS
(right)
or two neighboring joints). Put in another way, it is exactly the number of singular-value decompositions performed by the SBS algorithm. This number participates considerably on the difference between times for LBS and SBS. Theoretically, the number of different non-trivial joint sets could be very high. Fortunately, this number is surprisingly small in practice, because the joint influences tend to be local (e.g. it is unlikely to find vertices influenced by both left and right wrist). The additional memory needed for SBS is dominated by caching the computed centers of rotation. However, this amount of memory is negligible, considering the number of different non-trivial joint sets. In order to test the accuracy of QLERP, we experimented with spherical weighted averages presented in
[Buss and Fillmore 2001]. The algorithm proposed in
[Buss and Fillmore 2001] behaves like SLERP for the case of two rotations (in contrast to QLERP, which only approximates SLERP results). On the one hand, the difference in the deformed skin was barely observable, according to the results from section 4.1. On the other hand, the increase in the execution time was quite substantial. For the woman model, the time increased from original 4.54ms to 22.74ms. This only confirmed our choice of QLERP.
Hand
Woman
Creature
LBS time
3.28
3.59
9.0
SBS time
4.43
4.54
11.37
SVD executions
38
37
56
Hand Woman Creature LBS time 3.28 3.59 9.0 SBS time 4.43 4.54 11.37 SVD executions 38 37 56
Table 2: First two rows: run-time of LBS and SBS algorithms in milliseconds; last row: number of SVD executions
<H2>5.1 Conclusion and Future Work</H2>
The proposed skin deformation system is by no means perfect; it cannot compete with complex, layered models. However, the SBS algorithm offers reasonable price for elimination of the notorious LBS artifacts. The time and memory complexity of both algorithms is comparable. The overhead of replacing an existing LBS implementation by SBS is minimal, because the input data, as well as the internal data structures, are the same. In contrast to other methods, the SBS does not need any additional information, such as the example skins. The presented algorithm opens many questions and suggests several directions of future work. First of all, we worked only with vertex weights optimised for LBS. These weights are designed to suppress the LBS artifacts, even though they cannot remove them. It would be interesting to find out how much can be the SBS results improved by a set of weights especially designed for SBS. In order to accomplish this, a tool to explore the space of SBS deformations would help considerably. This tool has been presented for LBS in
[Mohr et al. 2003], but the situation of SBS is somewhat more complex, because our interpolation method is non-linear. Similarly, it would be possible to estimate the SBS vertex weights from examples, as was done for LBS in [Mohr and Gleicher 2003]. This could also cover additional effects like muscle bulging.
<H1>6 Acknowledgments</H1>
This work has been partly supported by the Ministry of Education, Youth and Sports of the Czech Republic under research program No.Y04/98:212300014 (Research in the area of information technologies and communications). We thank to Samuel Buss for providing the algorithm for spherical weighted averages [Buss and Fillmore 2001] and to LAPACK developers for their software. We would also like to thank to Jaroslav Seman cı k and the anonymous reviewers for valuable comments and to Adam J. Sporka for help with the accompanying video.
<H1>A Interpolation of Rotations with an Arbitrary Center</H1>
In this appendix we derive the formula (9), which describes the interpolation of rotations with respect to r c – a custom center of rotation. Let us denote by K the coordinate system with origin in r c and identical basis vectors as the world coordinate system. Then
the matrix T (7) can be interpreted as a transformation from K to the world coordinate system. By analogy, the inverse matrix T −1 = 0 I T −r 1 c
represents the transformation from the world coordinate system to K. It follows that T −1 C j i T is the transformation C j i expressed with respect to K. By interpolating these matrices with QLERP q(W ; T −1 C j 1 T, . . . , T −1 C j n T ) we obtain a matrix working also on vectors in K coordinates. We can express this matrix with respect to the world coordinate system easily T q(W ; T −1 C j 1 T, . . . , T −1 C j n T )T −1 which is exactly the formula (8). Recall that the matrix C j i has structure C j i = C 0 rot j T i C 1 tr j i
which enables us to write out T −1 C j i T = C 0 rot j T i C j i r c 1 − r c
as can be simply verified. Please note that the change of the coordinate system did not influence the rotation part C rot j i at all. Therefore the result of QLERP will be, according to equation (6)
q(W ; T −1 C j 1 T, . . . , T −1 C j n T ) = 0 Q T −r c + ∑ i=1 n 1 w i C j i r c
where Q stands for the interpolation of pure rotations, computed as indicated in section 4.2. Using T −1 v = v − r c and T x = x + r c , we see that v = T q(W ; T −1 C j 1 T, . . . , T −1 C j n T )T −1 v = T 0 Q T −r c + ∑ i=1 n 1 w i C j i r c v − 1 r c n = Q(v − r c ) + ∑ w i C j i r c i=1
is true for any vector v. This is exactly the equation (9).
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"text": "['Other aspects are quite important as well']",
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"text": "['the amount of artists work necessary to design the model']",
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2158,
2235
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"text": "['there exist many algorithms for modeling deformable objects in the literature']",
"type": "background_claim",
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"text": "['They differ by the intended area of application and generality of allowed models']",
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"text": "['Its most popular representative, known generally as the skeletal animation, is based on simple but versatile structure']",
"type": "background_claim",
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2498,
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"text": "['It consists of joints']",
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2562,
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"text": "['The segments connecting the joints are conveniently interpreted as bones']",
"type": "background_claim",
"id": "T8"
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2636,
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],
"text": "['The skeleton is']",
"type": "background_claim",
"id": "T9"
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2672,
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"text": "['a tree whose nodes are identified with the joints and edges with the bones']",
"type": "background_claim",
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{
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2748,
2852
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"text": "['The only displayed element is a skin, a 3D polygonal mesh, usually equipped with normal and texture data']",
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2863,
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"text": "['the terminology is adopted from the virtual humanoid modeling']",
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2926,
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"text": "['the skeletal animation is not limited to character animation']",
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"id": "T13"
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2989,
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"text": "['it can be applied to a wide range of soft objects']",
"type": "background_claim",
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3050,
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"text": "['imaginary (cartoon) creatures']",
"type": "data",
"id": "T15"
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"offsets": [
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3081,
3087
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],
"text": "['plants']",
"type": "data",
"id": "T16"
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"offsets": [
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3089,
3098
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],
"text": "['furniture']",
"type": "data",
"id": "T17"
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{
"offsets": [
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3105,
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],
"text": "['This is an apparent advantage over complex systems which rely on explicit anatomy']",
"type": "background_claim",
"id": "T18"
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{
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"text": "['The skeleton simplifies the animation task considerably']",
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"id": "T19"
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3245,
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"text": "['instead of animating each vertex individually, it is sufficient to manipulate the skeleton, and the skin deforms automatically']",
"type": "background_claim",
"id": "T20"
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3373,
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"text": "['The skeletal animation in general does not specify how exactly the skeleton posture should be propagated to the skin']",
"type": "background_claim",
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{
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3500,
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],
"text": "['there is an established standard used in majority of real-time 3D applications']",
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{
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3580,
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],
"text": "['It comes by many names, all relating to the same algorithm: linear blend skinning (LBS), skeleton subspace deformation, vertex blending, enveloping, or simply skinning']",
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3909,
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],
"text": "['LBS is very fast and advantageous to graphics hardware']",
"type": "background_claim",
"id": "T24"
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{
"offsets": [
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3965,
4084
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],
"text": "['it suffers from inherent artifacts, known as ”collapsing joints”, ”twisting elbow problem” or a ”candy-wrapper artifact']",
"type": "background_claim",
"id": "T25"
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4099,
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],
"text": "['the mesh deformed by LBS loses volume as the joint rotation increases']",
"type": "background_claim",
"id": "T26"
},
{
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4672,
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],
"text": "['An early contribution concerning the animation of deformable objects is']",
"type": "background_claim",
"id": "T27"
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4777,
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],
"text": "['which considers the movement of a human hand']",
"type": "background_claim",
"id": "T28"
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{
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4745,
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],
"text": "['Magnenat-Thalmann et al. 1988']",
"type": "data",
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},
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"offsets": [
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4823,
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],
"text": "['First 3D characters used in numerous computer games were animated by simple, often unpublished algorithms']",
"type": "background_claim",
"id": "T30"
},
{
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4940,
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],
"text": "['the basic principles of LBS were described by the game development community']",
"type": "background_claim",
"id": "T31"
},
{
"offsets": [
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5018,
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],
"text": "['Lander 1998']",
"type": "data",
"id": "T32"
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{
"offsets": [
[
5031,
5042
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],
"text": "['Lander 1999']",
"type": "data",
"id": "T33"
},
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"offsets": [
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5045,
5086
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],
"text": "['The artifacts of LBS were discovered soon']",
"type": "background_claim",
"id": "T34"
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{
"offsets": [
[
5089,
5099
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],
"text": "['Weber 2000']",
"type": "data",
"id": "T35"
},
{
"offsets": [
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],
"text": "['An improvement based on addition of auxiliary joints has been also proposed in']",
"type": "background_claim",
"id": "T36"
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{
"offsets": [
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5183,
5193
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],
"text": "['Weber 2000']",
"type": "data",
"id": "T37"
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5205,
5231
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],
"text": "['this reduces the artifacts']",
"type": "background_claim",
"id": "T38"
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{
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5233,
5305
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],
"text": "['the skin to joints relationship must be re-designed after joint addition']",
"type": "background_claim",
"id": "T39"
},
{
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5307,
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],
"text": "['The number and location of the additional joints remains questionable']",
"type": "background_claim",
"id": "T40"
},
{
"offsets": [
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5378,
5482
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],
"text": "['Another problem is how the movement of the original skeleton should be propagated into the augmented one']",
"type": "background_claim",
"id": "T41"
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{
"offsets": [
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5484,
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],
"text": "['More formal articles consider skin deformation as an interpolation problem']",
"type": "background_claim",
"id": "T42"
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{
"offsets": [
[
5570,
5587
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],
"text": "['Lewis et al. 2000']",
"type": "data",
"id": "T43"
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"offsets": [
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5682,
5712
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],
"text": "['Similar method is presented in']",
"type": "background_claim",
"id": "T44"
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{
"offsets": [
[
5715,
5732
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],
"text": "['Sloan et al. 2001']",
"type": "data",
"id": "T45"
},
{
"offsets": [
[
5739,
5754
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],
"text": "['Kry et al. 2002']",
"type": "data",
"id": "T46"
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{
"offsets": [
[
5757,
5898
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],
"text": "['The latter de-correlates the deformation displacements using principal component analysis, which reduces the memory requirements considerably']",
"type": "background_claim",
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{
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5900,
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],
"text": "['The advantage of example based methods is that they capture the designed shape']",
"type": "background_claim",
"id": "T48"
},
{
"offsets": [
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6019,
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],
"text": "['The drawback is the necessity of acquiring the example skins']",
"type": "background_claim",
"id": "T49"
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{
"offsets": [
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5980,
6017
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],
"text": "['including effects like muscle bulging']",
"type": "background_claim",
"id": "T50"
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{
"offsets": [
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6081,
6151
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],
"text": "['An interesting generalization of LBS is called multi-weight enveloping']",
"type": "background_claim",
"id": "T51"
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{
"offsets": [
[
6154,
6176
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],
"text": "['Wang and Phillips 2002']",
"type": "data",
"id": "T52"
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"offsets": [
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6223,
6271
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],
"text": "['greater flexibility to the deformation algorithm']",
"type": "background_claim",
"id": "T53"
},
{
"offsets": [
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6179,
6208
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],
"text": "['It introduces more parameters']",
"type": "background_claim",
"id": "T54"
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{
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6458,
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],
"text": "['The disadvantage is obvious']",
"type": "background_claim",
"id": "T55"
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{
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6493,
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],
"text": "['the LBS models can be weighted manually by artists']",
"type": "background_claim",
"id": "T56"
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{
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6560,
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],
"text": "['this is questionable with multiweight enveloping']",
"type": "background_claim",
"id": "T57"
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{
"offsets": [
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6546,
6556
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],
"text": "['Steed 2002']",
"type": "data",
"id": "T58"
},
{
"offsets": [
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6610,
6681
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],
"text": "['Tools that help animators to design the vertex weights are described in']",
"type": "background_claim",
"id": "T59"
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{
"offsets": [
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6684,
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],
"text": "['Mohr et al. 2003']",
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6703,
6770
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],
"text": "['This article is interesting also from the theoretical point of view']",
"type": "background_claim",
"id": "T61"
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{
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6780,
6850
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],
"text": "['it describes how to explore the space of all possible LBS deformations']",
"type": "background_claim",
"id": "T62"
},
{
"offsets": [
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6852,
6881
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],
"text": "['Another deformation algorithm']",
"type": "background_claim",
"id": "T63"
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{
"offsets": [
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6902,
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],
"text": "['uses a complex auxiliary structure – a medial']",
"type": "background_claim",
"id": "T64"
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{
"offsets": [
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6884,
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],
"text": "['Bloomenthal 2002']",
"type": "data",
"id": "T65"
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"offsets": [
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6949,
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],
"text": "['An idea similar to spherical blend skinning (SBS) is bones blending proposed by']",
"type": "background_claim",
"id": "T66"
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{
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7030,
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],
"text": "['Kavan and Zára ˇ 2003']",
"type": "data",
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7063,
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"text": "['bones blending is limited to vertices attached to only two joints']",
"type": "background_claim",
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7143,
7188
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"text": "['it requires hand-tuning of special parameters']",
"type": "background_claim",
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7190,
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],
"text": "['Another algorithm removes the LBS artifacts by adding additional joints, and computes the vertex weights automatically using examples']",
"type": "background_claim",
"id": "T70"
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{
"offsets": [
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7325,
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],
"text": "['Mohr and Gleicher 2003']",
"type": "data",
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7351,
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"text": "['A recent skin deformation algorithm presented in']",
"type": "background_claim",
"id": "T72"
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7433,
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"text": "['seems to give results competitive to SBS']",
"type": "background_claim",
"id": "T73"
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"text": "['it is based on a different mathematical fundament']",
"type": "background_claim",
"id": "T74"
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{
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7402,
7431
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],
"text": "['Magnenat-Thalmann et al. 2004']",
"type": "data",
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7536,
7546
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],
"text": "['Alexa 2002']",
"type": "data",
"id": "T76"
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"offsets": [
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7559,
7602
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],
"text": "['this method is considerably slower than LBS']",
"type": "background_claim",
"id": "T77"
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{
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7650,
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],
"text": "['recommends to use rather the standard LBS']",
"type": "background_claim",
"id": "T78"
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{
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7695,
7724
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],
"text": "['the joint rotations are small']",
"type": "data",
"id": "T79"
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{
"offsets": [
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7619,
7648
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],
"text": "['Magnenat-Thalmann et al. 2004']",
"type": "data",
"id": "T80"
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"offsets": [
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7859,
7953
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],
"text": "['the linear blend skinning is still widely used in many applications, in spite of the artifacts']",
"type": "background_claim",
"id": "T82"
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{
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7739,
7792
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],
"text": "['there are many methods correcting the problems of LBS']",
"type": "background_claim",
"id": "T81"
},
{
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7798,
7844
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],
"text": "['none of them is superior to LBS in all aspects']",
"type": "background_claim",
"id": "T83"
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{
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8025,
8121
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],
"text": "['the artifacts of LBS are caused by the straightforward, linear interpolation of vertex positions']",
"type": "own_claim",
"id": "T84"
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{
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8136,
8213
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],
"text": "['a linear blending is not suitable to capture deformations induced by skeleton']",
"type": "own_claim",
"id": "T85"
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{
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8223,
8255
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],
"text": "['their nature is rather spherical']",
"type": "own_claim",
"id": "T86"
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8468,
8513
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],
"text": "['we suggest to use a quaternion representation']",
"type": "own_claim",
"id": "T87"
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{
"offsets": [
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8398,
8466
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],
"text": "['we consider transformations consisting of a translation and rotation']",
"type": "data",
"id": "T88"
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"offsets": [
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8775,
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],
"text": "['the choice of the center of rotation influences the result of interpolation considerably']",
"type": "own_claim",
"id": "T89"
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{
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8614,
8644
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],
"text": "['we cope with two main problems']",
"type": "own_claim",
"id": "T90"
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8646,
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"text": "['determination of the center of rotation']",
"type": "own_claim",
"id": "T91"
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8691,
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"text": "['interpolation of multiple quaternions']",
"type": "own_claim",
"id": "T92"
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{
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8935,
8994
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],
"text": "['The second problem is simple in the case of two quaternions']",
"type": "background_claim",
"id": "T93"
},
{
"offsets": [
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9017,
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],
"text": "['gets considerably harder for more than two rotations']",
"type": "background_claim",
"id": "T94"
},
{
"offsets": [
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8997,
9010
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],
"text": "['Shoemake 1985']",
"type": "data",
"id": "T95"
},
{
"offsets": [
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9072,
9094
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],
"text": "['Buss and Fillmore 2001']",
"type": "data",
"id": "T96"
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{
"offsets": [
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9096,
9112
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],
"text": "['Park et al. 2002']",
"type": "data",
"id": "T97"
},
{
"offsets": [
[
9114,
9124
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],
"text": "['Alexa 2002']",
"type": "data",
"id": "T98"
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{
"offsets": [
[
9198,
9241
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],
"text": "['we use a simple linear quaternion averaging']",
"type": "own_claim",
"id": "T99"
},
{
"offsets": [
[
9135,
9196
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],
"text": "['the previous methods are not efficient enough for our purpose']",
"type": "background_claim",
"id": "T100"
},
{
"offsets": [
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9297,
9369
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],
"text": "['this solution is appropriate for our task (and probably for many others)']",
"type": "own_claim",
"id": "T101"
},
{
"offsets": [
[
9397,
9491
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],
"text": "['we obtain a skin animation algorithm that deforms the mesh in much more plausible way then LBS']",
"type": "own_claim",
"id": "T102"
},
{
"offsets": [
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9371,
9395
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],
"text": "['Resolving those problems']",
"type": "own_claim",
"id": "T103"
},
{
"offsets": [
[
9501,
9563
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],
"text": "['we change only the interpolation domain and not the input data']",
"type": "own_claim",
"id": "T104"
},
{
"offsets": [
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9565,
9618
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],
"text": "['our program works with exactly the same models as LBS']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
9620,
9735
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],
"text": "['The proposed algorithm improves a deformed shape even of models that have been designed and carefully tuned for LBS']",
"type": "own_claim",
"id": "T106"
},
{
"offsets": [
[
9795,
9847
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],
"text": "['it provides an attractive alternative to classic LBS']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
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9737,
9793
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],
"text": "['Considering the high speed and low memory demands of SBS']",
"type": "background_claim",
"id": "T108"
},
{
"offsets": [
[
31637,
31700
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],
"text": "['We compare the shape of the deformed skin on the model of woman']",
"type": "own_claim",
"id": "T109"
},
{
"offsets": [
[
31710,
31771
]
],
"text": "['human eye is most sensitive to the deformations of human body']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
31938,
31977
]
],
"text": "['both algorithms produce similar results']",
"type": "own_claim",
"id": "T111"
},
{
"offsets": [
[
31914,
31936
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],
"text": "['For small deformations']",
"type": "data",
"id": "T112"
},
{
"offsets": [
[
31989,
32011
]
],
"text": "['second row of Figure 6']",
"type": "data",
"id": "T113"
},
{
"offsets": [
[
32094,
32123
]
],
"text": "['the results of SBS are better']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
32136,
32197
]
],
"text": "['the models have been optimised to work with the LBS algorithm']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
32968,
33013
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],
"text": "['this number is surprisingly small in practice']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
33023,
33060
]
],
"text": "['the joint influences tend to be local']",
"type": "own_claim",
"id": "T117"
},
{
"offsets": [
[
33067,
33138
]
],
"text": "['it is unlikely to find vertices influenced by both left and right wrist']",
"type": "own_claim",
"id": "T118"
},
{
"offsets": [
[
33245,
33340
]
],
"text": "['this amount of memory is negligible, considering the number of different non-trivial joint sets']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
33141,
33234
]
],
"text": "['The additional memory needed for SBS is dominated by caching the computed centers of rotation']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
33652,
33709
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],
"text": "['the difference in the deformed skin was barely observable']",
"type": "own_claim",
"id": "T121"
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{
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33728,
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],
"text": "['results from section 4.1']",
"type": "data",
"id": "T122"
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{
"offsets": [
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33773,
33829
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"text": "['the increase in the execution time was quite substantial']",
"type": "own_claim",
"id": "T123"
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{
"offsets": [
[
33831,
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],
"text": "['For the woman model, the time increased from original 4.54ms to 22.74ms']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
33904,
33943
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],
"text": "['This only confirmed our choice of QLERP']",
"type": "own_claim",
"id": "T125"
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{
"offsets": [
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33471,
33496
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],
"text": "['The algorithm proposed in']",
"type": "own_claim",
"id": "T126"
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{
"offsets": [
[
33523,
33571
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],
"text": "['behaves like SLERP for the case of two rotations']",
"type": "own_claim",
"id": "T127"
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{
"offsets": [
[
33499,
33521
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],
"text": "['Buss and Fillmore 2001']",
"type": "data",
"id": "T128"
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"offsets": [
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33573,
33632
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],
"text": "['in contrast to QLERP, which only approximates SLERP results']",
"type": "own_claim",
"id": "T129"
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{
"offsets": [
[
32696,
32784
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],
"text": "['it is exactly the number of singular-value decompositions performed by the SBS algorithm']",
"type": "own_claim",
"id": "T130"
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{
"offsets": [
[
32249,
32256
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],
"text": "['Table 2']",
"type": "data",
"id": "T131"
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{
"offsets": [
[
32873,
32953
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],
"text": "['Theoretically, the number of different non-trivial joint sets could be very high']",
"type": "own_claim",
"id": "T132"
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{
"offsets": [
[
32786,
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],
"text": "['This number participates considerably on the difference between times for LBS and SBS']",
"type": "own_claim",
"id": "T133"
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{
"offsets": [
[
34897,
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],
"text": "['The proposed skin deformation system is by no means perfect']",
"type": "own_claim",
"id": "T134"
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{
"offsets": [
[
34958,
35004
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],
"text": "['it cannot compete with complex, layered models']",
"type": "own_claim",
"id": "T135"
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{
"offsets": [
[
35015,
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"text": "['the SBS algorithm offers reasonable price for elimination of the notorious LBS artifacts']",
"type": "own_claim",
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{
"offsets": [
[
35170,
35243
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"text": "['The overhead of replacing an existing LBS implementation by SBS is minima']",
"type": "own_claim",
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{
"offsets": [
[
35254,
35323
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],
"text": "['the input data, as well as the internal data structures, are the same']",
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"offsets": [
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35359,
35403
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"text": "['SBS does not need any additional information']",
"type": "own_claim",
"id": "T139"
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{
"offsets": [
[
35688,
35812
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"text": "['It would be interesting to find out how much can be the SBS results improved by a set of weights especially designed for SBS']",
"type": "own_claim",
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{
"offsets": [
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35843,
35914
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],
"text": "['a tool to explore the space of SBS deformations would help considerably']",
"type": "own_claim",
"id": "T142"
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"offsets": [
[
35981,
36026
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],
"text": "['the situation of SBS is somewhat more complex']",
"type": "own_claim",
"id": "T143"
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{
"offsets": [
[
36036,
36074
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],
"text": "['our interpolation method is non-linear']",
"type": "data",
"id": "T144"
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{
"offsets": [
[
35916,
35952
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],
"text": "['This tool has been presented for LBS']",
"type": "background_claim",
"id": "T145"
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{
"offsets": [
[
35958,
35974
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],
"text": "['Mohr et al. 2003']",
"type": "data",
"id": "T146"
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"offsets": [
[
36087,
36156
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],
"text": "['it would be possible to estimate the SBS vertex weights from examples']",
"type": "own_claim",
"id": "T147"
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{
"offsets": [
[
36158,
36177
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],
"text": "['as was done for LBS']",
"type": "background_claim",
"id": "T148"
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{
"offsets": [
[
36182,
36204
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],
"text": "['Mohr and Gleicher 2003']",
"type": "data",
"id": "T149"
},
{
"offsets": [
[
36207,
36267
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],
"text": "['This could also cover additional effects like muscle bulging']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
10586,
10658
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],
"text": "['The polygonal mesh and the skeleton are designed in a reference position']",
"type": "background_claim",
"id": "T151"
},
{
"offsets": [
[
10665,
10723
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],
"text": "['virtual characters are often posed in the da Vinci posture']",
"type": "background_claim",
"id": "T152"
},
{
"offsets": [
[
10726,
10736
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],
"text": "['Steed 2002']",
"type": "data",
"id": "T153"
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{
"offsets": [
[
11167,
11328
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],
"text": "['To compute the shape of the deformed skin, we need yet another set of matrices, describing the position and orientation of joints in the actual, animated posture']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
11579,
11647
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],
"text": "['The most simple skin deformation algorithm computes v = F j A −1 j v']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
11791,
11915
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],
"text": "['the first matrix A −1 j transforms v to the position with joint j’s coordinate system aligned to the world coordinate system']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
11917,
12025
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],
"text": "['The following transformation F j returns the vertex to its current position induced by the animated skeleton']",
"type": "own_claim",
"id": "T157"
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{
"offsets": [
[
12035,
12079
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],
"text": "['these transformations usually occur together']",
"type": "own_claim",
"id": "T158"
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{
"offsets": [
[
12081,
12129
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],
"text": "['we define the ”complete” matrix C j = F j A −1 j']",
"type": "own_claim",
"id": "T159"
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{
"offsets": [
[
12132,
12189
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],
"text": "['Some older computer games animated characters in this way']",
"type": "background_claim",
"id": "T160"
},
{
"offsets": [
[
12203,
12248
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],
"text": "['it does not produce nice, smooth deformations']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
12250,
12324
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],
"text": "['The linear blend skinning allows assignment of one vertex to multiple bone']",
"type": "background_claim",
"id": "T162"
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{
"offsets": [
[
12816,
12821
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],
"text": "['n = 2']",
"type": "data",
"id": "T163"
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"offsets": [
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12827,
12891
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],
"text": "['vertex v lies on the line segment connecting C j 1 v and C j 2 v']",
"type": "own_claim",
"id": "T164"
},
{
"offsets": [
[
13018,
13068
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],
"text": "['the SBS works on a circular arc instead of segment']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
13074,
13082
]
],
"text": "['Figure 1']",
"type": "data",
"id": "T166"
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{
"offsets": [
[
13088,
13117
]
],
"text": "['the joint rotations are large']",
"type": "data",
"id": "T167"
},
{
"offsets": [
[
13119,
13160
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],
"text": "['the LBS produces non-natural deformations']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
13162,
13250
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],
"text": "['In the extremal case of rotation by 180 degrees, the skin can collapse to a single point']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
13252,
13296
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],
"text": "['It is the notorious ”candy-wrapper” artifact']",
"type": "own_claim",
"id": "T170"
},
{
"offsets": [
[
13323,
13331
]
],
"text": "['Figure 2']",
"type": "data",
"id": "T171"
},
{
"offsets": [
[
13448,
13541
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],
"text": "['To understand why this undesirable effect occurs, it is sufficient to re-arrange the equation']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
13827,
13857
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],
"text": "['This formula is less efficient']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
13867,
13904
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],
"text": "['it blends matrices instead of vectors']",
"type": "data",
"id": "T174"
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{
"offsets": [
[
13910,
13937
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],
"text": "['gives us a valuable insight']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
13939,
14026
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],
"text": "['It is well known that the component-wise interpolation of matrices produces odd results']",
"type": "background_claim",
"id": "T176"
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{
"offsets": [
[
14028,
14103
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],
"text": "['it does not preserve the orthogonality of the rotational part of the matrix']",
"type": "own_claim",
"id": "T177"
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{
"offsets": [
[
14105,
14188
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],
"text": "['In some situations, it does not preserve even the rank of the interpolated matrices']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
14190,
14339
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],
"text": "['This is exactly what happens in the ”candywrapper” problem: the single point the skin collapses to is a result of transformation by a singular matrix']",
"type": "own_claim",
"id": "T179"
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{
"offsets": [
[
14341,
14420
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],
"text": "['A similar defect is visible also in the proximity of the singular configuration']",
"type": "own_claim",
"id": "T180"
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{
"offsets": [
[
14431,
14452
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],
"text": "['the matrix is regular']",
"type": "data",
"id": "T181"
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{
"offsets": [
[
14454,
14590
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],
"text": "['it involves a non-uniform scaling and skewing, which is responsible for the loss of volume of the deformed skin even for small rotations']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
14711,
14763
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],
"text": "['we propose to change the interpolation method in (2)']",
"type": "own_claim",
"id": "T183"
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{
"offsets": [
[
14891,
14978
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],
"text": "['An established interpolation of two rotations is spherical linear interpolation (SLERP)']",
"type": "background_claim",
"id": "T184"
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{
"offsets": [
[
14981,
14994
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],
"text": "['Shoemake 1985']",
"type": "data",
"id": "T185"
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{
"offsets": [
[
14810,
14889
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],
"text": "['the linear interpolation of the translation part of C j i matrices is all right']",
"type": "own_claim",
"id": "T186"
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{
"offsets": [
[
14997,
15064
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],
"text": "['Its key of success is the use of quaternions to represent rotations']",
"type": "background_claim",
"id": "T187"
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{
"offsets": [
[
15369,
15459
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],
"text": "['The more important problem is to compute a convenient center of the interpolated rotations']",
"type": "own_claim",
"id": "T188"
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{
"offsets": [
[
15474,
15509
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],
"text": "['this is really an important problem']",
"type": "own_claim",
"id": "T189"
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{
"offsets": [
[
16623,
16716
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],
"text": "['it is obvious that the center of rotation choice in the left column is much more advantageous']",
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"id": "T190"
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{
"offsets": [
[
15081,
15187
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],
"text": "['it is not possible to simply replace matrices C j i in (2) with corresponding pairs quaternion-translation']",
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"id": "T191"
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{
"offsets": [
[
15189,
15283
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"text": "['One of the problems is that the linear interpolation of quaternions is not equivalent to SLERP']",
"type": "own_claim",
"id": "T192"
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{
"offsets": [
[
15294,
15333
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],
"text": "['this is not the most serious difficulty']",
"type": "own_claim",
"id": "T193"
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{
"offsets": [
[
16268,
16336
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],
"text": "['the transformed rotation center is constant during the interpolation']",
"type": "own_claim",
"id": "T194"
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{
"offsets": [
[
16229,
16250
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],
"text": "['C j 1 r c = C j 2 r c']",
"type": "data",
"id": "T195"
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"offsets": [
[
16447,
16468
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],
"text": "['C j 1 r c = C j 2 r c']",
"type": "data",
"id": "T196"
},
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"offsets": [
[
16471,
16555
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"text": "['the transformed rotation center is linearly interpolated from C j 1 r c to C j 2 r c']",
"type": "own_claim",
"id": "T197"
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{
"offsets": [
[
16732,
16789
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],
"text": "['the interpolation of every single point is a circular arc']",
"type": "own_claim",
"id": "T198"
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{
"offsets": [
[
16797,
16805
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],
"text": "['Figure 1']",
"type": "data",
"id": "T199"
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{
"offsets": [
[
16816,
16884
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],
"text": "['a disturbing drift is inherent to any other choice of rotation cente']",
"type": "own_claim",
"id": "T200"
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{
"offsets": [
[
16895,
16898
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],
"text": "['r c']",
"type": "data",
"id": "T201"
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{
"offsets": [
[
16917,
16977
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],
"text": "['the condition of zero translation cannot be always satisfied']",
"type": "own_claim",
"id": "T202"
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{
"offsets": [
[
16989,
17025
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"text": "['for more than two influencing joints']",
"type": "data",
"id": "T203"
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{
"offsets": [
[
17039,
17126
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],
"text": "['the vertex is attached to only two joints k and l that are not neighbours of each other']",
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"offsets": [
[
17128,
17162
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"text": "['some translation may be inevitable']",
"type": "own_claim",
"id": "T205"
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{
"offsets": [
[
17190,
17328
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],
"text": "['there is no relative rotation between C k and C l , but there is a relative translation induced by the joints in the chain between k and l']",
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{
"offsets": [
[
17338,
17400
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"text": "['no choice of the center of rotation can avoid this translation']",
"type": "own_claim",
"id": "T208"
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{
"offsets": [
[
17410,
17434
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],
"text": "['the rotation is identity']",
"type": "data",
"id": "T209"
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{
"offsets": [
[
17444,
17571
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],
"text": "['it is possible to define the rotation center as the point whose transformations by associated matrices are as close as possible']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
17573,
17668
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],
"text": "['This minimizes the drift and works even if the vertex is assigned to n joints j 1 , . . . , j n']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
17671,
17712
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],
"text": "['We find the center of rotation r c as the']",
"type": "own_claim",
"id": "T211"
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{
"offsets": [
[
18168,
18294
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],
"text": "['least-squares solution of the system of n 2 linear vector equations C a r c = C b r c , a < b, a, b ∈ { j 1 , . . . , j n }']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
[
18830,
18915
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],
"text": "['we cannot make any assumptions about the rank of matrix D, which can vary from 0 to 3']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
18926,
18961
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],
"text": "['for example n = 2 and C j 1 = C j 2']",
"type": "data",
"id": "T214"
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{
"offsets": [
[
19031,
19087
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],
"text": "['there are multiple solutions giving the minimal Dr c − e']",
"type": "data",
"id": "T215"
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{
"offsets": [
[
19090,
19129
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],
"text": "['the r c with the minimal norm is chosen']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
19355,
19384
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],
"text": "['LAPACK routines are efficient']",
"type": "background_claim",
"id": "T217"
},
{
"offsets": [
[
19386,
19481
]
],
"text": "['computation of the center of rotation per each vertex would not result in a real-time algorithm']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
19320,
19340
]
],
"text": "['Anderson et al. 1999']",
"type": "data",
"id": "T219"
},
{
"offsets": [
[
19711,
19773
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],
"text": "['we can re-use the center of rotation computed formerly (cached']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
19628,
19708
]
],
"text": "['we encounter another vertex assigned to the same set of joints j 1 , . . . , j n']",
"type": "data",
"id": "T221"
},
{
"offsets": [
[
19496,
19612
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],
"text": "['the center of rotation depends only on the transformations of the joints j 1 , . . . , j n and not the vertex itself']",
"type": "data",
"id": "T222"
},
{
"offsets": [
[
19789,
19859
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],
"text": "['there is only one, or two neighboring joints that influence the vertex']",
"type": "data",
"id": "T223"
},
{
"offsets": [
[
19861,
19994
]
],
"text": "['we can determine the center of rotation precisely (as indicated in the beginning of this section) and omit the SVD computation at all']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
20169,
20207
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],
"text": "['This enables the real-time performance']",
"type": "own_claim",
"id": "T225"
},
{
"offsets": [
[
20014,
20146
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],
"text": "['the number of different non-trivial joint sets, and therefore the number of running the SVD, is surprisingly small for common models']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
20149,
20167
]
],
"text": "['about several tens']",
"type": "data",
"id": "T227"
},
{
"offsets": [
[
20322,
20397
]
],
"text": "['the interpolation of multiple rotations has already received some attention']",
"type": "background_claim",
"id": "T228"
},
{
"offsets": [
[
20441,
20501
]
],
"text": "['as well as interpolation of multiple general transformations']",
"type": "background_claim",
"id": "T229"
},
{
"offsets": [
[
20399,
20421
]
],
"text": "['Buss and Fillmore 2001']",
"type": "data",
"id": "T230"
},
{
"offsets": [
[
20423,
20439
]
],
"text": "['Park et al. 2002']",
"type": "data",
"id": "T231"
},
{
"offsets": [
[
20503,
20513
]
],
"text": "['Alexa 2002']",
"type": "data",
"id": "T232"
},
{
"offsets": [
[
20531,
20622
]
],
"text": "['all these methods are substantially slower then the simple linear interpolation used in LBS']",
"type": "background_claim",
"id": "T233"
},
{
"offsets": [
[
20695,
20756
]
],
"text": "['we propose an approximate but fast linear quaternion blending']",
"type": "own_claim",
"id": "T234"
},
{
"offsets": [
[
20630,
20693
]
],
"text": "['our goal is an algorithm with comparable time complexity as LBS']",
"type": "own_claim",
"id": "T235"
},
{
"offsets": [
[
20969,
21002
]
],
"text": "['this correspondence is not unique']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
21012,
21065
]
],
"text": "['both quaternions q and −q represent the same rotation']",
"type": "data",
"id": "T237"
},
{
"offsets": [
[
21067,
21147
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"text": "['The SLERP of two unit quaternions p, q assumes that their dot product (p, q) ≥ 0']",
"type": "background_claim",
"id": "T238"
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{
"offsets": [
[
21152,
21181
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"text": "['the dot product (p, q) < 0']",
"type": "data",
"id": "T239"
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{
"offsets": [
[
21183,
21205
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"text": "['we use −p instead of p']",
"type": "own_claim",
"id": "T240"
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{
"offsets": [
[
21233,
21274
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"text": "['both p and −p represent the same rotation']",
"type": "data",
"id": "T241"
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{
"offsets": [
[
21720,
21754
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],
"text": "['The difference to SLERP is obvious']",
"type": "own_claim",
"id": "T242"
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{
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[
21756,
21890
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"text": "['QLERP interpolates along the shortest segment, and then projects to arc, which does not result in the uniform interpolation of the arc']",
"type": "own_claim",
"id": "T243"
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{
"offsets": [
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21924,
21956
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"text": "['QLERP is sufficient for our task']",
"type": "own_claim",
"id": "T244"
},
{
"offsets": [
[
22125,
22130
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],
"text": "['t = 0']",
"type": "data",
"id": "T245"
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{
"offsets": [
[
22132,
22171
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],
"text": "['both QLERP and SLERP return of course p']",
"type": "own_claim",
"id": "T246"
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{
"offsets": [
[
22177,
22186
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"text": "['t > 0,']",
"type": "data",
"id": "T247"
},
{
"offsets": [
[
22207,
22307
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],
"text": "['both QLERP and SLERP work by concatenating p with some rotation (multiplying p with some quaternion)']",
"type": "own_claim",
"id": "T248"
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{
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22358,
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"text": "['It can be expressed as p ∗ s(t; p, q)']",
"type": "own_claim",
"id": "T249"
},
{
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[
22405,
22444
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"text": "['pr s (t) = pp ∗ s(t; p, q) = s(t; p, q)']",
"type": "data",
"id": "T250"
},
{
"offsets": [
[
23356,
23474
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"text": "['the SLERP can be written as pr s (t) and QLERP as pr l (t), where the rotations r s (t) and r l (t) have the same axis']",
"type": "own_claim",
"id": "T251"
},
{
"offsets": [
[
23486,
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"text": "['this axis is constant, i.e. independent on the interpolation parameter t']",
"type": "own_claim",
"id": "T252"
},
{
"offsets": [
[
23576,
23668
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"text": "['the only difference between QLERP and SLERP is in the angle of rotations r s (t) and r l (t)']",
"type": "own_claim",
"id": "T253"
},
{
"offsets": [
[
24729,
24839
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"text": "['both SLERP and QLERP interpolate by multiplying the first quaternion with a rotation with the same, fixed axis']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
24841,
25014
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"text": "['The difference between SLERP and QLERP is only in the angle of this rotation, and is strictly less then 0.143 radians (8.15 degrees) for any interpolation parameter t ∈ 0, 1']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
25017,
25140
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],
"text": "['This is an upper bound; practical results are much smaller and could hardly cause an observable defect in the deformed skin']",
"type": "own_claim",
"id": "T256"
},
{
"offsets": [
[
25142,
25239
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],
"text": "['The big advantage of QLERP is that it can be easily generalized to interpolate multiple rotations']",
"type": "own_claim",
"id": "T257"
},
{
"offsets": [
[
23954,
23986
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],
"text": "['SLERP assumes cos θ = (p, q) ≥ 0']",
"type": "data",
"id": "T258"
},
{
"offsets": [
[
23988,
24018
]
],
"text": "['the angle θ cannot exceed π /2']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
[
23680,
23765
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],
"text": "['both r s (t) and r l (t) have a form of linear combination of quaternions 1 and p ∗ q']",
"type": "data",
"id": "T260"
},
{
"offsets": [
[
23781,
23864
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],
"text": "['the results of both r s (t) and r l (t) always end up in certain 2D subspace of R 4']",
"type": "own_claim",
"id": "T261"
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{
"offsets": [
[
22815,
22887
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"text": "['we can express it as p ∗ q = cos θ + u sin θ for some axis of rotation u']",
"type": "own_claim",
"id": "T262"
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{
"offsets": [
[
22787,
22813
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"text": "['p ∗ q is a unit quaternion']",
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"id": "T263"
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{
"offsets": [
[
22892,
22928
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"text": "['we substitute this into equation (5)']",
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{
"offsets": [
[
22930,
23003
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"text": "['we obtain sin((1 − t) θ ) + sin(t θ ) cos θ r s (t) = sin θ + u sin(t θ )']",
"type": "own_claim",
"id": "T265"
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{
"offsets": [
[
23021,
23068
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"text": "['the direction of the axis u is independent on t']",
"type": "own_claim",
"id": "T266"
},
{
"offsets": [
[
26275,
26323
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],
"text": "['(q j 1 , q j i ) < 0 for any i = 2, . . . , n']",
"type": "data",
"id": "T267"
},
{
"offsets": [
[
26325,
26375
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"text": "['we replace q j i with −q j i (by analogy to SLERP)']",
"type": "own_claim",
"id": "T268"
},
{
"offsets": [
[
26377,
26483
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"text": "['Then the QLERP computes s = w 1 q j 1 + . . . + w n q j n , which is subsequently normalized to s n = s/ s']",
"type": "own_claim",
"id": "T269"
},
{
"offsets": [
[
27002,
27127
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"text": "['the shift of the center of rotation does not influence the interpolated rotation – it manifests only in the translation part.']",
"type": "own_claim",
"id": "T270"
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{
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[
27534,
27568
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"text": "['should be as efficient as possible']",
"type": "own_claim",
"id": "T271"
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{
"offsets": [
[
27461,
27518
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"text": "['The equation (9) has to be evaluated once per each vertex']",
"type": "own_claim",
"id": "T272"
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{
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[
27643,
27758
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"text": "['they do not depend on the actual vertex – only on the joint’s transformation, similarly as the rotation centers r c']",
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"id": "T273"
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{
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[
27570,
27632
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"text": "['The basic optimization is to pre-compute the quaternions q j i']",
"type": "own_claim",
"id": "T274"
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{
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[
27774,
27814
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"text": "['QLERP has to be executed for each vertex']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
27822,
27856
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],
"text": "['weights w 1 , . . . , w n can vary']",
"type": "data",
"id": "T276"
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{
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[
27926,
27993
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],
"text": "['The vertex v can be represented by a quaternion with zero real part']",
"type": "own_claim",
"id": "T277"
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{
"offsets": [
[
28019,
28126
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"text": "['its rotation by quaternion q can be expressed as q vq ∗ , which is a quaternion with zero real part as well']",
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"id": "T278"
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{
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[
28129,
28140
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"text": "['Eberly 2001']",
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"id": "T279"
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[
28152,
28200
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],
"text": "['this expression is not efficient for computation']",
"type": "own_claim",
"id": "T280"
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{
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[
28202,
28243
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"text": "['because of slow quaternion multiplication']",
"type": "own_claim",
"id": "T281"
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{
"offsets": [
[
28246,
28353
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"text": "['it enables us to write out the rotation of v by quaternion s n as s n vs n ∗ = s 1 2 svs ∗ = (s, 1 s) svs ∗']",
"type": "own_claim",
"id": "T282"
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{
"offsets": [
[
28354,
28461
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"text": "['This suggests to convert already the quaternion s to matrix Q and normalize subsequently by dividing (s, s)']",
"type": "own_claim",
"id": "T283"
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{
"offsets": [
[
28474,
28553
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"text": "['we can compute the Q matrix from (9) as Q = (s,s) Q and save the sqrt operation']",
"type": "own_claim",
"id": "T284"
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{
"offsets": [
[
29092,
29123
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"text": "['we substitute r c in place of v']",
"type": "data",
"id": "T285"
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{
"offsets": [
[
29125,
29143
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"text": "['no rotation occurs']",
"type": "own_claim",
"id": "T286"
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{
"offsets": [
[
29162,
29196
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"text": "['r c is indeed a center of rotation']",
"type": "own_claim",
"id": "T287"
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{
"offsets": [
[
29209,
29240
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"text": "['n = 2 and C j 1 r c = C j 2 r c']",
"type": "data",
"id": "T288"
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{
"offsets": [
[
29277,
29430
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"text": "['the translation part becomes w 1 C j 1 r c + w 2 C j 2 r c = (w 1 + w 2 )C j 1 r c = C j 1 r c which is independent of interpolation parameters (weights)']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
29437,
29492
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],
"text": "['the translation during interpolation is constant indeed']",
"type": "own_claim",
"id": "T290"
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{
"offsets": [
[
29501,
29605
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"text": "['the equation (9) is nothing but a generalization of LBS to an arbitrary method of rotation interpolation']",
"type": "own_claim",
"id": "T291"
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{
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[
29607,
29651
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],
"text": "['The choice of QLERP is not important for (9)']",
"type": "own_claim",
"id": "T292"
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{
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"text": "['the matrix Q can be replaced by matrix resulting from any other interpolation scheme']",
"type": "own_claim",
"id": "T293"
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{
"offsets": [
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29749,
29771
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"text": "['Buss and Fillmore 2001']",
"type": "data",
"id": "T294"
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29777,
29810
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"text": "['we substitute Q = ∑ w i C rot j i']",
"type": "data",
"id": "T295"
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{
"offsets": [
[
29868,
30070
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"text": "['we obtain v = Q(v − r c ) + ∑ w i C j i r c = ∑ w i C rot j i v − ∑ w i C rot j i r c + ∑ w i C rot j i r c + ∑ w i C tr j i = ∑ w i C rot j i v + ∑ w i C tr j i = ∑ w i C j i v which is exactly the LBS']",
"type": "own_claim",
"id": "T296"
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{
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[
30071,
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"text": "['equation (1)']",
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"id": "T297"
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30106,
30175
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"text": "['LBS is a special case, which is independent of the center of rotation']",
"type": "own_claim",
"id": "T298"
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{
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13004,
13016
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"text": "['next section']",
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<Document xmlns:gate="http://www.gate.ac.uk" name="A19_M11_Optimizing_Locomotion_Controllers_Using_Biologically-Based_Actuators_and_Objectives_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/2185520.2335376
<Title>Optimizing Locomotion Controllers Using Biologically-Based Actuators and Objectives</Title>
Jack M. Wang Samuel R. Hamner Scott L. Delp Vladlen Koltun Stanford University
<Abstract>We present a technique for automatically synthesizing walking and running controllers for physically-simulated 3D humanoid characters. The sagittal hip, knee, and ankle degrees-of-freedom are actuated using a set of eight Hill-type musculotendon models in each leg, with biologically-motivated control laws. The parameters of these control laws are set by an optimization procedure that satisfies a number of locomotion task terms while minimizing a biological model of metabolic energy expenditure. We show that the use of biologically-based actuators and objectives measurably increases the realism of gaits generated by locomotion controllers that operate without the use of motion capture data, and that metabolic energy expenditure provides a simple and unifying measurement of effort that can be used for both walking and running control optimization.</Abstract>
Keywords: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation Keywords: physics-based character animation, biomechanics, musculoskeletal simulation
<H1>1 Introduction</H1>
The development of physics-based locomotion controllers de novo, independent from stock motion data, has been a long-standing objective in computer graphics research and has seen resurgence in recent years. Despite impressive progress, the gaits produced by existing controllers fall short of the natural appearance of human locomotion. For example, physics-based walking controllers that do not rely on motion capture data commonly produce walking motion with exaggerated hip flexion which appears more crouched and less fluid than typical human walking. One likely cause of these differences is the control force generation mechanism. Biological control systems output neural excitation signals, which then generate musculotendon forces that lead to joint torques. The mapping from excitation to torque is highly complex due to variable moment arms, biarticular muscles, and the dependence of musculotendon forces on fiber length and contraction velocity
[Zajac 1989]. On the other hand, state-of-the-art bipedal locomotion control methods directly output joint torques, which ignore constraints and energetic costs imposed by muscle anatomy and physiology. Consequently, to accomplish a motion task, controllers often employ torque patterns that are inefficient or even impossible for humans. These biologically implausible torque patterns diminish the naturalness of the resulting gaits. The goal of our work is to enhance the realism of locomotion gaits exhibited by physically-simulated humanoids without dependence on motion capture data. To this end, we augment the jointactuated humanoid model with a set of Hill-type musculotendon units (MTUs). These musculotendon units generate torques for the most important degrees-of-freedom (DOFs) during locomotion— the sagittal plane hip, knee, and ankle DOFs. To actuate these muscles, we define biologically-motivated control functions that map the current state of the body (joint angles, muscle fiber lengths, etc.) to excitation signals. The parameters of these functions are optimized to yield gaits that move the character forward without falling down. While many sets of parameters are capable of achieving this task, the quality of the resulting motion varies significantly among them. To produce gaits that have a high degree of realism, we employ an objective based on minimization of metabolic energy expenditure, thus choosing the most effortless gait that achieves the task [Alexander 2003]. In living humans and animals, metabolic energy expenditure can be estimated by oxygen consumption. In contrast, it is less clear how metabolic energy expenditure should be modeled for simulated characters. A common substitute is the sum of squared joint torques [Schultz and Mombaur 2010]
, which does not account for the different effort levels required to generate torques in different joints, directions, and body configurations. More nuanced objectives can be learned from inverse optimization
[Liu et al. 2005]
, but are dependent on training data. Our use of biologically-based actuators enables the estimation of metabolic energy expenditure based on the internal state of the MTUs
[Anderson 1999]. The result is a locomotion control optimization procedure that minimizes a physiologically-based objective within a parameter space restricted to biologically plausible torque patterns. We demonstrate the presented approach by optimizing locomotion controllers for a wide range of speeds. For quantitative evaluation, we collected experimental ground truth data from 20 human subjects walking and running at eight speeds on an instrumented treadmill. Much like human locomotion, our controllers utilize significant ankle torque and generate smooth torque trajectories. The resulting gaits match human ground truth to a greater extent than state-of-the-art walking controllers that do not rely on motion capture data. Furthermore, we show that by simply changing the initialization and target velocity, the same optimization procedure leads to running controllers.
<H1>2 Related Work</H1>
Animation researchers have been interested in the control of locomotion for 3D humanoid characters for almost 20 years
[Hodgins et al. 1995; Laszlo et al. 1996; Faloutsos et al. 2001]. One important recent contribution is SIMBICON
[Yin et al. 2007], a remarkably robust 3D humanoid locomotion controller based on the balance control of Raibert and Hodgins [1991].
A num- ber of projects have since focused on expanding the controller repertoire for simulated bipeds
[Jain et al. 2009; Coros et al. 2010; de Lasa et al. 2010] and on locomotion in complex environments
[Mordatch et al. 2010; Wu and Popović 2010]. At the same time, efforts have been made to make the synthesized motions more human-like, or “natural.” As discussed by Wang et al. [2009]
, the original SIMBICON-style controllers tend to produce gaits lacking hip extension with a constant foot orientation. Knee angles lack flexion during swing, but lack extension at heelstrike. More recent controllers improve motions by designing better target trajectories in joint or feature space
[Coros et al. 2009; Coros et al. 2010; de Lasa et al. 2010]. While more human-like ankle motions have been produced, differences in the hip and knee angles persist ( Figure 6a ). Perhaps more importantly, controllers relying on hand-tuned trajectories cannot be easily used to investigate how the control strategies change with respect to new constraints. For example, how would the character’s motion style change given a physical disability? Can we synthesize appropriate gaits for older or younger characters? Impressive results have also been achieved by controllers based on tracking motion capture data
[da Silva et al. 2008; Muico et al. 2009; Kwon and Hodgins 2010; Lee et al. 2010; Ye and Liu 2010]. However, as with methods that tune joint trajectories or controller parameters by hand, motion capture driven controllers have a limited ability to predict changes in gait. Alternatively, de novo controller optimization has been used to capture features of human walking
[Wang et al. 2009; Wang et al. 2010]. While these methods were shown to produce gaits for a variety of characters and environmental conditions, they do not employ realistic effort measures or biologicallyplausible control torques. The resulting torque patterns are highly unnatural ( Figure 6b ), leading to artifacts such as excessive plantarflexion and sharp changes in kinematics ( Figure 6a ). In contrast, our approach is to actuate key DOFs using Hill-type MTUs and to measure effort based on metabolic energy expenditure. We demonstrate significantly more human-like kinematic and torque trajectories and show that the same control parameterization and effort objective produce both walking and running. While locomotion controllers discussed above all operate on joint-actuated models, musculoskeletal models have also been investigated in computer graphics.
Such models have been used in facial animation
[Waters 1987; Lee et al. 1995; Sifakis et al. 2005], simulation of the human hand
[Sueda et al. 2008], neck
[Lee and Terzopoulos 2006], torso
[Zordan et al. 2006], and the complete upper body
[Lee et al. 2009]. Hase et al. [2003]
optimize a CPG-based (central pattern generator) locomotion controller [Taga 1995] for 3D musculoskeletal models without tendon or activation dynamics, but their results were not compared to human kinematic and dynamic gait patterns. Moreover, full musculoskeletal models are significantly more difficult to construct than joint-actuated models. Our work demonstrates that measurable increase in locomotion realism can be produced by employing musculotendon actuators for a small subset of the body DOFs. In the biomechanics literature, abstract planar models have been used to study high-level principles of human locomotion.
For example, energy minimization has been suggested as the criterion for humans in determining step length given walking speed
[Kuo 2001], as well as in selecting between walking and running
[Srinivasan and Ruina 2006]. The spring-loaded inverted pendulum (SLIP) model [Blickhan 1989]
has been used as a basis for predicting center-of-mass (COM) movements of human runners [Full and Koditschek 1999]. However, in the absence of knee joints, these models cannot be used to simulate accurate gait pat- terns. Using a 2D model with knees and musculotendon actuators,
Geyer and Herr [2010]
showed that patterns of human walking can be generated by a set of simple control laws motivated by muscle reflexes, which inspired our work. We show how their basic ideas can be embedded in a 3D humanoid model and extended to running.
Similar 2D models have been used for gait prediction
[Ackermann and van den Bogert 2010], and to generate human-like responses to disturbances
[Murai and Yamane 2011]. Simulation studies on detailed 3D musculoskeletal models have been employed to understand muscle functions during locomotion tasks
[Anderson and Pandy 2001; Liu et al. 2008; Hamner et al. 2010]. In particular, Anderson and Pandy [2001] showed that human-like lower body motor patterns can be found by minimizing metabolic energy expenditure per distance travelled, and we adopt their proposed model of metabolic energy in our work. However, these biomechanical simulations only recovered muscle activation trajectories, and did not produce locomotion controllers that can function beyond the duration of input data. Finally, our work is complementary of the recent work of Jain and Liu [2011]
, who showed that simulating soft tissue deformation at contact sites could lead to more robust and realistic character motion. We demonstrate how musculotendon actuators, biologicallymotivated control laws, and a more realistic effort term can be used to produce more human-like locomotion gaits.
HFL GLU RF HAM VAS SOL GAS TA
(a) (b) (c)
Figure 1: Humanoid model. (a) Sixteen Hill-type MTUs, shown in red, generate torques for the hips, knees, and ankles. Note that the back joint is not rendered for aesthetic reasons. (b) Five uniarticular muscles in each leg produce flexion or extension torques at single joints. (c) Three biarticular muscles in each leg generate torques at pairs of joints. See Section 3 for details.
<H1>3 Humanoid Model</H1>
Our 3D humanoid model has 30 joint DOFs and mass distributions approximating a 180 cm, 70 kg male
[Wang et al. 2010]. From the original model, we adjust the lower-body joint locations and mass distributions to better match human data
[Hamner et al. 2010]. We use cylinders to approximate the heel and ball of the foot, which allows for some amount of foot rolling after heel-strike. Unlike previous work, where the model is actuated by setting torques to all joints, we use a model that is partially actuated by Hill-type MTUs ( Figure 1 ). Specifically, control torques for the hip, knee, and ankle joint DOFs in the sagittal plane—key DOFs for gait analysis
[Perry and Burnfield 2010]—are exclusively generated by eight MTUs in each leg. In addition, soft joint limit torques as defined by
Geyer and Herr [2010]
are applied to these DOFs. Figure 1b depicts locations of the uniarticular MTUs and the joints they actuate. The hip joint is extended by the gluteal muscles (GLU) and flexed by the hip flexor muscles (HFL), while the knee joint is extended by the vasti (VAS). The tibialis anterior (TA) and the soleus (SOL) generate dorsiflexion and plantarflexion torques at the ankle, respectively. The biarticular MTUs ( Figure 1c ) supply torques to two joints simultaneously. We include the hamstring (HAM), which extends the hip and flexes the knee, the rectus femoris (RF), which flexes the hip and extends the knee, and the gastrocnemius (GAS), which flexes the knee and plantarflexes the ankle. The choice of muscles is based on the planar model proposed by
Geyer and Herr [2010]. We have added the rectus femoris since we found that it improves the walking knee flexion profile during swing when compared to human data.
controller simulator activation contraction moment dynamics dynamics arms musculoskeletal model
Figure 2: Relationship between musculoskeletal model, controller, and simulator. The controller takes as input the simulation state (body position, joint configuration, contact state, etc.), denoted by s and outputs neural excitation signals (u) and torques ( τ ). Excitation signals are converted to muscle activations (a), which are then converted to torques τ for the hip, knee, and ankle sagittal DOFs. The remaining DOFs are directly actuated by τ . The excitation to torque mapping is a function of the contractile element kinematics (l CE , v CE ) and hip, knee, and ankle joint configuration (θ).
<H2>3.1 Musculotendon Model</H2>
We employ a Hill-type model
[Zajac 1989]
, where each MTU consists of three elements: contractile, parallel-elastic, and serialelastic. Conceptually, the contractile element (CE) models muscle fibers that can actively generate force (F CE ) depending on the current activation level (a). The parallel-elastic element (PE) models passive forces (F PE ) generated by the muscle fibers, while the serial-elastic element (SE) models the tendon. In particular, given the length and velocity of CE (l CE , v CE ), as well as the current muscle activation level (a), we can compute the MTU force (F MTU ) as follows: F MTU = F CE + F PE , F CE = aF 0 f l ( ̃ l CE )f v ( v CE ), where ̃ l CE = l CE /l opt and v CE = v CE /l opt . F 0 and l opt are musclespecific maximum isometric force and optimal fiber length parameters. f l and f v are the force-length and force-velocity curves (Figure 3). The computation of F PE and the analytic forms of f l and f v are described in the supplemental material. Intuitively, f l models the fact that muscles can generate force more efficiently near l opt , and f v captures how the muscle loses its ability to generate force as the contraction velocity increases
[Zajac 1989]. As to be discussed in Section 4.1, the nonlinearity introduced by these relations is crucial for how simple control laws for muscle excitation can lead to complex force and torque trajectories. Figure 2 illustrates how the musculotendon model interacts with the controller and the simulator. The controller outputs neural excitation signals (u), which are converted to muscle activations (a). The conversion does not occur instantaneously and is referred to as activation dynamics. The dynamics is modeled by a first-order differential equation
[Zajac 1989; Geyer et al. 2003]
, which can be integrated by a t+1 = 100h(u t − a t ) + a t , where h is the stepsize (1/2400 s) and a t and u t are the muscle activation and excitation values at the t-th timestep. A step-response graph for the activation dynamics, as well as details on the l CE and v CE computations (contraction dynamics) are given in the supplemental material. The joint torques generated by a given MTU is a function of the current body configuration. A simple variable moment arm model is assumed for MTUs attached to the knee or ankle: τ = r j cos(θ − φ j M )F MTU , where θ is the current knee or ankle angle in the sagittal plane, and r j is the maximum MTU-joint moment arm, which occurs at the joint angle φ j M . MTUs attached to the hip are assumed to have a constant moment arm: τ = r j F MTU . The total lower extremity joint torques in the sagittal plane are obtained by summing over contributions from all relevant muscles:
force-length force-velocity 1 1.5 force force muscle muscle 1 0.5 normalized normalized 0.5 0 0 0.5 1 1.5 2 −10 −5 0 5 10 normalized ber length normalized ber velocity
Figure 3: Muscle force-length and force-velocity curves used in our model. The force generating capacity of a muscle is dependent on the length of muscle fibers (force-length relationship) and the velocity of muscle fibers (force-velocity relationship). The force-length curve shows that muscles can generate force more efficiently near l opt , and the force-velocity curve shows that muscles lose ability to generate force as the magnitude of contraction velocity increases.
τ hip = τ GLU + τ HAM hip − τ HFL − τ RF hip , τ knee = τ RF knee + τ VAS − τ HAM knee − τ GAS knee , τ ankle = τ GAS knee + τ SOL − τ TA .
<H1>4 Control Parameterization</H1>
The main part of our control algorithm consists of functions that determine muscle excitation values for each of the lower body MTUs, which actuate the hip, knee, and ankle DOFs in the sagittal plane. For the upper body and the remaining DOFs in the lower body, we rely on a pose-graph controller
[Yin et al. 2007].
Stance ~ ~ d > d SI or contact on double stance SP SI ~ ~ d < d SP contact o Swing
Figure 4: High-level control states for each leg. The stance and swing phases are triggered by ground contact conditions. The signed horizontal distance between the COM and the ankle (d) normalized by leg length ( d) ̃ is compared against two threshold parameters ( d ̃ SI , d ̃ SP ) to start swing initiation (SI) and stance preparation
(SP), respectively. SI can also be started when the opposing leg makes ground contact (double stance).
<H2>4.1 Muscle Control</H2>
Our control laws for the actuators are based on the muscle-reflex controller introduced by
Geyer and Herr [2010]. We will describe the basic formulation and our modifications in this section. Two different sets of control laws apply for each muscle, depending on whether the leg is in stance or swing phase (i.e., foot is on the ground or not). We further define a swing initiation state within the stance phase, and a stance preparation state within the swing phase, where control laws for a subset of MTUs are modified ( Figure 4 ). The control laws map time-delayed features of the body to muscle excitation signals. The time-delay (∆t) models the time for neural signal propagation, set to 5 ms for MTUs connected to the hip, 20 ms for MTUs connected to the ankle, 10 ms for the VAS and ground contact
[Geyer and Herr 2010]. Body features include MTU force, fiber length, joint angle, and segment orientation. Depending on the input feature, three different mappings are defined: positive force feedback, positive length feedback, and muscle-driven proportional derivative (PD) control. These mappings serve as building blocks for the control laws, and we discuss each in turn in this section.
Positive force feedback. Given MTU m, the positive force feedback law is defined as u F m = G m F ̃ m MTU (t − ∆t m ),
where F ̃ m MTU (t − ∆t m ) is the MTU force normalized by F m 0 with a time-delay of ∆t m . The only free parameter is a positive gain constant G m , which is different for each MTU. Note that F ̃ m MTU cannot increase indefinitely since the muscle’s force generation capacity depends nonlinearly on the length and contraction velocity of the muscle fiber. As F ̃ m MTU starts to decrease due to muscle physiology, u F m starts to decrease as well. The force feedback is the main source of activation to the SOL, GAS, and VAS muscles during the stance phase. Figure 5 shows the activation and fiber length of GAS during the stance phase. We can see that u GAS F produces a positive feedback during mid-stance, when the muscle activation does not produce a significant change in muscle fiber length, as the foot is planted on the ground. As the heel loses ground contact in late stance, the same muscle activation rapidly shortens the fiber length, which reduces force output and the activation through u F GAS .
0.5 1.5 activation ber length activation 1 0 0.5 0 20 40 60 % gait cycle
Figure 5: Effects of muscle physiology on activation illustrated by GAS activation and normalized fiber length during the stance phase. Note the nonlinearity of the activation curve generated by the linear force feedback control law (Section 4.1). While the foot is flat on the ground during mid-stance, GAS activation does not significantly change the fiber length, and force feedback leads to an activation build-up. As heel loses contact during late-stance, the fiber rapidly shortens and reduces f l and f v ( Figure 3 ). As the gen-
erated force decreases, the same force feedback leads to a drop-off in activation. Positive length feedback. Positive length feedback is defined as
u m L = G m ̃ l m CE (t − ∆t m ) − H m , + where ̃ l m CE (t − ∆t m ) is the length of the muscle fiber normalized by the l m opt with a time-delay of ∆t m . G m and H m are free positive parameters and {} ± means only positive or negative values (0 otherwise). The positive length feedback effectively models a stretch reflex, which activates the muscle when the fiber is stretched beyond a fixed length. u L m is most useful during the swing phase, as the TA must be activated to dorsiflex so that toe-stubbing can be avoided. In addition, the HFL relies on length feedback to generate hip flexion torque during early swing, especially during running. Muscle-driven PD control. We also define a muscle-driven PD control law with respect to an angular feature θ as where K m , D m , θ m are free parameters of the PD-controller. The braces sign is positive if torque generated by m is in the opposing direction of θ—e.g., if m is the hip extensor and θ is the hip flexion angle—and negative otherwise. Much like the standard torquebased PD-controller, the muscle-driven PD control aims to adjust θ towards the target angle θ m while damping its velocity. However, unlike the standard PD-controller, muscles can only activate after a time-delay and each muscle can only generate forces to rotate the angular DOF in one direction. The PD-control laws are employed by the hip muscles during the stance phase to maintain the global upper body orientation, as well as during stance preparation to prepare for ground contact.
u m θ = K m (θ(t − ∆t m ) − θ m ) + D m θ(t ̇ − ∆t m ) , ±
<H2>4.2 Stance Phase</H2>
Each muscle has an initial constant excitation, or pre-stimulation value p m . These values are initialized close to zero, but are then optimized. The SOL and GAS both rely on positive force feedback and are the main sources of torque during walking. The TA ensures foot clearance during swing using a length feedback (u L TA ), but the activation is suppressed during stance in proportion to the current force generated from SOL. The suppression allows the generated TA activation patterns to better match human data during locomotion. The force feedback on the VAS creates a strong knee extension torque following ground contact, but excitation is suppressed when the knee flexion angle (θ k ) is extended below an offset (θ k off ) with an extension velocity ( θ ̇ k < 0). The suppression prevents hyperextension of the knee during mid-stance. Using muscle-driven PD control laws, the HAM, GLU, and HFL are responsible for maintaining the global orientation of the upper body (Θ), defined as the vector between the COM of the upper body and the COM of the pelvis projected onto the sagittal plane. During double stance, these control laws are only active for the leading leg, denoted as u Θ m lead . Specifically, control laws during the stance phase are as follows: Towards the end of the stance phase, the controller enters into the swing initiation, which begins when either the signed horizontal distance between the COM and the ankle normalized by leg length exceeds a constant threshold d ̃ > d ̃ SI or if the opposing leg has entered into stance phase (double stance). During swing initiation, constant excitation values between 0 and 1 (set during optimization) are added and subtracted to the VAS, RF, GLU, and HFL: The combination of HFL and GLU excitations creates a large hip flexion torque, while the VAS and RF excitations effectively allow the optimizer to adjust the initial knee swing angle and velocity. Two main differences between our stance phase control laws compared to
Geyer and Herr [2010]
lie in how the swing initiation state functions. First, for running we found it necessary to enter into swing initiation using the d ̃ > d ̃ SI condition, rather than just wait for double stance. Second, we found it unnecessary to modulate the muscle-driven PD-control laws in the hip by ground reaction forces. Instead, the responsibility to maintain upper body orientation is always assigned to the lead leg.
u SOL = p SOL + u SOL F , u TA = p TA + u TA L − u F SOL , u GAS = p GAS + u GAS F , u VAS = p VAS + u F VAS + k θ (θ k (t − ∆t VAS ) − θ k off ) , −, θ ̇ k <0 Θ u HAM = p HAM + {u HAM } lead , u RF = p RF , Θ u GLU = p GLU + {u GLU } lead , Θ u HFL = p HFL + {u HFL } lead .
u VAS = u VAS − s VAS , u RF = u RF + s RF , u GLU = u GLU − s GLU , u HFL = u HFL + s HFL .
<H2>4.3 Swing Phase</H2>
Much like in the stance phase, each muscle has an initial constant excitation value (q m ). The leg motion relies significantly on passive dynamics during the swing phase
[Collins et al. 2005], as most muscles are only excited at low levels. The main exceptions are the TA, which maintains the length feedback (u L TA ) to avoid toestubbing, and the HAM, which is activated at late swing phase to prevent the knee from being overextended before landing. The HFL introduces a hip flexion torque through a length feedback, which is suppressed when the HAM is stretched in during late swing. The amount of excitation in the HFL also depends on the value of upper body lean at the beginning of the swing phase (Θ lto ): the further the upper body leans forward compared to the reference lean angle (Θ d ), the more excitation is supplied from the HFL during the swing phase. Note that Θ d is the same as the target angle in u Θ HFL . Non-constant control laws during the swing phase are as follows: The controller enters into the stance preparation when d ̃ < d ̃ SP , where the swing leg enters into a PD-control mode. The GLU, HFL, and VAS work to guide the hip and knee joints toward a desired pose to prepare for ground contact: u VAS = q VAS + u VAS θ k , u GLU = q GLU + u θ GLU h , u HFL = q HFL + u θ HFL h . A single desired hip target angle (θ h ) is adjusted according to the SIMBICON balance feedback law
[Yin et al. 2007] and is shared by both the GLU and HFL. We found the addition of the stance preparation state to be important for discovering running gaits. The balance feedback law allows robust control strategies to be found in difficult environments (e.g., being pushed by random forces).
u TA = q TA + u L TA , u HAM = q HAM + u HAM F , u GLU = q GLU + u F GLU , u HFL = q HFL + u L HFL − u L HAM + k Θ (Θ lto − Θ d ).
<H2>4.4 Out-of-Plane and Upperbody Control</H2>
The rest of the DOFs are controlled using standard joint-space PDcontrollers with state-dependent parameters. Following Wang et al. [2010]
, the target features for the ankle and hip joints in the coronal plane are the global foot and pelvis orientations, respectively. The coronal swing hip target angles follow the same feedback law as θ h . Additionally, we set the toe joint to be a spring with spring constant of 30 Nm/rad, target angle 0, and no damping. Unlike in previous work, where a gait cycle is broken down into four states, only two are needed (triggered by left/right foot-strike) since DOFs with the most complex activities are actuated by muscles. Our upper body control also largely follows
Wang et al. [2010]
, with the exception that the target feature of our back joint in the coronal plane is the global orientation of the torso instead of the local joint angle between the torso and the pelvis. This global target allows our model to better keep the head upright during locomotion. We fix the spring and damper constants for all arm joints to 30 Nm/rad and 3 Nms/rad, respectively, with target angles set to 0. We found that more human-like arm swing can be generated by relating the elbow and shoulder target angles as θ s l = α arm θ h l − θ h r + βθ e d and φ l s = γθ e d , where θ s l and φ l s are the shoulder angles in the sagittal and transverse planes, respectively; θ h l and θ h r are the current left and right sagittal hip angles; θ e d is the desired elbow angle, β, γ are constants chosen based on human motion data (see supplemental material), and α arm is a scale constant that determines the magnitude of the arm swing. This formulation captures the tendency to rotate the shoulder backwards and inwards while bending the elbow. The scale constant and the desired elbow angle are among the parameters set by optimization, as described in the next section.
<H1>5 Optimization</H1>
The control algorithm specified in Section 4 has a large number of parameters, which we set by optimization
[Wang et al. 2010]. More specifically, each of the u m F , u m L , and u m θ laws have one, two, and three parameters, respectively. There are 56 parameters in total (30 stance, 26 swing) for the MTU control laws. For the upper body and the non-sagittal DOFs in the lower body, we optimize the PDcontrol parameters (spring-damper constants, target angle, balance feedback) for all joints except for arms, where only a target elbow angle and a swing scale parameter are optimized (Section 4.4). When combined with 33 free parameters describing the initial state of the simulation, 124 parameters (w) fully define a simulated motion {s 1 . . . s T } over T timesteps. We optimize control parameters and the initial state using Covariance Matrix Adaptation (CMA)
[Hansen 2006], with stepsize σ = 0.005 and 50 samples per iteration. The optimization aims to maximize the following return function: T R (w) = r(s t ) − w e J effort . t=1 Here r is a scalar reward function of the current state s t , J effort measures the effort of the synthesized motion, and w e is set to 0.004 divided by the mass of the model, motivated by
Wang et al. [2010]. The reward is defined as the negative sum of a number of task terms (i.e., r(s t ) = − i K i (s t )), which can be thought of as high-priority goals that the controller must satisfy while minimizing effort. In practice, these terms are weighed more heavily than the effort term. The tasks include moving the COM forward at a target velocity while not falling down for 10 seconds, and maintaining head stability and upper body orientation. The task terms are based on
Wang et al. [2010] and are defined in the supplemental material. Note that unlike in previous work, we did not need to include human-like speed to step-length ratio and minimal angular momentum about the COM as task terms.
<H2>5.1 Effort Term</H2>
The main contribution to our effort measurement is the total rate of metabolic energy expenditure ( E) ̇ over all MTUs.
To quantify E, ̇ we implement a model described by
Anderson [1999], which is later expanded by
Bhargava et al. [2004]. The rate of metabolic energy expenditure for a given muscle can be modeled as the sum of heat released and mechanical work done by the muscle: E ̇ = A ̇ + M ̇ + S ̇ + W, ̇ where A ̇ is the muscle activation heat rate, M ̇ is the muscle maintenance heat rate, S ̇ is the muscle shortening heat rate, and W ̇ is the positive mechanical work rate. The muscle activation heat rate models the rate of energy that is converted to heat by a muscle given a certain level of activation, and is a function of both the mass of the muscle and the excitation signal. The maintenance heat rate similarly models the heat rate for the muscle to maintain contraction at a certain level, and depends additionally on the current fiber length. Specifically, A ̇ = mass · f A (u) and M ̇ = mass · g( ̃ l CE )f M (a), where mass is the muscle mass and ̃ l CE is the normalized muscle fiber length. The forms of f A , f M , and g are described in the supplemental material. The dependence on muscle mass captures the fact that while larger muscles are generally capable of generating more force, they are also more costly to use. The muscle shortening heat rate models the heat generated by the shortening of muscle fibers and is proportional to the current force generated by the muscle and the shortening velocity: Finally, the positive mechanical work rate is the mechanical power produced by the active element of the MTU during contraction: Note that S ̇ is close to one-quarter of W ̇ . The difference is that F MTU is the net force (both active and passive) produced in the MTU, while F CE is only the active force. Let E ̇ m,t denote the rate of metabolic energy expenditure computed for MTU m at timestep t. We define the average rate of metabolic expenditure due to MTUs as where B ̇ is the basal metabolic energy rate, set to 1.51 times body mass
[Anderson 1999]. M is the set of all sixteen muscles defined in the model. Additionally, torques generated by the PD-controllers in the rest of the DOFs are penalized by the average sum of torque squared objective: T 1 2 J R = τ j,t , T t=1 j∈Q r where Q r is the set of all joint DOFs except for the sagittal hips, knees, and ankles. We similarly define J L to penalize the average sum of squared soft joint limit torques for the hip, knee, and ankle joints, specified in
Geyer and Herr [2010]. The overall effort of a particular motion is defined as J effort = w M J M + w R J R + w L J L , a weighted sum between the terms. We empirically set w M = 100, w R = 1, and w L = 0.5 for all experiments.
S ̇ = 0.25F MTU {−v CE } + .
W ̇ = F CE {−v CE } + .
T J M = B ̇ + 1 E ̇ m,t , T t=1 m∈M
<H1>6 Experiments</H1>
The simulations were implemented using Open Dynamics Engine (ODE) with a frequency of 2400 Hz. We simulate for T = 24000 timesteps (10 s) in each evaluation. The optimization is terminated after 3000 iterations, which takes approximately 10 hours using 50 compute cores on a cluster of Dell PowerEdge 1950 servers. An optimized controller can be simulated at interactive rates using standard hardware. We initialize walking parameters of the MTU control laws based on hand-tuned values for 2
D walking from Geyer and Herr [2010]. For running, we double the initial gain parameters of GAS and SOL, and initialize θ e d to set the elbow in a bent position. The precise initialization values are provided in the supplemental material. Human joint moment (torque) curves during locomotion can be computed from motion capture and ground reaction force data. In this work we are particularly interested in comparing our results to the mean and standard deviation curves for the sagittal hip, knee, and ankle joints for multiple subjects over multiple walking and running speeds.
While such data for walking is readily available
[Perry and Burnfield 2010], only scattered data are available for running
[Novacheck 1998; Yokozawa et al. 2007; Hamner et al. 2010]. Instead, we acquired our own ground truth data using an instrumented treadmill with 20 subjects. This data is available from http://graphics.stanford.edu/projects/bio-locomotion . We acquired kinematics and dynamics data for a range of walking and running speeds (from 1.0 m/s to 5.0 m/s). The supplemental material includes angle and moment plots for all speeds, as well as details on our data collection. Comparing the mean curves for walking speeds from 1.0 m/s to 1.75 m/s, we found that the range of hip angles in our subjects during walking increased by approx◦ imately 10 , while the location of maximum ankle plantarflexion shifted slightly earlier in the gait cycle. More pronounced differences are present between running data at different speeds. The hip ◦ angle range and maximum knee flexion both increased by 30 as running speed increased from 2.0 m/s to 5.0 m/s, while locations of both the maximum hip extension and ankle plantarflexion shifted earlier by 5% and 10%, respectively. Both the hip and ankle torque outputs increased with speed, though the ankle torque curves did not differ significantly between 4.0 m/s and 5.0 m/s.
<H2>6.1 Ground Truth Data</H2>
(a) Joint Angles 60 exion (degrees) 30 0 hip −30 120 exion (degrees) 60 knee 0 exion 40 dorsi (degrees) 0 ankle −40 0 20 40 60 80 100 % gait cycle
Figure 6: Comparison between walking controllers with speeds near 1.0 m/s. The shaded areas represent one standard deviation of the ground truth human data at 1.0 m/s. The hand-tuned SIMBICON-style controller [Coros et al. 2009] matches joint angle data relatively well, but lacks hip extension and relies primarily on hip torques. The robust feature-based controller [Mordatch et al. 2010] walks in a crouch and uses large knee torques. A controller optimized for human-like torque distributions [Wang et al. 2010] fails to generate human-like joint angles and torques. Our result (swalk in Table 1 ), optimized with a target velocity of 1.0 m/s, best matches human data.
<H2>6.2 Walking Controllers</H2>
We first optimized for a normal walking controller (referred to below as nwalk) with a target velocity of 1.25 m/s, which is approx- imately the human self-selected walking speed. Initializing with the normal controller, we then optimized for a 1.0 m/s slow walk controller (swalk) and a 1.5 m/s fast walk controller (fwalk). A 1.75 m/s very fast walk controller (vfwalk) is optimized by initializing from fwalk. Comparison to ground truth. Supplemental figures indicate that our kinematic patterns generally agree with data over a range of speeds and especially at lower speeds. Two main discrepancies are the timing of knee flexion during stance, and ankle dorsiflexion before heel-strike. For higher speeds, the maximum knee flexion angle is lower than human data, and the location of maximum ankle plantarflexion occurs earlier in the gait cycle. All angle and moment curves shown are averaged over multiple cycles. Note that we found time-delays to be important for generating human-like motion given our control model. Optimizing without activation dynamics and with ∆t m = 0 for all MTUs results in a solution where ankle torques build up too quickly in the stance phase, leading to shorter step-lengths compared to human data. Figure 6a shows the hip, knee, and ankle angles of walking data generated by our 1.0 m/s controller (swalk) compared to controllers of similar speeds presented by previous contributions
[Coros et al. 2009; Mordatch et al. 2010; Wang et al. 2010]
, as well as human data at 1.0 m/s. A major artifact from all of the previous works is the lack of hip extension during mid-gait, which does not occur in our result. The feature-based controller of
Mordatch et al. [2010]
is robust and flexible, but their basic walking gait shows an obvious crouch. Our result also exhibits a range of knee motion more similar to humans compared to previous works. However, all four controllers show excessive dorsiflexion before heel-strike. An important advantage of optimization over hand-tuning is the ability to create controllers based on high-level objectives such as walking speed. As demonstrated in supplemental material, our controllers generate more human-like gaits compared to optimized controllers from
Wang et al. [2010] at faster walking speeds (Wang10f, Wang10vf ) as well. An obvious artifact of all controllers from
Wang et al. [2010]
is the excessive plantarflexion in the early swing phase, which is not present in our result. 1 Examining differences in torque generation, we can see that the controller presented by Coros et al. [2009]
does not employ a human-like torque distribution between the joints ( Figure 6b ). In particular, as was the case in SIMBICON
[Yin et al. 2007]
, the gait is largely hip-driven, as can be seen by the large hip torques and small ankle torques compared to human data. In turn, controllers from Wang et al. [2010]
generated larger amounts of ankle torque by optimizing for a human-like torque ratio, but did not come close to matching the shapes of human torque data. Note that our work does not exhibit unnatural torque spikes due to state switching that are present in the previous works. Table 1 shows quantitative comparisons between the controllers. We compute the mean standard score against human data over 100 evenly spaced points on the curves. Note that our results show the lowest average standard score for all speeds. Evaluation of objective. We evaluate the metabolic energy expenditure objective described in Section 5 against the simple sum of squared torques objective, by redefining where Q s is the set of sagittal hip, knee, and ankle DOFs (with w M = 5). Controllers optimized for each of the two objectives (nwalk, min torque) are demonstrated in the accompanying video. For this comparison, we use a target speed of 1.25 m/s, which is the same as nwalk. The gait resulting from torque minimization exhibits too much knee flexion during the swing phase and too much dorsiflexion before heel-strike. Closer examination reveals 1 Wang et al. [2009; 2010] provided comparisons against the global thigh and foot orientation which, unlike the hip and angle ankles, do not capture the relative orientations of the body links. that the TA muscle, responsible for dorsiflexion, is highly activated throughout the gait when only torque is being minimized. Since the foot is a relatively light link, the actual magnitude of the dorsiflexion torque is not large even when the TA is fully activated, therefore it does not incur a large penalty in the torque objective. In contrast, the metabolic energy objective captures the fact that activating and maintaining contraction of TA generates significant heat and should therefore be discouraged. Note that unlike dorsiflexion torques, large ankle plantarflexion torques can be generated with relative ease. Simply increasing the penalty on ankle torques does not account for the effort difference between generating torques in different directions. A simple objective that could approximate effort given a musculoskeletal model is the sum of squared muscle activations, which is commonly used in static optimization—a technique for recovering activations given motion capture and force plate data [Anderson 1999]. However, as demonstrated in the accompanying video, this objective also does not lead to faithful walking kinematics. Here we define where M is the set of MTUs, and a m,t is the activation level of MTU m at timestep t (with w M = 60000). In the gait produced by the controller that minimizes this objective (min act), activations from the GAS/SOL are significantly lowered, while activations from VAS are increased. While the total amount of activations is reduced, the resulting gait walks in a crouch and relies heavily on the knee. Table 1 includes average standard score values compared against human data at 1.25 m/s. Controllers optimized using the torque and activation objectives both exhibit large errors compared to nwalk, especially at the ankle joint. While noticeable kinematic differences are seen in the gaits produced by different objectives, the torque curves are smooth due to the muscle model and the control parameterization. Changing muscle properties. The plantarflexors (GAS and SOL) are largely responsible for forward propulsion in normal walking
[Liu et al. 2008]. We found that weakening the GAS and SOL to a quarter of their original strength, while keeping all other objectives identical (target speed 1.25 m/s), results in a mild crouch gait characterized by excessive knee flexion (see accompanying video). Our result suggests that under the condition of weakened plantarflexors, the mild crouch gait may be metabolically efficient compared to other gait choices. The crouch gait is commonly found in cerebral palsy patients, and weakness in the plantarflexors is one of many factors thought to contribute to the gait abnormality [Steele et al. 2010]. Knee hyperextension, another common gait abnormality, causes patients to vault the body forward over the extended stance limb, and can result from hamstring lengthening surgery in cerebral palsy patients
[Kay et al. 2002]. In the accompanying video, we show that our optimization indeed results in a mild hyperextension gait after weakening HAM to a quarter of its original strength, with a mini◦ mum knee flexion angle of 2 . Note that the same angle for the gait ◦ generated by nwalk is 9 . Another cause of knee hyperextension is weakened quadriceps, which can be simulated by weakening the VAS in our model. We found that weakening the VAS to one-tenth of its original strength leads to a motion similar to quadriceps avoidance gait, which is seen in patients with quadriceps weakness and anterior cruciate ligament (ACL) deficiency
[Timoney et al. 1993].
(b) Normalized Joint Moments 2 extension (Nm/kg) 0 hip −2 2 extension (Nm/kg) 0 knee −2 exion 4 Mordatch Coros 2009 2010 Wang 2010 plantar (Nm/kg) 2 our human result data ankle 0 0 20 40 60 80 100 % gait cycle
ref. speed
controller
hip
knee
ankle
average
1.0 m/s
Coros09
1.63
2.04
1.54
1.73
Mordatch10
5.80
8.10
6.36
6.75
Wang10s
4.17
3.71
2.40
3.42
swalk
0.41
1.28
1.41
1.04
1.25 m/s
min torque
1.35
1.79
4.32
2.49
min act
1.51
3.20
3.74
2.82
nwalk
0.43
1.79
1.34
1.19
1.5 m/s
Wang10f
3.03
3.38
3.46
3.29
fwalk
0.79
2.53
1.28
1.28
1.75 m/s
Wang10vf
1.37
2.60
3.32
2.43
vfwalk
1.31
3.19
1.69
2.06
ref. speed controller hip knee ankle average 1.0 m/s Coros09 1.63 2.04 1.54 1.73 Mordatch10 5.80 8.10 6.36 6.75 Wang10s 4.17 3.71 2.40 3.42 swalk 0.41 1.28 1.41 1.04 1.25 m/s min torque 1.35 1.79 4.32 2.49 min act 1.51 3.20 3.74 2.82 nwalk 0.43 1.79 1.34 1.19 1.5 m/s Wang10f 3.03 3.38 3.46 3.29 fwalk 0.79 2.53 1.28 1.28 1.75 m/s Wang10vf 1.37 2.60 3.32 2.43 vfwalk 1.31 3.19 1.69 2.06
Table 1: Quantitative comparison of walking controllers with human kinematics data in standard score (average number of standard deviations away from the mean).
J M = T 1 T t=1 j∈Q s τ j,t 2 ,
J M = T 1 T t=1 m∈M a 2 m,t ,
(a) Joint Angles 60 exion (degrees) 30 0 hip −30 120 exion (degrees) 60 knee 0 exion 40 dorsi (degrees) 0 ankle −40 0 20 40 60 80 100 % gait cycle
Figure 7: Comparison of running. Our result is optimized with a target velocity of 4.0 m/s. The shaded regions represent one standard deviation of the human running data at 4.0 m/s.
<H2>6.3 Running Controllers</H2>
Our controller architecture and objective function is not limited or specific to walking alone. By simply changing the target velocity and initialization (changing the initial velocity from 1.3 m/s to 3.05 m/s, doubling the initial force feedback gains for GAS and SOL, and bending the elbow), the same procedure yields running controllers, without any modifications to the control parameterization. In contrast, previous optimization-based control synthesis methods required including torque ratios specific to walking as part of the objective [Wang et al. 2009] or adding spring elements for running [Wu and Popović 2010]. Our unified approach to both walking and running is consistent with the view that humans select between walking and running by minimizing energy at different speeds
[Srinivasan and Ruina 2006]. We compare running motions generated by our controller at 4.0 m/s with human running data in Figure 7 . Our running kinematic results do not match human data as well as walking, though the basic features of the curves are still present. A main discrepancy is that our hip and knee joints both reach maximum extension earlier than human running data. Similar to our walking results, our knee joint flexes less during the stance phase compared to humans. Our maximum knee flexion is also lower than human data. Figure 7b reveals possible causes for the kinematic differences. Our knee extension torque reaches maximum earlier than human data, which can cause the knee to extend too quickly during the stance phase. On the other hand, our plantarflexion torques have a lower peak than human data, resulting in a strategy that relies on the knees more than the ankles. In the supplemental material and the video, we include results for running at speeds ranging from 3.0 m/s to 5.0 m/s. The faster running results are optimized sequentially in 0.5 m/s increments (e.g., 4.0 m/s initialized from 3.5 m/s). As the target velocity increases, finding a satisfactory local minimum appears more difficult. We use 100 samples per iteration and a 0.25 m/s optimization increment for speeds over 4.0 m/s.
(b) Normalized Joint Moments 2 extension (Nm/kg) 0 hip −2 extension (Nm/kg) 2 knee 0 −2 exion 4 our result human data plantar (Nm/kg) 2 ankle 0 0 20 40 60 80 100 % gait cycle
<H2>6.4 Robustness</H2>
In this work, we have chosen to focus on reproducing humanlike kinematics and torque trajectories. Likely due to our modeling of human-like torque generation and activation delays, our controllers cannot tolerate nearly as much external force as recently developed controllers for purely joint-actuated characters
[Mordatch et al. 2010; Wang et al. 2010]. However, we can still follow
Wang et al. [2010]
and optimize explicitly for controllers that can deal with external forces. In particular, we optimized controllers that can tolerate 100 N, 0.4 s pushes to the torso. These controllers chose to walk in a stiff crouch gait, with lowered COM and a constantly dorsiflexed ankle to ensure foot clearance (see accompanying video). Note that 100 N is approximately the weight of a 10 kg object, a significant push to a human. Comparatively, the corresponding 100 N controller presented by Wang et al. [2010]
, who did not model biological torque generation constraints, did not employ a gait that is significantly different from the undisturbed baseline controller. We also optimized for a 4.0 m/s running controller tolerant of 50 N, 0.4 s pushes, as shown in the video.
<H1>7 Discussion</H1>
We have presented a biologically-motivated control parameterization that can be used to automatically generate 3D human-like walking and running controllers of different speeds. Controllers are optimized to satisfy a set of high-level task terms while minimizing an effort term based on modeling the rate of metabolic energy expenditure. Notably, walking and running emerge from the same optimization process simply by changing the target velocity and initialization. Through comparisons to kinematic and torque data of human walking, we show that our results adopt a human-like torque generation strategy while producing kinematic data significantly closer to humans than previous work. Our work demonstrates the importance of modeling constraints on torque generation due to muscle physiology, both in restricting the space of possible torque trajectories and in providing a realistic model of effort. We chose to focus on generating human-like locomotion in a straight line and on flat ground.
A natural extension is to investigate whether our control parameterization and effort term can be combined with the popular task-space controllers
[Coros et al. 2010; de Lasa et al. 2010; Wu and Popović 2010] and higher-level planning
[Coros et al. 2009; Mordatch et al. 2010] to create humanlike motions on uneven terrains
[Wu and Popović 2010] or obstacle courses
[Mordatch et al. 2010; Ye and Liu 2010]—scenarios that have only been addressed using purely joint-actuated characters. Finally, an exciting area for future work is to automatically synthesize locomotion controllers for more detailed, fully muscle-actuated human models [Weinstein et al. 2008; Lee et al. 2009]. As we have touched on in Section 6.2, our approach can be used to develop predictive biomechanical models to investigate the effects of muscle and control properties on gait. However, more scientific validation of our simulation results is needed before we can conclude that our results apply to real humans. One clear aspect for improvement is to adopt a more physically-accurate simulation engine
[Sherman et al. 2011], as ODE “emphasizes speed and stability over physical accuracy” [Smith 2006]. More accurate simulations and detailed models present additional computational challenges both in simulation speed and in parameter optimization, but are crucial for potential scientific and medical applications.
<H1>Acknowledgements</H1>
We thank Stelian Coros and Igor Mordatch for making their simulation data available. Thanks to Jared Duke for early contributions, Tim Dorn for comments on the manuscript, and Ajay Seth, Edith Arnold, and Katherine Steele for inspiring technical discussions. We thank the anonymous reviewers for their helpful and constructive comments. This work was funded in part by NSF grants IIS-1017938 and CNS-0619926, NIH grants U54 GM072970, R24 HD065690, and R01 HD033929, and an AWS research grant. This research was conducted in conjunction with the Intel Science and Technology Center for Visual Computing.
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"text": "['Muico et al. 2009']",
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7520,
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"text": "['Kwon and Hodgins 2010']",
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7543,
7558
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"text": "['Lee et al. 2010']",
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7560,
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"text": "['Ye and Liu 2010']",
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7587,
7749
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"text": "['as with methods that tune joint trajectories or controller parameters by hand, motion capture driven controllers have a limited ability to predict changes in gait']",
"type": "background_claim",
"id": "T76"
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7766,
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"text": "['de novo controller optimization has been used to capture features of human walking']",
"type": "background_claim",
"id": "T77"
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{
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7851,
7867
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"text": "['Wang et al. 2009']",
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7869,
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"text": "['Wang et al. 2010']",
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7894,
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"text": "['these methods were shown to produce gaits for a variety of characters and environmental conditions']",
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"text": "['they do not employ realistic effort measures or biologicallyplausible control torques']",
"type": "background_claim",
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"text": "['The resulting torque patterns are highly unnatural']",
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8147,
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"text": "['leading to artifacts such as excessive plantarflexion and sharp changes in kinematics']",
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8134,
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"text": "['Figure 6b']",
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"text": "['Figure 6a']",
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"text": "['our approach is to actuate key DOFs using Hill-type MTUs and to measure effort based on metabolic energy expenditure']",
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8379,
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"text": "['We demonstrate significantly more human-like kinematic and torque trajectories']",
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8462,
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"text": "['locomotion controllers discussed above all operate on joint-actuated models']",
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"text": "['musculoskeletal models have also been investigated in computer graphics']",
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8721,
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"text": "['Such models have been used in facial animation']",
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8822,
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"text": "['simulation of the human hand']",
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8873,
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"text": "['neck']",
"type": "background_claim",
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8907,
8912
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"text": "['torso']",
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8936,
8963
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"text": "['and the complete upper body']",
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8770,
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"text": "['Waters 1987']",
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8783,
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],
"text": "['Lee et al. 1995']",
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"offsets": [
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8800,
8819
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],
"text": "['Sifakis et al. 2005']",
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8853,
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"text": "['Sueda et al. 2008']",
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[
8880,
8904
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"text": "['Lee and Terzopoulos 2006']",
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"offsets": [
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8915,
8933
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],
"text": "['Zordan et al. 2006']",
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"offsets": [
[
8966,
8981
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"text": "['Lee et al. 2009']",
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9004,
9024
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],
"text": "['optimize a CPG-based']",
"type": "background_claim",
"id": "T103"
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{
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[
9053,
9074
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],
"text": "['locomotion controller']",
"type": "background_claim",
"id": "T104"
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"offsets": [
[
9087,
9154
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],
"text": "['for 3D musculoskeletal models without tendon or activation dynamics']",
"type": "background_claim",
"id": "T105"
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{
"offsets": [
[
9160,
9236
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"text": "['their results were not compared to human kinematic and dynamic gait patterns']",
"type": "background_claim",
"id": "T106"
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{
"offsets": [
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8984,
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"text": "['Hase et al.']",
"type": "data",
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"offsets": [
[
8997,
9001
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"text": "['2003']",
"type": "data",
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9076,
9085
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"text": "['Taga 1995']",
"type": "data",
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"offsets": [
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9248,
9348
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],
"text": "['full musculoskeletal models are significantly more difficult to construct than joint-actuated models']",
"type": "background_claim",
"id": "T110"
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{
"offsets": [
[
9350,
9507
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],
"text": "['Our work demonstrates that measurable increase in locomotion realism can be produced by employing musculotendon actuators for a small subset of the body DOFs']",
"type": "own_claim",
"id": "T111"
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{
"offsets": [
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9509,
9629
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"text": "['In the biomechanics literature, abstract planar models have been used to study high-level principles of human locomotion']",
"type": "background_claim",
"id": "T112"
},
{
"offsets": [
[
9645,
9758
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],
"text": "['energy minimization has been suggested as the criterion for humans in determining step length given walking speed']",
"type": "background_claim",
"id": "T113"
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{
"offsets": [
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9772,
9823
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],
"text": "['as well as in selecting between walking and running']",
"type": "background_claim",
"id": "T114"
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[
9761,
9769
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"text": "['Kuo 2001']",
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9826,
9851
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"text": "['Srinivasan and Ruina 2006']",
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"offsets": [
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9854,
9889
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"text": "['The spring-loaded inverted pendulum']",
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"id": "T117"
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{
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[
9897,
9902
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],
"text": "['model']",
"type": "background_claim",
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{
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9920,
9974
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],
"text": "['has been used as a basis for predicting center-of-mass']",
"type": "background_claim",
"id": "T119"
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{
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9981,
10007
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"text": "['movements of human runners']",
"type": "background_claim",
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10045,
10140
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"text": "['in the absence of knee joints, these models cannot be used to simulate accurate gait pat- terns']",
"type": "background_claim",
"id": "T121"
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{
"offsets": [
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9904,
9917
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],
"text": "['Blickhan 1989']",
"type": "data",
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{
"offsets": [
[
10009,
10033
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],
"text": "['Full and Koditschek 1999']",
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{
"offsets": [
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10142,
10197
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"text": "['Using a 2D model with knees and musculotendon actuators']",
"type": "background_claim",
"id": "T124"
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{
"offsets": [
[
10235,
10338
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"text": "['patterns of human walking can be generated by a set of simple control laws motivated by muscle reflexes']",
"type": "background_claim",
"id": "T125"
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{
"offsets": [
[
10200,
10214
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],
"text": "['Geyer and Herr']",
"type": "data",
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"offsets": [
[
10216,
10220
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"text": "['2010']",
"type": "data",
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10340,
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"text": "['which inspired our work']",
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10377,
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],
"text": "['their basic ideas can be embedded in a 3D humanoid model and extended to running']",
"type": "own_claim",
"id": "T129"
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{
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10460,
10512
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"text": "['Similar 2D models have been used for gait prediction']",
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10551,
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"text": "['and to generate human-like responses to disturbances']",
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10515,
10548
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"text": "['Ackermann and van den Bogert 2010']",
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10606,
10627
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"text": "['Murai and Yamane 2011']",
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10630,
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"text": "['Simulation studies on detailed 3D musculoskeletal models have been employed to understand muscle functions during locomotion tasks']",
"type": "background_claim",
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{
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10763,
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"text": "['Anderson and Pandy 2001']",
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10788,
10803
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],
"text": "['Liu et al. 2008']",
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10805,
10823
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"text": "['Hamner et al. 2010']",
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"offsets": [
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10879,
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"text": "['human-like lower body motor patterns can be found by minimizing metabolic energy expenditure per distance travelled']",
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10861,
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"text": "['2001']",
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10841,
10859
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"text": "['Anderson and Pandy']",
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11000,
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"text": "['we adopt their proposed model of metabolic energy in our work']",
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"id": "T141"
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{
"offsets": [
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11072,
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],
"text": "['these biomechanical simulations only recovered muscle activation trajectories']",
"type": "background_claim",
"id": "T142"
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{
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11155,
11245
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],
"text": "['did not produce locomotion controllers that can function beyond the duration of input data']",
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{
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11256,
11300
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"text": "['our work is complementary of the recent work']",
"type": "own_claim",
"id": "T144"
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{
"offsets": [
[
11326,
11450
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],
"text": "['who showed that simulating soft tissue deformation at contact sites could lead to more robust and realistic character motion']",
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{
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11452,
11620
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],
"text": "['We demonstrate how musculotendon actuators, biologicallymotivated control laws, and a more realistic effort term can be used to produce more human-like locomotion gaits']",
"type": "own_claim",
"id": "T146"
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{
"offsets": [
[
11304,
11316
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],
"text": "['Jain and Liu']",
"type": "data",
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{
"offsets": [
[
11318,
11322
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],
"text": "['2011']",
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"offsets": [
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12790,
12816
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"text": "['key DOFs for gait analysis']",
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{
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12845,
12896
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"text": "['are exclusively generated by eight MTUs in each leg']",
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"id": "T150"
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12712,
12789
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"text": "['control torques for the hip, knee, and ankle joint DOFs in the sagittal plane']",
"type": "own_claim",
"id": "T151"
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{
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[
12482,
12538
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"text": "['allows for some amount of foot rolling after heel-strike']",
"type": "own_claim",
"id": "T152"
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{
"offsets": [
[
12819,
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"text": "['Perry and Burnfield 2010']",
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"offsets": [
[
12413,
12474
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],
"text": "['We use cylinders to approximate the heel and ball of the foot']",
"type": "data",
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"offsets": [
[
13663,
13722
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],
"text": "['The choice of muscles is based on the planar model proposed']",
"type": "own_claim",
"id": "T155"
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{
"offsets": [
[
13727,
13741
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"text": "['Geyer and Herr']",
"type": "data",
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[
13743,
13747
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],
"text": "['2010']",
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14826,
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"text": "['Conceptually, the contractile element']",
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"text": "['models muscle fibers that can actively generate force']",
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14931,
14972
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"text": "['depending on the current activation level']",
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"id": "T160"
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{
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15686,
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"text": "['Intuitively, f l models the fact that muscles can generate force more efficiently near l opt']",
"type": "own_claim",
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{
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[
15080,
15106
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],
"text": "['the serial-elastic element']",
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"id": "T162"
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{
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15112,
15129
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"text": "['models the tendon']",
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15042,
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"text": "['generated by the muscle fibers']",
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15012,
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"text": "['models passive forces']",
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[
15937,
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"text": "['the nonlinearity introduced by these relations is crucial for how simple control laws for muscle excitation can lead to complex force and torque trajectories']",
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"offsets": [
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15785,
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"text": "['f v captures how the muscle loses its ability to generate force as the contraction velocity increases']",
"type": "own_claim",
"id": "T167"
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{
"offsets": [
[
14978,
15006
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"text": "['The parallel-elastic element']",
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15924,
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"text": "['Section 4.1']",
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"text": "['The controller outputs neural excitation signals']",
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"text": "['which are converted to muscle activations']",
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16295,
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"text": "['The conversion does not occur instantaneously']",
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"id": "T172"
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{
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17274,
17404
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"text": "['The total lower extremity joint torques in the sagittal plane are obtained by summing over contributions from all relevant muscles']",
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{
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18319,
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],
"text": "['The main part of our control algorithm consists of functions that determine muscle excitation values for each of the lower body MTUs, which actuate the hip, knee, and ankle DOFs in the sagittal plane']",
"type": "own_claim",
"id": "T174"
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{
"offsets": [
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18581,
18615
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"text": "['we rely on a pose-graph controller']",
"type": "own_claim",
"id": "T175"
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"offsets": [
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18524,
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"text": "['the upper body and the remaining DOFs in the lower body']",
"type": "data",
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"offsets": [
[
18618,
18633
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],
"text": "['Yin et al. 2007']",
"type": "data",
"id": "T177"
},
{
"offsets": [
[
19281,
19371
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],
"text": "['Our control laws for the actuators are based on the muscle-reflex controller introduced by']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
19373,
19387
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],
"text": "['Geyer and Herr']",
"type": "data",
"id": "T179"
},
{
"offsets": [
[
19389,
19393
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],
"text": "['2010']",
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"offsets": [
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20197,
20265
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"text": "['Depending on the input feature, three different mappings are defined']",
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"id": "T181"
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{
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[
20267,
20290
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],
"text": "['positive force feedback']",
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"id": "T182"
},
{
"offsets": [
[
19474,
19588
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],
"text": "['Two different sets of control laws apply for each muscle, depending on whether the leg is in stance or swing phase']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
19596,
19624
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],
"text": "['foot is on the ground or not']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
20292,
20316
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],
"text": "['positive length feedback']",
"type": "data",
"id": "T185"
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{
"offsets": [
[
20322,
20359
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],
"text": "['muscle-driven proportional derivative']",
"type": "data",
"id": "T186"
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{
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[
20365,
20372
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],
"text": "['control']",
"type": "data",
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20374,
20434
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],
"text": "['These mappings serve as building blocks for the control laws']",
"type": "own_claim",
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{
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[
20816,
20855
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],
"text": "['F ̃ m MTU cannot increase indefinitely']",
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"id": "T189"
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{
"offsets": [
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20862,
20979
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],
"text": "['the muscle’s force generation capacity depends nonlinearly on the length and contraction velocity of the muscle fiber']",
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"id": "T190"
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{
"offsets": [
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21279,
21333
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],
"text": "['u GAS F produces a positive feedback during mid-stance']",
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{
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21427,
21460
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"text": "['the foot is planted on the ground']",
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},
{
"offsets": [
[
21579,
21634
]
],
"text": "['reduces force output and the activation through u F GAS']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
21511,
21571
]
],
"text": "['the same muscle activation rapidly shortens the fiber length']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
21465,
21509
]
],
"text": "['the heel loses ground contact in late stance']",
"type": "data",
"id": "T195"
},
{
"offsets": [
[
21340,
21422
]
],
"text": "['the muscle activation does not produce a significant change in muscle fiber length']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
22737,
22879
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],
"text": "['The positive length feedback effectively models a stretch reflex, which activates the muscle when the fiber is stretched beyond a fixed length']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
[
22881,
22924
]
],
"text": "['u L m is most useful during the swing phase']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
22929,
23002
]
],
"text": "['the TA must be activated to dorsiflex so that toe-stubbing can be avoided']",
"type": "data",
"id": "T199"
},
{
"offsets": [
[
23017,
23127
]
],
"text": "['the HFL relies on length feedback to generate hip flexion torque during early swing, especially during running']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
23478,
23629
]
],
"text": "['Much like the standard torquebased PD-controller, the muscle-driven PD control aims to adjust θ towards the target angle θ m while damping its velocity']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
23640,
23719
]
],
"text": "['unlike the standard PD-controller, muscles can only activate after a time-delay']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
23724,
23803
]
],
"text": "['each muscle can only generate forces to rotate the angular DOF in one direction']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
23805,
23926
]
],
"text": "['The PD-control laws are employed by the hip muscles during the stance phase to maintain the global upper body orientation']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
23939,
23994
]
],
"text": "['during stance preparation to prepare for ground contact']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
24130,
24206
]
],
"text": "['Each muscle has an initial constant excitation, or pre-stimulation value p m']",
"type": "own_claim",
"id": "T206"
},
{
"offsets": [
[
24209,
24275
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],
"text": "['These values are initialized close to zero, but are then optimized']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
24277,
24329
]
],
"text": "['The SOL and GAS both rely on positive force feedback']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
[
24334,
24379
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],
"text": "['are the main sources of torque during walking']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
24381,
24447
]
],
"text": "['The TA ensures foot clearance during swing using a length feedback']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
24463,
24559
]
],
"text": "['the activation is suppressed during stance in proportion to the current force generated from SOL']",
"type": "own_claim",
"id": "T211"
},
{
"offsets": [
[
24910,
24979
]
],
"text": "['The suppression prevents hyperextension of the knee during mid-stance']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
[
24561,
24665
]
],
"text": "['The suppression allows the generated TA activation patterns to better match human data during locomotion']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
26064,
26134
]
],
"text": "['Two main differences between our stance phase control laws compared to']",
"type": "own_claim",
"id": "T214"
},
{
"offsets": [
[
26159,
26206
]
],
"text": "['lie in how the swing initiation state functions']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
26227,
26358
]
],
"text": "['we found it necessary to enter into swing initiation using the d ̃ > d ̃ SI condition, rather than just wait for double stance']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
26136,
26150
]
],
"text": "['Geyer and Herr']",
"type": "data",
"id": "T217"
},
{
"offsets": [
[
26152,
26156
]
],
"text": "['2010']",
"type": "data",
"id": "T218"
},
{
"offsets": [
[
26219,
26226
]
],
"text": "['running']",
"type": "data",
"id": "T219"
},
{
"offsets": [
[
26485,
26573
]
],
"text": "['the responsibility to maintain upper body orientation is always assigned to the lead leg']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
26368,
26474
]
],
"text": "['we found it unnecessary to modulate the muscle-driven PD-control laws in the hip by ground reaction forces']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
27062,
27121
]
],
"text": "['each muscle has an initial constant excitation value (q m )']",
"type": "own_claim",
"id": "T222"
},
{
"offsets": [
[
27123,
27201
]
],
"text": "['The leg motion relies significantly on passive dynamics during the swing phase']",
"type": "background_claim",
"id": "T223"
},
{
"offsets": [
[
27229,
27272
]
],
"text": "['most muscles are only excited at low levels']",
"type": "data",
"id": "T224"
},
{
"offsets": [
[
27204,
27223
]
],
"text": "['Collins et al. 2005']",
"type": "data",
"id": "T225"
},
{
"offsets": [
[
28508,
28607
]
],
"text": "['We found the addition of the stance preparation state to be important for discovering running gaits']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
27274,
27341
]
],
"text": "['The main exceptions are the TA, which maintains the length feedback']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
27352,
27372
]
],
"text": "['to avoid toestubbing']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
27374,
27484
]
],
"text": "['and the HAM, which is activated at late swing phase to prevent the knee from being overextended before landing']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
27621,
27737
]
],
"text": "['The amount of excitation in the HFL also depends on the value of upper body lean at the beginning of the swing phase']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
27748,
27825
]
],
"text": "['the further the upper body leans forward compared to the reference lean angle']",
"type": "data",
"id": "T231"
},
{
"offsets": [
[
27834,
27901
]
],
"text": "['the more excitation is supplied from the HFL during the swing phase']",
"type": "data",
"id": "T232"
},
{
"offsets": [
[
27913,
27959
]
],
"text": "['Θ d is the same as the target angle in u Θ HFL']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
28609,
28704
]
],
"text": "['The balance feedback law allows robust control strategies to be found in difficult environments']",
"type": "background_claim",
"id": "T234"
},
{
"offsets": [
[
28712,
28741
]
],
"text": "['being pushed by random forces']",
"type": "data",
"id": "T235"
},
{
"offsets": [
[
28971,
29079
]
],
"text": "['The rest of the DOFs are controlled using standard joint-space PDcontrollers with state-dependent parameters']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
29112,
29239
]
],
"text": "['the target features for the ankle and hip joints in the coronal plane are the global foot and pelvis orientations, respectively']",
"type": "background_claim",
"id": "T237"
},
{
"offsets": [
[
29091,
29102
]
],
"text": "['Wang et al.']",
"type": "data",
"id": "T238"
},
{
"offsets": [
[
29104,
29108
]
],
"text": "['2010']",
"type": "data",
"id": "T239"
},
{
"offsets": [
[
29241,
29312
]
],
"text": "['The coronal swing hip target angles follow the same feedback law as θ h']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
29509,
29528
]
],
"text": "['only two are needed']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
29530,
29565
]
],
"text": "['triggered by left/right foot-strike']",
"type": "data",
"id": "T242"
},
{
"offsets": [
[
29432,
29455
]
],
"text": "['Unlike in previous work']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
29573,
29634
]
],
"text": "['DOFs with the most complex activities are actuated by muscles']",
"type": "data",
"id": "T244"
},
{
"offsets": [
[
29636,
29679
]
],
"text": "['Our upper body control also largely follows']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
29681,
29691
]
],
"text": "['Wang et al']",
"type": "data",
"id": "T246"
},
{
"offsets": [
[
29694,
29698
]
],
"text": "['2010']",
"type": "data",
"id": "T247"
},
{
"offsets": [
[
29463,
29507
]
],
"text": "['a gait cycle is broken down into four states']",
"type": "data",
"id": "T248"
},
{
"offsets": [
[
29726,
29888
]
],
"text": "['the target feature of our back joint in the coronal plane is the global orientation of the torso instead of the local joint angle between the torso and the pelvis']",
"type": "own_claim",
"id": "T249"
},
{
"offsets": [
[
29890,
29975
]
],
"text": "['This global target allows our model to better keep the head upright during locomotion']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
30634,
30741
]
],
"text": "['This formulation captures the tendency to rotate the shoulder backwards and inwards while bending the elbow']",
"type": "own_claim",
"id": "T251"
},
{
"offsets": [
[
30743,
30834
]
],
"text": "['The scale constant and the desired elbow angle are among the parameters set by optimization']",
"type": "own_claim",
"id": "T252"
},
{
"offsets": [
[
30856,
30868
]
],
"text": "['next section']",
"type": "data",
"id": "T253"
},
{
"offsets": [
[
30933,
30954
]
],
"text": "['The control algorithm']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
30978,
31040
]
],
"text": "['has a large number of parameters, which we set by optimization']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
30968,
30977
]
],
"text": "['Section 4']",
"type": "data",
"id": "T256"
},
{
"offsets": [
[
31043,
31059
]
],
"text": "['Wang et al. 2010']",
"type": "data",
"id": "T257"
},
{
"offsets": [
[
31081,
31173
]
],
"text": "['each of the u m F , u m L , and u m θ laws have one, two, and three parameters, respectively']",
"type": "own_claim",
"id": "T258"
},
{
"offsets": [
[
31175,
31207
]
],
"text": "['There are 56 parameters in total']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
[
31230,
31254
]
],
"text": "['for the MTU control laws']",
"type": "own_claim",
"id": "T260"
},
{
"offsets": [
[
31209,
31218
]
],
"text": "['30 stance']",
"type": "data",
"id": "T261"
},
{
"offsets": [
[
31220,
31228
]
],
"text": "['26 swing']",
"type": "data",
"id": "T262"
},
{
"offsets": [
[
31622,
31679
]
],
"text": "['124 parameters (w) fully define a simulated motion {s 1 .']",
"type": "own_claim",
"id": "T263"
},
{
"offsets": [
[
31684,
31707
]
],
"text": "['s T } over T timesteps.']",
"type": "own_claim",
"id": "T264"
},
{
"offsets": [
[
31541,
31620
]
],
"text": "['combined with 33 free parameters describing the initial state of the simulation']",
"type": "data",
"id": "T265"
},
{
"offsets": [
[
31872,
31969
]
],
"text": "['The optimization aims to maximize the following return function: T R (w) = r(s t ) − w e J effort']",
"type": "own_claim",
"id": "T266"
},
{
"offsets": [
[
32186,
32286
]
],
"text": "['The reward is defined as the negative sum of a number of task terms (i.e., r(s t ) = − i K i (s t ))']",
"type": "own_claim",
"id": "T267"
},
{
"offsets": [
[
32315,
32391
]
],
"text": "['high-priority goals that the controller must satisfy while minimizing effort']",
"type": "data",
"id": "T268"
},
{
"offsets": [
[
32393,
32463
]
],
"text": "['In practice, these terms are weighed more heavily than the effort term']",
"type": "own_claim",
"id": "T269"
},
{
"offsets": [
[
32729,
32875
]
],
"text": "['unlike in previous work, we did not need to include human-like speed to step-length ratio and minimal angular momentum about the COM as task terms']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
32465,
32623
]
],
"text": "['The tasks include moving the COM forward at a target velocity while not falling down for 10 seconds, and maintaining head stability and upper body orientation']",
"type": "own_claim",
"id": "T271"
},
{
"offsets": [
[
32625,
32652
]
],
"text": "['The task terms are based on']",
"type": "background_claim",
"id": "T272"
},
{
"offsets": [
[
32654,
32665
]
],
"text": "['Wang et al.']",
"type": "data",
"id": "T273"
},
{
"offsets": [
[
32667,
32671
]
],
"text": "['2010']",
"type": "data",
"id": "T274"
},
{
"offsets": [
[
32681,
32717
]
],
"text": "['defined in the supplemental material']",
"type": "data",
"id": "T275"
},
{
"offsets": [
[
32931,
33028
]
],
"text": "['The main contribution to our effort measurement is the total rate of metabolic energy expenditure']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
33037,
33050
]
],
"text": "['over all MTUs']",
"type": "own_claim",
"id": "T277"
},
{
"offsets": [
[
33071,
33104
]
],
"text": "['we implement a model described by']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
33123,
33149
]
],
"text": "['which is later expanded by']",
"type": "own_claim",
"id": "T279"
},
{
"offsets": [
[
33106,
33114
]
],
"text": "['Anderson']",
"type": "data",
"id": "T280"
},
{
"offsets": [
[
33053,
33066
]
],
"text": "['To quantify E']",
"type": "data",
"id": "T281"
},
{
"offsets": [
[
33116,
33120
]
],
"text": "['1999']",
"type": "data",
"id": "T282"
},
{
"offsets": [
[
33151,
33166
]
],
"text": "['Bhargava et al.']",
"type": "data",
"id": "T283"
},
{
"offsets": [
[
33168,
33172
]
],
"text": "['2004']",
"type": "data",
"id": "T284"
},
{
"offsets": [
[
33529,
33660
]
],
"text": "['The muscle activation heat rate models the rate of energy that is converted to heat by a muscle given a certain level of activation']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
33666,
33736
]
],
"text": "['is a function of both the mass of the muscle and the excitation signal']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
33738,
33852
]
],
"text": "['The maintenance heat rate similarly models the heat rate for the muscle to maintain contraction at a certain level']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
33858,
33906
]
],
"text": "['depends additionally on the current fiber length']",
"type": "own_claim",
"id": "T288"
},
{
"offsets": [
[
33922,
33943
]
],
"text": "['A ̇ = mass · f A (u)']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
33948,
33981
]
],
"text": "['M ̇ = mass · g( ̃ l CE )f M (a)']",
"type": "own_claim",
"id": "T290"
},
{
"offsets": [
[
34017,
34062
]
],
"text": "[' ̃ l CE is the normalized muscle fiber length']",
"type": "data",
"id": "T291"
},
{
"offsets": [
[
33989,
34012
]
],
"text": "['mass is the muscle mass']",
"type": "data",
"id": "T292"
},
{
"offsets": [
[
34198,
34259
]
],
"text": "['larger muscles are generally capable of generating more force']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
34261,
34293
]
],
"text": "['they are also more costly to use']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
36230,
36315
]
],
"text": "['An optimized controller can be simulated at interactive rates using standard hardware']",
"type": "own_claim",
"id": "T295"
},
{
"offsets": [
[
36781,
36986
]
],
"text": "['we are particularly interested in comparing our results to the mean and standard deviation curves for the sagittal hip, knee, and ankle joints for multiple subjects over multiple walking and running speeds']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
37000,
37037
]
],
"text": "['data for walking is readily available']",
"type": "background_claim",
"id": "T297"
},
{
"offsets": [
[
36675,
36766
]
],
"text": "['curves during locomotion can be computed from motion capture and ground reaction force data']",
"type": "own_claim",
"id": "T298"
},
{
"offsets": [
[
36647,
36665
]
],
"text": "['Human joint moment']",
"type": "own_claim",
"id": "T299"
},
{
"offsets": [
[
37040,
37064
]
],
"text": "['Perry and Burnfield 2010']",
"type": "data",
"id": "T300"
},
{
"offsets": [
[
37067,
37112
]
],
"text": "['only scattered data are available for running']",
"type": "background_claim",
"id": "T301"
},
{
"offsets": [
[
37183,
37269
]
],
"text": "['we acquired our own ground truth data using an instrumented treadmill with 20 subjects']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
37115,
37129
]
],
"text": "['Novacheck 1998']",
"type": "data",
"id": "T303"
},
{
"offsets": [
[
37131,
37151
]
],
"text": "['Yokozawa et al. 2007']",
"type": "data",
"id": "T304"
},
{
"offsets": [
[
37153,
37171
]
],
"text": "['Hamner et al. 2010']",
"type": "data",
"id": "T305"
},
{
"offsets": [
[
37299,
37351
]
],
"text": "['http://graphics.stanford.edu/projects/bio-locomotion']",
"type": "data",
"id": "T306"
},
{
"offsets": [
[
37666,
37752
]
],
"text": "['the range of hip angles in our subjects during walking increased by approx◦ imately 10']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
37761,
37848
]
],
"text": "['the location of maximum ankle plantarflexion shifted slightly earlier in the gait cycle']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
37581,
37650
]
],
"text": "['Comparing the mean curves for walking speeds from 1.0 m/s to 1.75 m/s']",
"type": "data",
"id": "T309"
},
{
"offsets": [
[
37850,
37930
]
],
"text": "['More pronounced differences are present between running data at different speeds']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
38003,
38050
]
],
"text": "['running speed increased from 2.0 m/s to 5.0 m/s']",
"type": "data",
"id": "T311"
},
{
"offsets": [
[
38058,
38156
]
],
"text": "['locations of both the maximum hip extension and ankle plantarflexion shifted earlier by 5% and 10%']",
"type": "own_claim",
"id": "T312"
},
{
"offsets": [
[
37932,
37999
]
],
"text": "['The hip ◦ angle range and maximum knee flexion both increased by 30']",
"type": "own_claim",
"id": "T313"
},
{
"offsets": [
[
38172,
38230
]
],
"text": "['Both the hip and ankle torque outputs increased with speed']",
"type": "own_claim",
"id": "T314"
},
{
"offsets": [
[
38239,
38319
]
],
"text": "['the ankle torque curves did not differ significantly between 4.0 m/s and 5.0 m/s']",
"type": "own_claim",
"id": "T315"
},
{
"offsets": [
[
39719,
39856
]
],
"text": "['Supplemental figures indicate that our kinematic patterns generally agree with data over a range of speeds and especially at lower speeds']",
"type": "own_claim",
"id": "T316"
},
{
"offsets": [
[
39858,
39968
]
],
"text": "['Two main discrepancies are the timing of knee flexion during stance, and ankle dorsiflexion before heel-strike']",
"type": "own_claim",
"id": "T317"
},
{
"offsets": [
[
39989,
40044
]
],
"text": "['the maximum knee flexion angle is lower than human data']",
"type": "own_claim",
"id": "T318"
},
{
"offsets": [
[
40050,
40127
]
],
"text": "['the location of maximum ankle plantarflexion occurs earlier in the gait cycle']",
"type": "own_claim",
"id": "T319"
},
{
"offsets": [
[
39974,
39987
]
],
"text": "['higher speeds']",
"type": "data",
"id": "T320"
},
{
"offsets": [
[
40208,
40301
]
],
"text": "['we found time-delays to be important for generating human-like motion given our control model']",
"type": "own_claim",
"id": "T321"
},
{
"offsets": [
[
40303,
40455
]
],
"text": "['Optimizing without activation dynamics and with ∆t m = 0 for all MTUs results in a solution where ankle torques build up too quickly in the stance phase']",
"type": "own_claim",
"id": "T322"
},
{
"offsets": [
[
40457,
40511
]
],
"text": "['leading to shorter step-lengths compared to human data']",
"type": "own_claim",
"id": "T323"
},
{
"offsets": [
[
40796,
40888
]
],
"text": "['A major artifact from all of the previous works is the lack of hip extension during mid-gait']",
"type": "background_claim",
"id": "T324"
},
{
"offsets": [
[
40896,
40924
]
],
"text": "['does not occur in our result']",
"type": "own_claim",
"id": "T325"
},
{
"offsets": [
[
40983,
41005
]
],
"text": "['is robust and flexible']",
"type": "background_claim",
"id": "T326"
},
{
"offsets": [
[
40926,
40957
]
],
"text": "['The feature-based controller of']",
"type": "background_claim",
"id": "T327"
},
{
"offsets": [
[
40976,
40980
]
],
"text": "['2010']",
"type": "data",
"id": "T328"
},
{
"offsets": [
[
40959,
40974
]
],
"text": "['Mordatch et al.']",
"type": "data",
"id": "T329"
},
{
"offsets": [
[
41011,
41059
]
],
"text": "['their basic walking gait shows an obvious crouch']",
"type": "background_claim",
"id": "T330"
},
{
"offsets": [
[
41061,
41158
]
],
"text": "['Our result also exhibits a range of knee motion more similar to humans compared to previous works']",
"type": "own_claim",
"id": "T331"
},
{
"offsets": [
[
41169,
41236
]
],
"text": "['all four controllers show excessive dorsiflexion before heel-strike']",
"type": "own_claim",
"id": "T332"
},
{
"offsets": [
[
41238,
41361
]
],
"text": "['An important advantage of optimization over hand-tuning is the ability to create controllers based on high-level objectives']",
"type": "own_claim",
"id": "T333"
},
{
"offsets": [
[
41370,
41383
]
],
"text": "['walking speed']",
"type": "data",
"id": "T334"
},
{
"offsets": [
[
41427,
41512
]
],
"text": "['our controllers generate more human-like gaits compared to optimized controllers from']",
"type": "own_claim",
"id": "T335"
},
{
"offsets": [
[
41533,
41557
]
],
"text": "['at faster walking speeds']",
"type": "own_claim",
"id": "T336"
},
{
"offsets": [
[
41579,
41586
]
],
"text": "['as well']",
"type": "own_claim",
"id": "T337"
},
{
"offsets": [
[
41404,
41425
]
],
"text": "['supplemental material']",
"type": "data",
"id": "T338"
},
{
"offsets": [
[
41514,
41525
]
],
"text": "['Wang et al.']",
"type": "data",
"id": "T339"
},
{
"offsets": [
[
41527,
41531
]
],
"text": "['2010']",
"type": "data",
"id": "T340"
},
{
"offsets": [
[
41559,
41566
]
],
"text": "['Wang10f']",
"type": "data",
"id": "T341"
},
{
"offsets": [
[
41568,
41576
]
],
"text": "['Wang10vf']",
"type": "data",
"id": "T342"
},
{
"offsets": [
[
41588,
41631
]
],
"text": "['An obvious artifact of all controllers from']",
"type": "background_claim",
"id": "T343"
},
{
"offsets": [
[
41653,
41709
]
],
"text": "['is the excessive plantarflexion in the early swing phase']",
"type": "background_claim",
"id": "T344"
},
{
"offsets": [
[
41717,
41745
]
],
"text": "['is not present in our result']",
"type": "own_claim",
"id": "T345"
},
{
"offsets": [
[
41633,
41644
]
],
"text": "['Wang et al.']",
"type": "data",
"id": "T346"
},
{
"offsets": [
[
41646,
41650
]
],
"text": "['2010']",
"type": "data",
"id": "T347"
},
{
"offsets": [
[
41749,
41836
]
],
"text": "['Examining differences in torque generation, we can see that the controller presented by']",
"type": "background_claim",
"id": "T348"
},
{
"offsets": [
[
41858,
41925
]
],
"text": "['does not employ a human-like torque distribution between the joints']",
"type": "background_claim",
"id": "T349"
},
{
"offsets": [
[
41837,
41849
]
],
"text": "['Coros et al.']",
"type": "data",
"id": "T350"
},
{
"offsets": [
[
41851,
41855
]
],
"text": "['2009']",
"type": "data",
"id": "T351"
},
{
"offsets": [
[
41928,
41937
]
],
"text": "['Figure 6b']",
"type": "data",
"id": "T352"
},
{
"offsets": [
[
42005,
42035
]
],
"text": "['the gait is largely hip-driven']",
"type": "background_claim",
"id": "T353"
},
{
"offsets": [
[
41975,
41983
]
],
"text": "['SIMBICON']",
"type": "data",
"id": "T354"
},
{
"offsets": [
[
42040,
42123
]
],
"text": "['can be seen by the large hip torques and small ankle torques compared to human data']",
"type": "data",
"id": "T355"
},
{
"offsets": [
[
41986,
42001
]
],
"text": "['Yin et al. 2007']",
"type": "data",
"id": "T356"
},
{
"offsets": [
[
42134,
42150
]
],
"text": "['controllers from']",
"type": "background_claim",
"id": "T357"
},
{
"offsets": [
[
42171,
42255
]
],
"text": "['generated larger amounts of ankle torque by optimizing for a human-like torque ratio']",
"type": "background_claim",
"id": "T358"
},
{
"offsets": [
[
42261,
42323
]
],
"text": "['did not come close to matching the shapes of human torque data']",
"type": "background_claim",
"id": "T359"
},
{
"offsets": [
[
42151,
42162
]
],
"text": "['Wang et al.']",
"type": "data",
"id": "T360"
},
{
"offsets": [
[
42164,
42168
]
],
"text": "['2010']",
"type": "data",
"id": "T361"
},
{
"offsets": [
[
42335,
42384
]
],
"text": "['our work does not exhibit unnatural torque spikes']",
"type": "own_claim",
"id": "T362"
},
{
"offsets": [
[
42392,
42446
]
],
"text": "['state switching that are present in the previous works']",
"type": "data",
"id": "T363"
},
{
"offsets": [
[
42621,
42686
]
],
"text": "['our results show the lowest average standard score for all speeds']",
"type": "own_claim",
"id": "T364"
},
{
"offsets": [
[
43134,
43276
]
],
"text": "['The gait resulting from torque minimization exhibits too much knee flexion during the swing phase and too much dorsiflexion before heel-strike']",
"type": "own_claim",
"id": "T365"
},
{
"offsets": [
[
43669,
43765
]
],
"text": "['the actual magnitude of the dorsiflexion torque is not large even when the TA is fully activated']",
"type": "own_claim",
"id": "T371"
},
{
"offsets": [
[
43777,
43834
]
],
"text": "['it does not incur a large penalty in the torque objective']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
43632,
43667
]
],
"text": "['the foot is a relatively light link']",
"type": "data",
"id": "T373"
},
{
"offsets": [
[
43849,
44010
]
],
"text": "['the metabolic energy objective captures the fact that activating and maintaining contraction of TA generates significant heat and should therefore be discouraged']",
"type": "own_claim",
"id": "T374"
},
{
"offsets": [
[
44051,
44121
]
],
"text": "['large ankle plantarflexion torques can be generated with relative ease']",
"type": "own_claim",
"id": "T375"
},
{
"offsets": [
[
44123,
44263
]
],
"text": "['Simply increasing the penalty on ankle torques does not account for the effort difference between generating torques in different directions']",
"type": "own_claim",
"id": "T376"
},
{
"offsets": [
[
44265,
44431
]
],
"text": "['A simple objective that could approximate effort given a musculoskeletal model is the sum of squared muscle activations, which is commonly used in static optimization']",
"type": "own_claim",
"id": "T377"
},
{
"offsets": [
[
44432,
44512
]
],
"text": "['a technique for recovering activations given motion capture and force plate data']",
"type": "background_claim",
"id": "T378"
},
{
"offsets": [
[
44514,
44527
]
],
"text": "['Anderson 1999']",
"type": "data",
"id": "T379"
},
{
"offsets": [
[
44582,
44646
]
],
"text": "['this objective also does not lead to faithful walking kinematics']",
"type": "own_claim",
"id": "T380"
},
{
"offsets": [
[
44542,
44580
]
],
"text": "['demonstrated in the accompanying video']",
"type": "data",
"id": "T381"
},
{
"offsets": [
[
45154,
45293
]
],
"text": "['Controllers optimized using the torque and activation objectives both exhibit large errors compared to nwalk, especially at the ankle joint']",
"type": "own_claim",
"id": "T382"
},
{
"offsets": [
[
45301,
45388
]
],
"text": "['noticeable kinematic differences are seen in the gaits produced by different objectives']",
"type": "own_claim",
"id": "T383"
},
{
"offsets": [
[
45390,
45475
]
],
"text": "['the torque curves are smooth due to the muscle model and the control parameterization']",
"type": "own_claim",
"id": "T384"
},
{
"offsets": [
[
43504,
43517
]
],
"text": "['the TA muscle']",
"type": "own_claim",
"id": "T366"
},
{
"offsets": [
[
43549,
43588
]
],
"text": "['is highly activated throughout the gait']",
"type": "own_claim",
"id": "T367"
},
{
"offsets": [
[
43594,
43624
]
],
"text": "['only torque is being minimized']",
"type": "data",
"id": "T368"
},
{
"offsets": [
[
43519,
43547
]
],
"text": "['responsible for dorsiflexion']",
"type": "data",
"id": "T369"
},
{
"offsets": [
[
45538,
45602
]
],
"text": "['are largely responsible for forward propulsion in normal walking']",
"type": "background_claim",
"id": "T370"
},
{
"offsets": [
[
45505,
45523
]
],
"text": "['The plantarflexors']",
"type": "background_claim",
"id": "T385"
},
{
"offsets": [
[
45605,
45620
]
],
"text": "['Liu et al. 2008']",
"type": "data",
"id": "T386"
},
{
"offsets": [
[
45943,
46025
]
],
"text": "['the mild crouch gait may be metabolically efficient compared to other gait choices']",
"type": "own_claim",
"id": "T387"
},
{
"offsets": [
[
45895,
45941
]
],
"text": "['under the condition of weakened plantarflexors']",
"type": "data",
"id": "T388"
},
{
"offsets": [
[
45637,
45748
]
],
"text": "['weakening the GAS and SOL to a quarter of their original strength, while keeping all other objectives identical']",
"type": "own_claim",
"id": "T389"
},
{
"offsets": [
[
45774,
45843
]
],
"text": "['results in a mild crouch gait characterized by excessive knee flexion']",
"type": "own_claim",
"id": "T390"
},
{
"offsets": [
[
45849,
45867
]
],
"text": "['accompanying video']",
"type": "data",
"id": "T391"
},
{
"offsets": [
[
45750,
45771
]
],
"text": "['target speed 1.25 m/s']",
"type": "data",
"id": "T392"
},
{
"offsets": [
[
46027,
46087
]
],
"text": "['The crouch gait is commonly found in cerebral palsy patients']",
"type": "background_claim",
"id": "T393"
},
{
"offsets": [
[
46093,
46192
]
],
"text": "['weakness in the plantarflexors is one of many factors thought to contribute to the gait abnormality']",
"type": "background_claim",
"id": "T394"
},
{
"offsets": [
[
46194,
46212
]
],
"text": "['Steele et al. 2010']",
"type": "data",
"id": "T395"
},
{
"offsets": [
[
46269,
46340
]
],
"text": "['causes patients to vault the body forward over the extended stance limb']",
"type": "background_claim",
"id": "T396"
},
{
"offsets": [
[
46215,
46234
]
],
"text": "['Knee hyperextension']",
"type": "background_claim",
"id": "T397"
},
{
"offsets": [
[
46346,
46418
]
],
"text": "['can result from hamstring lengthening surgery in cerebral palsy patients']",
"type": "background_claim",
"id": "T398"
},
{
"offsets": [
[
46244,
46267
]
],
"text": "['common gait abnormality']",
"type": "data",
"id": "T399"
},
{
"offsets": [
[
46421,
46436
]
],
"text": "['Kay et al. 2002']",
"type": "data",
"id": "T400"
},
{
"offsets": [
[
46479,
46640
]
],
"text": "['our optimization indeed results in a mild hyperextension gait after weakening HAM to a quarter of its original strength, with a mini◦ mum knee flexion angle of 2']",
"type": "own_claim",
"id": "T401"
},
{
"offsets": [
[
46446,
46464
]
],
"text": "['accompanying video']",
"type": "data",
"id": "T402"
},
{
"offsets": [
[
46709,
46826
]
],
"text": "['Another cause of knee hyperextension is weakened quadriceps, which can be simulated by weakening the VAS in our model']",
"type": "own_claim",
"id": "T403"
},
{
"offsets": [
[
46842,
46952
]
],
"text": "['weakening the VAS to one-tenth of its original strength leads to a motion similar to quadriceps avoidance gait']",
"type": "own_claim",
"id": "T404"
},
{
"offsets": [
[
46960,
47035
]
],
"text": "['is seen in patients with quadriceps weakness and anterior cruciate ligament']",
"type": "background_claim",
"id": "T405"
},
{
"offsets": [
[
47042,
47052
]
],
"text": "['deficiency']",
"type": "background_claim",
"id": "T406"
},
{
"offsets": [
[
47055,
47074
]
],
"text": "['Timoney et al. 1993']",
"type": "data",
"id": "T407"
},
{
"offsets": [
[
50896,
50990
]
],
"text": "['Our controller architecture and objective function is not limited or specific to walking alone']",
"type": "own_claim",
"id": "T408"
},
{
"offsets": [
[
50992,
51049
]
],
"text": "['By simply changing the target velocity and initialization']",
"type": "own_claim",
"id": "T409"
},
{
"offsets": [
[
51190,
51294
]
],
"text": "['the same procedure yields running controllers, without any modifications to the control parameterization']",
"type": "own_claim",
"id": "T410"
},
{
"offsets": [
[
51309,
51440
]
],
"text": "['previous optimization-based control synthesis methods required including torque ratios specific to walking as part of the objective']",
"type": "background_claim",
"id": "T411"
},
{
"offsets": [
[
51051,
51105
]
],
"text": "['changing the initial velocity from 1.3 m/s to 3.05 m/s']",
"type": "data",
"id": "T412"
},
{
"offsets": [
[
51107,
51164
]
],
"text": "['doubling the initial force feedback gains for GAS and SOL']",
"type": "data",
"id": "T413"
},
{
"offsets": [
[
51170,
51187
]
],
"text": "['bending the elbow']",
"type": "data",
"id": "T414"
},
{
"offsets": [
[
51442,
51458
]
],
"text": "['Wang et al. 2009']",
"type": "data",
"id": "T415"
},
{
"offsets": [
[
51460,
51497
]
],
"text": "['or adding spring elements for running']",
"type": "background_claim",
"id": "T416"
},
{
"offsets": [
[
51499,
51518
]
],
"text": "['Wu and Popović 2010']",
"type": "data",
"id": "T417"
},
{
"offsets": [
[
51521,
51685
]
],
"text": "['Our unified approach to both walking and running is consistent with the view that humans select between walking and running by minimizing energy at different speeds']",
"type": "own_claim",
"id": "T418"
},
{
"offsets": [
[
51688,
51713
]
],
"text": "['Srinivasan and Ruina 2006']",
"type": "data",
"id": "T419"
},
{
"offsets": [
[
51820,
51892
]
],
"text": "['Our running kinematic results do not match human data as well as walking']",
"type": "own_claim",
"id": "T420"
},
{
"offsets": [
[
51901,
51951
]
],
"text": "['the basic features of the curves are still present']",
"type": "own_claim",
"id": "T421"
},
{
"offsets": [
[
51953,
52064
]
],
"text": "['A main discrepancy is that our hip and knee joints both reach maximum extension earlier than human running data']",
"type": "own_claim",
"id": "T422"
},
{
"offsets": [
[
52066,
52167
]
],
"text": "['Similar to our walking results, our knee joint flexes less during the stance phase compared to humans']",
"type": "own_claim",
"id": "T423"
},
{
"offsets": [
[
52169,
52223
]
],
"text": "['Our maximum knee flexion is also lower than human data']",
"type": "own_claim",
"id": "T424"
},
{
"offsets": [
[
52290,
52427
]
],
"text": "['Our knee extension torque reaches maximum earlier than human data, which can cause the knee to extend too quickly during the stance phase']",
"type": "own_claim",
"id": "T425"
},
{
"offsets": [
[
52448,
52508
]
],
"text": "['our plantarflexion torques have a lower peak than human data']",
"type": "own_claim",
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"text": "['de Lasa et al. 2010']",
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"text": "['an exciting area for future work is to automatically synthesize locomotion controllers for more detailed, fully muscle-actuated human models']",
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A37 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A37_C07_Efficient_Simulation_of_Inextensible_Cloth_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1276377.1276438
<Title>Efficient Simulation of Inextensible Cloth</Title>
Rony Goldenthal
David Harmon 1 Raanan Fattal 3 Michel Bercovier 2 Eitan Grinspun 1 1 Columbia University 2 The Hebrew University of Jerusalem 3 University of California
2
Berkeley
<Abstract>Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The resulting algorithm acts as a velocity filter that easily integrates into existing simulation code.</Abstract>
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation I.6.8 [Simulation and Modeling]: Types of Simulation—Animation Keywords: Physically-based Modeling, Cloth simulation, Constrained Lagrangian Mechanics, Constraints, Stretching, Inextensibility, Isometry.
<H1>1 Introduction</H1>
Our eyes are very sensitive to the behavior of fabrics, to the extent that we can identify the kind of fabric simply from its shape and motion
[Griffiths and Kulke 2002]. One important fact is that most fabrics do not stretch under their own weight. Unfortunately, for many popular cloth solvers, a reduction of permissible stretching is synonymous with degradation in performance: for tractable simulation times one may settle for an unrealistic 10% or more strain (compare 1% and 10%, Figure 1 ). Our work alleviates this problem by introducing a numerical solver that excels at timestepping quasi-inextensible surfaces (stretching below 1%). The solver builds on a framework of Constrained Lagrangian Mechanics (CLM)
[Marsden 1999]. Warp and weft, the perpendicular sets of strands that make up a textile, are prohibited from stretching by enforcing constraint equations, not by integrating spring forces. We present numerical evidence supporting the observation that a constraint-based method is inherently well-suited to operate in the quasi-inextensible regime. In contrast, for this regime spring-based methods are known to experience a range of difficulties, leading to the adoption of various strain limiting
[Provot 1995] and strain rate limiting algorithms. We are motivated by the work of Bridson et al. [2002]
, who viewed strain limiting as one of multiple velocity filtering passes (another being collision handling). The velocity filter paradigm enables the design of modular systems with mix-and-match flexibility.
Figure 1: Importance of capturing inextensibility. For efficiency, many simulation methods allow 10% or more strain, whereas many fabrics do not visibly stretch. A 1m 2 patch, pinned at two corners
1m apart, is allowed to relax under gravity. We compare (left to right) three simulations of progressively smaller permissible strain with an actual denim patch.
Contributions We propose a novel CLM formulation that is implicit on the constraint gradient (§4.1). We prove that the implicit method’s nonlinear equations correspond to a minimization problem (§4.2): this result motivates a fast projection method for enforcing inextensibility (§4.3). We describe an implementation of fast projection as a simple and efficient velocity filter, as part of a framework that decouples timestepping, inextensibility, and collision passes (§4.4). Consequently, the fast projection method easily incorporates with a code’s existing bending, damping, and collision models, to yield accelerated performance (§5). Before discussing these contributions, we summarize the relevant literature (§2) and describe the basic discrete cloth model (§3).
<H1>2 Related Work</H1>
For brevity, we review work on stretch resistance; for broad surveys on cloth simulation see
[House and Breen 2000; Choi and Ko 2005]. The most general approach is to treat cloth as an elastic material [Terzopoulos et al. 1987; Breen et al. 1994; Eberhardt et al. 1996; Baraff and Witkin 1998; Choi and Ko 2002]. To reduce visible stretching, elastic models typically adopt large elastic moduli or stiff springs, degrading numerical stability
[Hauth et al. 2003]. To address the stiffness of the resulting differential equations, Baraff and Witkin [1998]
proposed implicit integration, allowing for large, stable timesteps; adaptive timestepping was required to prevent over-stretching. Eberhardt [2000] and Boxerman et al. [2003] adopted implicit-explicit (IMEX) formulations, which treat only a subset of forces implicitly. Our method is closely related to the IMEX approach, in the sense that stretching forces are singled out for special treatment. These works, and many of their sequels, improved performance by allowing some perceptible stretch of the fabric.
In the quasiinextensible regime, however, implicit methods encounter numerical limitations
[Volino and Magnenat-Thalmann 2001; Boxerman 2003; Hauth et al. 2003]: the condition number of the implicit system grows with the elastic material stiffness, forcing iterative solvers to perform many iterations; additionally, timestepping algorithms such as Backward Euler and BDF2 introduce undesirable numerical damping when the system is stiff
[Boxerman 2003]. Given a stiff differential equation, an alternative to implicit integration is to reduce the stiff component and reformulate it as a constraint [Hairer et al. 2002]. In the smooth setting, the penalty-force and constraint-based approaches are equivalent in the limit of an infinitely stiff penalty term
[Bercovier and Pat 1984]. In the discrete setting, the constraint-based approach may be implemented with various iterative or global algorithms, as surveyed below:
Iterative enforcement Provot [1995] corrected edge lengths by iteratively displacing the incident vertices on stretched springs. While simple to implement, this approach suffers from poor convergence since each displacement may stretch other incident springs. Therefore, Provot’s method is used in cases where tight tolerances are not required, e.g.,
[Desbrun et al. 1999; Meyer et al. 2001; Fuhrmann et al. 2003]. Bridson et al. [2002; 2003] used Provot’s approach in conjunction with strain rate limiting, bounding the rate of change of spring length per timestep to 10% of the current length. Müller et al. [2006] used a non-linear Gauss-Seidel approach to enforce inextensibility on each constraint separately. Bridson et al. observed that iterative strain limiting algorithms behave essentially as Jacobi or Gauss-Seidel solvers. In this light, it is not surprising that for finely-discretized quasi-inextensible fabrics, iterative constraint enforcement requires a prohibitive number of iterations (see §5). Global enforcement In contrast to iterative constraint enforcement,
House et al. [1996] used Lagrange multipliers with CLM to treat stretching, and presented a hierarchical treatment of the constraint forces. The Lagrange multiplier approach alleviates the difficulties associated with poor numerical conditioning and artificial damping. House et al. later encountered difficulties in handling collision response within the proposed framework [2000]. By building on the velocity-filter paradigm, our method handles both inextensibility and complex collisions. House et al. formulated constraints as in [Witkin et al. 1990], which is subject to numerical drift that may be exacerbated by the discontinuities introduced during collision response. Drift may be attenuated using constraint-restoring springs, but the authors reported difficulty in adjusting the spring coefficients. We postulate that one reason for their difficulties with drift was consequent to the linearization of the constraint equation, which permitted higher order errors to accumulate over time. Our method does not linearize the constraint equations, and therefore it is not subject to drift. Recently, Tsiknis [2006]
proposed triangle-based strain limiting together with a global stitching step for stable constraint enforcement. Hong et al. [2005]
used a linearized implicit formulation in order to improve stability of constrained dynamics. This allowed for larger timesteps and reduced the need for springs to maintain the cloth on the constraint manifold. Both of these approaches enforce inextensibility only for strain exceeding 10%. In summary, when the tolerance for stretching is very small, modeling stretch response with spring-based or strain-limiting approaches is costly and even intractable; constraint-based methods present a promising alternative. The remainder of this paper discusses algorithms that excel at simulating quasi-inextensible cloth.
<H1>3 Cloth Model</H1>
Woven fabrics are not a continuous material, rather they are a complex mechanical network of interleaving yarn [Breen et al. 1994]. Since the constituent yarn is often quasi-inextensible, the material’s warp and weft directions do not stretch perceptibly. In imposing inextensibility on all edges of a triangle mesh, one quickly runs into parasitic stiffness in the bending modes, or locking
[Zienkiewicz and Taylor 1989], since locally-convex regions of a triangle mesh are rigid under isometry. Instead, we consider warpweft aligned quadrilateral meshes with a sparse number of triangles (quad-dominant meshes). A degree of freedom (DOF) counting argument suggests that constraining all edges of a quad mesh may circumvent the rigidification that occurs with triangle meshes: Given n vertices, we have 3n positional DOFs; their triangulation (resp. quadrangulation) introduces approximately 3n (resp. 2n) edges, with corresponding inextensibility constraints. Subtracting constraints from positional DOFs leaves nearly zero DOFs for a triangulation. In the case of a quadrangulation, O(n) DOFs remain, and we see that in a flat configuration they correspond to the normal direction at each vertex. Furthermore, under general mesh positions, the constraints are linearly independent, with a full-rank Jacobian treatable by a direct solver (§4). We ask that a warpor weft-aligned quad edge, (p a , p b ), maintain its undeformed length, l, by enforcing
1
C(p a , p b ) = p b − p a 2 /l − l = 0 .
The solve will require the constraint gradient
2
∇ p b C(p a , p b ) = 2(p b − p a )/l .
Since shearing modes excite only a mechanical interaction of warp and weft, and not a stretching of yarn, fabric does indeed shear perceptibly. Therefore, we model shear using non-stiff stretch springs applied on both diagonals of each quad. The complete model of in-plane deformation is compatible with an existing code’s quador triangle-based treatment of bending and collisions. With this simple formulation of inextensibility constraints in place, what is needed is an efficient method for enforcing constraints. In the following, we develop such a method.
<H1>4 Constrained Dynamics</H1>
Given a quadrilateral mesh with n vertices and m edges, the numerical integration algorithm for constrained dynamics can be developed directly from the augmented Lagrange equation
[Marsden 1999], L(x, v) = 1 v T Mv −V (x) − C(x) T λ , 2 where x(t) is the time-varying 3n-vector of vertex positions, v(t) = x(t) is its time derivative, M is the 3n × 3n mass matrix, and V (x) is the stored energy (e.g., bending, shear, and gravity). C(x) is the m-vector of constraints, with the i th entry corresponding to the violation of inextensibility of the i th edge, as computed by (1); λ is the m-vector of Lagrange multipliers. The corresponding EulerLagrange equations are
4
M v = −∇V (x) − ∇C(x) T λ , C(x) = 0 ,
where ∇ ≡ ∇ x is the gradient with respect to position, and −∇V (x) is the potential force. The term −∇C(x) T λ may be viewed as the constraint-maintaining force, where the factors −∇C(x) T and λ determine the direction and scaling for the force, respectively. ∇C(x) is a rectangular matrix whose dimensions are m × 3n. For simulation, we must discretize (3) and (4) in time using one of various schemes, each with benefits and drawbacks. One may choose differing explicit or implicit schemes for the potential and the constraint forces (similarly, potential forces are split and separately discretized in [Ascher et al. 1997]). The discrete equations replace x(t) and v(t) with {x 0 , x 1 , x 2 , . . .} and {v 0 , v 1 , v 2 , . . .}, where x n and v n are the position and velocity of the mesh at time t = nh, and h is the size of the timestep. One widely-used family of discretizations includes SHAKE and RATTLE, which extend the (unconstrained) Verlet scheme
[Hairer et al. 2002] by considering a constraint force direction, −∇C(x) T , evaluated at the beginning of the timestep. Unfortunately, enforcing length-preserving constraints with SHAKE fails for four common geometric configurations, which we refer to as (Q1)–(Q4) and depict in Figure 2 . This figure is a reproduction from [Barth et al. 1994]
, which discusses these drawbacks in SHAKE but does not offer a solution. In the figure, solid and hollow dots represent edge endpoints at the start and end of the timestep, as the particles would evolve if no constraints were applied. If the constraint direction, −∇C(x) T , is evaluated at the beginning of the timestep, x n , as in SHAKE, then no scaling, λ , of the constraint direction yields a satisfied end-of-timestep constraint, C(x n+1 ) = 0. Numerically, for (Q2)–(Q4) this observation manifests as a singular Jacobian in Newton’s method. These four cases correspond to rapid change in edge length or orientation; in practice, they occur often.
(Q1) (Q2) (Q3) (Q4)
Figure 2: Failure modes of methods using an explicit constraint direction. Reproduced from a discussion of SHAKE in [Barth et al. 1994].
<H2>4.1 Implicit constraint direction (ICD)</H2>
Consider evaluating the constraint direction, −∇C(x) T , at the end of the timestep. We observe (and prove in Appendix A) that this resolves (Q1), (Q2) and (Q4); (Q3) remains, but is automatically remedied by decreasing the timestep. Consider the ICD timestep, which treats potential forces explicitly 1 : v n+1 = v n − hM −1 ∇V (x n ) + ∇C(x n+1 ) T λ n+1 , x n+1 = x n + hv n+1 , C(x n+1 ) = 0 . Define x 0 n+1 =x n +hv n −h 2 M −1 ∇V (x n ), i.e., x n+1 0 is the position at the end of an unconstrained timestep; define δ x n+1 = x n+1 − x 0 n+1 , i.e., δ x n+1 is the correction of the unconstrained step. Next, eliminate v n+1 by rewriting the above system as two equations, F( δ x n+1 , λ n+1 ) = 0 and C(x n+1 ) = 0, in the free variables δ x n+1 and λ n+1 , keeping in mind that x n+1 is a linear function in δ x n+1 , and defining F( δ x n+1 , λ n+1 ) = δ x n+1 + h 2 M −1 ∇C(x n+1 ) T λ n+1 . F( δ x n+1 , λ n+1 ) and C(x n+1 ) are the residuals of the discretization of (3) and (4), respectively. In particular, F measures the deviation 1 For an implicit treatment, write ∇V (x n+1 ) in place of ∇V (x n ). of the trajectory away from that dictated by the governing (potential and constraint) forces; equivalently, it states that the correction of the unconstrained step is due to the constraint forces. C measures the deviation from the constraint manifold (in our case, the extensibility of the material). To implement ICD, we solve for the roots of F and C up to a desired tolerance using Newton’s method. Solving for an ICD step is costly, because there are many unknowns (≈ 5n), and each Newton step requires the solution of an indefinite linear system, whose matrix is costly to assemble. In §4.3, we develop an approximation to ICD that addresses these drawbacks without sacrificing constraint accuracy or robustness. To arrive at this fast projection method, the following section considers ICD from an alternative, geometric viewpoint.
<H2>4.2 Step and project (SAP)</H2>
Consider for a moment an alternative approach to constrained integration in two steps: (a) step forward only the potential forces to arrive at the unconstrained position, x n+1 0 ; (b) enforce the constraints by projecting onto the constraint manifold M = {x n+1 |C(x n+1 ) = 0}. Methods of this form are known as manifold-projection methods
[Hairer et al. 2002]. To define a specific method, we must choose a projection operator. In the method we refer to as SAP, we write the projection of the unconstrained point onto the constraint manifold as x n+1 0 + δ x n+1 , so that the projected point extremizes the objective function W ( δ x n+1 , λ n+1 ) = 1 ( δ x n+1 ) T M( δ x n+1 ) + C(x n+1 ) T λ n+1 , 2h 2 with respect to the free variables δ x n+1 and λ n+1 . Simply put, we choose the point on the constraint manifold closest to x n+1 0 . To define closest, we need a measure of distance. Take M as the physical mass matrix (usually arising from a finite-basis representation of x and a surface mass density). Then the choice ( δ x n+1 ) T M( δ x n+1 ) corresponds to the L 2 norm of the mass-weighted displacement of the mesh as it moves from x 0 n+1 to x n+1 . Formally, it is a discretization of the smooth integral x n+1 − x 0 n+1 2 ρ d A , S evaluated over the reference (material) domain, S. Here x n+1 and x 0 n+1 are the piecewise linear immersion functions mapping each point of S into R 3 , and ρ is the (possibly nonuniform) surface mass density. We use · to denote the Euclidean norm in R 3 .
Theorem 1: ICD ≡ SAP . Proof: The stationary equations for W ( δ x n+1 , λ n+1 ) are the ICD equations, F( δ x n+1 , λ n+1 ) = 0 and C(x n+1 ) = 0.
Corollary In 4.1, we interpreted the roots of C and F from the ICD view. We can interpret these roots from the SAP view as follows: C(x n+1 ) = 0 corresponds to finding some point on the constraint manifold. C(x n+1 ) = 0 with F( δ x n+1 , λ n+1 ) = 0 corresponds to finding the closest point on the constraint manifold.
<H2>4.3 Fast projection method</H2>
To solve SAP, one might extremize W ( δ x n+1 , λ n+1 ) using Newton’s method: each iteration would improve upon a guess for the shortest step, δ x n+1 that projects x n+1 0 onto the constraint manifold. Algorithm 1 Fast projection is a velocity filter that enforces constraints. It combines the robustness of using an implicit constraint direction with the efficiency of approximate manifold projection. Input: v // candidate velocity Input: x // known start-of-step position 1: j ← 0 2: x 0 ← x + h v // unconstrained timestep 3: while strain of x j exceeds threshold do 4: Solve linear system (7) for δ λ j+1 5: Evaluate (5) to obtain δ x j+1 6: x j+1 ← x j + δ x j+1 7: j ← j + 1 8: end while Output: 1 h (x j − x) // constraint-enforcing velocity Fast projection also uses a sequence of iterations, but it relaxes the requirement of SAP: starting with the unconstrained position, x n+1 0 , we propose to find a close, but not necessarily closest, point on the constraint manifold, by taking a sequence of “smallest” steps. Fast projection starts at x n+1 0 , and takes a sequence of steps, δ x n+1 j , j = 1, 2, . . ., toward the constraint manifold, with each step as short as possible. A step of fast projection Projection onto the constraint manifold occurs at a fixed instant in time. Therefore, we omit the superscripts (n + 1), which refer to time, in order to emphasize the subscripts, j, which refer to a specific iteration of fast projection, e.g., we write the input position, x n+1 0 , as x 0 , and progressively closer approximations to the constrained position as x 1 , x 2 , . . .. Formally, the ( j + 1) th step of fast projection, x j+1 = x j + δ x j+1 , extremizes the objective function
W ( δ x j+1 , δ λ j+1 ) = 2h 1 2 ( δ x j+1 ) T M( δ x j+1 ) + C(x j+1 ) T δ λ j+1 ,
with respect to the step increment, δ x j+1 , and the auxiliary variable δ λ j+1 . Expanding the constraint to first order, we obtain a quadratic objective function, whose stationary equations with respect to δ x j+1 and δ λ j+1 are
C(x j+1 ) = C(x j + δ x j+1 ) ≈ C(x j ) + ∇C(x j ) δ x j+1 ,
5
δ x j+1 = −h 2 M −1 ∇C(x j ) T δ λ j+1 ,
6
∇C(x j ) δ x j+1 = −C(x j ) .
Substituting (5) into (6), we eliminate δ x j+1 and solve a linear system in δ λ j+1 :
7
h 2 ∇C(x j )M −1 ∇C(x j ) T δ λ j+1 = C(x j ) .
Since the linear system matrix involves M −1 , the assembly of this system is most efficient for diagonal (e.g., lumped) mass matrices. Finally, we compute the increment (5) to obtain x j+1 = x j + δ x j+1 . As with ICD/SAP, a fast projection step requires a linear solve. However, fast projection’s system, (7), is smaller (≈ 2n × 2n compared to ≈ 5n × 5n), positive definite (compared to indefinite) and sparser. As a result it is considerably cheaper to evaluate, assemble, and solve than its ICD/SAP counterpart. Fast projection algorithm We repeatedly take fast projection steps until the maximal strain is below a threshold, i.e., the constraint may be satisfied up to a given tolerance. This process is summarized in Algorithm 1. Fast projection finds a manifold point, x n+1 , that is close, but not closest, to the unconstrained point, x 0 n+1 . Referring to the Corollary, we conclude that fast projection exactly solves C = 0 while it approximates F = 0. One important question is whether the fast projection’s error in F is acceptable. Compare a sequence of fast projection iterations to ICD/SAP’s sequence of Newton iterations. The first iteration of these methods is identical. At the end of this first iteration, F, C ∈ O(h 2 ). Additional fast projection iterations seek C → 0, and since C ∈ O(h 2 ), increments in x are O(h 2 ), therefore F remains in O(h 2 ). Observe that F ∈ O(h 2 ) is considered acceptable in many contexts, e.g., [Baraff and Witkin 1998; Choi and Ko 2002] halt the Newton process after a single iteration. To verify this claim, we measured F throughout the ballet dancer sequence. As recorded in Figure 3 , the first iteration of the fast projection method eliminates first-order error. The remaining iterations perturb F only to higher-order (often decreasing the error further).
-4 3.5 x 10 error 2.5 after after last first iteration iteration F vertex 1.5 per 0.5 0.0 0 0.2 0.4 0.6 0.8 1 simulation time
Figure 3: Effect of fast projection on the residual. Using the ballet dancer sequence, at each timestep (horizontal axis) we measured the residual, F (vertical axis), after the first and last iterations of fast projection (dashed-red and solid-blue curves, respectively).
2 10 40 Fast Projection Implicit Spring (seconds) (seconds) 30 1 10 20 time time 10 0 10 0 10 1 10 0 10 -1 20 40 60 80 allowed strain (%) number of vertices (a) (b)
Figure 4: Performance of fast projection vs. implicit springs. For a 1D chain simulated in MATLAB, we plot the computation time of one simulated second, as a function (a) of permissible strain (loglog plot for 80 vertices), and (b) of discretization resolution (linear plot for 1% permissible strain).
<H2>4.4 Implementation</H2>
We implement fast projection as a velocity filter, enabling easy integration into our existing cloth simulation system; refer to Algorithm 1. Step 3 requires solving a sparse symmetric positive definite linear system; we use the PARDISO
[Schenk and Gärtner 2006] solver. Each row of ∇C(x n+1 j ) corresponds to one edge, and is computed using (2). The right-hand side, C(x n+1 j ), is given by (1).
10 4 x10 2 15 Fast-Projection ICD Shake (seconds) 10 3 (seconds) 10 SL-Jacobi SL-Gauss-Seidel 2 time 10 time 5 1 10 0 10 1 10 0 10 -1 0 5000 10000 allowed strain (%) number of vertices (a) (b)
Figure 5: Performance of several constraint-enforcing methods. For a 2D cloth, simulated in C++, we plot the computation time of one simulated second, as a function (a) of permissible strain (log-log plot for 5041 vertices), and (b) of discretization resolution (linear plot for 1% permissible strain).
(a) (b)
Figure 6:
Qualitative visual comparison. Snapshot of a cloth draped using (a) fast projection and (b) implicit constraint direction.
<H1>5 Results</H1>
We describe several experiments comparing various stretchenforcement methods. All timings are with reference to a single process on a 2.66GHz Intel Core 2 Duo. One-dimensional chain Our first experiment compares the performance of fast projection against an implicit treatment of stiff springs. We observe the scaling of computational cost as a function of (a) permissible strain and (b) mesh resolution. The physical setup consists of a chain pinned at the top node and released to free fall under gravity. The simple 1D chain resists stretching, but not bending. In this didactic example, timings refer to MATLAB’s (sparse) direct solver. Our method shows asymptotically better performance as permissible strain vanishes (see Figure 4a ). Likewise, our algorithm exhibits favorable performance as mesh resolution increases (see Figure 4b ). Using 80 vertices and 1% strain, the fast projection method achieves a 25× speedup. Note that there exists considerable difficulty in setting spring coefficients a priori to satisfy a given strain limit. For settings more pragmatic than a simple chain, such as the following draping experiment, we are unable (despite considerable effort) to set spring coefficients that achieve a prescribed small strain. This explains why spring methods are often treated with strain-limiting procedures. Draping cloth The next experiment compares fast projection, ICD, SHAKE, and the strain limiting approach. We evaluate how the spatial discretization and permissible strain affect performance of these four algorithms. The setup consists of draping a cloth over a polygonal model of a sphere. We measure strain before the collision reaction pass. For the strain limiting algorithms (both Jacobi and Gauss-Siedel), we iterate until strain is in the permissible range. With GaussSiedel, we apply a random permutation to reduce bias resulting from the particular edge ordering. For SHAKE, we use the acceleration suggested in
[Barth et al. 1994] to rebuild the matrix once per step or when it fails to converge. As a consequence, the algorithm requires extremely small timesteps to converge, but each timestep is relatively inexpensive, as matrix re-assembly and re-factoring is infrequent. ICD is able to use larger timesteps than SHAKE and still converge, however, since each timestep is substantially more expensive than a SHAKE step, the overall time is higher. Figure 5a shows a timing comparison of these methods, and Figure 5b compares performance as the stiffness is increased for a cloth mesh with approximately 5000 vertices. All CLM methods scale equally well, asymptotically better than the strain limiting approach, with the fast projection being the fastest. As we refine the resolution, and allow strain of 1% ( Figure 5b ), the fast projection method outperforms the other methods. Figure 6 shows the same frame from simulations that use the fast projection and ICD methods, with qualitatively similar results. Figures 7 and 8 show still frames from more complex simulations demonstrating that fast projection is capable of producing complex, realistic simulations of cloth.
Figure 7: Inextensibility and dynamics. Inextensibility ensures that the tight-fitting pants do not drop past the dancer’s narrow waist. Using fast projection, an implicit treatment of shear and bending, and a mesh with 10600 vertices, the average simulation time per (30Hz) frame was 9 seconds.
<H1>6 Discussion</H1>
Our experiments focus on measuring the performance of enforcing inextensibility using CLM compared to strain limiting and stiff springs. In addition to the direct benefit of fast projection on computation times, further benefits can be reaped from the resulting inextensibility. For example, the work of
Bergou et al. [2006] assumes inextensibility in order to accelerate bending computation. In adopting the velocity-filtering viewpoint, we gain speed, simplicity, and software modularity—all key to a practical and maintainable implementation. However, this comes at a theoretical cost: there is no longer an efficient way to perfectly enforce both ideal inextensibility and ideal collision handling, since one filter must execute before the other, and both ideals correspond to sharp constraints. To enforce both perfectly would require combining them in a single pass, an elegant and exciting prospect from the standpoint of theory, but one which is likely to introduce considerable complexity and convergence challenges. Practically, we observe that this drawback does not cause artifacts in our simulation, for several reasons: first, we execute collision-handling last, to avoid glaring collision artifacts, yet we assert that empirically our strain remains negligible, as required. Second, unlike constraint-enforcement approaches such as
[Witkin et al. 1990], the inextensibility filter does not assume that the constraint is maintained at the beginning of the timestep and errors are not accumulated during the simulation. Conclusion Despite the fact that the most common fabrics do not visibly stretch when draped over the body, the trend in our community is to favor stretching formulations based on penalty-springs. The consequent numerical difficulties are then addressed by a combination of (a) relaxing realism by allowing 10% strain, and (b) adopting simple iterative strain and strain-rate algorithms that have poor convergence behavior. With Constrained Lagrangian Mechanics as our alternative point of departure, we demonstrate a straightforward filter, with good convergence behavior, for enforcing inextensibility. We provide one immediate and pragmatic approach to fast and realistic fabric simulation using CLM, and we hope that it will spur a renaissance of activity along this direction. Acknowledgments We are grateful for the valuable feedback from our reviewers, and in particular for the keen eyes and diligent guidance of the primary reviewer. We thank OptiTex for providing the 3D garment geometry as well as the animated figurines. We are grateful to David Ismailov for setting up the cloth models, and Ruzz Oved and Yaniv Gorali for lighting and shading our scenes. Our work benefited from the valuable insights of Jerrold E. Marsden, Ari Stern, and Max Wardetzky, and from the generous support of the NSF (MSPA 0528402, CSR 0614770, CAREER 0643268), Autodesk, mental images, NVIDIA, and Elsevier.
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Figure 8: Enforcing inextensibility using fast projection yields lively motion with detailed wrinkles and folds. Frames from ballet and runway sequences simulated using fast projection. The elastic term was integrated implicitly (top) and explicitly (bottom), respectively. The cloth contains 8325 (top) and 10688 (bottom) vertices, with average simulation time per (30Hz) frame of 5.2 and 7.8 seconds, respectively.
Appendix A
We briefly explain why ICD and fast projection (FP) are not troubled by configurations (Q1), (Q2), and (Q4), and are resilient to (Q3). Facts about the behavior of SHAKE are taken from [Barth et al. 1994]. Q1 SHAKE’s force ∇C(x n ) T λ n+1 cannot reduce the single edge’s length back to l; our force ∇C(x n+1 j ) T λ n+1 can reduce that edge’s length back to l. Q2 ∇C(x n+1 j ) and ∇C(x n ) T are both full-rank, yet SHAKE fails since ∇C(x n+1 j )M −1 ∇C(x n ) T is singular; FP uses ∇C(x n+1 j )M −1 ∇C(x n+1 j ) T , and ICD uses ∇C(x n+1 j )D∇C(x n+1 j ) T , where D is a symmetric full-rank matrix; in both cases this product is not singular. Q3 ICD and FP may fail if ∇C(x n+1 j ) is rank-deficient; for sufficiently small timestep, h, this case is always avoidable. Q4 ∇C(x n ) is rank-deficient, so SHAKE fails; ICD and FP do not use ∇C(x n ).
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"text": "['In the smooth setting, the penalty-force and constraint-based approaches are equivalent in the limit of an infinitely stiff penalty term']",
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"text": "['In the discrete setting, the constraint-based approach may be implemented with various iterative or global algorithms']",
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"text": "['Provot']",
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"text": "['each displacement may stretch other incident springs']",
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"text": "['Desbrun et al. 1999']",
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6620,
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"text": "['Meyer et al. 2001']",
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"text": "['Fuhrmann et al. 2003']",
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6688
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"text": "['Müller et al']",
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6991,
7080
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"text": "['iterative strain limiting algorithms behave essentially as Jacobi or Gauss-Seidel solvers']",
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6962,
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7127,
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"text": "['finely-discretized quasi-inextensible fabrics']",
"type": "data",
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7256,
7258
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"text": "['§5']",
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7350,
7404
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"text": "['used Lagrange multipliers with CLM to treat stretching']",
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7410,
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7330,
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"text": "['House et al']",
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7344,
7348
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"text": "['1996']",
"type": "data",
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"text": "['The Lagrange multiplier approach alleviates the difficulties associated with poor numerical conditioning and artificial damping']",
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"text": "['later encountered difficulties in handling collision response within the proposed framework']",
"type": "background_claim",
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7600,
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"text": "['House et al.']",
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7706,
7710
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"text": "['2000']",
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"text": "['By building on the velocity-filter paradigm, our method handles both inextensibility and complex collisions']",
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7886,
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"text": "['which is subject to numerical drift that may be exacerbated by the discontinuities introduced during collision response']",
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"text": "['formulated constraints as in']",
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"text": "['Witkin et al. 1990']",
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7822,
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"text": "['House et al']",
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"text": "['Drift may be attenuated using constraint-restoring springs']",
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8071,
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"text": "['the authors reported difficulty in adjusting the spring coefficients']",
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8159,
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"text": "['one reason for their difficulties with drift was consequent to the linearization of the constraint equation, which permitted higher order errors to accumulate over time']",
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8329,
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"text": "['Our method does not linearize the constraint equations']",
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8399,
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"text": "['it is not subject to drift']",
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8453,
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"text": "['proposed triangle-based strain limiting together with a global stitching step for stable constraint enforcement']",
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8437,
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"text": "['Tsiknis']",
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"text": "['2006']",
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8566,
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"text": "['Hong et al']",
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8579,
8583
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"text": "['2005']",
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8795
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"text": "['This allowed for larger timesteps and reduced the need for springs to maintain the cloth on the constraint manifold']",
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"text": "['Both of these approaches enforce inextensibility only for strain exceeding 10%']",
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"text": "['the tolerance for stretching is very small']",
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8938,
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"text": "['modeling stretch response with spring-based or strain-limiting approaches is costly and even intractable']",
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"text": "['constraint-based methods present a promising alternative']",
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"text": "['the material’s warp and weft directions do not stretch perceptibly']",
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"text": "['In imposing inextensibility on all edges of a triangle mesh, one quickly runs into parasitic stiffness in the bending modes, or locking']",
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"text": "['Zienkiewicz and Taylor 1989']",
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"text": "['constraining all edges of a quad mesh may circumvent the rigidification that occurs with triangle meshes']",
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10932,
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"text": "['shearing modes excite only a mechanical interaction of warp and weft, and not a stretching of yarn']",
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"text": "['The complete model of in-plane deformation is compatible with an existing code’s quador triangle-based treatment of bending and collisions']",
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"text": "['With this simple formulation of inextensibility constraints in place, what is needed is an efficient method for enforcing constraints']",
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"text": "['the numerical integration algorithm for constrained dynamics can be developed directly from the augmented Lagrange equation']",
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11555,
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"text": "['a quadrilateral mesh with n vertices and m edges']",
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11731,
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"text": "['we must discretize (3) and (4) in time using one of various schemes, each with benefits and drawbacks']",
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"text": "['One may choose differing explicit or implicit schemes for the potential and the constraint forces']",
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12911,
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"text": "['Ascher et al. 1997']",
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13151,
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"text": "['One widely-used family of discretizations includes SHAKE and RATTLE, which extend the']",
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13253,
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"text": "['Verlet scheme']",
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13289,
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"text": "['by considering a constraint force direction, −∇C(x) T , evaluated at the beginning of the timestep']",
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"text": "['Hairer et al. 2002']",
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13404,
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"text": "['enforcing length-preserving constraints with SHAKE fails for four common geometric configurations']",
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13548,
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"text": "['Figure 2']",
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13595,
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"text": "['Barth et al. 1994']",
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"text": "['These four cases correspond to rapid change in edge length or orientation']",
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"text": "['resolves (Q1), (Q2) and (Q4)']",
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"text": "['Appendix A']",
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"text": "['(Q3) remains, but is automatically remedied by decreasing the timestep']",
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"text": "['there are many unknowns']",
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"text": "['we develop an approximation to ICD that addresses these drawbacks without sacrificing constraint accuracy or robustness']",
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"text": "['§4.3']",
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"text": "['Fast projection also uses a sequence of iterations, but it relaxes the requirement of SAP']",
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"text": "['Projection onto the constraint manifold occurs at a fixed instant in time']",
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"text": "['the assembly of this system is most efficient for diagonal']",
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"text": "['mass matrices']",
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21194,
21200
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"text": "['lumped']",
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"text": "['the linear system matrix involves M −1']",
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21289,
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21363,
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"text": "['fast projection’s system']",
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21394,
21404
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"text": "['is smaller']",
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21457
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"text": "['positive definite']",
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21483,
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"text": "['and sparser']",
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21391
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"text": "['7']",
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"text": "['≈ 2n × 2n compared to ≈ 5n × 5n']",
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21508,
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"text": "['it is considerably cheaper to evaluate, assemble, and solve than its ICD/SAP counterpart']",
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21981,
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"text": "['fast projection exactly solves C = 0 while it approximates F = 0']",
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22047,
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"text": "['One important question is whether the fast projection’s error in F is acceptable']",
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22472,
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"text": "['F ∈ O(h 2 ) is considered acceptable in many contexts']",
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22576,
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"text": "['halt the Newton process after a single iteration']",
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22534,
22556
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"text": "['Baraff and Witkin 1998']",
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"text": "['Algorithm 1']",
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"text": "['settings more pragmatic than a simple chain']",
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"text": "['the following draping experiment']",
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"text": "['This explains why spring methods are often treated with strain-limiting procedures']",
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"text": "['the algorithm requires extremely small timesteps to converge']",
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"text": "['each timestep is relatively inexpensive']",
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"text": "['matrix re-assembly and re-factoring is infrequent']",
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"text": "['ICD is able to use larger timesteps than SHAKE and still converge']",
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"text": "['the overall time is higher']",
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"text": "['each timestep is substantially more expensive than a SHAKE step']",
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"text": "['All CLM methods scale equally well, asymptotically better than the strain limiting approach, with the fast projection being the fastest']",
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"text": "['we refine the resolution']",
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"text": "['8']",
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"text": "['Figures 7']",
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"text": "['In addition to the direct benefit of fast projection on computation times, further benefits can be reaped from the resulting inextensibility']",
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"text": "['the work of']",
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"text": "['Bergou et al.']",
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"text": "['In adopting the velocity-filtering viewpoint, we gain speed, simplicity, and software modularity—all key to a practical and maintainable implementation']",
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"text": "['this comes at a theoretical cost']",
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"text": "['there is no longer an efficient way to perfectly enforce both ideal inextensibility and ideal collision handling']",
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"text": "['one filter must execute before the other']",
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"text": "['both ideals correspond to sharp constraints']",
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"text": "['To enforce both perfectly would require combining them in a single pass']",
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"text": "['an elegant and exciting prospect from the standpoint of theory']",
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"text": "['one which is likely to introduce considerable complexity and convergence challenges']",
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"text": "['Practically, we observe that this drawback does not cause artifacts in our simulation, for several reasons']",
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"text": "['we assert that empirically our strain remains negligible']",
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"text": "['the inextensibility filter does not assume that the constraint is maintained at the beginning of the timestep and errors are not accumulated during the simulation']",
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"text": "['unlike constraint-enforcement approaches']",
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"text": "['Witkin et al. 1990']",
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"text": "['Despite the fact that the most common fabrics do not visibly stretch when draped over the body, the trend in our community is to favor stretching formulations based on penalty-springs']",
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"text": "['With Constrained Lagrangian Mechanics as our alternative point of departure, we demonstrate a straightforward filter, with good convergence behavior, for enforcing inextensibility']",
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<Title>Interaction of Fluids with Deformable Solids</Title>
Matthias M uller ̈ Simon Schirm Matthias Teschner Bruno Heidelberger Markus Gross
ETH Zürich, Switzerland
<Abstract>In this paper, we present a method for simulating the interaction of fluids with deformable solids. The method is designed for the use in interactive systems such as virtual surgery simulators where the real-time interplay of liquids and surrounding tissue is important. In computer graphics, a variety of techniques have been proposed to model liquids and deformable objects at interactive rates. As important as the plausible animation of these substances is the fast and stable modeling of their interaction. The method we describe in this paper models the exchange of momentum between Lagrangian particle-based fluid models and solids represented by polygonal meshes. To model the solid-fluid interaction we use virtual boundary particles. They are placed on the surface of the solid objects according to Gaussian quadrature rules allowing the computation of smooth interaction potentials that yield stable simulations. We demonstrate our approach in an interactive simulation environment for fluids and deformable solids.</Abstract>
Keywords: smoothed particle hydrodynamics (SPH), finite element method (FEM), fluid-solid interaction
<H1>1 Introduction</H1>
Interactive physically based simulation is a rapidly growing research area with an increasing number of applications, e. g. in games and computational surgery. In these simulation environments, deformable objects play an important role. For the simulation of deformable solids, a variety of models have been proposed ranging from efficient mass-spring approaches to methods based on the physically more accurate Finite Element Method (FEM). Some of these methods allow the simulation of elastically and plastically deformable solids at interactive speed. More recently, there has been an increased interest in efficient methods for the realistic simulation of fluids. These approaches can be employed to represent blood or other liquids. Besides deformable models, they play an essential role in applications such as surgery simulation. So far, only a few interactive methods for the simulation of fluids with free surfaces have been proposed. With the ability to simulate both, deformable solids and fluids, a new problem has been introduced, namely the mod- eling of the interaction of these structures. An interaction model suitable for the use in interactive environments needs to be computationally efficient and the generated interaction forces must not induce any instabilities to the dynamic simulation. In this paper, we present a new technique to model interactions between particle based fluids and mesh based deformable solids which meets these constraints. We present our interaction model with fluids represented by a Smoothed Particle Hydrodynamics approach (SPH) and with deformable solids represented by a Finite Element approach. However, the general interaction model we propose works with any type of deformation technique as long as the object surface is represented by a polygonal mesh and the fluid by Lagrangian particles.
<H1>2 Related Work</H1>
The majority of publications in the area of physically based animation focuses on physical systems of one single type. Deformable objects are interesting to study in their own right. In fluid simulation, on the other hand, boundary conditions are often considered a necessary but not a central issue. They are typically derived from simple geometric primitives. Our method connects these two areas of research. In the field of computer graphics, a large number of mesh based methods for the physically based simulation of deformable objects have been proposed since the pioneering paper of Terzopoulos
[ 1 ]. Early techniques were mostly based on mass-spring systems, which are still popular for cloth simulation
[ 2 , 3 ]. More recent methods discretize continuous elasticity equations via the Boundary Element Method (BEM) [ 4 ] or the Finite Element Method (FEM) [ 5 , 6 , 7 ]. Since T. Reeves
[ 8 ]
introduced particle systems as a technique for modeling fuzzy objects twenty years ago, a variety of special purpose, partice based fluid simulation techniques have been developed in the field of computer graphics.
Desbrun and Cani
[ 9 ] where among the first to use Smoothed Particle Hydrodynamics (SPH)
[ 10 ]
to derive interaction forces for particle systems. They added space-adaptivity in
[ 11 ]. Later, Stora et al.
[ 12 ]
used a similar particle based model to animate lava flows. In [ 13 ]
, Müller et al. derived inter particle forces from SPH and the Navier Stokes equation to simulate water with free surfaces at interactive rates. Recently, Premoze et al.
[ 14 ]
introduced the Moving-Particle SemiImplicit (MPS) method to computer graphics for the simulation of fluids. As a mesh-free method, it is closely related to SPH but in contrast to standard SPH, it allows the simulation of incompressible fluids. In all these papers, boundary conditions are not treated explicitly. The fluids typically interact with solid walls or the ground. Genevaux et al.
[ 15 ]
address the interaction problem explicitly. They propose a method to simulate the interaction between solids represented by mass-spring networks and an Eulerian fluid grid by applying spring forces to the mass-less marker particles in the fluid and the nodes of the mass-spring network. However, solids are typically represented by coarse meshes, especially in interactive simulations. Thus, the nodes of a mass-spring network are not very well suited for the application of interaction forces. Therefore, Monaghan, one of the founders of the SPH formalism, uses special boundary or ghost particles on fixed borders to model interactions
[ 16 ]. The idea of ghost particles was picked up in several following projects including our own. The key contribution of our paper is to place these ghost particles onto boundary triangles of deformable objects and to derive their locations and weights according to Gauss integration [ 17 ]
, which allows to model fluid-solid interactions stably at interactive rates.
1
Figure 1: A box falls into a pool and generates a shock wave that causes the pool to fracture.
<H1>3 Physical Problem Description</H1>
In physically based animation, we are interested in the simulation of macroscopic effects at interactive speed. Therefore, we consider macroscopic models for both, solids and fluids. Materials, which are homogeneous at the macroscopic level, can mathematically be described as a continuum
[ 18 ]. Thereby, quantities such as the density ρ, viscosity μ, deformation u or velocity v are all mathematically expressed by continuous functions over space and time. A physical model relates these quantities to each other via partial differential equations (PDEs). The mechanical behavior of an elastic solid can be described by the following equation
1
ρ ∂t ∂ 2 2 u = ∇ · σ s (u) + f ,
which expresses Newton’s equation of motion, namely that the change of momentum on the left hand side is equal to the internal elastic forces due to the stresses σ s plus the externally applied body forces f . The stresses σ s are functions of the displacements u. The equation is in Lagrangian form since the displacement vectors u follow the material points. Similarly, mechanical properties of incompressible Newtonian fluids can be described by the following two equations in Eulerian form where fluid quantities are observed in a fixed coordinate frame
2
ρ ∂v + v · ∇v = ∇ · σ f (v) + f ∂t
3
∇ · v = 0.
Equation (2) again states that the change of momentum equals the internal forces derived from the stresses σ f plus the externally applied body forces f . The stress tensor σ f = 2μ (v) − pI is composed of the viscosity stress and the pressure stress. The viscosity stress is dependent on the viscosity μ and the strain rate tensor while the pressure stress only depends on the scalar pressure p. For an incompressible fluid, the velocity field is divergence free (Eq. (3)). Comparison of the right hand side of the two equations of motion (1) and (2) reveals, that the Eulerian description makes the additional convection term v · ∇v necessary. In Sec. 4, we discuss numerical methods to solve both equations of motion. For fluids we focus on particle based methods such as SPH for which this term can be omitted.
<H2>3.1 Boundary Conditions</H2>
Materials such as fluids or solids are bounded by spatial limits. The behavior of materials at these limits is defined by boundary conditions. The boundary conditions relate the quantities of the two adjacent materials to each other at the interface. In the case of fluid-solid interaction, the geometrical domain of the interface Γ is defined as a surface between the volumetric solid continuum and the volumetric fluid continuum (see Fig. 2(a) ). We focus on three main types of boundary conditions.
<H2>3.2 No-Penetration Condition</H2>
If the solid is considered to be impermeable, no fluid element is allowed to cross the boundary, which is described in the following equation:
4
( ∂ u − v) · n = 0 at the boundary Γ, ∂t
where n is the normal on Γ (see Fig. 2(b) ). The equation states that the components of the velocities of the fluid and the deformable object perpendicular to Γ are equal.
<H2>3.3 No-Slip Condition</H2>
The no-slip condition models friction between the fluid and the solid (see Fig. 2(c) ). It holds for most fluids-solid surfaces and it states that the velocity components tangential to the fluid surface have to be equal, thus
5
( ∂ u − v) × n = 0 at the boundary Γ. ∂t
If both independent boundary conditions (4) and (5) hold, we simply have ∂t ∂ u = v at the boundary, i.e. both materials have the same velocity at the boundary.
2
<H2>3.4 Actio = Reactio</H2>
Newton’s Third Law demands the continuity of stresses σ s and σ f throughout the boundary (see Fig. 2(d) ). In other words, the traction forces of the solid g f must be opposite to the traction forces of the fluid g s on the boundary Γ
6
g s = σ s n = σ f (−n) = −g f ,
where n is the outward normal on the solid and −n the outward normal on the fluid.
u . v σ s σ f v g s u v n n -n . n u g f solid Γ fluid (a) (b) (c) (d)
Figure 2: Boundary conditions: (a) A solid object is deformed by a displacement field u and interacts with a fluid whose velocity field is v. The no-penetration condition is shown in image (b) where u ̇ is the time derivative of u. Image (c) illustrates the no-slip condition and image (d) Newton’s Third Law.
<H1>4 Computational Model</H1>
The continuous equations and boundary conditions described in the previous section need to be discretized in space and time via a numerical method before they can be used in a computer simulation. We do not go into the details of how equation (1) for elastic objects can be solved numerically.
For possible solutions using the Finite Element Method (FEM) we refer the reader to
[ 19 ], [ 6 ] or
[ 7 ]. All we require for our interaction method to work is • that the solid object is represented by a mesh and • that the displacements, velocities and forces are carried by the nodes of the mesh. Most of the methods used in computer graphics to simulate deformable objects meet these constraints including massspring systems, the Finite Volume Method (FVM) and the Boundary Element Method (BEM). For the simulation of fluids, two main numerical methods have been used in the field of physically based animation so far, namely Eulerian grid-based approaches
[ 20 , 21 , 22 ] and Lagrangian methods based on particles (see Sec. 2). In this paper we concentrate on Lagrangian methods because they allow fluids with free surfaces to move freely in space while in the Eulerian case fluid computations are restricted to a spatially fixed and bounded grid. From the fluid simulation method we require • that the fluid is represented by a set of particles and • that positions, velocities and internal forces are carried by the particles. Interaction modeling, thus, reduces to the problem of simulating the interaction between particles and triangulated surfaces.
(a) (b)
(c) (d)
Figure 3: (a) Iso surfaces of the Euclidean distance field of a piecewise linear curve (blue) with discontinuous first derivatives near concavities. (b) Weighted sums yield smooth iso surfaces with bulges. (c) Normalization does not remove the artifact. (d) Convolution yields bulge-free smooth iso surfaces.
<H2>4.1 Interaction of Particles with Triangles</H2>
In physics, interaction potentials of two objects always depend on the distance between them. While the Euclidean distance between two points is uniquely defined, the distance between a point and a triangle or a point and a triangulated surface needs to be defined. Let us define the distance of a point p from a triangle t as
7
d(p, t) = min |p − x|,
7
x∈t
and the distance of a point p from a triangulated surface T as
8
d(p, T ) = min d(p, t). t∈T
Figure 3(a) shows several isosurfaces of the resulting distance field which is C 0 continuous everywhere. Unfortunately, concavities as well as close disconnected meshes generate discontinuous first derivatives of the distance field. Those discontinuities lead to discontinuous derivatives in forces since the forces depend on the distance field. A force field with discontinuous first derivatives, in turn, yields artifacts such as the so called cooking of particles in concave regions and reduced stability of the simulation. The source of the discontinuity in the first derivatives is the minimum operator in Eqn. (8). One way to remove the problem is to replace the minimum by a weighted sum. Let the kernel W (d, h) ∈ C 1 be a positive smooth monotonously decreasing function which is zero for d ≥ h and has a vanishing derivative at d = h. We can then define the potential Φ of a point p with respect to a triangulated surface T which is not a Eucledian distance anymore
9
Φ(p, T ) = d(p, t)W (d(p, t), h), t∈T
but which is C 0 and C 1 continuous everywhere. However, as Fig. 3(b) shows, the resulting field is distorted near triangle boundaries. This effect can be removed by normalization
10
Φ(p, ̄ T ) = 0 w 1 Φ(p, T ) if w > 0
3
where w = t∈T W (d(p, t), h). Unfortunately, normalization just distributes the distortions to adjacent regions of triangle interfaces as Fig. 3(c) shows. Another difficulty introduced by the weighted field method is the choice of the support radius h with respect to the size of the features of the boundary T . For large supports, small features are smoothed out while small supports reduce the interaction range of T .
<H2>4.2 Convolution Surfaces</H2>
The problems mentioned in the previous section are well known in the field of implicit surface modeling introduced by Blinn
[ 23 ]. An elegant way to generate a bulge-free surface around a skeleton S, is to define a scalar function F S as the convolution
11
F S (p) = W (p − x)dx. x∈S
The implicit surface is defined by selecting an iso-surface of F S . By replacing the skeleton S with the triangulated surface T we get a smooth potential field around T (see Fig. 3(d) ). The problem with the weighted sum approach arises when when multiple triangles meet. In this case, all triangles contribute as a whole to the sum and generate bulges. In contrast, the convolution integral sums up infinitesimal parts of the skeleton each properly weighted (see Fig. 4 ). When the convolution integral is used, the interaction of p with the surface T is modeled as the interaction of p with all the infinitesimal points in T . For skeletal elements other than points, the integral in Eqn. (11) yields complex computations.
Approaches to approximate this integral were proposed by Bloomenthal
[ 24 ] and Sherstyuk
[ 25 ]. Bloomenthal uses radial Gauss kernels which can be separated with respect to different dimensions. The separation allows post evaluation of the convolution in 3D space, only considering the distance to the triangle plane. Sherstyuk discovered a special kernel which can be analytically convoluted over a triangle domain. Neither method is suitable for computing physical interactions because we are not free in the choice of the kernel. The potential function is given by physical laws.
p p p h (a) (b) (c)
Figure 4: (a) 2D cut through a 3D mesh.
Fluid particles within interaction range h from the surface interact with the triangles (shown in red). (b) The convolution integral sums up the contributions of infinitesimal parts of the triangles properly weighted. (c) Interactions with Gaussian particles (yellow) approximate the convolution in an optimal way.
<H2>4.3 Gaussian Boundary Particles</H2>
Our idea to solve the convolution integral is to use Gauss quadrature rules
[ 17 ]. For the interaction potential of a parti-
cle p with a single triangle t we get
12
U (p, t) = U (p − x)dx x∈t
13
≈ A w i U (p − x i ), i
where A is the surface area of t, x i the sampling points and w i their weights according to a chosen quadrature rule. We use the seven point rule which has convergence order O(L 6 ) with respect to the triangle size L. (see Fig. 5(a) and Tab. 1). These sampling points can be interpreted as boundary particles, which are placed and weighted according to the chosen Gauss quadrature rule. The weighted summation of their potentials approximates the convolution of the potential over the domain of the boundary triangle in an optimal way. Although the seven point rule yields good approximations of the convolution integral, triangles that are large in comparison to the interaction range of the surface would induce a poor sampling of the boundary field. Therefore, we subdivide the boundary triangle until a sufficient sampling rate is provided. We define a threshold for the maximal acceptable distance between boundary particles. This threshold is chosen relative to the maximal interaction radius of the fluid particles and can be regulated by the user. The boundary particles are generated by subdividing the triangle domain and by application of the Gauss quadrature rule to the resulting triangles (see Fig. 5(b) ). This has to be done at every time step, because triangles on the boundary are moved and deformed. Therefore, an efficient scheme is needed. We compute the relative vectors from the triangle nodes (shown in blue) to the boundary particles (shown in red) only once because they are the same for all subdivision triangles. These vectors are then added to the blue nodes to generate the complete set of boundary particles. Analog to positions, the velocities of boundary particles are interpolated from the velocities of the triangle nodes. Now that we have replaced the triangulated surface by a set of particles, the problem of triangle-particle interaction reduces to particle-particle interaction. We can, thus, use SPH-based approaches to approximate the boundary conditions stated in Sections 3.2, 3.3 and 3.4.
(a) (b)
Figure 5: (a) Boundary particles on a triangle according to the seven point rule. (b) Large triangles are subdivided and boundary particles are generated for each resulting triangle.
<H2>4.4 Boundary Repulsion and Adhesion</H2>
The no-penetration condition stated in Sec. 3.2 prevents fluid particles from penetrating the solid object. Monaghan
[ 16 ]
uses a Lennard-Jones-like force to generate repulsive forces which approximate the no-penetration condition. We propose a Lennard-Jones-like force that models both repulsion and adhesion to the contact surface. We define the force acting on
4
Point
Barycentric coordinates
Weights
1
(1/3, (a, 1/3, 1/3)
9/40
2
b, b)
e
3
(b, (b, a, b)
e
4
(c, b, a)
e
5
d, d)
f
6
(d, (d, c, d)
f
7
d, c)
f
Point Barycentric coordinates Weights 1 (1/3, 1/3, 1/3) 9/40 2 (a, b, b) e 3 (b, a, b) e 4 (b, b, a) e 5 (c, d, d) f 6 (d, c, d) f 7 (d, d, c) f
Table 1: Barycentric coordinates and weights of the seven point Gauss quadrature rule for triangles, where a = 0.05971587, b = 0.47014206, √ c = 0.79742699, d = 0.10128651, √ e = (155 + 15)/1200 and f = (155 − 15)/1200.
particle p due to triangle t by the convolution
14
f ra (p, t) = τ ra (|p − x|) dx. x∈t
The traction τ ra is dependent on the distance of the surface element from the particle p and has unit force per area in order to yield a force when integrated over the triangle. To model repulsion and adhesion, we use the following traction function
15
τ ra (r) = k (h−r) h 4 2 −(h−r r 0 (2h−r 0 ) 2 0 (h−r) ) 2 if r < h , 0 otherwise
where h is the interaction range and k controls the stiffness of the interaction. The traction has an order four repulsion term and an order two attraction term. It is designed to be zero for r = r 0 which is the preferred distance of fluid particles from the interface. The fact that for r = 0 the traction is finite (τ ra (0) = k) and that both, traction and first derivative vanish for r = h are important for robust real time simulations. Using Gaussian boundary particles, the force acting on a particle p is computed as
16
f ra (p) ≈ A i w ij τ ra (|p − x ij |), i j
where i iterates over all triangles within distance h of particle p. For each triangle the contributions of its boundary particles are summed up according to equation (13).
<H2>4.5 Boundary Friction</H2>
The no-slip condition (Eq. (5)) can be approximated by including the boundary particles into the viscosity evaluation of the SPH particles
[ 16 ]. We use the normalized kernel W visc proposed in
[ 13 ]
for viscosity computations. To evaluate the viscosity force f visc (p) on a fluid particle, the velocities of the boundary particles have to be interpolated from the velocity of mesh nodes (see Sec. 4.3). The traction τ visc depends on the velocity v b of the boundary particle, the v p of the fluid particle and the distance r between them
17
τ visc (r) = μ(v b − v p )∇ 2 W visc (r, h),
where the scalar μ controls the boundary viscosity and
18
∇ 2 W visc (r, h) = 0 πh 45 6 (h − r) if 0 ≤ r ≤ h
The kernel W visc is designed such that its Laplacian ∇ 2 W visc takes the linear form above, but satisfies the normalization criterion on the kernel itself. The normalization warrants second order interpolation convergence. The numerical approximation of the convolution integral over the triangle surface defines the final form of the viscosity force
19
f visc (p) = A i w ij τ visc (|p − x ij |). i j
<H2>4.6 Actio = Reactio</H2>
So far, we have applied forces to fluid particles only. However, according to Newton’s Third Law, proper reaction forces need to be applied to the deformable solid as well. The force contributions of boundary particles have to be distributed among the boundary triangle vertices so they can be picked up by the simulator of the deformable object. Bridson et al.
[ 26 ]
solve a similar problem in the context of cloth simulation. To resolve vertex-triangle collisions, an impulse is applied to the colliding vertex. Then, a distribution scheme is used to compute the corresponding reaction impulses for the three vertices of the triangle. We use the same scheme to distribute the forces to the vertices of the triangle surface. Given the force contribution f b computed for one boundary particle we compute the force contributions to the triangle nodes and the fluid particle as
20
f k triangle = 1 + w 1 2 2w + k w f b 2 2 + w 3 2
21
f particle = 1 + w 1 2 −2f + w b 2 2 + w 3 2 ,
where the w k are the barycentric coordinates of the boundary particle with respect to the triangle and k ∈ (1 . . . 3). According to
[ 26 ]
this distribution scheme provides continuity across triangle boundaries and introduces appropriate torques for off-center interactions. However, the scheme is not completely error free. Force magnitudes can get amplified – at most by a factor of 8/7 – at the triangle center. However, this error did not cause any artifacts or stability problems in our simulations.
<H1>5 Implementation</H1>
At every time step of the solid and fluid simulator, the following five steps are executed: 1. Surface triangle extraction: Boundary triangle references are stored in a flat list. 2. Particle grid hashing: A grid index on the fluid particles is created. 3. Neighbor search: For each boundary triangle a list of possible fluid interaction partners is generated. 4. Boundary sampling: For every boundary triangle with possible interaction partners, boundary particles are generated. 5. Interaction computation: For every interaction pair, composed of a boundary particle and a fluid particle, forces are computed and applied to related triangles and fluid particles. Processing the five phases one after the other would have a negative impact on storage requirements. Neighbor references and boundary particles for all triangles would have to be stored at the same time. If the computations of steps three to five are grouped around single triangles, only data relevant for the current triangle has to be stored at a time ( Fig. 6 ). The output of step 3 is a list, containing all fluid particles within interaction range h of a triangle t. To speed up the search for these particles we use a regular grid with spatial hashing
[ 27 ]. There is a trade-off between computation time for the neighbor search and the quality of the neighbor list. We extend the axes aligned bounding box (AABB) of t along all axes about the interaction range. Then, we query all grid cells intersecting the extended box. We also tested tighter queries which generate fewer neighbor candidates but their increased time complexity was not compensated by the reduced cost of interaction computations. In step 4, boundary particles are only generated for those triangles that have fluid particle neighbors. The boundary particles for a triangle t are kept only temporarily for the interaction computation. After t is processed, they are discarded. In this step, positions and velocities are interpolated from the triangle nodes for each boundary particle. To compute interaction forces in step 5 we iterate over all the boundary particles of a triangle. For each fluid particle within the interaction radius of the boundary particle, we compute the interaction forces as described in Sec. 4 and distribute them among the fluid particle and the triangle nodes according to Eqns. (20) and (21).
5
... ... Neighbor search Sampling Interaction computation
Figure 6: Algorithm overview: Triangles are processed separately. This avoids the storage of fluid particle neighbor lists and boundary particles for all triangles simultaneously.
<H1>6 Results</H1>
All experiments described in this section have been performed on an AMD Athlon 1.8 GHz PC with 512 MB RAM and a GeForce Ti 4400 graphics card with 128 MB RAM. Note that most of the simulations are recorded in a real-time interactive environment. Thus, we cannot afford several seconds or even minutes per frame for the reconstruction and rendering of the free fluid surface as in off-line simulations [ 14 , 15 ]
which explains the simplistic renderings of the fluids.
(a) (b)
Figure 7: (a) When the user pulls the pool wall, water flows out. (b) Boxes float on the water surface.
<H2>6.1 A Pool Filled with Water</H2>
To demonstrate the stability of our model in connection with concave surfaces, we filled a pool composed of 800 tetrahedral elements with 2000 fluid particles (see Fig. 7(a) ). The simulation runs at 20 frames per second. By pulling the pool wall, the user indirectly influences the water. The generated waves, in turn, deform the pool walls. Deformable boxes float freely on the water surface (see Fig. 7(b) ). Fig. 8 shows the fluid and boundary particles used in the simulation.
<H2>6.2 Floating Boxes</H2>
(a) (b)
Figure 8: A box floats in a pool: (a) The fluid particles are shown in blue. (b) For the interaction computation virtual boundary particles (white) are placed on the surfaces of the deformable objects.
We dropped an additional large box into the pool (see Fig. 1 ). When it touches the water, it emits a wave that hits the pool boundary and causes it to fracture. This scene demonstrates the interplay of various physical phenomena provided by the fluid simulator, the solid simulator and the interaction model.
<H2>6.3 Simulation of Blood Vessels</H2>
An important application of our method is the simulation of bleeding during virtual operations. Our simulation of a blood vessel is a first step into this direction. We simulate the flow of 3000 particles through a virtual vessel, consisting of a deformable mesh composed of 560 tetrahedra. The simulation took about 70 ms per time step. Fig. 9 shows the resulting blood flow. The velocity of the fluid particles is color coded visualizing the friction of the fluid with the boundary. In the experiment shown in Fig. 10 , we turned on fracture of the Finite Element mesh. Now, the vessel is torn open when the elastic stresses caused by blood pressure exceed the material threshold. The free surface of the particle system is rendered using the Marching Cubes algorithm. The animation of the mesh and the particles are possible in real time at 60 ms per time step, while surface reconstruction took about half a second per frame. On today’s hardware only a limited number of fluid particles can be simulated in real-time which yields a relatively coarse fluid surface.
6
Figure 9: Blood flow through a vessel. The image shows subsequent time slices of an interactive animation. The velocity of the fluid particles is color coded. Yellow colored particles are fast, while red ones are slow. Pulsation waves and viscosity at the vessel boundary can be observed.
Figure 10: Vessel injury. The Finite Element mesh fractures due to pressure forces in the blood stream.
<H1>7 Conclusion</H1>
We have presented a new method for the simulation of interactions of deformable solids with fluids. Our interaction model simulates repulsion, adhesion and friction near the fluid-solid interface. The smoothness of the force fields is important for the stability of the simulation. The core idea to get smooth interaction fields is to place boundary particles onto the surface triangles according to Gauss quadrature rules. This idea might be useful in other graphic domains as well. We mentioned the application to modeling with implicit surfaces. Character skinning is another application where bulges or knees are known problems in regions where several close bones meet. We demonstrated the usability of our method in an interactive simulation environment with several scenes. A difficulty in connection with the interactive simulation of fluids is the extraction and rendering of a plausible fluid surface in real time. Thus, ongoing work focusses on fast algorithms for surface reconstruction.
<H1>Acknowledgements</H1>
This project was funded by the Swiss National Commission for Technology and Innovation (KTI) project no. 6310.1 KTS-ET.
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[22] D. Enright, S. Marschner, and R. Fedkiw. Animation and rendering of complex water surfaces. In Proceedings of the 29th annual conference on Computer graphics and interactive techniques, pages 736–744. ACM Press, 2002.
[23] J. Blinn. A generalization of algebraic surface drawing. ACM Transactions on Graphics, 1(3):235–256, 1982.
[24] J. Bloomenthal. Skeletal Design of Natural Forms. PhD thesis, University of Calgary, Canada, 1995.
[25] A. Sherstyuk. Fast ray tracing of implicit surfaces. In Implicit Sufaces ’98, pages 145–153, 1998.
[26] R. Bridson, R. Fedkiw, and J. Anderson. Robust treatment of collisions, contact and friction for cloth animation. In ACM Transactions on Graphics, volume 21, pages 594–603. ACM SIGGRAPH 2002, August 2002.
[27] M. Teschner, B. Heidelberger, M. Mller, D. Pomeranerts, and M. Gross. Optimized spatial hashing for collision detection of deformable objects. In Proc. Vision, Modeling, Visualization VMV, pages 47–54, 2003.
7
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"text": "['the Eulerian description makes the additional convection term v · ∇v necessary']",
"type": "own_claim",
"id": "T116"
},
{
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[
8822,
8898
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],
"text": "['Comparison of the right hand side of the two equations of motion (1) and (2)']",
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},
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"offsets": [
[
9224,
9288
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],
"text": "['Materials such as fluids or solids are bounded by spatial limits']",
"type": "own_claim",
"id": "T118"
},
{
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[
9290,
9365
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],
"text": "['The behavior of materials at these limits is defined by boundary conditions']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
9367,
9473
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],
"text": "['The boundary conditions relate the quantities of the two adjacent materials to each other at the interface']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
9515,
9654
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],
"text": "['the geometrical domain of the interface Γ is defined as a surface between the volumetric solid continuum and the volumetric fluid continuum']",
"type": "own_claim",
"id": "T121"
},
{
"offsets": [
[
9660,
9669
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],
"text": "['Fig. 2(a)']",
"type": "data",
"id": "T122"
},
{
"offsets": [
[
9475,
9513
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],
"text": "['In the case of fluid-solid interaction']",
"type": "data",
"id": "T123"
},
{
"offsets": [
[
9805,
9846
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],
"text": "['the solid is considered to be impermeable']",
"type": "data",
"id": "T124"
},
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"offsets": [
[
9848,
9897
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],
"text": "['no fluid element is allowed to cross the boundary']",
"type": "own_claim",
"id": "T125"
},
{
"offsets": [
[
9982,
10022
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],
"text": "['( ∂ u − v) · n = 0 at the boundary Γ, ∂t']",
"type": "data",
"id": "T126"
},
{
"offsets": [
[
10050,
10070
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],
"text": "['n is the normal on Γ']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
10076,
10085
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],
"text": "['Fig. 2(b)']",
"type": "data",
"id": "T128"
},
{
"offsets": [
[
10114,
10215
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],
"text": "['the components of the velocities of the fluid and the deformable object perpendicular to Γ are equal.']",
"type": "own_claim",
"id": "T129"
},
{
"offsets": [
[
10285,
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],
"text": "['The no-slip condition models friction between the fluid and the solid']",
"type": "own_claim",
"id": "T130"
},
{
"offsets": [
[
10360,
10369
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],
"text": "['Fig. 2(c)']",
"type": "data",
"id": "T131"
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{
"offsets": [
[
10373,
10412
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],
"text": "['It holds for most fluids-solid surfaces']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
[
10417,
10504
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],
"text": "['it states that the velocity components tangential to the fluid surface have to be equal']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
10548,
10584
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],
"text": "['( ∂ u − v) × n = 0 at the boundary Γ']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
10683,
10709
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],
"text": "['∂t ∂ u = v at the boundary']",
"type": "own_claim",
"id": "T135"
},
{
"offsets": [
[
10613,
10666
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],
"text": "['both independent boundary conditions (4) and (5) hold']",
"type": "data",
"id": "T136"
},
{
"offsets": [
[
10716,
10769
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],
"text": "['both materials have the same velocity at the boundary']",
"type": "data",
"id": "T137"
},
{
"offsets": [
[
10850,
10939
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],
"text": "['Newton’s Third Law demands the continuity of stresses σ s and σ f throughout the boundary']",
"type": "own_claim",
"id": "T138"
},
{
"offsets": [
[
10945,
10954
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],
"text": "['Fig. 2(d)']",
"type": "data",
"id": "T139"
},
{
"offsets": [
[
10974,
11085
]
],
"text": "['the traction forces of the solid g f must be opposite to the traction forces of the fluid g s on the boundary Γ']",
"type": "own_claim",
"id": "T140"
},
{
"offsets": [
[
11123,
11152
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],
"text": "['g s = σ s n = σ f (−n) = −g f']",
"type": "own_claim",
"id": "T141"
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{
"offsets": [
[
11755,
11803
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],
"text": "['The continuous equations and boundary conditions']",
"type": "own_claim",
"id": "T142"
},
{
"offsets": [
[
11838,
11950
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],
"text": "['need to be discretized in space and time via a numerical method before they can be used in a computer simulation']",
"type": "own_claim",
"id": "T143"
},
{
"offsets": [
[
11989,
12047
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],
"text": "['equation (1) for elastic objects can be solved numerically']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
12050,
12104
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],
"text": "['For possible solutions using the Finite Element Method']",
"type": "own_claim",
"id": "T145"
},
{
"offsets": [
[
12137,
12139
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],
"text": "['19']",
"type": "data",
"id": "T146"
},
{
"offsets": [
[
12145,
12146
]
],
"text": "['6']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
12155,
12156
]
],
"text": "['7']",
"type": "data",
"id": "T148"
},
{
"offsets": [
[
12160,
12212
]
],
"text": "['All we require for our interaction method to work is']",
"type": "own_claim",
"id": "T149"
},
{
"offsets": [
[
12215,
12261
]
],
"text": "['that the solid object is represented by a mesh']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
12268,
12350
]
],
"text": "['that the displacements, velocities and forces are carried by the nodes of the mesh']",
"type": "own_claim",
"id": "T151"
},
{
"offsets": [
[
12352,
12480
]
],
"text": "['Most of the methods used in computer graphics to simulate deformable objects meet these constraints including massspring systems']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
12482,
12506
]
],
"text": "['the Finite Volume Method']",
"type": "data",
"id": "T153"
},
{
"offsets": [
[
12517,
12544
]
],
"text": "['the Boundary Element Method']",
"type": "data",
"id": "T154"
},
{
"offsets": [
[
12552,
12673
]
],
"text": "['For the simulation of fluids, two main numerical methods have been used in the field of physically based animation so far']",
"type": "background_claim",
"id": "T155"
},
{
"offsets": [
[
12675,
12712
]
],
"text": "['namely Eulerian grid-based approaches']",
"type": "background_claim",
"id": "T156"
},
{
"offsets": [
[
12735,
12772
]
],
"text": "['Lagrangian methods based on particles']",
"type": "background_claim",
"id": "T157"
},
{
"offsets": [
[
12778,
12784
]
],
"text": "['Sec. 2']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
12716,
12718
]
],
"text": "['20']",
"type": "data",
"id": "T159"
},
{
"offsets": [
[
12721,
12723
]
],
"text": "['21']",
"type": "data",
"id": "T160"
},
{
"offsets": [
[
12726,
12728
]
],
"text": "['22']",
"type": "data",
"id": "T161"
},
{
"offsets": [
[
12819,
12837
]
],
"text": "['Lagrangian methods']",
"type": "background_claim",
"id": "T162"
},
{
"offsets": [
[
12851,
12906
]
],
"text": "['allow fluids with free surfaces to move freely in space']",
"type": "background_claim",
"id": "T163"
},
{
"offsets": [
[
12934,
13005
]
],
"text": "['fluid computations are restricted to a spatially fixed and bounded grid']",
"type": "background_claim",
"id": "T164"
},
{
"offsets": [
[
12913,
12933
]
],
"text": "['in the Eulerian case']",
"type": "data",
"id": "T165"
},
{
"offsets": [
[
13058,
13104
]
],
"text": "['the fluid is represented by a set of particles']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
13116,
13186
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],
"text": "['positions, velocities and internal forces are carried by the particles']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
13188,
13208
]
],
"text": "['Interaction modeling']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
13216,
13312
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],
"text": "['reduces to the problem of simulating the interaction between particles and triangulated surfaces']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
13857,
13937
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],
"text": "['interaction potentials of two objects always depend on the distance between them']",
"type": "background_claim",
"id": "T170"
},
{
"offsets": [
[
13939,
14006
]
],
"text": "['While the Euclidean distance between two points is uniquely defined']",
"type": "background_claim",
"id": "T171"
},
{
"offsets": [
[
13845,
13855
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],
"text": "['In physics']",
"type": "data",
"id": "T172"
},
{
"offsets": [
[
14008,
14109
]
],
"text": "['the distance between a point and a triangle or a point and a triangulated surface needs to be defined']",
"type": "background_claim",
"id": "T173"
},
{
"offsets": [
[
14576,
14687
]
],
"text": "['concavities as well as close disconnected meshes generate discontinuous first derivatives of the distance field']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
14689,
14754
]
],
"text": "['Those discontinuities lead to discontinuous derivatives in forces']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
14761,
14800
]
],
"text": "['the forces depend on the distance field']",
"type": "data",
"id": "T176"
},
{
"offsets": [
[
14802,
14852
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],
"text": "['A force field with discontinuous first derivatives']",
"type": "own_claim",
"id": "T177"
},
{
"offsets": [
[
14863,
14879
]
],
"text": "['yields artifacts']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
14902,
14941
]
],
"text": "['cooking of particles in concave regions']",
"type": "data",
"id": "T179"
},
{
"offsets": [
[
14946,
14981
]
],
"text": "['reduced stability of the simulation']",
"type": "data",
"id": "T180"
},
{
"offsets": [
[
14983,
15075
]
],
"text": "['The source of the discontinuity in the first derivatives is the minimum operator in Eqn. (8)']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
15077,
15150
]
],
"text": "['One way to remove the problem is to replace the minimum by a weighted sum']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
15605,
15662
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],
"text": "['the resulting field is distorted near triangle boundaries']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
15664,
15707
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],
"text": "['This effect can be removed by normalization']",
"type": "own_claim",
"id": "T184"
},
{
"offsets": [
[
15865,
15954
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],
"text": "['normalization just distributes the distortions to adjacent regions of triangle interfaces']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
15958,
15967
]
],
"text": "['Fig. 3(c)']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
15975,
16130
]
],
"text": "['Another difficulty introduced by the weighted field method is the choice of the support radius h with respect to the size of the features of the boundary T']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
16153,
16184
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],
"text": "['small features are smoothed out']",
"type": "own_claim",
"id": "T188"
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{
"offsets": [
[
16191,
16239
]
],
"text": "['small supports reduce the interaction range of T']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
16133,
16151
]
],
"text": "['For large supports']",
"type": "data",
"id": "T190"
},
{
"offsets": [
[
16314,
16437
]
],
"text": "['The problems mentioned in the previous section are well known in the field of implicit surface modeling introduced by Blinn']",
"type": "background_claim",
"id": "T191"
},
{
"offsets": [
[
16441,
16443
]
],
"text": "['23']",
"type": "data",
"id": "T192"
},
{
"offsets": [
[
16447,
16569
]
],
"text": "['An elegant way to generate a bulge-free surface around a skeleton S, is to define a scalar function F S as the convolution']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
16608,
16629
]
],
"text": "['F S (p) = W (p − x)dx']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
16656,
16722
]
],
"text": "['The implicit surface is defined by selecting an iso-surface of F S']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
16844,
16893
]
],
"text": "['The problem with the weighted sum approach arises']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
16904,
16927
]
],
"text": "['multiple triangles meet']",
"type": "data",
"id": "T197"
},
{
"offsets": [
[
16943,
17009
]
],
"text": "['all triangles contribute as a whole to the sum and generate bulges']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
16929,
16941
]
],
"text": "['In this case']",
"type": "data",
"id": "T199"
},
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"offsets": [
[
17024,
17115
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],
"text": "['the convolution integral sums up infinitesimal parts of the skeleton each properly weighted']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
17121,
17127
]
],
"text": "['Fig. 4']",
"type": "data",
"id": "T201"
},
{
"offsets": [
[
17170,
17283
]
],
"text": "['the interaction of p with the surface T is modeled as the interaction of p with all the infinitesimal points in T']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
17136,
17168
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],
"text": "['the convolution integral is used']",
"type": "data",
"id": "T203"
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"offsets": [
[
17327,
17380
]
],
"text": "['the integral in Eqn. (11) yields complex computations']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
17286,
17325
]
],
"text": "['For skeletal elements other than points']",
"type": "data",
"id": "T205"
},
{
"offsets": [
[
17383,
17451
]
],
"text": "['Approaches to approximate this integral were proposed by Bloomenthal']",
"type": "background_claim",
"id": "T206"
},
{
"offsets": [
[
17455,
17457
]
],
"text": "['24']",
"type": "data",
"id": "T207"
},
{
"offsets": [
[
17464,
17473
]
],
"text": "['Sherstyuk']",
"type": "background_claim",
"id": "T208"
},
{
"offsets": [
[
17477,
17479
]
],
"text": "['25']",
"type": "data",
"id": "T209"
},
{
"offsets": [
[
17483,
17580
]
],
"text": "['Bloomenthal uses radial Gauss kernels which can be separated with respect to different dimensions']",
"type": "background_claim",
"id": "T210"
},
{
"offsets": [
[
17582,
17650
]
],
"text": "['The separation allows post evaluation of the convolution in 3D space']",
"type": "background_claim",
"id": "T211"
},
{
"offsets": [
[
17705,
17802
]
],
"text": "['Sherstyuk discovered a special kernel which can be analytically convoluted over a triangle domain']",
"type": "background_claim",
"id": "T212"
},
{
"offsets": [
[
17804,
17866
]
],
"text": "['Neither method is suitable for computing physical interactions']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
17875,
17918
]
],
"text": "['we are not free in the choice of the kernel']",
"type": "data",
"id": "T214"
},
{
"offsets": [
[
17920,
17968
]
],
"text": "['The potential function is given by physical laws']",
"type": "own_claim",
"id": "T215"
},
{
"offsets": [
[
18478,
18553
]
],
"text": "['Our idea to solve the convolution integral is to use Gauss quadrature rules']",
"type": "own_claim",
"id": "T216"
},
{
"offsets": [
[
18557,
18559
]
],
"text": "['17']",
"type": "data",
"id": "T217"
},
{
"offsets": [
[
19060,
19199
]
],
"text": "['These sampling points can be interpreted as boundary particles, which are placed and weighted according to the chosen Gauss quadrature rule']",
"type": "own_claim",
"id": "T218"
},
{
"offsets": [
[
19201,
19348
]
],
"text": "['The weighted summation of their potentials approximates the convolution of the potential over the domain of the boundary triangle in an optimal way']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
19359,
19434
]
],
"text": "['the seven point rule yields good approximations of the convolution integral']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
19436,
19565
]
],
"text": "['triangles that are large in comparison to the interaction range of the surface would induce a poor sampling of the boundary field']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
19578,
19657
]
],
"text": "['we subdivide the boundary triangle until a sufficient sampling rate is provided']",
"type": "own_claim",
"id": "T222"
},
{
"offsets": [
[
19745,
19868
]
],
"text": "['This threshold is chosen relative to the maximal interaction radius of the fluid particles and can be regulated by the user']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
19870,
20016
]
],
"text": "['The boundary particles are generated by subdividing the triangle domain and by application of the Gauss quadrature rule to the resulting triangles']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
20022,
20031
]
],
"text": "['Fig. 5(b)']",
"type": "data",
"id": "T225"
},
{
"offsets": [
[
20035,
20073
]
],
"text": "['This has to be done at every time step']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
20083,
20131
]
],
"text": "['triangles on the boundary are moved and deformed']",
"type": "data",
"id": "T227"
},
{
"offsets": [
[
20144,
20173
]
],
"text": "['an efficient scheme is needed']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
20306,
20353
]
],
"text": "['they are the same for all subdivision triangles']",
"type": "own_claim",
"id": "T229"
},
{
"offsets": [
[
20355,
20452
]
],
"text": "['These vectors are then added to the blue nodes to generate the complete set of boundary particles']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
20475,
20570
]
],
"text": "['the velocities of boundary particles are interpolated from the velocities of the triangle nodes']",
"type": "own_claim",
"id": "T231"
},
{
"offsets": [
[
21166,
21194
]
],
"text": "['The no-penetration condition']",
"type": "own_claim",
"id": "T232"
},
{
"offsets": [
[
21214,
21272
]
],
"text": "['prevents fluid particles from penetrating the solid object']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
21205,
21213
]
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"text": "['Sec. 3.2']",
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A15 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A15_M06_Interactive_Motion_Generation_from_Examples_CITATION_PURPOSE_M_v1.xml">
3315e831b4f59b30e8bdad883c77df9ed5f301dbb0e70be6b8443d0ad5d2b311
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http://dx.doi.org/10.1145/566570.566606
To appear in the ACM SIGGRAPH conference proceedings
<Title>Interactive Motion Generation from Examples</Title>
Okan Arikan
University of California, Berkeley
<Abstract>There are many applications that demand large quantities of natural looking motion. It is difficult to synthesize motion that looks natural, particularly when it is people who must move. In this paper, we present a framework that generates human motions by cutting and pasting motion capture data. Selecting a collection of clips that yields an acceptable motion is a combinatorial problem that we manage as a randomized search of a hierarchy of graphs. This approach can generate motion sequences that satisfy a variety of constraints automatically. The motions are smooth and human-looking. They are generated in real time so that we can author complex motions interactively. The algorithm generates multiple motions that satisfy a given set of constraints, allowing a variety of choices for the animator. It can easily synthesize multiple motions that interact with each other using constraints. This framework allows the extensive re-use of motion capture data for new purposes.</Abstract>
CR Categories: I.2.7 [Artificial Intelligence]: Problem Solving, Control Methods and Search—Graph and tree search strategies I.3.7 [COMPUTER GRAPHICS ]: Three-Dimensional Graphics and Realism—Animation Keywords: Motion Capture, Motion Synthesis, Human motion, Graph Search, Clustering, Animation with Constraints
D. A. Forsyth
<H1>1 Introduction</H1>
Motion is one of the most important ingredients of CG movies and computer games. Obtaining realistic motion usually involves key framing, physically based modelling or motion capture. Creating natural looking motions with key framing requires lots of effort and expertise. Although physically based modelling can be applied to simple systems successfully, generating realistic motion on a computer is difficult, particularly for human motion. A standard solution is motion capture: motion data for an approximate skeletal hierarchy of the subject is recorded and then used to drive a reconstruction on the computer. This allows other CG characters to be animated with the same motions, leading to realistic, “human looking” motions for use in movies or games. The biggest drawbacks of motion capture are:
1. Most motion capture systems are very expensive to use, because the process is time consuming for actors and technicians and motion data tends not to be re-used.
2. It is very hard to obtain motions that do exactly what the animator wants. Satisfying complex timed constraints is difficult and may involve many motion capture iterations. Examples include being at a particular position at a particular time accurately or synchronizing movement to a background action that had been shot before.
In order to make motion capture widely available, the motion data needs to be made re-usable. This may mean using previous motion capture data to generate new motions so that certain requirements are met, transferring motions from one skeletal configuration to another so that we can animate multiple figures with the same motion without it looking “funny”, or changing the style of the motion so that the directors can have higher level control over the motion. There are three natural stages of motion synthesis:
1. Obtaining motion demands involves specifying constraints on the motion, such as the length of the motion, where the body or individual joints should be or what the body needs to be doing at particular times. These constraints can come from an interactive editing system used by animators, or from a computer game engine itself. 2. Generating motion involves obtaining a rough motion that satisfies the demands. In this paper, we describe a technique that cuts and pastes bits and pieces of example motions together to create such a motion. 3. Post processing involves fixing small scale offensive artifacts. An example would involve fixing the feet so that they do not penetrate or slide on the ground, lengthening or shortening strides and fixing constraint violations.
In this paper, we present a framework that allows synthesis of new motion data meeting a wide variety of constraints. The synthesized motion is created from example motions at interactive speeds.
<H1>2 Related Work</H1>
In the movie industry, motion demands are usually generated by animators. However, automatic generation of motion demands is required for autonomous intelligent robots and characters
[Funge et al. 1999].
An overview of the automatic motion planning can be found in
[Latombe 1999; O’Rourke 1998]. Generating motion largely follows two threads: using examples and using controllers. Example based motion synthesis draws on an analogy with texture synthesis where a new texture (or motion) that looks like an example texture (or motion example) needs to be synthesized
[Efros and Leung 1999; Heeger and Bergen 1995]. Pullen and Bregler used this approach to create cyclic motions by sampling motion signals in a “signal pyramid” [2000]. They also used a similar approach to fetch missing degrees of freedom in a motion from a motion capture database
[Pullen and Bregler 2002]. The sampling can also be done in the motion domain to pick clips of motions to establish certain simple constraints
[Lamouret and van de Panne 1996; Schodl et al. 2000].
A roadmap of all the motion examples can be constructed and searched to obtain a desired motion
[Choi et al. 2000; Lee et al. 2002; Kovar et al. 2002].
The clips in this roadmap can also be parameterized for randomly sampling different motion sequences
[Li et al. 2002]. The motion signals can also be clustered. The resulting Markov chain can be searched using dynamic programming to find a motion that connects two keyframes
[Molina-Tanco and Hilton 2000]
or used in a variable length Markov model to infer behaviors [Galata et al. 2001]
or directly sampled from to create new motions [Bowden 2000].
This is similar to our work. However, our clustering method does not operate on body configurations and our probabilistic search strategy is more effective than dynamic programming as it will be explained below. Types of probabilistic search algorithms have also been used in physically based animation synthesis
[Chenney and Forsyth 2000] and rendering [Veach and Guibas 1997]. Controller based approaches use physical models of systems and controllers that produce outputs usually in the form of forces and torques as a function of the state of the body. These controllers can be designed specifically to accomplish particular tasks
[Brogan et al. 1998; Hodgins et al. 1995] or they can be learned automatically using statistical tools [Grzeszczuk and Terzopoulos 1995; Grzeszczuk et al. 1998; Mataric 2000]. The motion data can also be post processed to fix problems such as feet sliding on the ground or some constraints not being satisfied
[Gleicher 1998; Lee and Shin 1999; Popovic 1999; Rose et al. 1996]. This usually involves optimization of a suitable displacement function on the motion signal. Different body sizes move according to different time scales, meaning that motion cannot simply be transferred from one body size to another; modifying motions appropriately is an interesting research problem [Hodgins and Pollard 1997].
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To appear in the ACM SIGGRAPH conference proceedings
<H1>3 Synthesis as Graph Search</H1>
We assume there is a set of N motion sequences forming our dataset, each belonging to the same skeletal configuration. Every motion is discretely represented as a sequence of frames each of which has the same M degrees of freedom. This is required to be able to compare two motions and to be able to put clips from different motion sequences together. We write the i’th frame of s’th motion as s i .
<H2>3.1 Motion Graph</H2>
The collection of motion sequences could be represented as a directed graph. Each frame would be a node. There would be an edge from every frame to every frame that could follow it in an acceptable splice. In this graph, there would be (at least) an edge from the k’th frame to the k + 1’th frame in each sequence. This graph is not a particularly helpful representation because it is extremely large — we can easily have tens of thousands of nodes and hundreds of thousands of edges — and it obscures the structure of the sequences. Instead, we collapse all the nodes (frames) belonging to the same motion sequence together. This yields a graph G where the nodes of G are individual motion sequences and there is an edge from s to t for every pair of frames where we can cut from s to t. Since edges connect frames, they are labelled with the frames in the incident nodes (motion sequences) that they originate from and they point to. We also assume that the edges in G are attached a cost value which tells us the cost of connecting the incident frames. If cutting from one sequence to another along an edge introduces a discontinuous motion, then the cost attached to the edge is high. Appendix A introduces the cost function that we used. The collapsed graph still has the same number of edges. For an edge e from s i to t j , let f romMotion(e) = s, toMotion(e) = t, f romFrame(e) = i, toFrame(e) = j and cost(e) be the cost associated with the edge (defined in Appendix A). In this setting, any sequence of edges e 1 · · · e n where toMotion(e i ) = f romMotion(e i+1 ) and toFrame(e i ) < f romFrame(e i+1 ), ∀i, 1≤i < n is a valid path and defines a legal sequence of splices. (figure 1).
Path Edge 1 Edge 5 Edge 2 Edge 4 Edge 3 Edge 1 Sequences Edge 2 Edge 4 Edge 3 Motion Edge 5 Corresponding Motion Time
Figure 1: We wish to synthesize human motions by splicing together pieces of existing motion capture data. This can be done by representing the collection of motion sequences by a directed graph (top). Each sequence becomes a node; there is an edge between nodes for every frame in one sequence that can be spliced to a frame in another sequence or itself. A valid path in this graph represents a collection of splices between sequences, as the middle shows. We now synthesize constrained motion sequences by searching appropriate paths in this graph using a randomized search method.
<H2>3.2 Constraints</H2>
We wish to construct paths in the motion graph that satisfy constraints. Many constraints cannot be satisfied exactly. For example, given two positions, there may not be any sequence of frames in the collection that will get us from the first position to the second position exactly. We define hard constraints to be those that can (and must) be satisfied exactly. Typically, a hard constraint involves using a particular frame in a particular time slot. For example, instead of considering all valid paths, we can restrict ourselves to valid paths that pass through particular nodes at particular times. This way, we can constrain the moving figure to be at a specific pose at a specific time. This enables us to search for motions such as jumping, falling, or pushing a button at a particular time. A soft constraint cannot generally be met exactly. Instead we score sequences using an objective function that reflects how well the constraint has been met and attempt to find extremal sequences. One example is the squared distance between the position of the constraint and the actual position of the body at the time of the constraint. Example soft constraints include:
1. The total number of frames should be a particular number.
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To appear in the ACM SIGGRAPH conference proceedings
2. The motion should not penetrate any objects in the environment. 3. The body should be at a particular position and orientation at a particular time. 4. A particular joint should be at a particular position (and maybe having a specific velocity) at a specific time. 5. The motion should have a specified style (such as happy or energetic) at a particular time.
Finding paths in the motion graph that satisfy the hard constraints and optimize soft constraints involves a graph search. Unfortunately, for even a small collection of motions, the graph G has a large number of edges and straightforward search of this graph is computationally prohibitive. The main reason is the need to enumerate many paths. There are, in general, many perfectly satisfactory motions that satisfy the constraints equally well. For example, if we require only that the person be at one end of a room at frame 0 and near the other end at frame 5000, unless the room is very large, there are many motions that satisfy these constraints.
<H1>4 Randomized Search</H1>
The motion graph is too hard to search with dynamic programming as there are many valid paths that satisfy the constraints equally well. There may be substantial differences between equally valid paths — in the example above, whether you dawdle at one side of the room or the other is of no significance. This suggests summarizing the graph to a higher level and coarser presentation that is easier to search. Branch and bound algorithms are of no help here, because very little pruning is possible. In order to search the graph G in practical times, we need to do the search at a variety of levels where we do the large scale motion construction first and then “tweak” the details so that the motion is continuous and satisfies the constraints as well as possible. Coarser levels should have less complexity while allowing us to explore substantially different portions of the path space. In such a representation, every level is a summary of the one finer level. Let G ← G ← G ← · · · ← G n ← G be such a hierarchical representation where G is the coarsest level and G is the finest. We will first find a path in G and then push it down the hierarchy to a path in G for synthesis.
<H2>4.1 Summarizing the Graph</H2>
All the edges between two nodes s and t can be represented in a matrix P st . The (i, j)’th entry of P st contains the weight of the edge connecting s i to t j and infinity if there is no such edge. In the appendix A, we give one natural cost function C(s i ,t j ) for edge weights. We now have: (P st ) i j = C(s ∞ i ,t j ) otherwise. if there is an edge from s i to t j The cost function explained in section A causes the P matrices to have non-infinite entries to form nearly elliptical groups ( figure 2 ). This is due to the fact that if two frames are similar, most probably their preceding and succeeding frames also look similar. In order to summarize the graph, we cluster the edges of G. We now have G , whose nodes are the same as the nodes of G, and whose edges represent clusters of edges of G in terms of their f romFrame and toFrame labels. We require that, if there is a cut between two sequences represented by an edge between two nodes in G, there be at least one edge between the corresponding nodes in G . If this were not the case, our summary would rule out potential paths. In order to insure that this condition holds and because the graph is very large, we cluster edges connecting every pair of nodes in G separately. We cluster unconnected edge groups of G from the P matrices (defined between every pair of nodes) using k-means ma joraxislength
[Bishop 1995]. The number of clusters is chosen as minoraxislength for each group where the axis lengths refer to the ellipse that fits to the cluster (obtained through Principal Component Analysis). The nodes of G are the same as the nodes of G. The edges connecting nodes in G are cluster centers for clusters of edges connecting corresponding nodes in G. The centers are computed by taking the average of the edges in terms of f romFrame, toFrame and cost values. At this point, every edge in G represents many edges in G. We would like to have a tree of graph representations whose root is G , and whose leaves are G. We use k-means clustering to split each cluster of edges in half at each intermediate level and obtain a hierarchical representation G ← G ← G ← · · · ← G n ← G for the original graph G. This is an instance of Tree-Structured Vector Quantization
[Gersho and Gray 1992]. Thus, in our summarized graph G , each edge is the root of a binary tree and represents all the edges in close neighborhood in terms of the edge labels. Note that the leaf edges are the edges in the original graph and intermediate edges are the averages of all the leaf edges beneath them. A path in G represents a sequence of clips; so does a path in G , but now the positions of the clip boundaries are quantized, so there are fewer paths.
Frame i Frame j Walking Running Walking , frame i
Clustering
j frame Running,
Figure 2: Every edge between two nodes representing different motion clips can be represented as a matrix where the entries correspond to edges. Typically, if there is one edge between two nodes in our graph, there will be several, because if it is legal to cut from one frame in the first sequence to another in the second, it will usually also be legal to cut between neighbors of these frames. This means that, for each pair of nodes in the graph, there is a matrix representing the weights of edges between the nodes. The i, j’th entry in this matrix represents the weight for a cut from the i’th frame in the first sequence to the j’th frame in the second sequence. The weight matrix for the whole graph is composed as a collection of blocks of this form. Summarizing the graph involves compressing these blocks using clustering.
<H2>4.2 Searching the Summaries</H2>
While searching this graph, we would like to be able to generate different alternative motions that achieve the same set of constraints. During the search, we need to find paths close to optimal solutions but do not require exact extrema, because they are too hard to find. This motivates a random search. We used the following search strategy:
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To appear in the ACM SIGGRAPH conference proceedings
1. Start with a set of n valid random “seed” paths in the graph G 2. Score each path and score all possible mutations 3. Where possible mutations are: (a) Delete some portion of the path and replace it with 0 or 1 hops. (b) Delete some edges of the path and replace them with their children 4. Accept the mutations that are better than the original paths 5. Include a few new valid random “seed” paths 6. Repeat until no better path can be generated through mutations
Intuitively the first mutation strategy replaces a clip with a (hopefully) better one and the second mutation strategy adjusts the detailed position of cut boundaries. Since we start new random “seed” paths at every iteration, the algorithm does not get stuck at a local optimum forever. Section 4.2.2 explains these mutations in more detail. Hard constraints are easily dealt with; we restrict our search to paths that meet these constraints. Typically hard constraints specify the frame (in a particular node) to be used at a particular time. We do this by ensuring that “seed” paths meet these constraints, and mutations do not violate them. This involves starting to sample the random paths from the hard constraint nodes and greedily adding sequences that get us to the next hard constraint if any. Since the path is sampled at the coarse level, a graph search can also be performed between the constraint nodes. At every iteration we check if the proposed mutation deletes a motion piece that has a hard constraint in it. Such mutations are rejected immediately. Note that here we assume the underlying motion graph is connected. Section 4.2.1 explains the constraints that we used in more detail. Notice that this algorithm is similar to MCMC search (a good broad reference to application of MCMC is
[Gilks et al. 1996]). However, it is difficult to compute proposal probabilities for the mutations we use, which are strikingly successful in practice. This is an online algorithm which can be stopped at anytime. This is due to the fact that edges in intermediate graphs G · · · G n also represent connections and are valid edges. Thus we do not have to reach the leaf graph G to be able to create a path (motion sequence). We can stop the search iteration, take the best path found so far, and create a motion sequence. If the sequence is not good enough, we can resume the search from where we left off to get better paths through mutations and inclusion of random paths. This allows an intuitive computation cost vs. quality tradeoff.
<H3>4.2.1 Evaluating a Path</H3>
Since during the search all the paths live in a subspace implied by the hard constraints, these constraints are always satisfied. Given a sequence of edges e 1 · · · e n , we score the path using the imposed soft constraints. For each constraint, we compute a cost where the cost is indicative of the satisfaction of the constraint. Based on the scores for each of the constraints, we weight and sum them to create a final score for the path (The S function in equation 1). We also add the sum of the costs of the edges along the path to make sure we push the search towards paths that are continuous. The weights can be manipulated to increase/decrease the influence of a particular soft constraint. We now have an expression of the form:
n
<H3>4.2.2 Mutating a Path S(e · · · e ) = w ∗ ∑ cost(e ) + w ∗ F + w ∗ B + w ∗ J (1)</H3>
1
1 n c i f b j
i=1
Figure 3: The two mutations are: deleting some portion of the path (top-left, crossed out in red) and replacing that part with another set of edges (top-right), and deleting some edges in the path (bottomleft) and replacing deleted edges with their children in our hierarchy (bottom-right)
Where w c ,w f ,w b and w j are weights for the quality (continuity) of the motion, how well the length of the motion is satisfied, how well the body constraints are satisfied and how well the joints constraints are defined. We selected these weights such that an error of 10 frames increases the total score the same amount as an error of 30 centimeters in position and 10 degrees in orientation. The scores F, B and J are defined as:
1. F: For the number of frame constraints, we compute the squared difference between the actual number of frames in the path and the required number of frames. 2. B: For body constraints, we compute the distance between the position and orientation of the constraint versus the actual position and orientation of the torso at the time of the constraint and sum the squared distances. The position and orientation of the body at the constraint times are found by putting the motion pieces implied by the subsequent edges together ( figure 1 ). This involves taking all the frames of motion toMotion(e i ) between frames f romFrame(e i+1 ) and toFrame(e i ) and putting the sequence of frames starting from where the last subsequence ends or from the first body constraint if there is no previous subsequence. Note that we require that we have at least two body constraints enforcing the position/orientation of the body at the beginning of the synthesized motion (so that we know where to start putting the frames down) and at the end of the synthesized motion. The first body constraint is always satisfied, because we always start putting the motions together from the first body constraint. 3. J: For joint constraints, we compute the squared distance between the position of the constraint and the position of the constrained joint at the time of the constraint and sum the squared distance between the two. To determine the configuration of the body at the time at which the constraint applies, we must assemble the motion sequence up to the time of the constraint; in fact, most of the required information such as the required transformation between start and end of each cut is already available in the dataset.
We implemented two types of mutations which can be performed quickly on an active path.
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Domain of smoothing Discontinuity Magnitude
Smoothed Signal
Figure 4: In the synthesized motion, discontinuities in orientation are inevitable. We deal with these discontinuities using a form of localized smoothing. At the top left, a discontinuous orientation signal, with its discontinuity shown at the top right. We now construct an interpolant to this discontinuity, shown on the bottom right and add it back to the original signal to get the continuous version shown on the bottom left. Typically, discontinuities in orientation are sufficiently small that no more complex strategy is necessary.
1. Replace a sequence by selecting two edges e i and e i+ j where 0 ≤ j ≤ n − i, deleting all the edges between them in the path and connecting the unconnected pieces of the path using one or two edges in the top level graph G (if possible). Since in the summarized graph, there are relatively fewer edges, we can quickly find edges that connect the two unconnected nodes by checking all the edges that go out from toMotion(e i ), and enumerating all the edges that reach to f romMotion(e i+ j ) and generate a valid path. Note that we enumerate only 0 or 1 hop edges (1 edge or 2 edge connections respectively). 2. Demoting two edges to their children and replacing them with one of their children if they can generate a valid path. Doing this mutation on two edges simultaneously allows us to compensate for the errors that would happen if only one of them was demoted.
We check every possible mutation, evaluate them and take the best few. Since the summary has significantly fewer edges than the original graph, this step is not very expensive. If a motion sequence cannot generate a mutation whose score is lower that itself, we decide that the current path is a local minimum in the valid path space and record it as a potential motion. This way, we can obtain multiple motions that satisfy the same set of constraints.
<H3>4.2.3 Creating and Smoothing the Final Path</H3>
We create the final motion by taking the frames between toFrame(e i ) and f romFrame(e i+1 ) from each motion toMotion(e i ) where 1 ≤ i < n ( figure 1 ). This is done by rotating and translating every motion sequence so that each piece starts from where the previous one ended. In general, at the frames corresponding to the edges in the path, we will have C 0 discontinuities, because of the finite number of motions sampling an infinite space. In practice these discontinuities are small and we can distribute them within a smoothing window around the discontinuity. We do this by multiplying the magnitude of the discontinuity by a smoothing function and adding the result back to the signal ( figure 4 ). We choose the smoothing domain to be ±30 frames (or one second of animation) around the discontinuity and
Figure 5: Body constraints allow us to put “checkpoints” on the motion: in the figure, the arrow on the right denotes the required starting position and orientation and the arrow on the left is the required ending position and orientation. All constraints are also time stamped forcing the body to be at the constraint at the time stamp. For these two body constraints, we can generate many motions that satisfy the constraints in real-time.
Figure 6: We can use multiple “checkpoints” in a motion. In this figure, the motion is required to pass through the arrow (body constraint) in the middle on the way from the right arrow to the left.
y( f ) = 0 − 0 2 1 ∗ 2 1 ( ∗ f ( −d+s f s −d+s s ) 2 ) 2 + 2 ∗ ( f −d+s s ) − 2 d d f f − ≤ > < s d d f ≤ + − ≤ f s s d < + d s
as the smoothing function that gives the amount of displacement for every frame f , where d is the frame of the discontinuity and s if the smoothing window size (in our case 30). To make sure that we interpolate the body constraints (i.e. having a particular position/orientation at a particular frame), we take the difference between the desired constraint state, subtract the state at the time of the constraint and distribute this difference uniformly over the portion of the motion before the time of the constraint. Note that these “smoothing” steps can cause artifacts like feet penetrating or sliding on the ground. However, usually the errors made in terms of constraints and the discontinuities are so small that they are unnoticeable.
<H2>4.3 Authoring Human Motions</H2>
Using iterative improvements of random paths, we are able to synthesize human looking motions interactively. This allows interactive manipulation of the constraints. This is important, because motion synthesis is inherently ambiguous as there may be multiple motions that satisfy the same set of constraints. The algorithm can find these “local minimum” motions that adhere to the same constraints. The animator can choose between them or all the different motions can be used to create a variety in the environment. Since the algorithm is interactive, the animator can also see the ambiguity and guide the search by putting extra constraints ( figure 6 ). Currently, we can constrain the length of the motion, the body’s position and orientation at a particular frame ( figure 5 ,6), a joint (e.g. head, hand) to a particular state at a particular frame ( figure 7 ), or constrain the entire body’s pose at a particular frame (figure 8). Notice that we can synthesize multiple interacting motions independently using hard constraints ( figure 9 ); we simply select the poses, position and orientation at which the figures interact and this framework fills in the missing motion, in a sense, interpolating the constraints. These are only a few of the constraints that can be implemented. As long as the user specifies a cost function that evaluates a motion and attaches a score that is indicative of the animator’s satisfaction with the path, many more constraints can be implemented. For example, if the motions in our database are marked with their individual stylistic attributes, we can also constrain the style of the desired motion by penalizing motions that do not have the particular style. In a computer game environment, we can constrain the synthesized motion to avoid obstacles in the environment. In such a case, body position/orientation constraints can also come from an underlying path planner. Thus, given high level goals (such as going from point A to point B, say) human looking motions can be generated automatically.
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To appear in the ACM SIGGRAPH conference proceedings
Figure 7: In addition to body constraints, joint constraints can be used to further assign “checkpoints” to individual joints. In this figure, the head of the figure is also constrained to be high (indicated by the blue line), leading to a jumping motion.
<H1>5 Results</H1>
We have presented a framework that allows interactive synthesis of natural looking motions that adhere to user specified constraints. We assess our results using four criteria. Firstly, the motion looks human. Secondly, the motions generated by the method do not have unnatural artifacts such as slipping feet on the ground or jerky movement. Third, the user specified constraints are satisfied, i.e. the motion passes through the required spot at the required time, or the character falls to a particular position ( figure 8 ). Finally, motions are generated interactively — typically depending on the quality of the path desired, an acceptable 300 frame motion is found in between 3 and 10 seconds on an average PC (Pentium III at 800 Mhz). This speed allows interactive motion authoring. For example, we generated the real-time screen captures in the attached video using a dataset of 60-80 unorganized, short (below 300 frames each) motion capture fragments. The average precomputation time required for this many motions (computing the motion graph) is 5 hours on the same computer. On average, the results shown in the video contain 3-30 motion pieces cut from the original motions. This framework is completely automatic. Once the input motions are selected, the computation of the hierarchic motion graph does not require any user intervention and the resulting representation is searched in real-time. For many kinds of constraints the motion synthesis problem is underconstrained; there are many possible combinations of motion pieces that achieve the same set of constraints. Randomized search is well suited to find many different motions that satisfy the constraints. On the other hand, some constraints, may not be met by any motion. In this case, randomized search will try to minimize our objective motion and find the “closest” motion. For example, if the user asks for 100 meters in 5 seconds, the algorithm will tend to put fast running motions together but not necessarily satisfying the constraints. Similarly, if the set of motions to begin with do not form a connected graph, the algorithm will perform searches confined to the unconnected graphs. If there are hard constraints in different unconnected components, we will not even be able to find starting seed paths. From this perspective, the selection of the database to work with is important. In our system, we used 60-100 football motions that have a strong bias towards motions that run forward. However, as the attached video suggest, the randomized search has no problem finding rare motions that turn back to satisfy the constraints. The motion databases that we used were unorganized except that we excluded football warming up and tackling motions unless they were desired ( figure 9 ). The randomized search scales linearly as a function of the database size with a very small constant. We have tried datasets of 50-100 motions without a noticeable change in the running time of the algorithm. The linearity in the running time comes from the linear increase in the number of alternative mutations at every step. Note that as the database size gets larger, the constant τ (Appendix A) that is used to create the edges can get lower since more motions mean that we expect to find better connections between motions, decreasing the number of edges. This will lead to a sublinear increase in the running time. The framework can work on any motion dataset: it can be created by traditional key framing, physically based modelling or motion capture. For example, we can take the motion data for “Woody” – who may well have been key-framed, from “Toy Story” and create new “Woody” motions automatically. The framework is also appli- cable to non-human motion synthesis. For example, this framework can be used to generate control signals for robots to achieve a particular task by generating the motion graph for previously known motion-control signal pairs. During the synthesis we can not only synthesize the final robot motion but also the associated control signals that achieve specific goals. Since the generated motions are obtained by putting pieces of motions in the dataset, the resulting motions will also carry the underlying style of the data. This way, we can take the motion data for one character, and produce more motions with the intrinsic style of the character.
Figure 8: Using hard constraints, we can force the figure to perform specific activities. Here, we constrain the end of the motion to be lying flat on the ground at a particular position/orientation and time. Our framework generates the required tipping and tumbling motion in real-time.
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<H1>6 Future Work</H1>
During the construction of the final motion, better ways of smoothing between adjacent motions could be used to improve realism
[Popovic 1999]. Using better post processing, motions could also be synthesized on non-uniform surfaces which the current framework cannot handle. Additional post processing may involve physically based modelling to make sure the synthesized motions are also physically correct. Automatic integration of higher level stylistic constraints could be incorporated into the framework, avoiding the arduous job of labelling every motion with the intrinsic style by hand. By analyzing patterns in the motion dataset, we might also infer these styles or obtain higher level descriptions [Brand and Hertzmann 2001]. The synthesized motions are strictly bound to the motions that were available in the original dataset. However, it is conceivable that the motions that are very close to the dataset could also be incorporated in the synthesizable motions using learned stylistic variations. The integrity of the original dataset directly effects the quality of the synthesized motion. For example, if the incoming motion dataset does not contain any “turning left” motions, we will not be able to synthesize motions that involve “turning left”. An automatic way of summarizing the portions of the “possible human motions” space that have not been explored well enough by the dataset could improve the data gathering and eventually the synthesized motions. This could also serve as a palette for artists: some portions of the precomputed motion graph can be paged in and out of memory depending on the required motion. For example, the animator could interactively select the motions that need to be used during the synthesis, and only the portion of the motion graph involving the desired motions could be loaded. This would give animators a tool whereby they can select the set of motions to work with in advance and the new motions will be created only from the artist selected set. Furthermore this encourages comprehensive re-use of motion data.
<H1>7 Acknowledgements</H1>
This research was supported by Office of Naval Research grant no. N00014-01-1-0890, as part of the MURI program. We would like to thank Electronic Arts for supplying us with the motion data.
<H1>A Appendix: Similarity Metric</H1>
We define the torso coordinate frame to be the one where the body stands centered at origin on the xz plane and looks towards the positive z axis. Any point p in the torso coordinate frame can be transformed to the global coordinate frame by T (s i ) + R(s i ) · p , where T (s i ) is the 3 × 1 translation of the torso and R(s i ) is the 3 × 1 rotation of the torso and R(s i ) represents the rotation matrix associated with the rotation. We wish to have a weight on edges of the motion graph (section 3.1) that encodes the extent to which two frames can follow each other. If the weight of an edge is too high, it is dropped from the graph. To compute the weight of an edge, we use the difference between joint positions and velocities and the difference between the torso velocities and accelerations in the torso coordinate frame. Let P(s i ) be a 3 × n matrix of positions of n joints for s i in torso coordinate frame. Equation 2 gives us the difference in joint position and body velocity.
2
D s i ,t j = [(P(s i ) − P(t j )) (|T (s i )| − |T (t j )|) (|R(s i )| − |R(t j )|) ]
We then define the normalizing matrices O and L in equation 3 and 4.
3
O = max s,i (|D s T i ,s i D s i ,s i+1 |)
4
L = max s,i (|D s T i ,s i D s i ,s i+1 |)
Then the cost function function in equation 5 is used to relate s i to t j .
5
C(s i ,t j ) = trace(D s i ,t j MO −1 D s T i ,t j + D s i ,t j T L −1 D s T i ,t j )
Where diagonal (n + 2) × (n + 2) matrices M and T are used to weight different joints differently. For example, position differences in feet are much more noticeable than position differences of hands because the ground provides a comparison frame. We have found M and T matrices empirically by trying different choices. Unfortunately, defining a universal cost metric is a hard problem. The metric defined above produces visually acceptable results. Using this cost metric, we create edges from s i to t j where C(s i ,t j ) < τ . For an edge e from s i to t j , we set cost(e) = C(s i ,t j ). τ is a user specified quality parameter that influences the number of edges in G. We have fixed this value so that cuts created between motions along the edges do not have visible artifacts. Note that an error that is visible on a short person may not be visible on an extremely large person. Thus, in theory, the weights must be adjusted from person to person. However, in practice, possible size variation of adult people is small enough that we used the same weights for different people without creating a visible effect.
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Figure 9: Using combinations of constraints, we can generate multiple motions interacting with each other. Here, the skeletons are constrained to be tackling each other (using hard constraints) at specified times and specified positions/orientations. We can then generate the motion that connects the sequence of tackles.
8
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"text": "['The sampling can also be done in the motion domain to pick clips of motions to establish certain simple constraints']",
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"text": "['Lamouret and van de Panne 1996']",
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"text": "['Schodl et al. 2000']",
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"text": "['a new texture (or motion) that looks like an example texture (or motion example) needs to be synthesized']",
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"text": "['A roadmap of all the motion examples can be constructed and searched to obtain a desired motion']",
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"text": "['Choi et al. 2000']",
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"text": "['Lee et al. 2002']",
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],
"text": "['Kovar et al. 2002']",
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"text": "['The clips in this roadmap can also be parameterized for randomly sampling different motion sequences']",
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"text": "['Li et al. 2002']",
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"text": "['The motion signals can also be clustered']",
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"text": "['The resulting Markov chain can be searched using dynamic programming to find a motion that connects two keyframes']",
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"text": "['Molina-Tanco and Hilton 2000']",
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"text": "['used in a variable length Markov model to infer behaviors']",
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"text": "['directly sampled from to create new motions']",
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"text": "['Galata et al. 2001']",
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6516,
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"text": "['Bowden 2000']",
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"text": "['This is similar to our work']",
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"text": "['our clustering method does not operate on body configurations']",
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"text": "['our probabilistic search strategy is more effective than dynamic programming']",
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"text": "['Types of probabilistic search algorithms have also been used in physically based animation synthesis']",
"type": "background_claim",
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"text": "['and rendering']",
"type": "background_claim",
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"text": "['Chenney and Forsyth 2000']",
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6887,
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],
"text": "['Veach and Guibas 1997']",
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6911,
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"text": "['Controller based approaches use physical models of systems and controllers that produce outputs usually in the form of forces and torques as a function of the state of the body']",
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"text": "['These controllers can be designed specifically to accomplish particular tasks']",
"type": "background_claim",
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"text": "['they can be learned automatically using statistical tools']",
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"text": "['Brogan et al. 1998']",
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7189,
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],
"text": "['Hodgins et al. 1995']",
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"offsets": [
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7272,
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],
"text": "['Grzeszczuk and Terzopoulos 1995']",
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"text": "['Grzeszczuk et al. 1998']",
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"text": "['Generating motion largely follows two threads']",
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"text": "['using examples']",
"type": "background_claim",
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"text": "['using controllers']",
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],
"text": "['Mataric 2000']",
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"offsets": [
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"text": "['The motion data can also be post processed to fix problems such as feet sliding on the ground or some constraints not being satisfied']",
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"text": "['Gleicher 1998']",
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],
"text": "['Lee and Shin 1999']",
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],
"text": "['Popovic 1999']",
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"id": "T93"
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],
"text": "['Rose et al. 1996']",
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"text": "['This usually involves optimization of a suitable displacement function on the motion signal']",
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"text": "['modifying motions appropriately is an interesting research problem']",
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"id": "T96"
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],
"text": "['Hodgins and Pollard 1997']",
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"text": "['motion cannot simply be transferred from one body size to another']",
"type": "background_claim",
"id": "T98"
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"text": "['Different body sizes move according to different time scales']",
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"text": "['This graph is not a particularly helpful representation']",
"type": "own_claim",
"id": "T100"
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"text": "['The collection of motion sequences could be represented as a directed graph']",
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"text": "['it is extremely large']",
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"text": "['it obscures the structure of the sequences']",
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"text": "['the cost attached to the edge is high']",
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"text": "['cutting from one sequence to another along an edge introduces a discontinuous motion']",
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"text": "['any sequence of edges e 1 · · · e n']",
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"text": "['is a valid path and defines a legal sequence of splices']",
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"text": "['tells us the cost of connecting the incident frames']",
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"text": "['edges connect frames']",
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"text": "['that they originate from and they point to']",
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"text": "['toMotion(e i ) = f romMotion(e i+1 )']",
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"text": "['toFrame(e i ) < f romFrame(e i+1 )']",
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"text": "['∀i']",
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"text": "['1≤i < n']",
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"text": "['figure 1']",
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"text": "['In this setting']",
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"text": "['We wish to construct paths in the motion graph that satisfy constraints']",
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"text": "['Many constraints cannot be satisfied exactly']",
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"text": "['We define hard constraints to be those that can (and must) be satisfied exactly']",
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"text": "['given two positions']",
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"text": "['there may not be any sequence of frames in the collection that will get us from the first position to the second position exactly']",
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"text": "['we can restrict ourselves to valid paths that pass through particular nodes at particular times']",
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"text": "['we can constrain the moving figure to be at a specific pose at a specific time']",
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"text": "['This enables us to search for motions such as jumping, falling, or pushing a button at a particular time']",
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"text": "['A soft constraint cannot generally be met exactly']",
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"text": "['we score sequences using an objective function that reflects how well the constraint has been met']",
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"text": "['attempt to find extremal sequences']",
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"text": "['Finding paths in the motion graph that satisfy the hard constraints and optimize soft constraints involves a graph search']",
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"text": "['even a small collection of motions']",
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"text": "['Unfortunately']",
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"text": "['the graph G has a large number of edges']",
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"text": "['straightforward search of this graph is computationally prohibitive']",
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"text": "['many perfectly satisfactory motions that satisfy the constraints equally well']",
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"text": "['The main reason is the need to enumerate many paths']",
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"text": "['There are']",
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"text": "['Example soft constraints include']",
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"text": "['The total number of frames should be a particular number']",
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"text": "['To appear in the ACM SIGGRAPH conference proceedings']",
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"text": "['The motion should not penetrate any objects in the environment']",
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"text": "['The body should be at a particular position and orientation at a particular time']",
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"text": "['at a specific time']",
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"text": "['A particular joint should be at a particular position']",
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"text": "['at a particular time']",
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"text": "['The motion should have a specified style']",
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"text": "['the squared distance between the position of the constraint and the actual position of the body at the time of the constraint.']",
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"text": "['there are many motions that satisfy these constraints']",
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"text": "['we require only that the person be at one end of a room at frame 0 and near the other end at frame 5000']",
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"text": "['The motion graph is too hard to search with dynamic programming']",
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"text": "['there are many valid paths that satisfy the constraints equally well']",
"type": "own_claim",
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"text": "['There may be substantial differences between equally valid paths']",
"type": "own_claim",
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13622,
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"text": "['whether you dawdle at one side of the room or the other is of no significance']",
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"text": "['This suggests summarizing the graph to a higher level and coarser presentation that is easier to search']",
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"text": "['Branch and bound algorithms are of no help here']",
"type": "own_claim",
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13894
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"text": "['very little pruning is possible']",
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"text": "['In such a representation, every level is a summary of the one finer level']",
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"id": "T160"
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14284
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"text": "['Coarser levels should have less complexity while allowing us to explore substantially different portions of the path space']",
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"text": "['to search the graph G in practical times']",
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"text": "['we need to do the search at a variety of levels where we do the large scale motion construction first and then “tweak” the details']",
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"text": "['the motion is continuous and satisfies the constraints as well as possible']",
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"text": "['there be at least one edge between the corresponding nodes in G']",
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15519,
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"text": "['there is a cut between two sequences represented by an edge between two nodes in G']",
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15672,
15694
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"text": "['this were not the case']",
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"text": "['our summary would rule out potential paths']",
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"text": "['most probably their preceding and succeeding frames also look similar']",
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15186,
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"text": "['two frames are similar']",
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"text": "['The cost function']",
"type": "own_claim",
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15056,
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"text": "['causes the P matrices to have non-infinite entries to form nearly elliptical groups']",
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"text": "['section A']",
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"text": "['figure 2']",
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"text": "['We would like to have a tree of graph representations whose root is G , and whose leaves are G']",
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"text": "['This is an instance of Tree-Structured Vector Quantization']",
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"text": "['Gersho and Gray 1992']",
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"text": "['in our summarized graph G']",
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"text": "['each edge is the root of a binary tree and represents all the edges in close neighborhood in terms of the edge labels']",
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"text": "['we would like to be able to generate different alternative motions that achieve the same set of constraints']",
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"text": "['While searching this graph']",
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"text": "['This motivates a random search']",
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"text": "['During the search, we need to find paths close to optimal solutions but do not require exact extrema']",
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"text": "['they are too hard to find']",
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"text": "['better one']",
"type": "own_claim",
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"text": "['Intuitively the first mutation strategy replaces a clip with a']",
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"text": "['the second mutation strategy adjusts the detailed position of cut boundaries']",
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"text": "['we start new random “seed” paths at every iteration']",
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"text": "['the algorithm does not get stuck at a local optimum forever']",
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"text": "['Section 4.2.2']",
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"text": "['Hard constraints are easily dealt with']",
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"text": "['we restrict our search to paths that meet these constraints']",
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"text": "['hard constraints specify the frame']",
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"text": "['to be used at a particular time']",
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"text": "['We do this by ensuring that “seed” paths meet these constraints, and mutations do not violate them']",
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"text": "['a graph search can also be performed between the constraint nodes']",
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"text": "['the path is sampled at the coarse level']",
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"text": "['the underlying motion graph is connected']",
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"text": "['this algorithm is similar to MCMC search']",
"type": "own_claim",
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{
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],
"text": "['Gilks et al. 1996']",
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20366,
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"text": "['Such mutations are rejected immediately']",
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"text": "['it is difficult to compute proposal probabilities for the mutations we use, which are strikingly successful in practice']",
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"id": "T200"
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],
"text": "['This is an online algorithm which can be stopped at anytime']",
"type": "own_claim",
"id": "T201"
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{
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20888,
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],
"text": "['edges in intermediate graphs G · · · G n also represent connections and are valid edges']",
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20982,
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],
"text": "['we do not have to reach the leaf graph G to be able to create a path']",
"type": "own_claim",
"id": "T203"
},
{
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21070,
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],
"text": "['We can stop the search iteration, take the best path found so far, and create a motion sequence']",
"type": "own_claim",
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{
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21170,
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],
"text": "['the sequence is not good enough']",
"type": "data",
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21203,
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],
"text": "['we can resume the search from where we left off to get better paths through mutations and inclusion of random paths']",
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21320,
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"text": "['This allows an intuitive computation cost vs. quality tradeoff']",
"type": "own_claim",
"id": "T207"
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{
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21542,
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],
"text": "['these constraints are always satisfied']",
"type": "own_claim",
"id": "T208"
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21458,
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],
"text": "['during the search all the paths live in a subspace implied by the hard constraints']",
"type": "data",
"id": "T209"
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"offsets": [
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21624,
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],
"text": "['we score the path using the imposed soft constraints']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
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21582,
21621
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],
"text": "['Given a sequence of edges e 1 · · · e n']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
21699,
21783
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],
"text": "['we compute a cost where the cost is indicative of the satisfaction of the constraint']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
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21678,
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],
"text": "['For each constraint']",
"type": "data",
"id": "T213"
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"offsets": [
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22054,
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],
"text": "['The weights can be manipulated to increase/decrease the influence of a particular soft constraint']",
"type": "own_claim",
"id": "T214"
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{
"offsets": [
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24267,
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],
"text": "['The first body constraint is always satisfied']",
"type": "own_claim",
"id": "T215"
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24322,
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],
"text": "['we always start putting the motions together from the first body constraint']",
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"id": "T216"
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"text": "['Doing this mutation on two edges simultaneously allows us to compensate for the errors that would happen if only one of them was demoted']",
"type": "own_claim",
"id": "T217"
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{
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26824,
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],
"text": "['this step is not very expensive']",
"type": "own_claim",
"id": "T218"
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{
"offsets": [
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26757,
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],
"text": "['the summary has significantly fewer edges than the original graph']",
"type": "data",
"id": "T219"
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{
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],
"text": "['we can obtain multiple motions that satisfy the same set of constraints']",
"type": "own_claim",
"id": "T220"
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{
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27051,
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],
"text": "['This way']",
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"id": "T221"
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{
"offsets": [
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27527,
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],
"text": "['at the frames corresponding to the edges in the path, we will have C 0 discontinuities']",
"type": "background_claim",
"id": "T222"
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{
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27626,
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],
"text": "['the finite number of motions sampling an infinite space']",
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27683,
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],
"text": "['In practice these discontinuities are small and we can distribute them within a smoothing window around the discontinuity']",
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"id": "T224"
},
{
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29479,
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],
"text": "['these “smoothing” steps can cause artifacts like feet penetrating or sliding on the ground']",
"type": "own_claim",
"id": "T225"
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{
"offsets": [
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29580,
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],
"text": "['usually the errors made in terms of constraints and the discontinuities are so small that they are unnoticeable']",
"type": "own_claim",
"id": "T226"
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{
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29825,
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],
"text": "['we are able to synthesize human looking motions interactively']",
"type": "own_claim",
"id": "T227"
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{
"offsets": [
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29779,
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],
"text": "['Using iterative improvements of random paths']",
"type": "data",
"id": "T228"
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"offsets": [
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29888,
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],
"text": "['This allows interactive manipulation of the constraints']",
"type": "own_claim",
"id": "T229"
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{
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],
"text": "['This is important']",
"type": "own_claim",
"id": "T230"
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29972,
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],
"text": "['motion synthesis is inherently ambiguous as there may be multiple motions that satisfy the same set of constraints']",
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"text": "['The algorithm can find these “local minimum” motions that adhere to the same constraints']",
"type": "own_claim",
"id": "T232"
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{
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],
"text": "['The animator can choose between them or all the different motions can be used to create a variety in the environment']",
"type": "own_claim",
"id": "T233"
},
{
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30332,
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],
"text": "['the animator can also see the ambiguity and guide the search by putting extra constraints']",
"type": "own_claim",
"id": "T234"
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{
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30424,
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],
"text": "['figure 6']",
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30302,
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"text": "['the algorithm is interactive']",
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],
"text": "['Currently, we can constrain the length of the motion, the body’s position and orientation at a particular frame']",
"type": "own_claim",
"id": "T237"
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],
"text": "['a joint']",
"type": "own_claim",
"id": "T238"
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30590,
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"text": "['to a particular state at a particular frame']",
"type": "own_claim",
"id": "T239"
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"text": "['constrain the entire body’s pose at a particular frame']",
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"text": "['figure 5']",
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"text": "['6']",
"type": "data",
"id": "T242"
},
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"offsets": [
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30578,
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],
"text": "['head']",
"type": "data",
"id": "T243"
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"offsets": [
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30584,
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],
"text": "['hand']",
"type": "data",
"id": "T244"
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],
"text": "['figure 7']",
"type": "data",
"id": "T245"
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"text": "['figure 8']",
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"id": "T246"
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30730,
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"text": "['we can synthesize multiple interacting motions independently using hard constraints']",
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"id": "T247"
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"text": "['figure 9']",
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"offsets": [
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"text": "['we simply select the poses, position and orientation at which the figures interact']",
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30915,
30957
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"text": "['this framework fills in the missing motion']",
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"id": "T250"
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{
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30971,
31000
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],
"text": "['interpolating the constraints']",
"type": "own_claim",
"id": "T251"
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{
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31002,
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],
"text": "['These are only a few of the constraints that can be implemented']",
"type": "own_claim",
"id": "T252"
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{
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],
"text": "['many more constraints can be implemented']",
"type": "own_claim",
"id": "T253"
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"text": "['the user specifies a cost function that evaluates a motion']",
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31141,
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"text": "['attaches a score that is indicative of the animator’s satisfaction with the path']",
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{
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31364,
31477
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],
"text": "['we can also constrain the style of the desired motion by penalizing motions that do not have the particular style']",
"type": "own_claim",
"id": "T256"
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{
"offsets": [
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31281,
31362
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],
"text": "['the motions in our database are marked with their individual stylistic attributes']",
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],
"text": "['we can constrain the synthesized motion to avoid obstacles in the environment']",
"type": "own_claim",
"id": "T258"
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{
"offsets": [
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31606,
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],
"text": "['body position/orientation constraints can also come from an underlying path planner']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
[
31479,
31509
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],
"text": "['In a computer game environment']",
"type": "data",
"id": "T260"
},
{
"offsets": [
[
31590,
31604
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],
"text": "['In such a case']",
"type": "data",
"id": "T261"
},
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"offsets": [
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31765,
31817
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],
"text": "['human looking motions can be generated automatically']",
"type": "own_claim",
"id": "T262"
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31697,
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],
"text": "['given high level goals']",
"type": "data",
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],
"text": "['We have presented a framework that allows interactive synthesis of natural looking motions that adhere to user specified constraints']",
"type": "own_claim",
"id": "T264"
},
{
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],
"text": "['the motion looks human']",
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"id": "T265"
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{
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32462,
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],
"text": "['the motions generated by the method do not have unnatural artifacts such as slipping feet on the ground or jerky movement']",
"type": "own_claim",
"id": "T266"
},
{
"offsets": [
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32592,
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],
"text": "['the user specified constraints are satisfied']",
"type": "own_claim",
"id": "T267"
},
{
"offsets": [
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32643,
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],
"text": "['the motion passes through the required spot at the required time']",
"type": "data",
"id": "T268"
},
{
"offsets": [
[
32712,
32756
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],
"text": "['the character falls to a particular position']",
"type": "data",
"id": "T269"
},
{
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32759,
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],
"text": "['figure 8']",
"type": "data",
"id": "T270"
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{
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],
"text": "['motions are generated interactively']",
"type": "own_claim",
"id": "T271"
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{
"offsets": [
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32874,
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],
"text": "['an acceptable 300 frame motion is found in between 3 and 10 seconds on an average PC']",
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"id": "T272"
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{
"offsets": [
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32828,
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],
"text": "['depending on the quality of the path desired']",
"type": "data",
"id": "T273"
},
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"offsets": [
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32985,
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],
"text": "['This speed allows interactive motion authoring']",
"type": "own_claim",
"id": "T274"
},
{
"offsets": [
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33205,
33267
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],
"text": "['The average precomputation time required for this many motions']",
"type": "background_claim",
"id": "T275"
},
{
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33297,
33328
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],
"text": "['is 5 hours on the same computer']",
"type": "background_claim",
"id": "T276"
},
{
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33431,
33469
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],
"text": "['This framework is completely automatic']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
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33508,
33593
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],
"text": "['the computation of the hierarchic motion graph does not require any user intervention']",
"type": "own_claim",
"id": "T280"
},
{
"offsets": [
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33476,
33506
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],
"text": "['the input motions are selected']",
"type": "data",
"id": "T281"
},
{
"offsets": [
[
33653,
33731
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],
"text": "['For many kinds of constraints the motion synthesis problem is underconstrained']",
"type": "own_claim",
"id": "T282"
},
{
"offsets": [
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33598,
33651
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],
"text": "['the resulting representation is searched in real-time']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
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33733,
33827
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],
"text": "['there are many possible combinations of motion pieces that achieve the same set of constraints']",
"type": "data",
"id": "T284"
},
{
"offsets": [
[
33829,
33921
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],
"text": "['Randomized search is well suited to find many different motions that satisfy the constraints']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
33942,
33988
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],
"text": "['some constraints, may not be met by any motion']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
34004,
34093
]
],
"text": "['randomized search will try to minimize our objective motion and find the “closest” motion']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
33990,
34002
]
],
"text": "['In this case']",
"type": "data",
"id": "T288"
},
{
"offsets": [
[
34154,
34261
]
],
"text": "['the algorithm will tend to put fast running motions together but not necessarily satisfying the constraints']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
34111,
34152
]
],
"text": "['the user asks for 100 meters in 5 seconds']",
"type": "data",
"id": "T290"
},
{
"offsets": [
[
34341,
34411
]
],
"text": "['the algorithm will perform searches confined to the unconnected graphs']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
34277,
34339
]
],
"text": "['the set of motions to begin with do not form a connected graph']",
"type": "data",
"id": "T292"
},
{
"offsets": [
[
34416,
34478
]
],
"text": "['there are hard constraints in different unconnected components']",
"type": "data",
"id": "T293"
},
{
"offsets": [
[
34480,
34532
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],
"text": "['we will not even be able to find starting seed paths']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
34557,
34612
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],
"text": "['the selection of the database to work with is important']",
"type": "own_claim",
"id": "T295"
},
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"text": "['the randomized search has no problem finding rare motions that turn back to satisfy the constraints']",
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"text": "['figure 9']",
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"text": "['that is used to create the edges can get lower']",
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"text": "['Appendix A']",
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"text": "['more motions mean that we expect to find better connections between motions, decreasing the number of edges']",
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"text": "['it can be created by traditional key framing, physically based modelling or motion capture']",
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"text": "['we can take the motion data for “Woody”']",
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"text": "['from “Toy Story” and create new “Woody” motions automatically']",
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"text": "['The framework is also appli- cable to non-human motion synthesis']",
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"text": "['this framework can be used to generate control signals for robots to achieve a particular task by generating the motion graph for previously known motion-control signal pairs']",
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"text": "['During the synthesis we can not only synthesize the final robot motion but also the associated control signals that achieve specific goals']",
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"text": "['the generated motions are obtained by putting pieces of motions in the dataset']",
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"text": "['we can take the motion data for one character, and produce more motions with the intrinsic style of the character']",
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"text": "['During the construction of the final motion, better ways of smoothing between adjacent motions could be used to improve realism']",
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"text": "['Popovic 1999']",
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"text": "['motions could also be synthesized on non-uniform surfaces which the current framework cannot handle']",
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"text": "['Using better post processing']",
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"text": "['Additional post processing may involve physically based modelling to make sure the synthesized motions are also physically correct']",
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"text": "['By analyzing patterns in the motion dataset, we might also infer these styles or obtain higher level descriptions']",
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"text": "['Automatic integration of higher level stylistic constraints could be incorporated into the framework, avoiding the arduous job of labelling every motion with the intrinsic style by hand']",
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"text": "['Brand and Hertzmann 2001']",
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"text": "['The synthesized motions are strictly bound to the motions that were available in the original dataset']",
"type": "own_claim",
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"text": "['it is conceivable that the motions that are very close to the dataset could also be incorporated in the synthesizable motions using learned stylistic variations']",
"type": "own_claim",
"id": "T326"
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38056,
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"text": "['The integrity of the original dataset directly effects the quality of the synthesized motion']",
"type": "own_claim",
"id": "T327"
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"text": "['we will not be able to synthesize motions that involve “turning left”']",
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"text": "['the incoming motion dataset does not contain any “turning left” motions']",
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"text": "['An automatic way of summarizing the portions of the “possible human motions” space that have not been explored well enough by the dataset could improve the data gathering and eventually the synthesized motions']",
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"text": "['This could also serve as a palette for artists']",
"type": "own_claim",
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"text": "['the animator could interactively select the motions that need to be used during the synthesis']",
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"text": "['some portions of the precomputed motion graph can be paged in and out of memory depending on the required motion.']",
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"text": "['the new motions will be created only from the artist selected set']",
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A33 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A33_C03_Capturing_and_Animating_Occluded_Cloth_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/1275808.1276420
<Title>Capturing and Animating Occluded Cloth</Title>
Keenan Crane Ryan White† of California Berkeley
†University
Figure 1: We reconstruct a stationary sleeve using thousands of markers to estimate the geometry (texture added with bump mapping).
<Abstract>We capture the shape of moving cloth using a custom set of color markers printed on the surface of the cloth. The output is a sequence of triangle meshes with static connectivity and with detail at the scale of individual markers in both smooth and folded regions. We compute markers’ coordinates in space using correspondence across multiple synchronized video cameras. Correspondence is determined from color information in small neighborhoods and refined using a novel strain pruning process. Final correspondence does not require neighborhood information. We use a novel data driven hole-filling technique to fill occluded regions. Our results include several challenging examples: a wrinkled shirt sleeve, a dancing pair of pants, and a rag tossed onto a cup. Finally, we demonstrate that cloth capture is reusable by animating a pair of pants using human motion capture data.</Abstract>
D.A. Forsyth University of Illinois
Urbana Champaign
<H1>1 Introduction</H1>
We capture the motion of cloth using multiple video cameras and specially tailored garments. The resulting surface meshes have an isometric parameterization and maintain static connectivity over time. Over the course of roughly half a dozen papers on cloth capture a prevailing strategy has emerged. First, a pattern is printed on the cloth surface such that small regions of the pattern are unique. Next, correspondence is determined by matching regions across multiple views. The 3D location of a region is determined by intersecting rays through the corresponding observations in the image set ( figure 4 ). Reconstruction is done independently on a frame by frame basis and the resulting data is smoothed and interpolated. Previous work, such as [Scholz et al. 2005], yields pleasing results.
Little work has been done to capture garments with folds and scenes with occlusion. In this paper we use folding to refer to local phenomena such as wrinkles around a knee and occlusion to refer to large scale effects such as one limb blocking the view of another. Folds and occlusion are common, especially when dealing with real garments such as pants where limbs block interior views and cloth collects around joints. Both phenomena are symptoms of the same problem: views of the surface are blocked by other parts of the surface. However, there is a distinction in scale and different methods are required to solve each problem. When a surface is heavily folded, contiguous visible regions are often small and oddly shaped. In these regions correspondence is essential for detailed reconstruction yet can be challenging to identify. We solve the correspondence problem both by improving the pattern printed on the surface of the cloth and by improving the method used to match regions. Our method gets more information per pixel than previous methods by drawing from the full colorspace instead of a small finite set of colors in the printed pattern. Additionally, because cloth cannot stretch much before ripping, we use strain constraints to eliminate candidates in an iterative search for correspondence. In combination, these two modifications eliminate the need for neighborhood information in the final iteration of our algorithm. As a result, we determine correspondence using regions that are 25 times smaller than in previous work ( figure 6 ). Many regions on the surface are impossible to observe due to occlusion. We fill these holes using reconstructions of the same surface region taken from other points in time. We found that MeshIK ([Sumner et al. 2005]), a tool originally developed for mesh posing and animation, is appropriate for filling holes in cloth. In fact, MeshIK is well-suited to cloth data and we use it to bind reconstruction of our pants to motion capture data. We suggest two tools to evaluate marker-based capture systems. The first, markers per megapixel, is a measure of efficiency in capture systems. Efficiency is important because camera resolution and bandwidth are expensive: the goal is to get more performance from the same level of equipment. This metric is designed to predict scaling as technology moves from the research lab to the professional studio. The second tool is information theory: we look at the predictive power of different cues in a capture system. By doing simple bit calculations, we direct our design efforts more appropriately.
Acquisition (Section 4)
Mesh Processing (Section 5)
images Process color neighborhoods Match local with reconstruct 3D Prune strain with
Figure 2: We construct an animated sequence of surface meshes in two stages: acquisition and mesh processing. In acquisition, we convert raw images into a 3D point cloud. In mesh processing, we triangulate the mesh, fill the holes and apply temporal smoothing.
<H1>2 Previous Work</H1>
Previous work in cloth motion capture has focused on placing high density markers in correspondence between multiple views. The primary challenge is to increase marker density while correctly assigning correspondence between markers. We suggest markers per megapixel as an appropriate metric for comparison ( figure 3 ) because it measures the method instead of the equipment. Most high density full frame-rate capture has focused on cloth, however, there has been some recent work enhancing human motion capture
[Park and Hodgins 2006]. These methods have far fewer markers per megapixel because they affix individual markers. When working with cloth, markers are typically painted on the surface.
These markers can be broken into three categories: complex surface gradients
[Pritchard and Heidrich 2003; Scholz and Magnor 2004; Hasler et al. 2006] (typically detected using SIFT descriptors
[Lowe 2004]), intersecting lines
[Tanie et al. 2005] and regions of constant color
[Guskov and Zhukov 2002; Guskov et al. 2003; Scholz et al. 2005]. Our work falls in the third category: regions of contant color. We evaluate previous work by examining the quality of the reconstructed cloth in still images and video. The most common errors are marker mismatches and are observable in reconstructions by local strain in the reconstructed surface. Overall, we observe that constant color markers perform the best. [Pritchard and Heidrich 2003]
used cloth with unique line drawings as markers. Their work identifies parameterization as one of the key aspects of cloth capture. They use a stereo camera to acquire 3D and SIFT descriptors to establish correspondence. These descriptors are often mismatched and require significant pruning. They introduce a rudimentary strain metric, as measured along the surface, to rule out incorrect matches. While successful, their static reconstructions show numerous correspondence errors. The real-time system described in
[Guskov et al. 2003]
introduces markers of constant color, resulting in significantly fewer correspondence errors than in [Pritchard and Heidrich 2003]. This system uses a Kalman smoothing filter and is heavily damped. Additionally, the complexity of the color pattern limits the method to simple geometry. [Scholz et al. 2005]
improve upon [Guskov et al. 2003] by creating a non-repeating grid of color markers. Each marker has five possible colors and all three by three groups are unique. This allows substantially larger sections of cloth and virtually eliminates correspondence errors. Results include a human wearing a shirt and a skirt captured using eight 1K x 1K cameras. However, the range of motion is limited to avoid occlusion (e.g., arms are always held at 90 degrees to the torso). They use thin-plate splines to fill holes. [White et al. 2005]
introduce a combined strain reduction/bundle adjustment that improves the quality of the reconstruction by minimizing strain while reconstructing the 3D location of the points on the surface of the cloth. [White et al. 2006]
introduce the use of silhoutte cues to improve reconstruction of difficult to observe regions. While silhouette cues improve reconstruction, hole filling is
point static cloud Hole Temporally connectivity Mesh fill smooth triangle mesh Section 5.1 Section 5.2 Section 5.3
Markers per Work Megapixels Markers† Megapixel Park 2006 48 ≤ 350 ≤ 7.3 Tanie 2005 10 407.9 40 Guskov 2003 0.9 ≤ 136 ≤ 148 Scholz 2005 8 ≤ 3500 ≤ 434 Sleeve 15 7557 504 Pants 2.5 2405.3 979
more effective in many circumstances because it enforces an appropriate prior on the shape of the cloth. We make three main contributions: we improve the color pattern and matching procedure to get more information per marker, we introduce strain constraints to simplify correspondence and we create a data driven hole filling technique that splices previously captured cloth into the mesh. As a result, our system is capable of capturing a full range of motion with folding and occlusion.
<H1>3 Analyzing Acquisition Methods</H1>
To acquire a 3D point cloud of the cloth surface, we print a colored pattern on the cloth, sew it together, and record its motion using multiple synchronized cameras. We then reconstruct the 3D location of surface points by detecting corresponding points in multiple views ( figure 4 ). Our goal is high marker density in the 3D reconstruction – especially in regions with high curvature. To achieve this, we need markers that are both small in scale and highly discriminative. These two goals are in tension: small markers are less discriminative. In addition, we cannot increase camera resolution without bound because camera bandwidth becomes very expensive. As a result, we opt for the smallest markers that we can reliably detect and we make small markers more distinctive. We combine information from three cues to establish correspondence: marker color, neighboring markers and strain constraints in the reconstruction. Marker color and strain constraints are more useful than neighboring markers because they place fewer requirements on local cloth geometry. Specifically, neighboring markers are observed only when the cloth is relatively flat. When the surface is heavily curved only small portions of the surface are visible before the cloth curves out of view. In subsequent sections we adopt the following strategy: maximize information obtained from marker color and eliminate the information needed from neighbors.
parametric domain
<H2>3.1 Entropy as an Analytical Tool</H2>
We optimize our correspondence technique by analyzing the information provided by different cues. In this framework we can accurately minimize the number of neighbors required for correspondence and observe folds better. We can compare our work to previous methods using this framework ( figure 6 ). It takes log 2 M bits to determine the identity of each observed marker on a garment with M total markers. Because independent information adds linearly, we can compute the information needed to meet this threshold by adding information from the different cues: color, neighbors and strain. However, structural ambiguities in the pattern subtract information lost to determine which neighbor is which. As a result, we compute our information budget (I ) as:
N = number of observed neighbors C = color information per marker A = information lost to structural ambiguities S = information gained from strain constraints I = (N + 1) ∗C + S − A
As an example, imagine a rectangular grid of markers and a correspondence method that uses a single immediate neighbor. This neighbor is one of four possible neighbors – thus it takes two bits to specify which neighbor we found (A = 2). In this case, the equation reduces to I = 2 ∗C − 2 + S. Given almost any structured pattern, we can detect regions by increasing N until I > log 2 (M) bits. However, larger marker regions have the disadvantage that curvature can cause local occlusions and prevent observation of the entire region. Our best efforts are to improve C – the number of bits from each marker observation. We do this by picking marker color from the full colorspace instead of a small discrete set of colors.
<H2>3.2 Garment Design and Color Processing</H2>
We print a random colored pattern on the surface of cloth in an attempt to maximize the information available per pixel. While our pattern is composed of tesselated triangles ( figure 5 ), any shape that tiles the plane will work (squares and hexagons are also natural choices). To maximize the density of reconstructed points, we print the smallest markers that we can reliably detect. To maximize the information contained in the color of each marker, we print colors that span the gamut of the printer-camera response, then use a gaussian color model (section 4.1). From a system view, the printer-camera response is a sequence of lossy steps: we generate a color image on a computer, send the image to the printer, pose the cloth, and capture it with a camera. Our experiments suggest that loss is largely attributable to camera response because larger markers produced substantially more information. Illumination is also problematic and takes two forms: direct illumination on a lambertian surface and indirect illumination. To correct for variations in direct illumination, we remove the luminosity component from our color modelling. We do not correct for indirect illumination. Each marker in the printed pattern has a randomly chosen color, subject to the constraint that neighboring marker colors must be dissimilar. In the recognition stage, we detect markers by comparing colors to a known color. These comparisons must be made in the proper color space: we photograph the surface of the printed cloth with our video cameras to minimize the effect of non-linearities in the printing process.
<H1>4 Acquisition</H1>
The goal of our acquisition pipeline is to compute correspondence using minimal neighborhoods. We accomplish this through an iterative algorithm where we alternate between computing correspondence and pruning bad matches based on those correspondences. After each iteration we shrink the size of the neighborhood used to match. We start with N = 3 and end with N = 0. In the final iteration, markers are matched using color and strain alone. This iterative approach allows us to match without neighborhoods. This is better than label propagation methods. To be successful, propagation methods [Guskov et al. 2003; Scholz et al. 2005; Lin
parametric domain
1 st 2 nd 4 th [Scholz iteration iteration iteration 2005]
Relative Area 15.8 11.8 4.0 100
C
Color ≥ 5 ≥ 5 ≥ 5 1.93
Neighbors (N) 3 2 0 8
S
Strain 0 ∼ 7 ∼ 9 –
Ambiguities (A) 1.6 1.6 0 3 Total bits (I ) 18.4 20.4 14 14.4
Figure 6:
Our correspondence algorithm iterates from large to small regions. At each stage, the number of recovered bits must stay above the marker complexity (11.6 bits for our pants). We are able to obtain significantly more information per unit cloth surface area than previous work. See section 3.1 for the entropy equation and appendix B for detailed analysis.
and Liu 2006] require large sections of unoccluded cloth and must stop at occluding contours. As shown in figure 5 , occluding contours are both common and difficult to detect. In contrast, our iterative approach relies on strain constraints – which require computing the distance between a point and a line, and color detection – which requires averaging color within a marker. Both of these computations are easier than detecting occluding contours. We describe our acquisition pipeline, shown in figure 2 , below. Color Processing: We compare observed colors with stored values using a gaussian noise model. Our gaussian noise model has a single free parameter, the variance, which must be computed empirically for each recording setup. This variance determines the color response for the entire setup — smaller variances mean more bits from color. At this stage, we compute color information for each marker and eliminate hypothetical correspondences from further consideration that have large color differences. Neighborhood Matching: At each iteration, we match highly distinctive neighborhoods by combining information across cues. The size of the neighborhood is chosen so that we get more than enough bits to meet our information budget (log 2 M bits – typically 11 to 13). The analysis in figure 6 shows that we can set N = 3 at the start and continue until N = 0. Because the identity of the marker is overspecified, there are few mistakes. This approach works from flat regions in the first iteration to foldy regions in the later iterations. In the first iteration, we require three neighbors to make a match. In heavily folded regions, often neighboring markers on the image do not neighbor on the surface of the cloth. As such, these regions are not going to match. In contrast, in the last iteration, no neighbors are necessary. Occluding contours, which are common in heavily folded regions, no longer disrupt the matching procedure. 3D Reconstruction: Markers that are observed in multiple views (at least 2) are reconstructed in 3D using textbook methods
[Hartley and Zisserman 2000]. We use reprojection error to prune bad matches (reprojection errors average 0.3 pixels and we discard points with errors larger than 2 pixels). Pruning with Strain: We do two separate strain pruning steps: one on reconstructed 3D points and one on marker observations in each image. The first discards reconstructed points that cause physically unrealistic strain on the surface of the mesh and the second constrains our search for correspondence. Our strain constraint is based on the work of
[Provot 1995]
who noted that strain in cloth does not exceed 20% in practice. Relaxing the constraint to distances in 3D (surface distance is always more than the distance in 3D), we can use strain to exclude possible correspondences. Strain naturally fits in to our information theory framework: if strain excludes 87.5% of the possible correspondences, then strain has added 3 bits (because log 2 (1 − 0.875) = −3). The strain cue is described in figure 7 .
A A B = possible identities for B = locations too close to A
Figure 7: Top: we compute the shortest distance between a known point A and the eye ray through unidentified image point B. Bottom: in the parametric domain, this distance restricts the possible identities of B to the green region. The distance from A to B along the surface can be no shorter than the shortest distance in 3D.
<H2>4.1 Representation</H2>
To find correspondence, we match each image marker to a marker in the parametric domain. To do this, we define affinities a i, j between image marker i and parametric marker j. Each affinity is a product over different cues. We write c i, j ∈ [0, 1] for the color affinity, d(C i ,C j ) for the color distance between i and j, s i, j ∈ {0, 1} for the strain constraint, n i for the image neighbors of marker i and N j for the parametric neighbors of marker j:
a i, j = c i, j s i, j ∏ max c k,l l∈N j k∈n i d(C i ,C j ) 2 c i, j = exp (− 2 σ 2 ) 0 if a strain constraint is violated s i, j = 1 if not
When only one affinity for image marker i is above a theshold, then we declare a correspondence. Initially, we learned this threshold from labelled data, but we found that changing it by several orders of magnitude had little effect on our results. Subsequently, we use the value 10 −5(N+1) where N is the number of neighbors.
<H1>5 Mesh Processing</H1>
In the acquisition process, occlusion inevitably creates holes in the reconstructed mesh ( figure 8 ). One would like to fill these holes with real cloth. One of our major contributions is a data driven approach to hole filling: we fill holes with previously observed sections of cloth. Our work differs from
[Anguelov et al. 2005] because our hole filling procedure does not assume a skeleton that drives the surface and our procedure estimates a single coefficient per example. This hole filling procedure has a number of requirements: the missing section needs to be replaced by a section with the same topology; the new section needs to obey a number of point constraints around the edge of the hole, and the splicing method should respect properties of cloth (specifically strain). We select a reconstruction technique based on deformation gradients [Sumner and Popovic 2004]. In this approach, we fit deformation gradients for the missing section to a combination of deformation gradients in other observed sections. Then, we reconstruct the point locations from the deformation gradients. This procedure has a number of advantages. First, deformation gradients naturally yield cloth like properties. Deformation gradients are the transformation matrix between triangles in two poses of the mesh. By penalizing elements that deviate in this matrix, we have a fairly direct penalty on large changes in scale or strain. In contrast, methods based on the Laplacian of the mesh (
[Sorkine et al. 2004]
) do little to penalize these strains and can show many artifacts around the edge of the mesh. Second, deformation gradients can be converted into vertex locations by inverting a linear system, allowing us to specify vertex locations as constraints. Methods such as [Lipman et al. 2005]
don’t allow vertex constraints. Our subsequent discussion is divided into three sections: constructing a mesh from the point cloud, filling the holes in the mesh using deformation gradients, and temporally smoothing the results.
Example Meshes
constraints examples
triangles: multiview single view (unused) unobserved errors (discarded) seam backface
<H2>5.1 Meshing and Seams</H2>
We produce a mesh by forming equilateral triangles for sections of cloth that are printed with a contiguous pattern by referencing the triangle stucture of markers on the cloth. Our recovered markers are at the center of each triangle – so we average points to get out the vertices and subsequently the original mesh. We insert artificial points where two pieces of fabric come together. These points are created once per garment by hand clicking on photos of the each seam. The 3D locations of these points are recreated in each frame by averaging points near the seam.
<H2>5.2 Hole Filling</H2>
We use occlusion free meshes from other frames to automatically interpolate holes. For each hole in each frame, we cut out the missing region plus a ring of two triangles around the region. We select a set of examples of the enlarged region, then use MeshIK (
[Sumner et al. 2005]) to reconstruct the surface. MeshIK works by choosing a combination of deformation gradients from the examples and then solving for the missing point locations. We use the points from the ring of known triangles around the hole as constriants in MeshIK. The most restrictive aspect of MeshIK is that it requires example meshes without holes. In practice, we never observe complete ex- ample meshes – each mesh is missing some triangles. These holes appear in different places in different meshes and we create complete meshes in an iterative method. First, we fill all holes with a naive linear algorithm (specifically, we triangulate across gaps and use barycentric coordinates to place the missing points – this gets the job done, but works poorly). Then, we do another pass through all the data, where we replace the linear sections with sections created using MeshIK on the linearly filled data. To downweight the linear data, we select the examples with the highest percentage of viewed points in the missing section. These frames are then used as examples in MeshIK to hole fill in the rest of the sequence. For the pants capture, we iteratively refine a set of 27 extreme poses which were captured specifically for filling holes. The advantage of this apporach is that the example poses are chosen to capture the relevant degrees of freedom – yielding better results. For the cloth toss sequence, we chose the simpler approach: iteratively refine the entire sequence.
solution ? MeshIK
<H2>5.3 Smoothing</H2>
We introduce flexibility preserving smoothing – a method similar to anisotropic diffusion
[Perona and Malik 1990] that smoothes near-rigid movement without effecting flexible deformation. Typical temporal smoothing is dangerous because fast non-rigid movements can easily become physically implausible when blurred over time. However, because fast non-rigid regions of the cloth are complex, small temporal errors are often difficult to notice. In contrast, small errors in regions of the cloth that move rigidly are typically easy to observe. As a result we use flexibility preserving smoothing, a procedure that smoothes rigid movement more heavily than non-rigid movement. To do this, we take a local region around each vertex in the mesh (typically 25 points) and compute a rigid transformation to previous and subsequent frames. Aligning the regions with this transformation, we compute the movement of the vertices in this reference frame as a proxy for rigidity. Large variations in location indicate non-rigid movement and consequently receive little smoothing. Smaller variations indicates rigid movement and benefit from more substantial smoothing. We use a size adjusted gaussian to smooth in this reference frame.
<H1>6 Results and Applications</H1>
Our video sequences were taken with synchronized firewire cameras (Foculus FO214C) with a capture resolution of 640 x 480 and a capture rate of 24 frames per second. Our still captures were taken using a digital SLR camera and then downsampled to approximate available video resolutions. We use the automated calibration technique in
[White and Forsyth 2005], but any standard calibration will work ([Zhang 2002] and [Bouguet 2005]
are good choices). In the pants sequences, we used seven lights totalling 1550 Watts to illuminate the scene. Adequate lighting is critical: from our experience fewer lights degrade performance due to harsh shadows and dim lighting causes motion blur through slower shutter speeds. Our cloth was printed by a digital mail order fabric printing service. On a P4 2.4 GHz machine, acquisition takes roughly 6 minutes and mesh processing 2 minutes per frame. Code is written in MATLAB.
observed unobserved backface
<H2>6.1 Capture Results</H2>
Our capture results are best evaluated by looking at our video and figures 1,12,13. However, to compare against other capture techniques, it is also necessary to evaluate on several numerical criteria for each capture session: cloth pants table drop dance cloth sleeve† # cameras 6 8 18 10 resolution 640x480 640x480 900x600 1500x1000 total markers 853 3060 4793 13465 recovered 819 2405 4361 7557 percentage 96% 79% 91% 56% bits needed 9.7 11.6 12.2 13.7 color bits 6.1 5.1 6.4 4.5 strain bits 9.1 9.4 11.4 ∼ 6.6 †The sleeve example is unique because it was one of the first items we captured. Much of the cloth is in contact with the floor and unobservable – yielding fewer bits of strain. In addition, the camera images were not output in a linear color space, reducing the number of color bits. As a result, we terminated the correspondence algorithm at N = 2. Our pants animation is by far the most challenging, and we analyze some of the details a little more closely. With an average of 2405 observed markers, there were 979 3D markers per megapixel. If we factor out the pixels lost to background, we get 3500 3D markers per foreground megapixel or 282 foreground pixels per recovered 3D marker. Our marker observations average 56 pixels per marker per image. There are several reasons for the discrepancy: markers must be observed multiple times (approx 44% of 3D markers are observed in 3 or more views), some markers are observed but not reconstructed (due to errors or missing correspondence), and many pixels are not considered part of a marker: they lie in heavy shadow
or occupy the edge between two markers (approx 35% of pixels). 6.2 Retargeting Animations
We use a small set of captured frames (the previous basis of the 27 examples) in combination with MeshIK to skin skeletal human motion capture data ( figure 11 ). This approach covers a reasonably large range of motion, but ignores cloth dynamics. The largest challenge is that captured cloth meshes contain only points on the cloth surface, so we do not know joint locations. Instead, we insert proxy points for knee and hip joints in each of our basis meshes. These points are then connected to a small set of nearby triangles in the original mesh. For each frame of animation we set the proxy points’ locations according to joint angles in the skeletal mocap data. The resulting transformed joints are used as constraint points in MeshIK, which produces the final output meshes. Using our MATLAB implementation of MeshIK, this process takes around 5-10 seconds per frame. We use the same 27 bases poses for MeshIK based reconstruction. In order for a small basis to adequately express a full range of motion, each basis pose must be an extreme configuration. For simple objects such as a cylinder, a small bend (for example) is sufficient to extrapolate to a larger bend
[Sumner et al. 2005]. However, for pants the relationship is more complex: the fact that no folding occurs in a small bend does not imply that folding will be absent in a larger bend. Conversely, if a decent amount of folding occurs in a small bend, we do not expect extreme folds in a corresponding larger bend. As a result, MeshIK is most useful when a basis is carefully chosen to prevent extrapolation artifacts. One drawback to our approach is the loss of secondary kinematic motion, such as the sway of loose cloth. Because MeshIK does not use velocity information, the resulting animation appears damped.
<H1>7 Discussion</H1>
We have brought cloth capture from constrained laboratory examples to real settings by providing robust methods for dealing with occlusion and folding. Like human motion capture, this tool requires significant engineering effort. Camera setup and calibration are time consuming and the equipment is costly. However, once these obstacles have been overcome, capturing large amounts of data is relatively easy. So that other researchers can benefit from our work, we are releasing our capture data at http://www.ryanmwhite.com/data. In our video, we show some of the uses of this data, including editing using [Kircher and Garland 2006] and posing using
[Sumner et al. 2005]. Future work in cloth capture should involve more cameras, higher resolution (leading to smaller denser markers), different garments and different materials. We plan to pursue more tools to edit and repurpose captured data. Finally, we would like to conclude with a discussion about cloth capture in the context of other cloth animation techniques.
Simulation and image based rendering both provide methods to generate animation of cloth (a limited simulation list includes
[House and Breen 2000; Terzopoulos et al. 1987; Choi and Ko 2002; Bridson et al. 2003; Baraff et al. 2003] and a limited image based rendering list includes
[Bradley et al. 2005; White and Forsyth 2006; Lin and Liu 2006; Scholz and Magnor 2006]). These methods have several advantages: simulation gives significant user control and produces higher resolution meshes while image based rendering techniques produce more accurate illumination. However, capturing large amounts of data is far easier than simulating large amounts of data and provides more control than image based rendering. Common simulation complaints include long computation times, significant parameter tweaking and tangling. In contrast, capture is relatively quick (our code is 8 minutes per frame in MATLAB); parameters are set by selecting the type of cloth [Bhat et al. 2003] and tangling is relatively uncommon. Cloth capture makes it easy to capture large amounts of cloth, including fast light cloths that create instabilities in simulation. An added attraction of cloth capture is that complex interaction between the cloth and the body is recorded without complicated human models.
References
<H1>Acknowledgements</H1>
We thank Jai Vasanth and Anthony Lobay for early support of this project, Scott Kircher and Robert Sumner for providing mesh editing binaries, and Sony Computer Entertainment America for supplying human motion capture data. This work was supported in part by a Department of Homeland Security Fellowship and an ATI Graduate Research Fellowship.
<H1>References</H1>
- ANGUELOV, D., SRINIVASAN, P., KOLLER, D., THRUN, S.,RODGERS, J., AND DAVIS, J. 2005. Scape: shape completion and animation of people. In SIGGRAPH. BARAFF, D., WITKIN, A., AND KASS, M. 2003. Untangling cloth. In SIGGRAPH. BHAT, K., TWIGG, C., HODGINS, J. K., KHOSLA, P., POPOVIC,Z., AND SEITZ, S. 2003. Estimating cloth simulation parameters from video. In SCA. BOUGUET, J.-Y., 2005. Camera calibration toolbox for - matlab. http://www.vision.caltech.ed u/bouguetj/calib doc/. BRADLEY, D., ROTH, G., AND BOSE, P. 2005. Augmented clothing.In Graphics Interface. BRIDSON, R., MARINO, S., AND FEDKIW, R. 2003. Simulation of clothing with folds and wrinkles. In SCA. CHOI, K.-J., AND KO, H.-S. 2002. Stable but responsive cloth.In SIGGRAPH. FORSYTH, D., AND PONCE, J. 2002. Computer Vision: a modern approach. Prentice-Hall. GUSKOV, I., AND ZHUKOV, L. 2002. Direct pattern tracking on flexible geometry. In WSCG. GUSKOV, I., KLIBANOV, S., AND BRYANT, B. 2003. Trackable surfaces. In SCA. - HARTLEY, R., AND ZISSERMAN, A. 2000. Multiple View Geometry.Cambridge University Press. HASLER, N., ASBACH, M., ROSENHAHN, B., OHM, J.-R., AND SEIDEL, H.-P. 2006. Physically based tracking of cloth. In VMV. HOUSE, D., AND BREEN, D., Eds. 2000. Cloth Modelling and Animation. A.K. Peters. KIRCHER, S., AND GARLAND, M. 2006. Editing arbitrarily deforming surface animations. In SIGGRAPH. LIN, W.-C., AND LIU, Y. 2006. Tracking dynamic near-regular textures under occlusion and rapid movements. In ECCV. LIPMAN, Y., SORKINE, O., LEVIN, D., AND COHEN-OR, D. 2005. Linear rotation-invariant coordinates for meshes. In SIGGRAPH. LOWE, D. 2004. Distinctive image features from scale-invariant keypoints. IJCV. PARK, S. I., AND HODGINS, J. K. 2006. Capturing and animating skin deformation in human motion. In SIGGRAPH. PERONA, P., AND MALIK, J. 1990. Scale-space and edge detection using anisotropic diffusion. PAMI. PRITCHARD, D., AND HEIDRICH, W. 2003. Cloth motion capture. Eurographics. PROVOT, X. 1995. Deformation constraints in a mass-spring model to describe rigid cloth behavior. In Graphics Interface. SCHOLZ, V., AND MAGNOR, M. A. 2004. Cloth motion from optical flow. In VMV. SCHOLZ, V., AND MAGNOR, M. 2006. Texture replacement of garments in monocular video sequences. In Rendering Techniques. SCHOLZ, V., STICH, T., KECKEISEN, M., WACKER, M., AND MAGNOR, M. 2005. Garment motion capture using color-coded patterns. In Eurographics. SORKINE, O., COHEN-OR, D., LIPMAN, Y., ALEXA, M., R ̈OSSL, C., AND SEIDEL, H.-P. 2004. Laplacian surface editing. In Symposium of Geometry Processing. SUMNER, R. W., AND POPOVIC, J. 2004. Deformation transfer for triangle meshes. In SIGGRAPH. SUMNER, R. W., ZWICKER, M., GOTSMAN, C., AND POPOVIC, J. 2005. Mesh-based inverse kinematics. In SIGGRAPH. TANIE, H., YAMANE, K., AND NAKAMURA, Y. 2005. High marker density motion capture by retroreflective mesh suit. In ICRA. TERZOPOULOS, D., PLATT, J., BARR, A., AND FLEISCHER, K. 1987. Elastically deformable models. In SIGGRAPH. WHITE, R., AND FORSYTH, D., 2005. Deforming objects provide better camera calibration. UC Berkeley Technical Report. WHITE, R., AND FORSYTH, D. 2006. Retexturing single views using texture and shading. In ECCV. capture. UC Berkeley Technical Report. WHITE, R., FORSYTH, D., AND VASANTH, J., 2006. Capturing real folds in cloth. UC Berkeley Technical Report. ZHANG, Z. 2002. A flexible new technique for camera calibration. PAMI. WHITE, R., LOBAY, A., AND FORSYTH, D., 2005. Cloth
Figure 13: We reconstruct cloth being tossed over a cup, a tablecloth and a pair of pants (shown in the middle of a jump). See the video for a better view of the results.
A Image Processing
We do some pre-processing to get marker locations and connectivity from raw images. We recommend readers unfamiliar with these techniques refer to
[Forsyth and Ponce 2002]. We start by converting each image to HSV, disregarding the luminosity (V) and using polar coordinates to compute distances in hue and saturation. To detect markers, our code looks for uniformly colored blobs in two stages: first regions are built by growing neighborhoods based on similarity between pixels. This method is sensitive to image noise and can produce oversized regions when the color boundaries are smoothed. The second stage takes the center of mass of each blob from the first stage, computes the mean color and grows a region based on distance to the mean color (it is computationally intractable to use this as the first stage of the blob detection). The process is iterated for increasing thresholds on the affinity value in the first stage, using the portions of the image where detection failed in previous stages. Finally, blobs are thresholded based on size. Next, we need to determine the neighborhood relationships. For each marker, we construct a covariate neighborhood (a fitted ellipse) and vote for links to the three closest markers with similar covariate neighborhoods. This measures distances appropriately in parts of the scene where the cloth is receding from view and discourages links between markers with wildly different tilts. All links that receive two votes (one from either side) are kept while the rest are discarded. Links that bridge markers with conflicting color information are also discarded (typically on internal silhouettes).
B Entropy Comparison
For more reading on information theory, consult
[Cover and Thomas 1991]. Our analysis is based on the information entropy definition: H(X) = − ∑ n i=1 p(x i ) · log 2 x i . For [Scholz et al. 2005]
, the equation in section 3.1 is reduced to I = 9 ∗ C − A because they use 8 neighbors and no strain constraints. They use 5 colors which, without errors, yields C = log 2 5 bits per marker. They cite an error rate of five to ten percent. As a result, they recover anywhere from 1.65 to 2.04 bits per marker. In our comparison, we use C = 1.93 bits for color information from their method (five percent error, with equal probabilities for all remaining choices). Note that this is effectively less than four colors! Second, we compute structural ambiguities in their method which account for uncertainty in observations. The orientation of the surface is unknown, yielding four possible directions, or two bits of structural ambiguity. Second, in their paper, they say that oblique views cause another bit of uncertainty. As a result A = 3 bits. For our work, C is an empirical observation. Depending on the lighting and camera configuration, we get anywhere from 5 to 7 bits. We use the conservative estimate of C = 5 bits per marker. Second, our mesh is triangular and there are three possible neighborhood rotations, yielding A = log 2 3 = 1.59 bits of structural ambiguity. When neighborhoods are not used, there is no structural ambiguity. Strain information is difficult to compute and depends on the geometry of the surface and the orientation of the camera. In most cases, we observe more than 9 bits of strain information.
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"text": "['virtually eliminates correspondence errors']",
"type": "background_claim",
"id": "T71"
},
{
"offsets": [
[
8290,
8339
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],
"text": "['the range of motion is limited to avoid occlusion']",
"type": "background_claim",
"id": "T72"
},
{
"offsets": [
[
8347,
8394
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],
"text": "['arms are always held at 90 degrees to the torso']",
"type": "data",
"id": "T73"
},
{
"offsets": [
[
8461,
8664
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],
"text": "['introduce a combined strain reduction/bundle adjustment that improves the quality of the reconstruction by minimizing strain while reconstructing the 3D location of the points on the surface of the cloth']",
"type": "background_claim",
"id": "T74"
},
{
"offsets": [
[
8441,
8458
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],
"text": "['White et al. 2005']",
"type": "data",
"id": "T75"
},
{
"offsets": [
[
8687,
8780
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],
"text": "['introduce the use of silhoutte cues to improve reconstruction of difficult to observe regions']",
"type": "background_claim",
"id": "T76"
},
{
"offsets": [
[
8667,
8684
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],
"text": "['White et al. 2006']",
"type": "data",
"id": "T77"
},
{
"offsets": [
[
8788,
8826
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],
"text": "['silhouette cues improve reconstruction']",
"type": "background_claim",
"id": "T78"
},
{
"offsets": [
[
8828,
8843
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],
"text": "['hole filling is']",
"type": "background_claim",
"id": "T79"
},
{
"offsets": [
[
9173,
9209
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],
"text": "['more effective in many circumstances']",
"type": "background_claim",
"id": "T80"
},
{
"offsets": [
[
9218,
9276
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],
"text": "['it enforces an appropriate prior on the shape of the cloth']",
"type": "background_claim",
"id": "T81"
},
{
"offsets": [
[
9312,
9398
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],
"text": "['we improve the color pattern and matching procedure to get more information per marker']",
"type": "own_claim",
"id": "T82"
},
{
"offsets": [
[
9400,
9458
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],
"text": "['we introduce strain constraints to simplify correspondence']",
"type": "own_claim",
"id": "T83"
},
{
"offsets": [
[
9463,
9562
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],
"text": "['we create a data driven hole filling technique that splices previously captured cloth into the mesh']",
"type": "own_claim",
"id": "T84"
},
{
"offsets": [
[
9577,
9661
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],
"text": "['our system is capable of capturing a full range of motion with folding and occlusion']",
"type": "own_claim",
"id": "T85"
},
{
"offsets": [
[
10296,
10346
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],
"text": "['we cannot increase camera resolution without bound']",
"type": "own_claim",
"id": "T86"
},
{
"offsets": [
[
10355,
10394
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],
"text": "['camera bandwidth becomes very expensive']",
"type": "data",
"id": "T87"
},
{
"offsets": [
[
10021,
10121
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],
"text": "['Our goal is high marker density in the 3D reconstruction – especially in regions with high curvature']",
"type": "own_claim",
"id": "T88"
},
{
"offsets": [
[
10140,
10210
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],
"text": "['we need markers that are both small in scale and highly discriminative']",
"type": "own_claim",
"id": "T89"
},
{
"offsets": [
[
10244,
10281
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],
"text": "['small markers are less discriminative']",
"type": "data",
"id": "T90"
},
{
"offsets": [
[
10212,
10242
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],
"text": "['These two goals are in tension']",
"type": "own_claim",
"id": "T91"
},
{
"offsets": [
[
10409,
10468
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],
"text": "['we opt for the smallest markers that we can reliably detect']",
"type": "own_claim",
"id": "T92"
},
{
"offsets": [
[
10473,
10511
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],
"text": "['we make small markers more distinctive']",
"type": "own_claim",
"id": "T93"
},
{
"offsets": [
[
10661,
10737
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],
"text": "['Marker color and strain constraints are more useful than neighboring markers']",
"type": "own_claim",
"id": "T94"
},
{
"offsets": [
[
10746,
10799
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],
"text": "['they place fewer requirements on local cloth geometry']",
"type": "own_claim",
"id": "T95"
},
{
"offsets": [
[
10815,
10886
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],
"text": "['neighboring markers are observed only when the cloth is relatively flat']",
"type": "own_claim",
"id": "T96"
},
{
"offsets": [
[
11030,
11162
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],
"text": "['we adopt the following strategy: maximize information obtained from marker color and eliminate the information needed from neighbors']",
"type": "own_claim",
"id": "T97"
},
{
"offsets": [
[
11378,
11481
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],
"text": "['we can accurately minimize the number of neighbors required for correspondence and observe folds better']",
"type": "own_claim",
"id": "T98"
},
{
"offsets": [
[
11483,
11547
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],
"text": "['We can compare our work to previous methods using this framework']",
"type": "own_claim",
"id": "T99"
},
{
"offsets": [
[
11550,
11558
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],
"text": "['figure 6']",
"type": "data",
"id": "T100"
},
{
"offsets": [
[
11716,
11822
]
],
"text": "['we can compute the information needed to meet this threshold by adding information from the different cues']",
"type": "own_claim",
"id": "T101"
},
{
"offsets": [
[
11677,
11714
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],
"text": "['independent information adds linearly']",
"type": "background_claim",
"id": "T102"
},
{
"offsets": [
[
11862,
11962
]
],
"text": "['structural ambiguities in the pattern subtract information lost to determine which neighbor is which']",
"type": "own_claim",
"id": "T103"
},
{
"offsets": [
[
11977,
12019
],
[
12020,
12212
]
],
"text": "['we compute our information budget (I ) as:', 'N = number of observed neighbors C = color information per marker A = information lost to structural ambiguities S = information gained from strain constraints I = (N + 1) ∗C + S − A']",
"type": "own_claim",
"id": "T104"
},
{
"offsets": [
[
12398,
12450
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],
"text": "['it takes two bits to specify which neighbor we found']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
12343,
12390
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],
"text": "['This neighbor is one of four possible neighbors']",
"type": "own_claim",
"id": "T106"
},
{
"offsets": [
[
12452,
12457
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],
"text": "['A = 2']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
[
12474,
12514
]
],
"text": "['the equation reduces to I = 2 ∗C − 2 + S']",
"type": "own_claim",
"id": "T108"
},
{
"offsets": [
[
12463,
12472
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],
"text": "['this case']",
"type": "data",
"id": "T109"
},
{
"offsets": [
[
12553,
12618
]
],
"text": "['we can detect regions by increasing N until I > log 2 (M) bits']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
12516,
12551
]
],
"text": "['Given almost any structured pattern']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
12629,
12759
]
],
"text": "['larger marker regions have the disadvantage that curvature can cause local occlusions and prevent observation of the entire region']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
12761,
12794
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],
"text": "['Our best efforts are to improve C']",
"type": "own_claim",
"id": "T113"
},
{
"offsets": [
[
13211,
13258
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],
"text": "['our pattern is composed of tesselated triangles']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
13261,
13269
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],
"text": "['figure 5']",
"type": "data",
"id": "T115"
},
{
"offsets": [
[
13273,
13313
]
],
"text": "['any shape that tiles the plane will work']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
13878,
13925
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],
"text": "['loss is largely attributable to camera response']",
"type": "own_claim",
"id": "T117"
},
{
"offsets": [
[
13934,
13988
]
],
"text": "['larger markers produced substantially more information']",
"type": "data",
"id": "T118"
},
{
"offsets": [
[
13363,
13469
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],
"text": "['To maximize the density of reconstructed points, we print the smallest markers that we can reliably detect']",
"type": "own_claim",
"id": "T119"
},
{
"offsets": [
[
13471,
13637
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],
"text": "['To maximize the information contained in the color of each marker, we print colors that span the gamut of the printer-camera response, then use a gaussian color model']",
"type": "own_claim",
"id": "T120"
},
{
"offsets": [
[
13639,
13650
]
],
"text": "['section 4.1']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
13673,
13729
]
],
"text": "['the printer-camera response is a sequence of lossy steps']",
"type": "own_claim",
"id": "T122"
},
{
"offsets": [
[
13731,
13770
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],
"text": "['we generate a color image on a computer']",
"type": "data",
"id": "T123"
},
{
"offsets": [
[
13772,
13801
]
],
"text": "['send the image to the printer']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
13803,
13817
]
],
"text": "['pose the cloth']",
"type": "data",
"id": "T125"
},
{
"offsets": [
[
13823,
13847
]
],
"text": "['capture it with a camera']",
"type": "data",
"id": "T126"
},
{
"offsets": [
[
13990,
14022
]
],
"text": "['Illumination is also problematic']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
[
14027,
14042
]
],
"text": "['takes two forms']",
"type": "own_claim",
"id": "T128"
},
{
"offsets": [
[
14044,
14087
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],
"text": "['direct illumination on a lambertian surface']",
"type": "data",
"id": "T129"
},
{
"offsets": [
[
14092,
14113
]
],
"text": "['indirect illumination']",
"type": "data",
"id": "T130"
},
{
"offsets": [
[
14494,
14550
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],
"text": "['These comparisons must be made in the proper color space']",
"type": "own_claim",
"id": "T131"
},
{
"offsets": [
[
14552,
14687
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],
"text": "['we photograph the surface of the printed cloth with our video cameras to minimize the effect of non-linearities in the printing process']",
"type": "own_claim",
"id": "T132"
},
{
"offsets": [
[
14438,
14492
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],
"text": "['we detect markers by comparing colors to a known color']",
"type": "own_claim",
"id": "T133"
},
{
"offsets": [
[
14751,
14844
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],
"text": "['The goal of our acquisition pipeline is to compute correspondence using minimal neighborhoods']",
"type": "own_claim",
"id": "T134"
},
{
"offsets": [
[
14846,
15002
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],
"text": "['We accomplish this through an iterative algorithm where we alternate between computing correspondence and pruning bad matches based on those correspondences']",
"type": "own_claim",
"id": "T135"
},
{
"offsets": [
[
15193,
15257
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],
"text": "['This iterative approach allows us to match without neighborhoods']",
"type": "own_claim",
"id": "T136"
},
{
"offsets": [
[
15259,
15304
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],
"text": "['This is better than label propagation methods']",
"type": "own_claim",
"id": "T137"
},
{
"offsets": [
[
15143,
15191
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],
"text": "['markers are matched using color and strain alone']",
"type": "data",
"id": "T138"
},
{
"offsets": [
[
15966,
16065
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],
"text": "['We are able to obtain significantly more information per unit cloth surface area than previous work']",
"type": "own_claim",
"id": "T139"
},
{
"offsets": [
[
16071,
16082
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],
"text": "['section 3.1']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
16112,
16122
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],
"text": "['appendix B']",
"type": "data",
"id": "T141"
},
{
"offsets": [
[
15324,
15343
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],
"text": "['propagation methods']",
"type": "background_claim",
"id": "T142"
},
{
"offsets": [
[
16168,
16246
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],
"text": "['require large sections of unoccluded cloth and must stop at occluding contours']",
"type": "background_claim",
"id": "T143"
},
{
"offsets": [
[
16154,
16166
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],
"text": "['and Liu 2006']",
"type": "data",
"id": "T144"
},
{
"offsets": [
[
15385,
15388
]
],
"text": "['Lin']",
"type": "data",
"id": "T145"
},
{
"offsets": [
[
15345,
15363
]
],
"text": "['Guskov et al. 2003']",
"type": "data",
"id": "T146"
},
{
"offsets": [
[
15365,
15383
]
],
"text": "['Scholz et al. 2005']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
16271,
16329
]
],
"text": "['occluding contours are both common and difficult to detect']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
16260,
16268
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],
"text": "['figure 5']",
"type": "data",
"id": "T149"
},
{
"offsets": [
[
16344,
16395
]
],
"text": "['our iterative approach relies on strain constraints']",
"type": "own_claim",
"id": "T150"
},
{
"offsets": [
[
16533,
16604
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],
"text": "['Both of these computations are easier than detecting occluding contours']",
"type": "own_claim",
"id": "T151"
},
{
"offsets": [
[
17473,
17527
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],
"text": "['we can set N = 3 at the start and continue until N = 0']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
17441,
17461
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],
"text": "['analysis in figure 6']",
"type": "data",
"id": "T153"
},
{
"offsets": [
[
17582,
17604
]
],
"text": "['there are few mistakes']",
"type": "own_claim",
"id": "T154"
},
{
"offsets": [
[
17537,
17580
]
],
"text": "['the identity of the marker is overspecified']",
"type": "own_claim",
"id": "T155"
},
{
"offsets": [
[
17606,
17707
]
],
"text": "['This approach works from flat regions in the first iteration to foldy regions in the later iterations']",
"type": "own_claim",
"id": "T156"
},
{
"offsets": [
[
17971,
17997
]
],
"text": "['no neighbors are necessary']",
"type": "own_claim",
"id": "T157"
},
{
"offsets": [
[
17951,
17969
]
],
"text": "['the last iteration']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
17897,
17933
]
],
"text": "['these regions are not going to match']",
"type": "own_claim",
"id": "T159"
},
{
"offsets": [
[
18063,
18103
]
],
"text": "['no longer disrupt the matching procedure']",
"type": "own_claim",
"id": "T160"
},
{
"offsets": [
[
17999,
18017
]
],
"text": "['Occluding contours']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
18707,
18752
]
],
"text": "['Our strain constraint is based on the work of']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
18755,
18766
]
],
"text": "['Provot 1995']",
"type": "data",
"id": "T163"
},
{
"offsets": [
[
18784,
18831
]
],
"text": "['strain in cloth does not exceed 20% in practice']",
"type": "background_claim",
"id": "T164"
},
{
"offsets": [
[
18935,
18988
]
],
"text": "['we can use strain to exclude possible correspondences']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
18833,
18875
]
],
"text": "['Relaxing the constraint to distances in 3D']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
18990,
19050
]
],
"text": "['Strain naturally fits in to our information theory framework']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
19055,
19108
]
],
"text": "['strain excludes 87.5% of the possible correspondences']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
19115,
19138
]
],
"text": "['strain has added 3 bits']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
19148,
19170
]
],
"text": "['log 2 (1 − 0.875) = −3']",
"type": "data",
"id": "T170"
},
{
"offsets": [
[
19204,
19212
]
],
"text": "['figure 7']",
"type": "data",
"id": "T171"
},
{
"offsets": [
[
19720,
19783
]
],
"text": "['we match each image marker to a marker in the parametric domain']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
19873,
19919
]
],
"text": "['Each affinity is a product over different cues']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
20425,
20469
]
],
"text": "['we learned this threshold from labelled data']",
"type": "own_claim",
"id": "T174"
},
{
"offsets": [
[
20489,
20564
]
],
"text": "['changing it by several orders of magnitude had little effect on our results']",
"type": "own_claim",
"id": "T175"
},
{
"offsets": [
[
20738,
20798
]
],
"text": "['occlusion inevitably creates holes in the reconstructed mesh']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
20801,
20809
]
],
"text": "['figure 8']",
"type": "data",
"id": "T177"
},
{
"offsets": [
[
20813,
20863
]
],
"text": "['One would like to fill these holes with real cloth']",
"type": "background_claim",
"id": "T178"
},
{
"offsets": [
[
20865,
20937
]
],
"text": "['One of our major contributions is a data driven approach to hole filling']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
20939,
20995
]
],
"text": "['we fill holes with previously observed sections of cloth']",
"type": "own_claim",
"id": "T180"
},
{
"offsets": [
[
21051,
21128
]
],
"text": "['our hole filling procedure does not assume a skeleton that drives the surface']",
"type": "own_claim",
"id": "T181"
},
{
"offsets": [
[
21133,
21189
]
],
"text": "['our procedure estimates a single coefficient per example']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
20997,
21018
]
],
"text": "['Our work differs from']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
21021,
21041
]
],
"text": "['Anguelov et al. 2005']",
"type": "data",
"id": "T184"
},
{
"offsets": [
[
21191,
21247
]
],
"text": "['This hole filling procedure has a number of requirements']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
21249,
21325
]
],
"text": "['the missing section needs to be replaced by a section with the same topology']",
"type": "own_claim",
"id": "T186"
},
{
"offsets": [
[
21327,
21414
]
],
"text": "['the new section needs to obey a number of point constraints around the edge of the hole']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
21420,
21474
]
],
"text": "['the splicing method should respect properties of cloth']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
21807,
21848
]
],
"text": "['This procedure has a number of advantages']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
21857,
21916
]
],
"text": "['deformation gradients naturally yield cloth like properties']",
"type": "background_claim",
"id": "T190"
},
{
"offsets": [
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22148,
22190
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"text": "['deformation gradients can be converted into vertex locations by inverting a linear system, allowing us to specify vertex locations as constraints']",
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"text": "['Methods']",
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"text": "['Lipman et al. 2005']",
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"text": "['MeshIK works by choosing a combination of deformation gradients from the examples and then solving for the missing point location']",
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"text": "['Sumner et al. 2005']",
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"text": "['The most restrictive aspect of MeshIK is that it requires example meshes without holes']",
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"text": "['we never observe complete ex- ample meshes']",
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"text": "['each mesh is missing some triangles']",
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"text": "['we create complete meshes in an iterative method']",
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"text": "['works poorly']",
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"text": "['this gets the job done']",
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"text": "['The advantage of this apporach is that the example poses are chosen to capture the relevant degrees of freedom']",
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"text": "['we chose the simpler approach']",
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"text": "['the cloth toss sequence']",
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"text": "['iteratively refine the entire sequence']",
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"text": "['We introduce flexibility preserving smoothing']",
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"text": "['a method similar to anisotropic diffusion']",
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"text": "['Perona and Malik 1990']",
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"text": "['Typical temporal smoothing is dangerous']",
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"text": "['fast non-rigid movements can easily become physically implausible when blurred over time']",
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"text": "['small temporal errors are often difficult to notice']",
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"text": "['fast non-rigid regions of the cloth are complex']",
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"text": "['small errors in regions of the cloth that move rigidly are typically easy to observe']",
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"text": "['we use flexibility preserving smoothing']",
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26418,
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"text": "['Large variations in location indicate non-rigid movement and consequently receive little smoothing']",
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"text": "['Smaller variations indicates rigid movement and benefit from more substantial smoothing']",
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"text": "['a procedure that smoothes rigid movement more heavily than non-rigid movement']",
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"text": "['We use the automated calibration technique']",
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"text": "['any standard calibration will work']",
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"text": "['White and Forsyth 2005']",
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27151,
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"text": "['Zhang 2002']",
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27168,
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"text": "['Bouguet 2005']",
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"text": "['are good choices']",
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"text": "['Adequate lighting is critical']",
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27344,
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"text": "['fewer lights degrade performance']",
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"text": "['our experience']",
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"text": "['harsh shadows']",
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"text": "['dim lighting causes motion blur']",
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"text": "['slower shutter speeds']",
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"text": "['acquisition takes roughly 6 minutes and mesh processing 2 minutes per frame']",
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"text": "['a P4 2.4 GHz machine']",
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"text": "['Our capture results are best evaluated by looking at our video and figures 1,12,13']",
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"text": "['it is also necessary to evaluate on several numerical criteria for each capture session']",
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"text": "['The sleeve example is unique']",
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27882,
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"text": "['to compare against other capture techniques']",
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"text": "['it was one of the first items we captured']",
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"text": "['Much of the cloth is in contact with the floor and unobservable']",
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"text": "['yielding fewer bits of strain']",
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"text": "['we terminated the correspondence algorithm at N = 2']",
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"text": "['approx 44% of 3D markers are observed in 3 or more views']",
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"text": "['The largest challenge is that captured cloth meshes contain only points on the cloth surface']",
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"text": "['Using our MATLAB implementation of MeshIK, this process takes around 5-10 seconds per frame']",
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"text": "['for a small basis to adequately express a full range of motion']",
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"text": "['Sumner et al. 2005']",
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30584,
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"text": "['simple objects such as a cylinder']",
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"text": "['for pants the relationship is more complex']",
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"text": "['the fact that no folding occurs in a small bend does not imply that folding will be absent in a larger bend']",
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"text": "['we do not expect extreme folds in a corresponding larger bend']",
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"text": "['a decent amount of folding occurs in a small bend']",
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"text": "['MeshIK is most useful when a basis is carefully chosen to prevent extrapolation artifacts']",
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"text": "['One drawback to our approach is the loss of secondary kinematic motion']",
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"text": "['the sway of loose cloth']",
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"text": "['the resulting animation appears damped']",
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"text": "['MeshIK does not use velocity information']",
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"text": "['We have brought cloth capture from constrained laboratory examples to real settings by providing robust methods for dealing with occlusion and folding']",
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"text": "['this tool requires significant engineering effort']",
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"text": "['Like human motion capture']",
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"text": "['Camera setup and calibration are time consuming']",
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"text": "['the equipment is costly']",
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{
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31682,
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"text": "['once these obstacles have been overcome, capturing large amounts of data is relatively easy']",
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{
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"text": "['we are releasing our capture data']",
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"text": "['http://www.ryanmwhite.com/data']",
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"text": "['we show some of the uses of this data']",
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31900,
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"text": "['our video']",
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"text": "['editing using [Kircher and Garland 2006]']",
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32019,
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"text": "['posing using ', '[Sumner et al. 2005]']",
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"text": "['Future work in cloth capture should involve more cameras, higher resolution']",
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{
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"text": "['different garments and different materials']",
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32198,
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"text": "['We plan to pursue more tools to edit and repurpose captured data']",
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{
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"text": "['Simulation and image based rendering both provide methods to generate animation of cloth']",
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{
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"text": "['These methods have several advantages']",
"type": "background_claim",
"id": "T300"
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{
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32517,
32537
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],
"text": "['House and Breen 2000']",
"type": "data",
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{
"offsets": [
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32539,
32562
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],
"text": "['Terzopoulos et al. 1987']",
"type": "data",
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{
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32564,
32580
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],
"text": "['Choi and Ko 2002']",
"type": "data",
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},
{
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[
32582,
32601
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],
"text": "['Bridson et al. 2003']",
"type": "data",
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{
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32603,
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],
"text": "['Baraff et al. 2003']",
"type": "data",
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},
{
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[
32675,
32694
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],
"text": "['Bradley et al. 2005']",
"type": "data",
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},
{
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32696,
32718
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],
"text": "['White and Forsyth 2006']",
"type": "data",
"id": "T307"
},
{
"offsets": [
[
32720,
32736
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],
"text": "['Lin and Liu 2006']",
"type": "data",
"id": "T308"
},
{
"offsets": [
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32738,
32760
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],
"text": "['Scholz and Magnor 2006']",
"type": "data",
"id": "T309"
},
{
"offsets": [
[
32803,
32882
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],
"text": "['simulation gives significant user control and produces higher resolution meshes']",
"type": "background_claim",
"id": "T310"
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{
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<Document xmlns:gate="http://www.gate.ac.uk" name="A24_F04_FlowFixer_Using_BFECC_for_Fluid_Simulation_CITATION_PURPOSE_M_v1.xml">
f8ecf34c6738231f7f30a7e564bb73c1f9b7434e9bbc5cad3cb3a160eab9c904
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http://dx.doi.org/10.1145/1275808.1276500
Eurographics Workshop on Natural Phenomena (2005) E. Galin, P. Poulin (Editors)
<Title>FlowFixer: Using BFECC for Fluid Simulation</Title>
ByungMoon Kim Yingjie Liu Ignacio Llamas Jarek Rossignac
Georgia Institute of Technology
<Abstract>Back and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation by using the level set method. We show that it can be applied to reduce dissipation and diffusion encountered in various advection steps in fluid simulation such as velocity, smoke density and image advections. BFECC can be implemented easily on top of the first order upwinding or semi-Lagrangian integration of advection equations, while providing second order accuracy both in space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We combine these techniques with variable density projection and show that they yield a realistic animations of two-phase flows. We demonstrate the benefits of this approach on the image advection and on the simulation of smoke, of bubbles in water, and of a highly dynamic interaction between water, a solid, and air.</Abstract>
<H1>1. Introduction</H1>
Simulation of incompressible fluid involves several computation steps including diffusion, advection and pressure projection. Advection steps transport some quantities from one region to another along the fluid’s velocity field. In this paper, we explore four forms of advection encountered in fluid simulation: velocity, smoke density, image and level set advections. Velocity advection transports the velocity field along the velocity itself. This step is always needed in nonsteady flow simulation based on Navier-Stokes equation. Smoke density advection transports smoke along the velocity field. Sometimes, we may want to advect a colored image, which may be considered as colored smoke. We call this process image advection. When one uses a level set method
[OS88]
to simulate a free surface or a two-phase flow, for example a water surface simulation, the level set must be transported as well. We call it level set advection. Those advection steps can be computed by an upwind or a semi-Lagrangian method. The latter is often preferred due to its stability for large time step. The first order semi-Lagrangian method is popular in computer animation because of its simplicity. However, the first order semiLagrangian contains a significant amount of numerical diffusion and dissipation. In velocity advection, it yields damped fluid motion. In smoke density advection, it leads to a premature dilution of smoke, and is not able to simulate pure advec- tion. Therefore, higher order schemes, such as WENO or CIP [ TFK ∗ 03 ], are desired. We show that the implementation complexity of these schemes may be easily avoided by adding a very simple Back and Forth Error Compensation and Correction (BFECC) to an existing first order semiLagrangian schemes, thus improving its space and time accuracy to second order. We show that this approach reduces velocity damping and smoke density dilution and demonstrate its benefits on the four forms of advections discussed previously. BFECC was recently proposed in
[DL03, DL04]
as a level set interface computation method. As is mentioned in
[ELF05]
, high order methods may not prevent volume loss much. However, the authors of [DL03]
combined BFECC with their simple redistancing technique and applied it to the Zalesak’s problem, showing significantly reduced the volume loss. In the level set framework, the one with smallest volume error would be the particle level set method
[EMF02, ELF05] In this paper, we focus on applying BFECC to velocity and smoke advection steps rather than level set advection. We, however, use BFECC and the simple redistancing for level set advection of various fluid simulations and show that sufficiently realistic fluid animation can be obtained. It would be interesting to apply this to the level set advection part of the particle level set method
[ELF05]
for more demanding simulation.
The stability problems in the earlier works such as
[ FM96 ] were successfully remedied in
[ Sta99 ]
by introducing the pressure projection scheme to enforce incompressibility of the fluid and the semi-Lagrangian treatment of the advection term in the Navier-Stoke equation.
This solution is popular for the simulation of incompressible Fluids, such as smoke
[ FSJ01 ] and also for more challenging free surface flows
[ FF01 , EMF02 ]. The semi-Lagrangian velocity advection
[Sta99]
comes with built-in dissipation, i.e., the velocity is dissipated quickly since the linear interpolation in the first order semiLagrangian produces large error. While higher order interpolation can solve the problem, it involves more neighboring grid point values and increases the complexity, particularly when non-uniform mesh structures are used. In [FSJ01]
, vorticity is added to generate small scale fluid rolling motion. Recently,
[ SSK05 ]
addressed this built-in dissipation by enhancing advection itself. They adopted the CIP [ TFK ∗ 03 ] method that increases the order of accuracy in space by introducing the derivatives of velocity to build a sub-cell velocity profile. A nice feature of this CIP method is that it is local in the sense that only the grid point values of one cell are used in order to update a point value. However, in this CIP method, all components of velocity and their partial derivatives should be advected, increasing the implementation complexity and computation time, especially in 3D. It is also worth noting that CIP has higher order accuracy in space only. Therefore high order integration of characteristics is also necessary. In contrast, BFECC is easier to implement and exhibits second order accuracy both in space and time and is local during each of its operational steps. Song et al
[ SSK05 ]
focused on applying CIP to generate more dynamic water surface behavior. However, we believe that having less dissipative and diffusive advection provides significant benefits in smoke simulations as well. This is illustrated in the middle five images of Fig. 4 , where large amount of dissipation makes the smoke look very dark. In contrast, when BFECC is used, the smoke keeps full brightness throughout the simulation as is shown in the last five images. The introduction of the level set method to fluid animation in
[ FF01 ]
allowed realistic simulation of fluids with complex free surfaces. The problem left here was the volume loss in the level set method and the solution, known as the particle level set method, proposed subsequently in
[ EMF02 ]
, turned out to be very successful in volume preservation. The two phase fluid solver using variable density projection has been broadly studied in mathematics and fluid mechanics [ SSO94 , OKBG00 , HKLS04 ]. It has been used in graphics applications by
[ HK03 ]
, where the authors simulated air bubbles rising and merging and by [TFK ∗ 03, SSK05], where splash style interactions between water surface and air are studied.
c The Eurographics Association 2005.
ByungMoon Kim & Yingjie Liu & Ignacio Llamas & Jarek Rossignac / FlowFixer: Using BFECC for Fluid Simulation
2. Previous Work
3. Fluid Simulation
1
Consider the following Navier-Stokes equation ∂ ∂ u t = −u · ∇u + ν ∇ · (∇u) − ρ 1 ∇P + f.
We follow the operator splitting steps proposed in
[Sta99] except for the advection step, where we use BFECC and for the projection step for which we use the variable density pressure projection. We use the standard staggered grid
[FSJ01]. Suppose all terms in (1) except for − ρ 1 ∇P are treated and let the velocity obtained so far be u. ̃ The final step is applying the variable density pressure projection step to enforce the continuity equation ∇ · u = 0, i.e, solving the equation ∇ · ∆t ρ ∇P = ∇ · u. ̃ Its first order discretization is ∆x ∆t 2 P i, ρ j − i− P 2 1 i−1, , j j + P i, ρ j − i+ P 1 2 i+1, , j j + P i, ρ j − i, j− P i, 2 1 j−1 + P i, ρ j − i, j+ P i, 2 1 j+1 1 = ∆x u ̃ i+ 2 1 , j − u ̃ i− 2 1 , j + v ̃ i, j+ 1 2 − v ̃ i, j− 1 2 . (2) We assume ∆x = ∆y here and through the rest of the presentation. The extension to 3D is straightforward and hence omitted. This first order approximation is identical to
[ SSK05 ] and higher order formulations can be found in [ ABS96 , SAB ∗ 99 ]. Obviously, if ρ is constant, we have the pressure projection ∆t ρ ∇ 2 P = ∇ · u introduced in [ Sta99 ]. We also include a simple implementation of surface tension similar to [ SAB ∗ 99 ].
<H1>4. The BFECC Method</H1>
In this section, we review the BFECC method. Since we want to apply it to various advections, we use φ to denote a quantity that is advected and reserve the symbol φ for the level set function through the presentation of this paper. This φ can be the velocity components u, v, w, smoke density, RGB color of an image or level set function φ , satisfying
3
φ t + u · ∇φ = 0.
We briefly describe the BFECC method here. Let L be the first order upwinding or semi-Lagrangian integration steps to integrate (3), such that
4
φ n+1 = L(u, φ n ).
Using this notation, the BFECC can be written as the following three L-calling process
5
φ n+1 = L u, φ n + 1 φ n − φ ̄ 2 where φ ̄ = L (−u, L (u, φ n )) .
One may understand this method intuitively as follows. If the advection step L(·, ·) is exact, the first two forward and backward steps should return the value exactly the same as the original one, i.e., φ n = φ ̄ . However, this does not hold due to the error in the advection operation L. Suppose L contains an error e. Then the first two forward and backward steps will produce error 2e, i.e., φ ̄ = φ n + 2e. Therefore, the error can be computed as e = − 1 2 ( φ n − φ ̄ ). We subtract this error e before the final forward advection step. Then the equation (5) becomes φ n+1 = L(u, φ n − e). This step will add an additional e, which will be cancelled by the subtracted amount −e. This method is proven to be second order accurate both in space and time
[DL03, DL04].
c The Eurographics Association 2005.
ByungMoon Kim & Yingjie Liu & Ignacio Llamas & Jarek Rossignac / FlowFixer: Using BFECC for Fluid Simulation
Figure 1: On the right column, a highly dynamic behavior of water interaction with air, air bubbles, and solid is made possible by the two-phase formulation and the BFECCbased reduction of the dissipation in the velocity advection step. On the left column, the BFECC is turned off and the splash is lower.
<H2>4.1. Implementation of BFECC</H2>
In this section, we provide a pseudo code to show the simplicity of the BFECC implementation. First let the function SingleStep(u, v, φ n , φ n+1 ) implement upwind or semiLagrangian integration of the scalar field φ , which can be the velocity components u,v,w, the smoke density, RGB colors of an image or the level set function φ . Then BFECC is implemented as:
SingleStep(u, v, φ n , φ ̃ ) SingleStep(−u, −v, φ ̃ , φ ̄ ) φ ̃ := φ n + ( φ n − φ ̄ )/2 SingleStep(u, v, φ ̃ , φ n+1 )
Figure 2: Comparison of without (top) and with (bottom) BFECC in velocity advection on a 80×200 grid. Top images show damped fluid motion. Bottom ones show small scale details as well as large scale fluctuations thanks to BFECC applied to the velocity advection step.
<H2>4.2. BFECC for Velocity Advection</H2>
We can use (5) to implement the velocity advection step in solving the Navier-Stokes Equation. In this case, φ becomes u, v and w. We show that BFECC can improve the damping in the first order semi-Lagrangian implementation of velocity advection, which is a well known drawback of
[Sta99]. For multiphase flow, this BFECC needs to be turned off near the interface to prevent velocities of different fluids with different densities from being mixed, which creates momentum changes. We simply turn BFECC off, i.e., use the first order semi-Lagrangian, for the grid points where | φ | < 5∆x. We also turn it off near the boundary. Notice that reducing velocity dissipation is equally important in the entire fluid domain, not only near the interface. In other words, turning BFECC off near the interface has little effect since it is still turned on in most of the fluid domain. As is shown in Fig. 2 , applying BFECC adds details in smoke motion. Notice that these details cannot be obtained from the vorticity confinement method
[FSJ01]
, which only adds small scale rolling motions. We also performed the same test in a coarser grid of 100×40. In this case, the flow did not fluctuate at all around the obstacles with the first order semi-Lagrangian advection. However, when BFECC was added, the flow fluctuated as in the refined grid. We conclude that BFECC creates a physically correct fluctuations in a coarser grid. Velocity advection can also be important when rigid bodies are involved. In Fig. 7 , the cup does not tumble due to the velocity dissipation in the first order semi-Lagrangian method, while the cup tumbles when BFECC is applied to the velocity advection step.
c The Eurographics Association 2005.
ByungMoon Kim & Yingjie Liu & Ignacio Llamas & Jarek Rossignac / FlowFixer: Using BFECC for Fluid Simulation
Figure 3: Advection of an image along with the up-going flow field on 100×250 grid. The first image shows initial location of the image. The next six images are without BFECC where the dissipation/diffusion are significant. The last six images are with BFECC, where the dissipation is greatly reduced and the features of the image can be identified.
Figure 4: Simulation of weakly diffusive and dissipative smoke in bubble rising and bursting situation. The far left image shows the initial bubble. The next five are without BFECC, where the dissipation/diffusion in the semi-Lagrangian step deteriorate the density of smoke. The last five images simulated with BFECC show minimal dissipation/diffusion and the smoke is in full density throughout the simulation. Notice that all simulation parameters are exactly same except for the usage of BFECC in smoke advection. Therefore the only difference is the density of smoke. Also, notice that the simulation time differs by less than 1% since it is dominated by the pressure projection step.
<H2>4.3. BFECC for Smoke Density and Image Advection</H2>
We also apply BFECC to the advection of smoke density for the smoke simulation. In Fig. 3 and 4, we show that BFECC can reduce dissipation and diffusion significantly. As is shown in
[DL03]
, BFECC is linearly stable in l 2 sense, i.e., ||a|| l 2 = ∑ |a i j | 2 is bounded, when the velocity field is constant, where a is the smoke density. However, density values a i j can become negative or greater than 1.0 for some grid points. In our simulation, this problem was not significant and we simply clamped those values to stay in [0, 1]. To measure the diffusion/dissipation amount, we design a test problem similar to Zalesak’s problem. Instead of the notched disk, we place a color image and rotate it 360 degree and then compare it with the original image as is shown in Fig. 5 . As is shown in (d), the dissipation of the color is significantly reduced with BFECC. During the advection, the image is also diffused to neighboring region, even though it is not visible. To visualize the diffusion amount, we plot background pixels as blue to show the region where the image has been diffused into. As is shown in (d), the color of the object is little diffused into neighboring region when BFECC is used. Also notice that the size of the image looks smaller and its position is noticeably different from the original location in (c), which is again fixed in (d) where BFECC is used. The computation time was 0.156 sec (without BFECC) and 0.36 sec (with BFECC) per frame on a 3GHz Pentuim4. Advection is often used for scientific visualization, especially for various forms of flow visualization. For example,
[JEH02]
uses semi-Lagrangian advection of dye to visualize the vector field. [Wei04]
applied level set method to advect dye without diffusion. Only one dye color is allowed and the dye cannot be diffused at all. Also level set implementation is needed. In contrast, BFECC is trivial to implement and provides advection of fully colored pattern of dye, if necessary. As is shown in Fig. 3 , the dissipation/diffusion is very small. Thus, we believe that it can be used in flow visualization as well. This remains as a future work.
Figure 5: Test of dissipation and diffusion with image advection problem on a rotating vector field (800×800 grid, CFL = 6.29). (b) is the top center portion of the original image (a). (c) is obtained by rotating it 360 degree using the first order semi-Lagrangian scheme, where one can see large amount of dissipation, diffusion, shrinkage of image and position error. These are significantly improved in (d) where BFECC is used. The blue background region is in fact in black color but it is rendered as blue to illustrate the region where the color is not diffused into.
c The Eurographics Association 2005.
ByungMoon Kim & Yingjie Liu & Ignacio Llamas & Jarek Rossignac / FlowFixer: Using BFECC for Fluid Simulation
5. Results
Figure 6: The far left image shows an air bubble placed in olive oil at time zero. The next three images are first order semi-Lagrangian implementation of level set advection. The next three images are produced using BFECC and simple redistancing, showing significantly reduced volume loss.
<H2>
<H2>4.4. BFECC for Level Set Advection</H2>
</H2>
Even though, BFECC still has some volume loss in fluid simulation, especially for small droplets or thin filaments, it is still interesting to show how BFECC performs in the fluid simulation since it is trivial to implement and fast. When we use the BFECC for level set advection, i.e., φ = φ , redistancing is needed to keep the level set function as a signed distance function. We use the following redistancing equation
[SSO94]
6
∇φ φ τ + w · ∇φ = sgn( φ ) where w = sgn( φ ) |∇ φ | .
w is the velocity vector for redistancing. This equation can be solved by applying first order upwinding in discretizing the term w · ∇φ . An alternative is the semi-Lagrangian style integration, i.e., φ n+1 = φ n (x − w∆ τ ) + sgn( φ n )∆ τ , where x is the location of each grid point. Hence, φ n (x − w∆ τ ) is the φ value of previous location. When these integration formulae for (6) are combined with BFECC, the redistancing tends to spoil good φ values computed from the second order accurate BFECC. This leads to the idea of turning redistancing off near the interface to keep good φ values there. The conditions to turn off redistancing is provided in [DL03]
, where the significant enhancement were shown for the Zalesak’s problem. This simple redistancing is crucial in preserving volume [DL03]. It is also easy to implement since it simply requires to perform redistancing at the points where at least one of the following two conditions are met. • When the grid point is not close to the interface, i.e., when φ i, j has the same sign with its eight neighbors. • When the slope is sufficiently high, i.e., when | φ i, j − φ i±1, j | or | φ i, j − φ i, j±1 | ≥ 1.1∆x. We test BFECC in different fluid simulations. We simulate air-water and olive oil-air interactions. Properties of these fluids are provided in Table 1 . Water is rendered as bluish surface and olive oil is rendered in yellowish color. We use PovRay ( http://povray.org ) to render images. In Fig. 7 , we simulated a cup, air and water interaction. The cup is released upside down near the water surface. Due to its weight, the cup sinks deep into water but it soon rise again because of the air in it. However, in the top, we turned BFECC off for velocity advection and hence the water became dissipative, preventing the cup from tumbling. In the bottom, we use BFECC for velocity advection, where the velocity dissipation is small and hence the cup can tumble 180 degree. This example indicates that reducing velocity dissipation could be important in simulating fluid and rigid body interaction. We implement the rigid fluid method
[CMT04]
to simulate rigid body and fluid interaction in Fig. 1 and 7. We use multiple pressure projections to address the seeping problem mentioned in [CMT04]. The computation time varies in situations such as the complexity of fluid motions. In simple bubble rising situation without rigid body, it took a few seconds per time step using a 50 3 mesh. The cup example in Fig. 7 has multiple pressure projections and it took about 30 to 130 seconds per time step on a 70 3 grid.
ρ [kg/m 3 ]
ν [m 2 /sec]
Surf. Tension [N/m]
air
1.125
1.7×10 −5
water
1000
1.0×10 −6 −5
0.07
olive oil
910
9.2×10
0.035
ρ [kg/m 3 ] ν [m 2 /sec] Surf. Tension [N/m] air 1.125 1.7×10 −5 water 1000 1.0×10 −6 0.07 olive oil 910 9.2×10 −5 0.035
Table 1: Properties of fluids used in simulations
<H1>6. Conclusion</H1>
We have shown that the BFECC scheme can be used to improve the simulation of fluids. Once the simple first order upwinding or semi-Lagrangian steps for velocity, smoke density, image or level set advections are implemented, BFECC can be added with a trivial amount of code. We show that this simple extension yields significant enhancements in reducing diffusion and dissipation in velocity, smoke, image advection and in preserving volume under various situations including two-phase flows and rigid bodies. The benefits of the proposed approach are illustrated in the accompanying video.
<H1>7. Acknowledgement</H1>
This work was supported by the NSF under the ITR Digital clay grant 0121663.
c The Eurographics Association 2005.
ByungMoon Kim & Yingjie Liu & Ignacio Llamas & Jarek Rossignac / FlowFixer: Using BFECC for Fluid Simulation
Figure 7: Simulation of a sinking cup. The top row is simulated without the BFECC in velocity advection, where the cup fails to tumble. The bottom row is simulated with BFECC and the cup tumbles thanks to the reduced damping in velocity field.
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[Wei04] W EISKOPF D.: Dye advection without the blur: A levelset approach for texture-based visualization of unsteady flow. In EUROGRAPHICS (2004), vol. 23.
c The Eurographics Association 2005.
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http://dx.doi.org/10.1145/1507149.1507156
<Title>Controlling Deformable Material with Dynamic Morph Targets</Title>
Nico Galoppo ∗ Miguel A. Otaduy † William Moss ‡ Jason Sewall § Sean Curtis ¶ Ming C. Lin
∗
∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] ¶ e-mail: [email protected] e-mail: [email protected]
Intel Corporation URJC Madrid UNC Chapel Hill UNC Chapel Hill
<Abstract>We present a method to control the behavior of elastic, deformable material in a dynamic simulation. We introduce dynamic morph targets, the equivalent in dynamic simulation to the geometric morph targets in (quasi-static) modeling. Dynamic morph targets define the pose-dependent physical state of soft objects, including surface deformation and elastic and inertial properties. Given these morph targets, our algorithm then derives a dynamic model that can be simulated in time-pose-space, interpolating the dynamic morph targets at the input poses. Our method easily integrates with current modeling and animation pipelines: at different poses, an artist simply provides a set of dynamic morph targets. Whether these input states are physically plausible is completely up to the artist. The resulting deformable models expose fully dynamic, pose-dependent behavior, driven by the artist-provided morph targets, complete with inertial effects. Our deformable models are computationally efficient at runtime through modal reduction and pose-space polynomial interpola-</Abstract>
tion. These models can therefore be plugged into existing dynamic simulation engines, either forming interactive, deformable content in real-time games or providing secondary dynamic effects for kinematically-driven characters in feature animation films. Finally, our method also facilitates certain time-consuming rigging procedures, by providing a physically based approach to resolve co-articulation deficiencies in traditional skinning methods, such as in shoulder regions, fully automatically.
Keywords: physically based modeling, deformation, reduced dynamics, modal analysis, morph targets
(a)
(b)
Figure 1: When under influence of dynamic events such as jumping from a diving board or bouncing off a wall, our method using morph targets produces deformations consistent with Herbert’s morph targets defined in (a). The fat morph target is associated with a crunched pose to mimic a bulging belly (and only for that pose). As shown in (b), the simulation without dynamic morph targets does not show bulging, whereas our method shown in (c) does. Image (d) is a snapshot of our run-time diving demo.
e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Copyright © 2009 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions Dept, ACM Inc., fax +1 (212) 869-0481 or e-mail [email protected] .
UNC Chapel Hill UNC Chapel Hill UNC Chapel Hill
<H1>1 Introduction</H1>
Animation of skin and muscular deformations of human characters and other living creatures has long been one of the most important applications of deformable modeling in computer graphics, notably in feature animation and more recently in increasingly realistic computer games and interactive medical and training applications. Realistic deformation is a complex and subtle phenomenon due to the tightly coupled interplay of bones and musculature governing the deformations. Generally speaking, there are three common approaches for modeling surface deformation: purely kinematic, example-based and physically based. Purely algorithmic approaches for skeleton-driven
[Kavan and Zara 2005]
and facial deformations [Pighin and Lewis 2006]
are very fast, but have difficulty in capturing realistic skin deformation in areas with multiple influences.
Example-based approaches capture more realism by pose-space interpolation of desired skin shapes at different poses
[Magnenat-Thalmann et al. 1988; Mohr and Gleicher 2003; Kry et al. 2002; Lewis et al. 2000]. Physically based deformation algorithms, governed by the physics of muscle motion and tendon influences, provide automatic means to achieve dynamic deformations under influence of external forces and inertial effects, but are computationally more expensive
[Chadwick et al. 1989; Gourret et al. 1989; Chen and Zeltzer 1992; Scheepers et al. 1997; Wilhelms and Gelder 1997; Sifakis et al. 2005]. Our approach seeks to bridge the gap between geometric examplebased methods and physically based approaches. We introduce dynamic morph targets, i.e. predefined and possibly artist-authored physical descriptors of skin deformations and elastic material properties. Dynamic morph targets rely on three key contributions: • A pose-space method for interpolation of simple elastic deformation models that allows the artist to author complex nonlinear deformation behavior. • A compact way of interpolating skin geometry, elastic forces, and their derivatives, all in a unified manner. • The extension of the method to support modal reduction and therefore very efficient implementation that is linear in the number of coefficients of the force polynomial. The main advantages of our method over previous approaches are three-fold: quality of deformations, dynamic behavior and computational efficiency. Although our method is physically based, we avoid expensive modeling of musculature or tendon influences, and instead rely on physical constitutive models of deformable material to minimize skin pinching artifacts and bypass complex rigging requirements that are common to purely geometric approaches. The use of such constitutive material models also enables response to external forces and inertial effects in dynamic simulations. Due to performance requirements, one is commonly restricted to linear or quasi-linear models that cannot model pose-dependent effects such as bulging and wrinkling. Instead, we guide dynamic simulations by dynamic morph targets — discrete pose-space examples of skin properties and deformations.
I3D 2009, Boston, Massachusetts, February 27–March 1, 2009. © 2009 ACM 978-1-60558-429-4/09/0002 $5.00
39
<H1>2 Related Work</H1>
Due to its immense importance in character animation, there has been an extensive collection of work in the area of surface deformation in the last few decades. Here, we will primarily focus on significant work related to control of surface deformation of kinematic and dynamic characters. Readers are referred to extensive surveys for other important work
[Gibson and Mirtich 1997; Nealen et al. 2006]. Purely data-driven methods are an attractive choice for control purposes, as the input shapes provide guide examples of desired deformations. In its most essential form, one simply interpolates between character poses in a large database
[Maestri 2006], providing ample control of skin deformation to animators. However, many poses are required in the database to achieve good results. Purely data-driven methods lack a kinematic model, making them of limited use for animation and dynamic simulation. Purely kinematic approaches such as skeletal-subspace deformation (SSD)
[Magnenat-Thalmann et al. 1988]
model the deformation of the skin surface by linear blending of the animated bone transformations. This technique, also known as linear blend skinning, cannot capture complex deformations and typically has problems deforming skin near joints due to collapsing geometry (i.e. pinching), because the deformation is restricted to the subspace of the affine transformation of the joints.
Different methods have been proposed to address the problems of linear blend skinning by inserting additional joints tuned from examples
[Mohr and Gleicher 2003], or employing blending of transformations instead of weights
[Kavan and Zara 2005], among others.
Recent techniques have extended skinning to mesh deformations
[James and Twigg 2005], motion capture data without a predefined skeleton
[Park and Hodgins 2006], or interactive models
[Der et al. 2006]. Unlike shape interpolation and other data-driven methods, SSD does not permit direct sculpting or control. Instead, artists have to tweak vertex weights, giving SSD algorithms the reputation of being tedious to control. The first work to add control in a kinematic approach is that of pose-space deformations
[Lewis et al. 2000]. PSD is a hybrid method that combines SSD with morphing and employs scattered data interpolation to compute non-linear skin corrections in posespace, resulting in a kinematic model that also has artist-sculpted poses. When dealing with large pose-spaces that have many example poses, PSD becomes memory inefficient due to the large database of surface displacements. PSD can be extended to support per-vertex pose-space deformation (WPSD)
[Kurihara and Miyata 2004; Rhee et al. 2006]
, largely reducing the number of required example poses. The EigenSkin method
[Kry et al. 2002]
also provides a way to reduce per-vertex displacement memory footprint by computing an error-optimal set of eigenbases for approximating the original deformation model. Other recent methods
[Weber et al. 2007; Wang et al. 2007]
learn example-based corrections on sparse points and assume that these corrections can be smoothly interpolated. Pose space deformation and related example-based methods allow for direct sculpting of geometric morph targets, but are purely kinematic approaches to (quasi-)static deformation, without reference to underlying forces or mass. Our method builds on the concept of pose-space deformation and applies it to pose-space interpolation of dynamic morph targets to achieve not only (quasi-)static deformations, but a fully dynamic model in time-pose-space. Finally, physically based methods in graphics are based on biomechanical models of skin tissue and musculature. In terms of efficiency versus accuracy, these methods fall into two broad categories. The first category of algorithms aim for accuracy
[Chen and Zeltzer 1992; Scheepers et al. 1997; Wilhelms and Gelder 1997; Koch et al. 1996; Zordan et al. 2004; Sifakis et al. 2005; Sueda et al. 2008]
by simulating the actions of the individual muscles, bones and tendons in the skin. Interactive physically based approaches trade accuracy for performance
[Terzopoulos et al. 1987; Terzopoulos and Witkin 1988; Metaxas and Terzopoulos 1992; Picinbono et al. 2001; Capell et al. 2002; Müller and Gross 2004; Galoppo et al. 2007]. These methods use simplified (quasi-)linear elastic models that cannot capture complex non-linear behavior such as muscle bulging. Physically based methods can only provide control through the influence of forces. While methods that control global deformation modes have been around for a while
[Witkin and Welch 1990], providing control of sculpted deformations for simulation of deformable models has only recently caught attention in graphics research. A method for physically based rigging was proposed by
[Capell et al. 2005], using pose-dependent forces to guide the shape of the character. In contrast to our method, their approach does not support pose-dependent elastic properties and its performance is highly dependent on the resolution of the sculpted deformations.
Given an input animation, shape keyframes can be used to retarget the elastic deformations
[Kondo et al. 2005] or to enhance the surface deformations with physically simulated detail using subspace constraints
[Bergou et al. 2007]. The former provides good control of shapes but is restricted to a given input animation, while the latter achieves rich secondary surface detail but does not provide direct manipulation of the surface. Our method provides the ability to sculpt the dynamic morph targets directly and produces a dynamic model that is not restricted to a given animation; our model can be plugged into any simulated environment and be subject to external forces.
Modal reduction has proven useful to increase performance in posespace deformation methods
[Kry et al. 2002] as well as in physically based methods
[Hauser et al. 2003; Choi and Ko 2005]. Our work exploits the technique of
[Barbic and James 2005] that enables fast modal integration of St.Venant-Kirchoff elastic forces, where the performance depends mainly on the number of simulated eigenmodes, not on the resolution of the model. This technique makes our approach suitable for real-time applications. To the best of our knowledge, our method is the first to provide shape and surface behavior control of dynamic reduced models.
40
<H1>3 Method</H1>
The goal of our method is to simulate controllable non-linear deformations by interpolation of dynamic morph targets at runtime, kindred to geometric morph targets in static character modeling. In this section, we will describe the concept of dynamic morph targets, and explain how they can be used to simulate a pose-dependent elastic model that is fast enough for real-time applications.
<H2>3.1 Dynamic Morph Targets</H2>
Dynamic morph targets define pose-specific soft skin behavior. More formally, they are pairs of elastic models E i and poses s i , i.e. pairs {E i , s i } of elastic models in pose space. Similar to geometric morph targets, dynamic morph targets associate surface and volume deformation with character pose. In contrast to geometric morph targets, dynamic morph targets also define pose-specific elastic properties including stiffness and plasticity. The combination of surface deformation and elastic properties defines the elastic model E i . We represent a pose by a vector s ∈ S where posespace S ⊂ R k . Note that our implementation uses skeletal pose, but the concept of pose can easily be extended beyond the skeletal sense; in fact any notion of state of a character can be used, such as emotional state, velocity state, contact state, or muscle activation. Dynamic morph targets can easily be created in existing modeling packages; very similar to creating geometric morph targets. A modeler defines a set of m poses {s 1 , s 2 , . . . , s m } of the character and sculpts desired deformations that cannot be captured with traditional skinning methods
[Kavan and Zara 2005; MagnenatThalmann et al. 1988]. Elastic properties can be assigned for each pose, such that the same skin section can be defined stiff for one pose and flabby for another pose, e.g. to mimic contraction and relaxation of a muscle, or to exaggerate skin bulging. Dynamic characters enhanced with dynamic morph targets can react to external forces just as with other common physically based deformation algorithms, but they also expose non-linear deformations and elastic behavior as imposed by the dynamic morph targets.
<H2>3.2 Pose-dependent Elastic Model</H2>
Dynamic morph targets are used to build a pose-dependent elastic model E(x, s). For hyper-elastic materials, an elastic model can be defined as a material function E(u(x)) defining the internal elastic energy at material points x in function of the deformation u. For our experiments and in correspondence with the space in which the morph targets are defined, we choose to express elastic deformation in the skeletal bind pose as has been proposed in the past
[Lewis et al. 2000; Kry et al. 2002; Galoppo et al. 2007]. On the other hand, it is certainly possible to use other formulations of elastic strain to define a pose-dependent model with dynamic morph targets. Traditionally, the elastic energy is a pose-independent material potential that causes internal elastic forces R(u) in the material. We create a pose-dependent elastic model by taking into account the dynamic morph targets {E i , s i } as example inputs. We use scattered data interpolation to derive an expression for the internal elastic forces R(u, s) anywhere in pose-space S, given the expressions for the elastic forces R i (u) that are imposed by the dynamic morph targets at poses s i . Equations of motion Without loss of generality, we can use the finite element method to discretize the partial differential equations of solid continuum mechanics in space, leading to the common Euler-Lagrange motion equations, where we substitute the posedependent elastic forces R(u, s):
1
Mü + D(u, u, ̇ s) + R(u, s) = f ,
with M the mass matrix and f the external forces. Here we use the (local) Raleigh damping model D(u, u, ̇ s): Polynomial elastic forces The computation of pose-dependent elastic forces R(u, s) requires the interpolation of pose-specific forces R i (u). However, since forces are a function of the timevarying deformation u, they cannot simply be evaluated once and then interpolated at runtime. In our method, we have opted for elastic models for which R i (u) can be expressed as a (multivariate) polynomial function of the degrees of freedom u. Then, the interpolation of elastic models reduces to the interpolation of polynomial coefficients.
Common examples of such elastic models are the so-called ‘completely linear’ FEM deformation model (with or without stiffness warping
[Müller and Gross 2004]), or the ’seminon-linear’ St.Venant-Kirchoff model (StVK)
[Barbic and James 2005; Capell et al. 2005]. We have simulated our examples with both linear and semi-non-linear elastic models. However, because we express deformation in the skeletal bind pose, we did not see any noticeable quality difference between both elastic models in our experiments. Therefore, we have opted for the more efficient linear elasticity to produce most of the images and videos unless otherwise noted (see Section 5). In Section 3.4 we show that both models are amenable to modal reduction for efficiency. Each polynomial R i (u) is associated with a dynamic morph target at pose s i and is uniquely defined by its set of coefficients {a k } i which we collect in a vector a i . We can then determine the posedependent elastic force R(u, s), which is also uniquely defined by its set of coefficients a(s). At an arbitrary pose s in pose-space, a(s) can be interpolated from the example coefficients a i . This is described in more detail in Section 3.3. The interpolation of polynomial coefficients yields the interpolation of force values R(u, s) at all possible deformation values u. However, it also yields the interpolation of force derivatives, such as the ∂R(u,s) stiffness matrix . One subtle detail remains for defining a ∂u complete interpolation of elastic models. The rest configuration at each input pose may be different, therefore the deformation u may not be consistent across poses. We choose a certain pose as a reference, and express the deformation of all other poses by adding the difference between rest configurations, ∆u. The addition of this term simply modifies the coefficients of the force polynomials, which can then be safely interpolated, as all deformations are expressed w.r.t. a consistent configuration.
„ ∂R(u, s) « D(u, u, ̇ s) = αM + β u. ̇ ∂u
<H2>3.3 Interpolating Force Polynomials in Pose Space</H2>
Conceptually, a pose can be described in many ways, as long as it provides a description of the state of a model or character, and a metric to measure distances between poses. In our implementation, we choose the joint configuration of an articulated character as the pose descriptor. We define the pose descriptor s ∈ R 6k , where k is the number of joints. Each joint contributes 6 components to s, namely the 6-dof representation of its parent-relative coordinate frame. The first three components represent the local joint translation, whereas the last three components represent the local joint
41
(a) Morph targets (b) Simulation with all soft morph targets
Figure 2: Concept of Dynamic Morph Targets: Simple cylinder geometry mimicking an elbow joint with bulging skin, for which two morph targets (out of a total of four) are given in (a). The skin of the bulged morph target was chosen to be stiffer to mimick muscle contraction. On the right, we show runtime snapshots of simulations using our pose-dependent elastic model, under influence of identical downward forces. (b) was generated with four soft morph targets, whereas (c) has increasingly stiffer morph targets, to mimic muscle contraction. The dynamic skin behavior is identical for the straight joint (a relaxed muscle), because the elastic properties of the first morph target are identical for (b) and (c). But, for the bent joint, the force clearly causes more skin deformation in (b). This undesirable behavior can be fixed to mimic muscle contraction by making the fourth morph target stiffer, as shown in in (c). This simple example shows a dynamic bulging effect that can only be achieved with dynamic morph targets.
rotation. We define distance between two poses to be the inner product of the difference of its descriptors. As an articulated character moves between observed configurations, its elastic model should approximate the elastic models of the input poses. As mentioned in Section 3.2, we are looking for a way to interpolate internal elastic forces R i . Obviously, as the character moves from one pose to the other, the internal forces change continuously but highly non-linearly. In other words, elastic forces form a non-linear smooth field in pose-space. Radial base functions [Powell 1987] (RBF) are a common choice for interpolating scattered data of an underlying smooth field that is non-linear. Moreover, as our goal is to have as few input models E i as possible, RBFs are suited because they work well with sparse input data sets. RBFs extend easily to high dimensional domains, enabling the capture of multi-joint coupling effects. As mentioned in Section 3.2, we can determine the pose-dependent elastic forces R(u, s) by computing the polynomial coefficient vector a(s). Using RBFs, the interpolated coefficient vector is computed at runtime as m
2
a(s) = X w j φ( s − s j ) + Q(s) j=1
where m is the number of dynamic morph targets. In our experiments, it was sufficient to use a constant vector Q for the polynomial Q(s). We also employed the globally supported biharmonic RBF kernel φ(r) = r, since its optimal fairness allows for smoother interpolation of sparsely scattered example poses as compared to locally supported kernels
[Carr et al. 2001]. In our experience, locally supported kernels such as the Gaussian RBF kernel are harder to tune and are unable to extrapolate across dynamic morph targets that are far apart in pose-space. We also use the smoothing term from
[Carr et al. 2001]
to achieve smoother behavior across large gaps between input poses. The RBF weight vectors w j are given by the solution of m linear systems (one for each input pose i): m
3
a(s i ) = a i = X w j φ( s i − s j ) + Q(s i ). j=1
The combination of these m linear systems provides m vectorial equations with m vector unknowns w j , plus the additional constant unknown vector Q (in case of choosing a constant polyno-
(c) Simulation with stiffer bulged morph target
mial). The combined system is underdetermined (m + 1 vector unknowns for m vectorial equations), but it can be solved by imposing orthogonality conditions on the weights w i
[Carr et al. 2001]. In order to avoid redundancy in the pose descriptors s i , and to guarantee that Eqn. (3) is not singular, we perform a principal component analysis
[Golub and Loan 1996] on the set of input pose descriptors. By selecting modes with non-zero or large eigenvalues only, we reduce the dimension of s and define a mapping to the reduced pose descriptor s = U T s s. We replace s by s both in the preprocessing stage to solve Eqn. (3) and at runtime for Eqn. (2). Modal reduction of the pose descriptors is very effective for robustness, but is also useful when our method is used for facilitating rigging. In highly complex areas of skin deformation such as the shoulder area, the skin is under influence of many bones for which the skinbone relationships cannot easily be determined by a human rigger or technical director. Our system can automatically deduce these relationships and reduce them to only a few significant modes. At runtime, given a(s) from Eqn. (2), we can compute the elastic ∂R(u,s) forces R(u, s) and their Jacobian for implicit integration ∂u of Eqn. (1). Unfortunately, due to the large number of coefficients that need to be interpolated, the evaluation of Eqn. (2) is rather costly. The number of coefficients is proportional to the number of nodes n in the finite element mesh for linear elastic models, and O(S 3 n) for StVK materials, where S is the average size of the neighborhood of a node. Instead, we propose a way to increase performance and to reduce the dependency on the resolution of the input geometry by reducing the number of degrees of freedom, while still maintaining the non-linear behavior defined by the morph targets. We use a reduced model u = Uq to enable dynamic simulation that is independent from the input resolution of the geometry. Given mass-orthogonal basis U T MU = I, the ODE in Eqn. (1), is transformed into
3.4 Reduced Equations of Motion
4
q + D(q, ̃ q, ̇ s) + R(q, ̃ s) = ̃ f
where D, ̃ R ̃ and ̃ f are r-dimensional reduced forces, D(q, ̃ q, ̇ s) = U T D(Uq, U q, ̇ s) (5a) R(q, ̃ s) = U T R(Uq, s) (5b) ̃ f = U T f . (5c)
42
Similarly, one can form the dense reduced tangent stiffness matrix,
6
∂ R(q, ̃ s) = U T ∂R(Uq, s) U ∈ R (r,r) ∂q ∂q
When applying model reduction to (multivariate) polynomial elastic forces, it can be shown that the reduced forces are still (multivariate) polynomial elastic forces. In particular, reduced ‘completely-linear’ elastic forces are linear polynomials in terms of the reduced coordinates q, and [Barbic and James 2005] has shown that StVK internal forces and tangent stiffness matrices are multivariate cubic polynomials that can be evaluated in Θ(r 4 ) time, with r the number of reduced modes (typically 10-30), by simply evaluating polynomials in terms of the reduced coordinates q: R(q) ̃ = P i q i + Q ij q i q j + S ijk q i q j q k (7a) ∂ R(q) ̃ l li il lij ilj ijl = P + (Q + Q )q i + (S + S + S )q i q j , ∂q l (7b) Note that Einstein summation convention was used in Eqn. (7). Here, P i , Q ij , S ijk ∈ R r are constant vector polynomial coefficients. The polynomial coefficients can be precomputed, given the rest pose p i . For linear materials, the Q ij and S ijk terms are all zero. We can now combine scattered polynomial interpolation from Section 3.3 with the reduced motion equations by concatenating the reduced coefficients into a = [ P i ; Q ij ; S ijk ]. Just as in Section 3.3, each dynamic morph target defines a set of coefficients a i which can then be used to set up an interpolator for the posedependent coefficients a(s). This then yields all the necessary information to compute R(q, ̃ s) in Eqn. (5). Note that, because the number of reduced modes r is typically many orders of magnitude smaller than the number of vertices of the mesh, the cost of evaluating Eqn. (2) is significantly smaller than in the non-reduced case. The construction of the reduction basis U will be discussed in Section 4. The reduced equations of motion Eqn. (4) can be solved using a reduced implicit Newmark Solver, employing the aforementioned internal forces and tangent stiffness matrices evaluated at each time step. As described in Section 3.1, an artist begins by modeling the base model surface and a skeleton with associated SSD skinning weights, and defines a set of geometric morph targets. Using vertex painting, he can then assign stiffness parameters such as Young’s modulus and Poisson ratio to certain parts of the skin. This is where our preprocessing stage starts. Preprocessing We first tetrahedralize the base mesh and propagate the surface node SSD skinning weights to internal nodes. We do this by solving a homogeneous Poisson problem for the internal node weights, where the known surface node weights are set up as boundary conditions. Then, for each morph target, a corresponding tetrahedral rest-pose mesh is defined (still in the skeletal bind pose). This can be done by displacing the surface nodes of the base tetrahedral mesh with the morph target’s values. We then relax the internal nodes by performing a physical simulation, constraining the new surface positions and using the elastic model of the base mesh. Once the new internal node positions are determined, the morph target’s position offsets ∆u from the base mesh are computed and the force polynomials a i associated with R i (u) are precomputed (see Eqn. (3), Sections 3.2 and 3.4). For reduced elastic models, a modal subspace is also constructed (Section 4). Runtime skeletal constraints At runtime, the final positions of the material points are computed as the combination of linear blend skinning and elastic deformation computed in the skeletal bind pose. We also account for the inertial forces caused by the moving coordinate frames of the bones
[Galoppo et al. 2007]. Additionally, we conceptually constrain the material points that are attached to internal bones. This can be achieved by removing the elastic degrees of freedom that are associated with corresponding internal mesh nodes. Hence, the positions of these points are then completely governed by the linear blend skinning transformations only. We remove these degrees of freedom in our preprocessing step by identifying tetrahedral mesh elements that are intersected by skeletal bones. Degrees of freedom that are associated with these elements are removed (i.e. they are ’fixed’ in the pose-space), unless they lie on the model’s boundary surface.
3.5 Model Construction and Kinematic Constraints
<H1>4 Subspace Construction</H1>
We describe here our implementation of the reduced equations of motion, through morph target aware subspace construction. In the reduced model, the displacement vector u is expressed as u = Uq, where U ∈ R (3n,r) is the displacement basis matrix, and q ∈ R r is the vector of reduced displacement coordinates. Here, U is a time-independent matrix specifying a basis of some r-dimensional (r << 3n) linear subspace of R 3n . For each of the dynamic morph targets, we employ linear modal analysis (LMA), which provides the best deformation basis for small deformations away from the rest configuration. Intuitively, modal basis vectors are directions into which the model can be pushed with the smallest possible increase in elastic strain energy. To find the modal basis vectors U i , we solve the following symmetric generalized eigenproblem (for a small number k of eigenvectors)
8
0 K(x )U i = MU i Λ i ,
8
i
. . < λ k i . The stiffness matrix K = ∂R(x) ∂x is evaluated at x 0 i , the rest configuration for input pose i, which defines a ‘goal’ deformation for the input poses. Note that at this point, one could easily add modal derivatives, as in [Barbic and James 2005]. In the next step, we have to combine the basis matrices U i into a global basis matrix U. We have taken into account three requirements when choosing the basis.
1. Avoid redundancy in the basis set, i.e. find an orthogonal set that is as compact as possible. 2. The characteristic deformations of all the morph targets have to be well represented. 3. The input deformations of each of the dynamic morph targets have to be well represented in the reduced space, otherwise the sculpted deformations can’t be simulated. In other words, the basis has to be aware of the morph targets.
The most straight forward approach is to combine all eigenvectors together as U = ˆ U 1 U 2 . . . U m ̃ , and orthogonalize them (each set of eigenvectors U p is orthogonal, but eigenvectors between different sets are not). This approach takes care of the second requirement, but results in a large set of eigenmodes with relatively small contributions for many eigenmodes, because all the common first deformation modes (such as stretch, shear, . . . ) are represented in each of the U i . Instead, similar to [Barbic and James 2005], we can construct a low-dimensional motion subspace by applying mass-PCA. We scale the derivatives according to the eigenvalues of the corresponding linear modes. Namely, we obtain the low-dimensional deformation basis by applying mass-PCA on the set of vectors
43
9
{ λ λ i 1 j i U j i | i = 1, . . . , m; j = 1, . . . , k i }.
We select the first r principal modes to achieve the basis U . Scaling is necessary to put greater weight on the more important low-frequency modes, which would otherwise be masked by highfrequency modes. In case the eigenmodes in different poses show large variance, we can adapt the normalization factor such that the eigenvectors are normalized across morph targets, by replacing the scaling factor by λ λ i 1 i λ i . k j Finally, to make the basis aware of the morph targets, we add m − 1 rest pose deformations {x 0 p |i = 2, . . . , m} to the set U and remass-orthogonalize the set {U i } ∪ {x 0 p } into the final basis U ∈ R 3n×r with r = (r + m − 1). Figure 3 illustrates the entire process.
Figure 3 : Construction of a morph-target-driven, massorthogonal reduction basis U: For each dynamic morph target i, during LMA the r smallest eigenmodes are selected to construct eigenbases U i . Mass-PCA combines and compacts the U i , retaining only k most significant modes. Finally, we explicitly add morph target deformations x 0 i to the eigenbasis and and guarantee massorthogonality of the final basis U.
<H1>5 Results</H1>
By using pose-space efficient polynomial interpolation to achieve pose-dependent behavior, we are able to demonstrate rich nonlinear deformation effects at relatively small extra cost compared to simple simulation of linear or semi-non-linear materials. We have performed experiments with three different input models: a simple bulging cylinder with 4 bones (see Fig. 2 ), a shoulder model with 4 bones, and Herbert, our swimsuit model with 46 bones. For each
Figure 4:
Herbert jumps off on a diving board: Comparison of single (pose-independent) linear elasticity (left column), our method with dynamic morph targets (middle column), and our method with dynamic morph targets and modal reduction applied (right column). When balled up, Herbert’s back (top) and belly (bottom) bulge in correspondence with his morph targets defined in Fig. 1 . On the bottom left, Herbert’s belly looks very flabby, as if he swallowed a brick. However, Herbert’s ‘fetal pose’ morph target 3 was authored with a stiff belly. Our method (bottom right) shows the more desired behavior.
model, we qualitatively compare simulations with different elastic models. We compare a single (pose-independent) elastic model with our pose-dependent elastic model that employs multiple dynamic morph targets, both with and without modal reduction. Herbert model For the Herbert simulations, we used only 3 morph targets, illustrated in Fig. 1 . The first morph target is a skinny version of Herbert, in which his skin is very soft and flabby, the third target is a stiff, bulged Herbert in fetal position, while the second target has been chosen in between the first and the third. In our video, we show Herbert’s belly deformations as he cycles between upright and fetal poses. While the single elastic model shows little or no dynamic behavior, our pose-dependent elastic model adds a dramatic amount of realism due to the bulging behavior and inertial skin motion. Next, we drive the skinned Herbert model with a skeletal animation and add inertial forces due to the bone’s moving frames. As Herbert jumps off a diving board and flips through different poses, we show the advantage of our pose-dependent model from an artistic viewpoint. With single elastic models, the belly is flabby and skinny throughout the entire simulation. Using the aforementioned morph targets for Herbert, an animator can impose a stiff, bulged belly in balled-up poses, and softer, skinny belly behavior in upright poses. Our video demonstrates the imposed behavior as Herbert’s belly exposes bulging and non-flabby skin when he jumps from the diving board. Also, in Fig. 4 , we show the use of reduced models in our method achieves the same quality of desired deformations as the computationally more expensive unreduced model. Shoulder model Our method also provides a physically-based approach to resolving regions affected by multiple joints, such as a shoulder rig. Our approach facilitates complex rigging: a set of
44
Shoulder example morph targets:
(a) A few skinned input poses with associated target deformations as provided by an artist (targets morph targets in total, including one bent arm input with bicep bulging (target
6).
Bicep muscle bulge and self-intersection:
(b) At runtime, a pose-independent force model clearly shows undesired and self-intersecting deformations when bending the arm (left column). On the other hand, our method in the middle column shows deformations consistent with morph target (Fig. (a)), even after application of modal reduction for efficiency (right column).
(c)
Extrapolation to new shoulder poses:
Our method also extrapolates to non-input poses.
Figure 5: Shoulder Rig Simulation
skinning weights and a set of morph targets are sufficient to simulate complex co-articulation effects. There is no need for manual tweaking of the complex mapping of joint configuration to blending weights of geometric morph targets. In our shoulder example, we have 6 morph targets, shown in Fig. 5(a) . The shoulder example also demonstrates our method’s ability to simulate dynamic behavior at poses away from the morph target input poses ( Fig. 5 ). The input morph target set contains only one example of a folded elbow but we show two distinct folding scenarios in the full animation. Both folding scenarios show severe self-intersection in the single pose-independent model due to the effect of linear blend skinning. Our pose-dependent model resolves both automatically. Another
1, 3 6).
and We used
6
6
interesting co-articulation effect is the motion of the chest muscle as the arm makes a folding motion (see video). Whereas the chest seems to collapse for single elastic models, it bulges more realistically with our method. The shoulder model has 4899 degrees of freedom. After modal reduction, we were able to accelerate the simulation significantly by using only 19 eigenmodes ( Table 1 ) with almost no visible effect on the simulation quality ( Fig. 5 ). Performance In addition to qualitative comparisons, we have also compared simulation timings. We have performed all our experiments on a 2.4 GHz Intel Core 2 Duo Macbook Pro laptop (us- ing one of its two cores), with 2 GB of RAM and a NVidia GeForce 8600M GT graphics card. All rendering was done with the opensource Blender modeling package. Our timing results are summarized in Table 1 . All our methods achieve real-time performance due to efficient pose-space interpolation of low-complexity linear elastic forces and modal reduction of either linear or semi-nonlinear StVK forces. Comparing our method with the performance of single (pose-independent) elastic models, it is clear from Table 1 that our method has only a marginal extra cost, due to efficient polynomial interpolation of the dynamic morph target models. Finally, precomputation of the force polynomial coefficients in Eqn. (7) can be significant in case of StVK models, but never prohibitive. The precomputation times are shown in Table 1 . Quality versus artist effort and performance The quality of the output animation depends on the quality and number of the artist’s input poses. Too few input poses can cause slight popping of the animation towards the input shapes but the simulation of the pose-dependent elastic model will be stable nevertheless. Therefore, stable results can be achieved with little effort from the artist and poses can then be added incrementally to areas of the posespace where the behavior is not satisfactory. The effect of the number of poses on the run-time performance of the algorithm is very small as it doesn’t affect the number of degrees of freedom in the simulation (see comparison 6 vs. 9 shoulder DMTs in Table 1 ). Only the cost of the RBF interpolation in Eq. (2) is affected linearly in the number of poses m. Comparison with other methods While geometric morph targets enable control of non-linear deformations, these deformations are purely static and cannot react to external forces in a physical simulation. Our dynamic morph targets add dynamic behavior to non-linear deformations such that external and inertial forces can be applied, as shown in Fig. 2 . The method by Capell et al. [2005] also enables deformations under influence of external forces, corresponding to the behavior in Fig. 2(b) , but does not influence the underlying properties of the elastic material. As shown in Fig 2(c) , our method can correct such undesirable behavior by setting elastic properties for each of the individual morph targets, effectively mimicking muscle contraction.
45
Table 1: Model Statistics and Performance.
Model
Elasticity
# DOFS
# DMTs
Preproc.
Without
With
Type
(s)
DMTs
DMTs
(fps)
(fps)
Bulging
Lin.
918
4
/
28
25
Cylinder
Red. Lin.
13
4
/
93
89
Red. StVK
13
4
21.2
89
64
Herbert
Lin.
603
3
/
36
34
Red. Lin.
12
3
/
96
92
Red. StVK
12
3
16.8
90
58
Shoulder
Lin.
4899
6
/
3
3
Lin.
4899
9
/
3
3
Red. Lin.
19
6
/
87
85
Red. Lin.
19
9
/
83
81
Red. StVK
19
6
661
30
22
Red. StVK
19
9
1078
28
21
Model Elasticity # DOFS # DMTs Preproc. Without With Type (s) DMTs DMTs (fps) (fps) Bulging Lin. 918 4 / 28 25 Cylinder Red. Lin. 13 4 / 93 89 Red. StVK 13 4 21.2 89 64 Herbert Lin. 603 3 / 36 34 Red. Lin. 12 3 / 96 92 Red. StVK 12 3 16.8 90 58 Shoulder Lin. 4899 6 / 3 3 Lin. 4899 9 / 3 3 Red. Lin. 19 6 / 87 85 Red. Lin. 19 9 / 83 81 Red. StVK 19 6 661 30 22 Red. StVK 19 9 1078 28 21
<H1>6 Summary and Future Work</H1>
In this paper we have presented dynamic morph targets, posedependent elastic models that allow an artist to easily author and control the geometry and elastic behavior of dynamic characters. Dynamic morph targets avoid complex rigging requirements of purely geometric methods, and complex musculoskeletal modeling of purely physically-based methods. By extending our basic framework to support modal reduction, we also achieve high runtime performance. The current dynamic morph target framework can be enhanced with additional features to be included in the future, such as support for (contact) constraints, and an extension to weighted pose-space deformations [Kurihara and Miyata 2004] to allow for improved localized interpolation. Acknowledgements This work is supported in part by ARO, NSF, DARPA/RDECOM and Intel Corporation. We would like to thank the GAMMA group at UNC Chapel Hill and the CGL at ETH Zurich for their support. We would also like to thank the anonymous reviewers for their critical feedback to improve the paper.
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"text": "['Purely data-driven methods lack a kinematic model']",
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8223,
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"text": "['making them of limited use for animation and dynamic simulation']",
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"text": "['Purely kinematic approaches']",
"type": "background_claim",
"id": "T59"
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8394,
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"text": "['model the deformation of the skin surface by linear blending of the animated bone transformations']",
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"text": "['skeletal-subspace deformation']",
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8362,
8391
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"text": "['Magnenat-Thalmann et al. 1988']",
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8493,
8507
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"text": "['This technique']",
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"id": "T63"
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8546,
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"text": "['cannot capture complex deformations']",
"type": "background_claim",
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"text": "['typically has problems deforming skin near joints due to collapsing geometry']",
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"text": "['the deformation is restricted to the subspace of the affine transformation of the joints']",
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"text": "['Different methods have been proposed to address the problems of linear blend skinning by inserting additional joints tuned from examples']",
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8918,
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"text": "['Mohr and Gleicher 2003']",
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"text": "['employing blending of transformations instead of weights']",
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"text": "['Kavan and Zara 2005']",
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"text": "['Recent techniques have extended skinning to mesh deformations']",
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9107,
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"text": "['James and Twigg 2005']",
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9130,
9179
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"text": "['motion capture data without a predefined skeleton']",
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9182,
9203
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"text": "['Park and Hodgins 2006']",
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"text": "['interactive models']",
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9230,
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"text": "['Der et al. 2006']",
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"text": "['SSD does not permit direct sculpting or control']",
"type": "background_claim",
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9364,
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"text": "['artists have to tweak vertex weights']",
"type": "background_claim",
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9402,
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"text": "['giving SSD algorithms the reputation of being tedious to control']",
"type": "background_claim",
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"text": "['The first work to add control in a kinematic approach is that of pose-space deformations']",
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9559,
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"text": "['Lewis et al. 2000']",
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9579,
9726
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"text": "['PSD is a hybrid method that combines SSD with morphing and employs scattered data interpolation to compute non-linear skin corrections in posespace']",
"type": "background_claim",
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9728,
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"text": "['resulting in a kinematic model that also has artist-sculpted poses']",
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9862,
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"text": "['PSD becomes memory inefficient']",
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9801,
9860
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"text": "['dealing with large pose-spaces that have many example poses']",
"type": "data",
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"text": "['the large database of surface displacements']",
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9945,
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"text": "['PSD can be extended to support per-vertex pose-space deformation']",
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10019,
10043
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"text": "['Kurihara and Miyata 2004']",
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10045,
10061
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"text": "['Rhee et al. 2006']",
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10065,
10118
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"text": "['largely reducing the number of required example poses']",
"type": "background_claim",
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{
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10120,
10140
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"text": "['The EigenSkin method']",
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10143,
10158
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],
"text": "['Kry et al. 2002']",
"type": "data",
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"offsets": [
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10161,
10328
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],
"text": "['also provides a way to reduce per-vertex displacement memory footprint by computing an error-optimal set of eigenbases for approximating the original deformation model']",
"type": "background_claim",
"id": "T93"
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{
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10330,
10350
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],
"text": "['Other recent methods']",
"type": "background_claim",
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"offsets": [
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10391,
10502
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"text": "['learn example-based corrections on sparse points and assume that these corrections can be smoothly interpolated']",
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10353,
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"text": "['Weber et al. 2007']",
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10372,
10388
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],
"text": "['Wang et al. 2007']",
"type": "data",
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"offsets": [
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10504,
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"text": "['Pose space deformation and related example-based methods allow for direct sculpting of geometric morph targets']",
"type": "background_claim",
"id": "T98"
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{
"offsets": [
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10620,
10729
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],
"text": "['are purely kinematic approaches to (quasi-)static deformation, without reference to underlying forces or mass']",
"type": "background_claim",
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{
"offsets": [
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10731,
10951
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],
"text": "['Our method builds on the concept of pose-space deformation and applies it to pose-space interpolation of dynamic morph targets to achieve not only (quasi-)static deformations, but a fully dynamic model in time-pose-space']",
"type": "own_claim",
"id": "T100"
},
{
"offsets": [
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10962,
11063
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],
"text": "['physically based methods in graphics are based on biomechanical models of skin tissue and musculature']",
"type": "background_claim",
"id": "T101"
},
{
"offsets": [
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11065,
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],
"text": "['In terms of efficiency versus accuracy, these methods fall into two broad categories']",
"type": "background_claim",
"id": "T102"
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{
"offsets": [
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11151,
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],
"text": "['The first category of algorithms aim for accuracy']",
"type": "background_claim",
"id": "T103"
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{
"offsets": [
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11354,
11436
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],
"text": "['by simulating the actions of the individual muscles, bones and tendons in the skin']",
"type": "background_claim",
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{
"offsets": [
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11203,
11224
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],
"text": "['Chen and Zeltzer 1992']",
"type": "data",
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{
"offsets": [
[
11226,
11247
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],
"text": "['Scheepers et al. 1997']",
"type": "data",
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{
"offsets": [
[
11249,
11273
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],
"text": "['Wilhelms and Gelder 1997']",
"type": "data",
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{
"offsets": [
[
11275,
11291
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],
"text": "['Koch et al. 1996']",
"type": "data",
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},
{
"offsets": [
[
11293,
11311
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],
"text": "['Zordan et al. 2004']",
"type": "data",
"id": "T109"
},
{
"offsets": [
[
11313,
11332
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],
"text": "['Sifakis et al. 2005']",
"type": "data",
"id": "T110"
},
{
"offsets": [
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11334,
11351
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],
"text": "['Sueda et al. 2008']",
"type": "data",
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"offsets": [
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11438,
11508
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],
"text": "['Interactive physically based approaches trade accuracy for performance']",
"type": "background_claim",
"id": "T112"
},
{
"offsets": [
[
11511,
11534
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],
"text": "['Terzopoulos et al. 1987']",
"type": "data",
"id": "T113"
},
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"offsets": [
[
11536,
11563
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],
"text": "['Terzopoulos and Witkin 1988']",
"type": "data",
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"offsets": [
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11565,
11593
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],
"text": "['Metaxas and Terzopoulos 1992']",
"type": "data",
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11595,
11616
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],
"text": "['Picinbono et al. 2001']",
"type": "data",
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{
"offsets": [
[
11618,
11636
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],
"text": "['Capell et al. 2002']",
"type": "data",
"id": "T117"
},
{
"offsets": [
[
11638,
11659
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],
"text": "['Müller and Gross 2004']",
"type": "data",
"id": "T118"
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"offsets": [
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11661,
11680
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],
"text": "['Galoppo et al. 2007']",
"type": "data",
"id": "T119"
},
{
"offsets": [
[
11683,
11812
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],
"text": "['These methods use simplified (quasi-)linear elastic models that cannot capture complex non-linear behavior such as muscle bulging']",
"type": "background_claim",
"id": "T120"
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{
"offsets": [
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11814,
11895
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],
"text": "['Physically based methods can only provide control through the influence of forces']",
"type": "background_claim",
"id": "T121"
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{
"offsets": [
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11903,
11977
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],
"text": "['methods that control global deformation modes have been around for a while']",
"type": "background_claim",
"id": "T122"
},
{
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[
11980,
12001
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],
"text": "['Witkin and Welch 1990']",
"type": "data",
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},
{
"offsets": [
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12004,
12138
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],
"text": "['providing control of sculpted deformations for simulation of deformable models has only recently caught attention in graphics research']",
"type": "background_claim",
"id": "T124"
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{
"offsets": [
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12140,
12190
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],
"text": "['A method for physically based rigging was proposed']",
"type": "background_claim",
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"offsets": [
[
12196,
12214
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"text": "['Capell et al. 2005']",
"type": "data",
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12217,
12280
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"text": "['using pose-dependent forces to guide the shape of the character']",
"type": "background_claim",
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12309,
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"text": "['their approach does not support pose-dependent elastic properties']",
"type": "background_claim",
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"text": "['its performance is highly dependent on the resolution of the sculpted deformations']",
"type": "background_claim",
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12490,
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"text": "['shape keyframes can be used to retarget the elastic deformations']",
"type": "background_claim",
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{
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12464,
12488
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"text": "['Given an input animation']",
"type": "data",
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12557,
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"text": "['Kondo et al. 2005']",
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12576,
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"text": "['or to enhance the surface deformations with physically simulated detail using subspace constraints']",
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"text": "['Bergou et al. 2007']",
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"text": "['The former provides good control of shapes']",
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12745,
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],
"text": "['is restricted to a given input animation']",
"type": "background_claim",
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"text": "['the latter achieves rich secondary surface detail']",
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"id": "T137"
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{
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12847,
12898
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],
"text": "['does not provide direct manipulation of the surface']",
"type": "background_claim",
"id": "T138"
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12900,
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"text": "['Our method provides the ability to sculpt the dynamic morph targets directly and produces a dynamic model that is not restricted to a given animation']",
"type": "own_claim",
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13051,
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"text": "['our model can be plugged into any simulated environment and be subject to external forces']",
"type": "own_claim",
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{
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13143,
13233
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"text": "['Modal reduction has proven useful to increase performance in posespace deformation methods']",
"type": "background_claim",
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13236,
13251
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"text": "['Kry et al. 2002']",
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13253,
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"text": "['as well as in physically based methods']",
"type": "background_claim",
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13294,
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"text": "['Hauser et al. 2003']",
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13314,
13330
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],
"text": "['Choi and Ko 2005']",
"type": "data",
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13333,
13364
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],
"text": "['Our work exploits the technique']",
"type": "own_claim",
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13393,
13465
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],
"text": "['that enables fast modal integration of St.Venant-Kirchoff elastic forces']",
"type": "own_claim",
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{
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13370,
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"text": "['Barbic and James 2005']",
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13473,
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"text": "['the performance depends mainly on the number of simulated eigenmodes, not on the resolution of the model']",
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13579,
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"text": "['This technique makes our approach suitable for real-time applications']",
"type": "own_claim",
"id": "T150"
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13680,
13775
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"text": "['our method is the first to provide shape and surface behavior control of dynamic reduced models']",
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{
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13836,
14028
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"text": "['The goal of our method is to simulate controllable non-linear deformations by interpolation of dynamic morph targets at runtime, kindred to geometric morph targets in static character modeling']",
"type": "own_claim",
"id": "T152"
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{
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14118,
14224
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"text": "['they can be used to simulate a pose-dependent elastic model that is fast enough for real-time applications']",
"type": "own_claim",
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{
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14290,
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],
"text": "['Dynamic morph targets define pose-specific soft skin behavior']",
"type": "own_claim",
"id": "T154"
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14368,
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"text": "['they are pairs of elastic models E i and poses s i']",
"type": "own_claim",
"id": "T155"
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{
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14514,
14596
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"text": "['dynamic morph targets associate surface and volume deformation with character pose']",
"type": "own_claim",
"id": "T156"
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"offsets": [
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14638,
14739
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],
"text": "['dynamic morph targets also define pose-specific elastic properties including stiffness and plasticity']",
"type": "own_claim",
"id": "T157"
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14741,
14832
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],
"text": "['The combination of surface deformation and elastic properties defines the elastic model E i']",
"type": "own_claim",
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"offsets": [
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14952,
15020
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],
"text": "['the concept of pose can easily be extended beyond the skeletal sense']",
"type": "own_claim",
"id": "T159"
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14909,
14946
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"text": "['our implementation uses skeletal pose']",
"type": "own_claim",
"id": "T160"
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{
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15030,
15076
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],
"text": "['any notion of state of a character can be used']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
15086,
15101
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],
"text": "['emotional state']",
"type": "data",
"id": "T162"
},
{
"offsets": [
[
15103,
15117
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],
"text": "['velocity state']",
"type": "data",
"id": "T163"
},
{
"offsets": [
[
15119,
15132
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],
"text": "['contact state']",
"type": "data",
"id": "T164"
},
{
"offsets": [
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15137,
15154
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],
"text": "['muscle activation']",
"type": "data",
"id": "T165"
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{
"offsets": [
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15156,
15229
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],
"text": "['Dynamic morph targets can easily be created in existing modeling packages']",
"type": "own_claim",
"id": "T166"
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"offsets": [
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15343,
15450
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],
"text": "['of the character and sculpts desired deformations that cannot be captured with traditional skinning methods']",
"type": "own_claim",
"id": "T167"
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{
"offsets": [
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15281,
15315
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],
"text": "['A modeler defines a set of m poses']",
"type": "own_claim",
"id": "T168"
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{
"offsets": [
[
15453,
15472
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],
"text": "['Kavan and Zara 2005']",
"type": "data",
"id": "T169"
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{
"offsets": [
[
15474,
15502
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],
"text": "['MagnenatThalmann et al. 1988']",
"type": "data",
"id": "T170"
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{
"offsets": [
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15505,
15553
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],
"text": "['Elastic properties can be assigned for each pose']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
15565,
15648
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],
"text": "['the same skin section can be defined stiff for one pose and flabby for another pose']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
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15735,
15884
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],
"text": "['Dynamic characters enhanced with dynamic morph targets can react to external forces just as with other common physically based deformation algorithms']",
"type": "own_claim",
"id": "T173"
},
{
"offsets": [
[
15890,
15991
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],
"text": "['they also expose non-linear deformations and elastic behavior as imposed by the dynamic morph targets']",
"type": "own_claim",
"id": "T174"
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16073,
16151
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"text": "['Dynamic morph targets are used to build a pose-dependent elastic model E(x, s)']",
"type": "own_claim",
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16182,
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"text": "['an elastic model can be defined as a material function E(u(x)) defining the internal elastic energy at material points x in function of the deformation u']",
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{
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"text": "['hyper-elastic materials']",
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"text": "['we choose to express elastic deformation in the skeletal bind pose as has been proposed in the past']",
"type": "own_claim",
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"text": "['Lewis et al. 2000']",
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16555,
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"text": "['Kry et al. 2002']",
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"offsets": [
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16572,
16591
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"text": "['Galoppo et al. 2007']",
"type": "data",
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16613,
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"text": "['it is certainly possible to use other formulations of elastic strain to define a pose-dependent model with dynamic morph targets']",
"type": "own_claim",
"id": "T182"
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{
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16743,
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"text": "['Traditionally, the elastic energy is a pose-independent material potential that causes internal elastic forces R(u) in the material']",
"type": "background_claim",
"id": "T183"
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{
"offsets": [
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16876,
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],
"text": "['We create a pose-dependent elastic model by taking into account the dynamic morph targets {E i , s i } as example inputs']",
"type": "own_claim",
"id": "T184"
},
{
"offsets": [
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17258,
17464
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"text": "['Without loss of generality, we can use the finite element method to discretize the partial differential equations of solid continuum mechanics in space, leading to the common Euler-Lagrange motion equations']",
"type": "own_claim",
"id": "T185"
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"offsets": [
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17758,
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"text": "['The computation of pose-dependent elastic forces R(u, s) requires the interpolation of pose-specific forces R i (u)']",
"type": "own_claim",
"id": "T186"
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"offsets": [
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17890,
17944
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"text": "['forces are a function of the timevarying deformation u']",
"type": "own_claim",
"id": "T187"
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17946,
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"text": "['they cannot simply be evaluated once and then interpolated at runtime']",
"type": "own_claim",
"id": "T188"
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{
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18032,
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"text": "['we have opted for elastic models for which R i (u) can be expressed as a (multivariate) polynomial function of the degrees of freedom u']",
"type": "own_claim",
"id": "T189"
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"text": "['Common examples of such elastic models are the so-called ‘completely linear’ FEM deformation model']",
"type": "background_claim",
"id": "T190"
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{
"offsets": [
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18169,
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"text": "['Then, the interpolation of elastic models reduces to the interpolation of polynomial coefficients']",
"type": "own_claim",
"id": "T191"
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{
"offsets": [
[
18405,
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"text": "['Müller and Gross 2004']",
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18433,
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"text": "['the ’seminon-linear’ St.Venant-Kirchoff model']",
"type": "background_claim",
"id": "T193"
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18488,
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"text": "['Barbic and James 2005']",
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18511,
18529
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"text": "['Capell et al. 2005']",
"type": "data",
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"offsets": [
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18532,
18614
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"text": "['We have simulated our examples with both linear and semi-non-linear elastic models']",
"type": "own_claim",
"id": "T196"
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{
"offsets": [
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18633,
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"text": "['we express deformation in the skeletal bind pose']",
"type": "own_claim",
"id": "T197"
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{
"offsets": [
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18683,
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"text": "['we did not see any noticeable quality difference between both elastic models in our experiments']",
"type": "own_claim",
"id": "T198"
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{
"offsets": [
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18791,
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"text": "['we have opted for the more efficient linear elasticity to produce most of the images and videos unless otherwise noted']",
"type": "own_claim",
"id": "T199"
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18915,
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"text": "['Section 5']",
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18955,
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"text": "['both models are amenable to modal reduction for efficiency']",
"type": "own_claim",
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18930,
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"text": "['Section 3.4']",
"type": "data",
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"text": "['The interpolation of polynomial coefficients yields the interpolation of force values R(u, s) at all possible deformation values u']",
"type": "own_claim",
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19604,
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"text": "['it also yields the interpolation of force derivatives']",
"type": "own_claim",
"id": "T204"
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19667,
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"text": "['the ∂R(u,s) stiffness matrix']",
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"offsets": [
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19698,
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"text": "['One subtle detail remains for defining a ∂u complete interpolation of elastic models']",
"type": "own_claim",
"id": "T206"
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"offsets": [
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19784,
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"text": "['The rest configuration at each input pose may be different']",
"type": "own_claim",
"id": "T207"
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"text": "['the deformation u may not be consistent across poses']",
"type": "own_claim",
"id": "T208"
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"text": "['The addition of this term simply modifies the coefficients of the force polynomials, which can then be safely interpolated']",
"type": "own_claim",
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"text": "['all deformations are expressed w.r.t. a consistent configuration']",
"type": "own_claim",
"id": "T210"
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"text": "['a pose can be described in many ways']",
"type": "own_claim",
"id": "T211"
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20462,
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"text": "['it provides a description of the state of a model or character']",
"type": "data",
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20530,
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"text": "['a metric to measure distances between poses']",
"type": "data",
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"offsets": [
[
22381,
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"text": "['its elastic model should approximate the elastic models of the input poses']",
"type": "own_claim",
"id": "T214"
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"offsets": [
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"text": "['an articulated character moves between observed configurations']",
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22486,
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"text": "['we are looking for a way to interpolate internal elastic forces R i']",
"type": "own_claim",
"id": "T216"
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22473,
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"text": "['Section 3.2']",
"type": "data",
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"offsets": [
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22570,
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"text": "['the character moves from one pose to the other']",
"type": "data",
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"offsets": [
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22618,
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"text": "['the internal forces change continuously but highly non-linearly']",
"type": "own_claim",
"id": "T219"
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"text": "['elastic forces form a non-linear smooth field in pose-space']",
"type": "own_claim",
"id": "T220"
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{
"offsets": [
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22760,
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"text": "['Radial base functions']",
"type": "background_claim",
"id": "T221"
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"offsets": [
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22802,
22903
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"text": "['are a common choice for interpolating scattered data of an underlying smooth field that is non-linear']",
"type": "background_claim",
"id": "T222"
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{
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22783,
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"text": "['Powell 1987']",
"type": "data",
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22975,
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"text": "['RBFs are suited']",
"type": "own_claim",
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22999,
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"text": "['they work well with sparse input data sets']",
"type": "background_claim",
"id": "T225"
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"offsets": [
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22918,
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"text": "['our goal is to have as few input models E i as possible']",
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"offsets": [
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23043,
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"text": "['RBFs extend easily to high dimensional domains, enabling the capture of multi-joint coupling effects']",
"type": "background_claim",
"id": "T227"
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"offsets": [
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23174,
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],
"text": "['we can determine the pose-dependent elastic forces R(u, s) by computing the polynomial coefficient vector a(s)']",
"type": "own_claim",
"id": "T228"
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{
"offsets": [
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23161,
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"text": "['Section 3.2']",
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"text": "['Using RBFs, the interpolated coefficient vector is computed at runtime as']",
"type": "own_claim",
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"offsets": [
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23393,
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"text": "['a(s) = X w j φ( s − s j ) + Q(s) j=1']",
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"offsets": [
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"text": "['it was sufficient to use a constant vector Q for the polynomial Q(s)']",
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"id": "T232"
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"offsets": [
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"text": "['We also employed the globally supported biharmonic RBF kernel φ(r) = r']",
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"id": "T233"
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"text": "['its optimal fairness allows for smoother interpolation of sparsely scattered example poses as compared to locally supported kernels']",
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23797,
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"text": "['Carr et al. 2001']",
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23835,
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"text": "['locally supported kernels']",
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"text": "['are harder to tune and are unable to extrapolate across dynamic morph targets that are far apart in pose-space']",
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23869,
23892
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"text": "['the Gaussian RBF kernel']",
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24005,
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"text": "['We also use the smoothing term']",
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"text": "['to achieve smoother behavior across large gaps between input poses']",
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24043,
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"text": "['Carr et al. 2001']",
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"offsets": [
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24130,
24202
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"text": "['The RBF weight vectors w j are given by the solution of m linear systems']",
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"text": "['a(s i ) = a i = X w j φ( s i − s j ) + Q(s i ). j=1']",
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24334,
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"text": "['The combination of these m linear systems provides m vectorial equations with m vector unknowns w j']",
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"text": "['plus the additional constant unknown vector Q']",
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24494,
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"text": "['choosing a constant polyno-']",
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24600,
24604
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"text": "['mial']",
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"text": "['The combined system is underdetermined']",
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24701,
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"text": "['it can be solved by imposing orthogonality conditions on the weights w i']",
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24647,
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"text": "['m + 1 vector unknowns for m vectorial equations']",
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24776,
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"text": "['Carr et al. 2001']",
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"offsets": [
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24795,
24943
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"text": "['In order to avoid redundancy in the pose descriptors s i , and to guarantee that Eqn. (3) is not singular, we perform a principal component analysis']",
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"text": "['on the set of input pose descriptors']",
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24946,
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"text": "['Golub and Loan 1996']",
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"offsets": [
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25005,
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"text": "['By selecting modes with non-zero or large eigenvalues only, we reduce the dimension of s and define a mapping to the reduced pose descriptor s = U T s s']",
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"text": "['Modal reduction of the pose descriptors is very effective for robustness']",
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25334,
25348
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"text": "['is also useful']",
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25354,
25397
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"text": "['our method is used for facilitating rigging']",
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"offsets": [
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25470,
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"text": "['the skin is under influence of many bones for which the skinbone relationships cannot easily be determined by a human rigger or technical director']",
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"text": "['In highly complex areas of skin deformation']",
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25451,
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"text": "['the shoulder area']",
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25618,
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"text": "['Our system can automatically deduce these relationships and reduce them to only a few significant modes']",
"type": "own_claim",
"id": "T262"
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"offsets": [
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25761,
25869
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"text": "['we can compute the elastic ∂R(u,s) forces R(u, s) and their Jacobian for implicit integration ∂u of Eqn. (1)']",
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"offsets": [
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"text": "['At runtime']",
"type": "own_claim",
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25741,
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"text": "['a(s) from Eqn. (2)']",
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25893,
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"text": "['the large number of coefficients that need to be interpolated']",
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"text": "['the evaluation of Eqn. (2) is rather costly']",
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"offsets": [
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26001,
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"text": "['The number of coefficients is proportional to the number of nodes n in the finite element mesh for linear elastic models, and O(S 3 n) for StVK materials']",
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"offsets": [
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"text": "['we propose a way to increase performance and to reduce the dependency on the resolution of the input geometry by reducing the number of degrees of freedom, while still maintaining the non-linear behavior defined by the morph targets']",
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"id": "T269"
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"offsets": [
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"text": "['We use a reduced model u = Uq to enable dynamic simulation that is independent from the input resolution of the geometry']",
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26620,
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"text": "['the ODE in Eqn. (1), is transformed into']",
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"offsets": [
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26732,
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"text": "['q + D(q, ̃ q, ̇ s) + R(q, ̃ s) = ̃ f']",
"type": "own_claim",
"id": "T272"
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{
"offsets": [
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26586,
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"text": "['mass-orthogonal basis U T MU = I']",
"type": "data",
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26976,
27031
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"text": "['one can form the dense reduced tangent stiffness matrix']",
"type": "own_claim",
"id": "T274"
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"offsets": [
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27064,
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"text": "['∂ R(q, ̃ s) = U T ∂R(Uq, s) U ∈ R (r,r) ∂q ∂q']",
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"id": "T275"
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{
"offsets": [
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27203,
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],
"text": "['it can be shown that the reduced forces are still (multivariate) polynomial elastic forces']",
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"id": "T276"
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{
"offsets": [
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27133,
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"text": "['applying model reduction to (multivariate) polynomial elastic forces']",
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"offsets": [
[
27310,
27413
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"text": "['reduced ‘completely-linear’ elastic forces are linear polynomials in terms of the reduced coordinates q']",
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"id": "T278"
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{
"offsets": [
[
27420,
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"text": "['Barbic and James 2005']",
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{
"offsets": [
[
27458,
27582
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"text": "['StVK internal forces and tangent stiffness matrices are multivariate cubic polynomials that can be evaluated in Θ(r 4 ) time']",
"type": "background_claim",
"id": "T280"
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{
"offsets": [
[
27638,
27762
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],
"text": "['by simply evaluating polynomials in terms of the reduced coordinates q: R(q) ̃ = P i q i + Q ij q i q j + S ijk q i q j q k']",
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"id": "T281"
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{
"offsets": [
[
27768,
27844
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],
"text": "['∂ R(q) ̃ l li il lij ilj ijl = P + (Q + Q )q i + (S + S + S )q i q j , ∂q l']",
"type": "background_claim",
"id": "T282"
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{
"offsets": [
[
27988,
28034
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"text": "['The polynomial coefficients can be precomputed']",
"type": "own_claim",
"id": "T283"
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{
"offsets": [
[
28042,
28059
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],
"text": "['the rest pose p i']",
"type": "data",
"id": "T284"
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{
"offsets": [
[
28084,
28121
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],
"text": "['the Q ij and S ijk terms are all zero']",
"type": "own_claim",
"id": "T285"
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{
"offsets": [
[
28066,
28082
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],
"text": "['linear materials']",
"type": "data",
"id": "T286"
},
{
"offsets": [
[
28123,
28176
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],
"text": "['We can now combine scattered polynomial interpolation']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
28194,
28301
]
],
"text": "['with the reduced motion equations by concatenating the reduced coefficients into a = [ P i ; Q ij ; S ijk ]']",
"type": "own_claim",
"id": "T288"
},
{
"offsets": [
[
28327,
28475
]
],
"text": "['each dynamic morph target defines a set of coefficients a i which can then be used to set up an interpolator for the posedependent coefficients a(s)']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
28477,
28557
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],
"text": "['This then yields all the necessary information to compute R(q, ̃ s) in Eqn. (5)']",
"type": "own_claim",
"id": "T290"
},
{
"offsets": [
[
28578,
28693
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],
"text": "['the number of reduced modes r is typically many orders of magnitude smaller than the number of vertices of the mesh']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
28695,
28780
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],
"text": "['the cost of evaluating Eqn. (2) is significantly smaller than in the non-reduced case']",
"type": "own_claim",
"id": "T292"
},
{
"offsets": [
[
28856,
29055
]
],
"text": "['The reduced equations of motion Eqn. (4) can be solved using a reduced implicit Newmark Solver, employing the aforementioned internal forces and tangent stiffness matrices evaluated at each time step']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
29086,
29235
]
],
"text": "['an artist begins by modeling the base model surface and a skeleton with associated SSD skinning weights, and defines a set of geometric morph targets']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
29073,
29084
]
],
"text": "['Section 3.1']",
"type": "data",
"id": "T295"
},
{
"offsets": [
[
29237,
29370
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],
"text": "['Using vertex painting, he can then assign stiffness parameters such as Young’s modulus and Poisson ratio to certain parts of the skin']",
"type": "own_claim",
"id": "T296"
},
{
"offsets": [
[
30084,
30130
]
],
"text": "['the new internal node positions are determined']",
"type": "data",
"id": "T297"
},
{
"offsets": [
[
30132,
30272
]
],
"text": "['the morph target’s position offsets ∆u from the base mesh are computed and the force polynomials a i associated with R i (u) are precomputed']",
"type": "own_claim",
"id": "T298"
},
{
"offsets": [
[
30278,
30286
]
],
"text": "['Eqn. (3)']",
"type": "data",
"id": "T299"
},
{
"offsets": [
[
30288,
30308
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],
"text": "['Sections 3.2 and 3.4']",
"type": "data",
"id": "T300"
},
{
"offsets": [
[
30339,
30375
]
],
"text": "['a modal subspace is also constructed']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
30315,
30337
]
],
"text": "['reduced elastic models']",
"type": "data",
"id": "T302"
},
{
"offsets": [
[
30377,
30386
]
],
"text": "['Section 4']",
"type": "data",
"id": "T303"
},
{
"offsets": [
[
30418,
30588
]
],
"text": "['At runtime, the final positions of the material points are computed as the combination of linear blend skinning and elastic deformation computed in the skeletal bind pose']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
30590,
30681
]
],
"text": "['We also account for the inertial forces caused by the moving coordinate frames of the bones']",
"type": "own_claim",
"id": "T305"
},
{
"offsets": [
[
30684,
30703
]
],
"text": "['Galoppo et al. 2007']",
"type": "data",
"id": "T306"
},
{
"offsets": [
[
30720,
30801
]
],
"text": "['we conceptually constrain the material points that are attached to internal bones']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
30803,
30925
]
],
"text": "['This can be achieved by removing the elastic degrees of freedom that are associated with corresponding internal mesh nodes']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
30934,
31042
]
],
"text": "['the positions of these points are then completely governed by the linear blend skinning transformations only']",
"type": "own_claim",
"id": "T309"
},
{
"offsets": [
[
31044,
31184
]
],
"text": "['We remove these degrees of freedom in our preprocessing step by identifying tetrahedral mesh elements that are intersected by skeletal bones']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
31186,
31256
]
],
"text": "['Degrees of freedom that are associated with these elements are removed']",
"type": "own_claim",
"id": "T311"
},
{
"offsets": [
[
31300,
31347
]
],
"text": "['unless they lie on the model’s boundary surface']",
"type": "own_claim",
"id": "T312"
},
{
"offsets": [
[
31977,
32074
]
],
"text": "['which provides the best deformation basis for small deformations away from the rest configuration']",
"type": "own_claim",
"id": "T313"
},
{
"offsets": [
[
31899,
31969
]
],
"text": "['For each of the dynamic morph targets, we employ linear modal analysis']",
"type": "own_claim",
"id": "T314"
},
{
"offsets": [
[
32653,
32706
]
],
"text": "['at this point, one could easily add modal derivatives']",
"type": "own_claim",
"id": "T315"
},
{
"offsets": [
[
32715,
32736
]
],
"text": "['Barbic and James 2005']",
"type": "data",
"id": "T316"
},
{
"offsets": [
[
32829,
32898
]
],
"text": "['We have taken into account three requirements when choosing the basis']",
"type": "own_claim",
"id": "T317"
},
{
"offsets": [
[
32911,
33004
]
],
"text": "['Avoid redundancy in the basis set, i.e. find an orthogonal set that is as compact as possible']",
"type": "data",
"id": "T318"
},
{
"offsets": [
[
33009,
33093
]
],
"text": "['The characteristic deformations of all the morph targets have to be well represented']",
"type": "data",
"id": "T319"
},
{
"offsets": [
[
33098,
33206
]
],
"text": "['The input deformations of each of the dynamic morph targets have to be well represented in the reduced space']",
"type": "data",
"id": "T320"
},
{
"offsets": [
[
33336,
33465
]
],
"text": "['The most straight forward approach is to combine all eigenvectors together as U = ˆ U 1 U 2 . . . U m ̃ , and orthogonalize them']",
"type": "own_claim",
"id": "T321"
},
{
"offsets": [
[
33467,
33509
]
],
"text": "['each set of eigenvectors U p is orthogonal']",
"type": "own_claim",
"id": "T322"
},
{
"offsets": [
[
33515,
33558
]
],
"text": "['eigenvectors between different sets are not']",
"type": "own_claim",
"id": "T323"
},
{
"offsets": [
[
33561,
33611
]
],
"text": "['This approach takes care of the second requirement']",
"type": "own_claim",
"id": "T324"
},
{
"offsets": [
[
33617,
33709
]
],
"text": "['results in a large set of eigenmodes with relatively small contributions for many eigenmodes']",
"type": "own_claim",
"id": "T325"
},
{
"offsets": [
[
33719,
33757
]
],
"text": "['all the common first deformation modes']",
"type": "own_claim",
"id": "T326"
},
{
"offsets": [
[
33776,
33781
]
],
"text": "['shear']",
"type": "data",
"id": "T327"
},
{
"offsets": [
[
33767,
33774
]
],
"text": "['stretch']",
"type": "data",
"id": "T328"
},
{
"offsets": [
[
33791,
33825
]
],
"text": "['are represented in each of the U i']",
"type": "own_claim",
"id": "T329"
},
{
"offsets": [
[
33873,
33944
]
],
"text": "['we can construct a low-dimensional motion subspace by applying mass-PCA']",
"type": "own_claim",
"id": "T330"
},
{
"offsets": [
[
33849,
33870
]
],
"text": "['Barbic and James 2005']",
"type": "data",
"id": "T331"
},
{
"offsets": [
[
33946,
34033
]
],
"text": "['We scale the derivatives according to the eigenvalues of the corresponding linear modes']",
"type": "own_claim",
"id": "T332"
},
{
"offsets": [
[
34043,
34133
]
],
"text": "['we obtain the low-dimensional deformation basis by applying mass-PCA on the set of vectors']",
"type": "own_claim",
"id": "T333"
},
{
"offsets": [
[
34176,
34236
]
],
"text": "['{ λ λ i 1 j i U j i | i = 1, . . . , m; j = 1, . . . , k i }']",
"type": "own_claim",
"id": "T334"
},
{
"offsets": [
[
34318,
34458
]
],
"text": "['Scaling is necessary to put greater weight on the more important low-frequency modes, which would otherwise be masked by highfrequency modes']",
"type": "own_claim",
"id": "T335"
},
{
"offsets": [
[
34523,
34679
]
],
"text": "['we can adapt the normalization factor such that the eigenvectors are normalized across morph targets, by replacing the scaling factor by λ λ i 1 i λ i . k j']",
"type": "own_claim",
"id": "T336"
},
{
"offsets": [
[
34468,
34521
]
],
"text": "['the eigenmodes in different poses show large variance']",
"type": "data",
"id": "T337"
},
{
"offsets": [
[
34689,
34913
]
],
"text": "['to make the basis aware of the morph targets, we add m − 1 rest pose deformations {x 0 p |i = 2, . . . , m} to the set U and remass-orthogonalize the set {U i } ∪ {x 0 p } into the final basis U ∈ R 3n×r with r = (r + m − 1)']",
"type": "own_claim",
"id": "T338"
},
{
"offsets": [
[
34915,
34923
]
],
"text": "['Figure 3']",
"type": "data",
"id": "T339"
},
{
"offsets": [
[
35457,
35709
]
],
"text": "['By using pose-space efficient polynomial interpolation to achieve pose-dependent behavior, we are able to demonstrate rich nonlinear deformation effects at relatively small extra cost compared to simple simulation of linear or semi-non-linear materials']",
"type": "own_claim",
"id": "T340"
},
{
"offsets": [
[
35711,
35774
]
],
"text": "['We have performed experiments with three different input models']",
"type": "own_claim",
"id": "T341"
},
{
"offsets": [
[
35776,
35814
]
],
"text": "['a simple bulging cylinder with 4 bones']",
"type": "data",
"id": "T342"
},
{
"offsets": [
[
35820,
35826
]
],
"text": "['Fig. 2']",
"type": "data",
"id": "T343"
},
{
"offsets": [
[
35830,
35859
]
],
"text": "['a shoulder model with 4 bones']",
"type": "data",
"id": "T344"
},
{
"offsets": [
[
35865,
35906
]
],
"text": "['Herbert, our swimsuit model with 46 bones']",
"type": "data",
"id": "T345"
},
{
"offsets": [
[
36578,
36651
]
],
"text": "['model, we qualitatively compare simulations with different elastic models']",
"type": "own_claim",
"id": "T346"
},
{
"offsets": [
[
35908,
35916
]
],
"text": "['For each']",
"type": "own_claim",
"id": "T347"
},
{
"offsets": [
[
36842,
36899
]
],
"text": "['For the Herbert simulations, we used only 3 morph targets']",
"type": "own_claim",
"id": "T348"
},
{
"offsets": [
[
36916,
36922
]
],
"text": "['Fig. 1']",
"type": "data",
"id": "T349"
},
{
"offsets": [
[
37265,
37325
]
],
"text": "['the single elastic model shows little or no dynamic behavior']",
"type": "own_claim",
"id": "T350"
},
{
"offsets": [
[
37327,
37446
]
],
"text": "['our pose-dependent elastic model adds a dramatic amount of realism due to the bulging behavior and inertial skin motion']",
"type": "own_claim",
"id": "T351"
},
{
"offsets": [
[
37643,
37719
]
],
"text": "['we show the advantage of our pose-dependent model from an artistic viewpoint']",
"type": "own_claim",
"id": "T352"
},
{
"offsets": [
[
37575,
37641
]
],
"text": "['Herbert jumps off a diving board and flips through different poses']",
"type": "data",
"id": "T353"
},
{
"offsets": [
[
37454,
37538
]
],
"text": "['we drive the skinned Herbert model with a skeletal animation and add inertial forces']",
"type": "own_claim",
"id": "T354"
},
{
"offsets": [
[
37546,
37570
]
],
"text": "['the bone’s moving frames']",
"type": "data",
"id": "T355"
},
{
"offsets": [
[
37721,
37812
]
],
"text": "['With single elastic models, the belly is flabby and skinny throughout the entire simulation']",
"type": "own_claim",
"id": "T356"
},
{
"offsets": [
[
37814,
37981
]
],
"text": "['Using the aforementioned morph targets for Herbert, an animator can impose a stiff, bulged belly in balled-up poses, and softer, skinny belly behavior in upright poses']",
"type": "own_claim",
"id": "T357"
},
{
"offsets": [
[
38145,
38288
]
],
"text": "['the use of reduced models in our method achieves the same quality of desired deformations as the computationally more expensive unreduced model']",
"type": "own_claim",
"id": "T358"
},
{
"offsets": [
[
38128,
38134
]
],
"text": "['Fig. 4']",
"type": "data",
"id": "T359"
},
{
"offsets": [
[
38305,
38406
]
],
"text": "['Our method also provides a physically-based approach to resolving regions affected by multiple joints']",
"type": "own_claim",
"id": "T360"
},
{
"offsets": [
[
38416,
38430
]
],
"text": "['a shoulder rig']",
"type": "data",
"id": "T361"
},
{
"offsets": [
[
38432,
38472
]
],
"text": "['Our approach facilitates complex rigging']",
"type": "own_claim",
"id": "T362"
},
{
"offsets": [
[
38474,
38482
]
],
"text": "['a set of']",
"type": "own_claim",
"id": "T363"
},
{
"offsets": [
[
39445,
39547
]
],
"text": "['skinning weights and a set of morph targets are sufficient to simulate complex co-articulation effects']",
"type": "own_claim",
"id": "T364"
},
{
"offsets": [
[
39549,
39678
]
],
"text": "['There is no need for manual tweaking of the complex mapping of joint configuration to blending weights of geometric morph targets']",
"type": "own_claim",
"id": "T365"
},
{
"offsets": [
[
39680,
39728
]
],
"text": "['In our shoulder example, we have 6 morph targets']",
"type": "own_claim",
"id": "T366"
},
{
"offsets": [
[
39739,
39748
]
],
"text": "['Fig. 5(a)']",
"type": "data",
"id": "T367"
},
{
"offsets": [
[
39751,
39887
]
],
"text": "['The shoulder example also demonstrates our method’s ability to simulate dynamic behavior at poses away from the morph target input poses']",
"type": "own_claim",
"id": "T368"
},
{
"offsets": [
[
39890,
39896
]
],
"text": "['Fig. 5']",
"type": "data",
"id": "T369"
},
{
"offsets": [
[
39900,
39970
]
],
"text": "['The input morph target set contains only one example of a folded elbow']",
"type": "own_claim",
"id": "T370"
},
{
"offsets": [
[
39975,
40035
]
],
"text": "['we show two distinct folding scenarios in the full animation']",
"type": "own_claim",
"id": "T371"
},
{
"offsets": [
[
40037,
40126
]
],
"text": "['Both folding scenarios show severe self-intersection in the single pose-independent model']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
40134,
40169
]
],
"text": "['the effect of linear blend skinning']",
"type": "data",
"id": "T373"
},
{
"offsets": [
[
40171,
40223
]
],
"text": "['Our pose-dependent model resolves both automatically']",
"type": "own_claim",
"id": "T374"
},
{
"offsets": [
[
40225,
40232
]
],
"text": "['Another']",
"type": "own_claim",
"id": "T375"
},
{
"offsets": [
[
40312,
40414
]
],
"text": "['interesting co-articulation effect is the motion of the chest muscle as the arm makes a folding motion']",
"type": "own_claim",
"id": "T376"
},
{
"offsets": [
[
40420,
40425
]
],
"text": "['video']",
"type": "data",
"id": "T377"
},
{
"offsets": [
[
40436,
40489
]
],
"text": "['the chest seems to collapse for single elastic models']",
"type": "own_claim",
"id": "T378"
},
{
"offsets": [
[
40491,
40535
]
],
"text": "['it bulges more realistically with our method']",
"type": "own_claim",
"id": "T379"
},
{
"offsets": [
[
40585,
40691
]
],
"text": "['After modal reduction, we were able to accelerate the simulation significantly by using only 19 eigenmodes']",
"type": "own_claim",
"id": "T380"
},
{
"offsets": [
[
40694,
40701
]
],
"text": "['Table 1']",
"type": "data",
"id": "T381"
},
{
"offsets": [
[
40704,
40759
]
],
"text": "['with almost no visible effect on the simulation quality']",
"type": "own_claim",
"id": "T382"
},
{
"offsets": [
[
40762,
40768
]
],
"text": "['Fig. 5']",
"type": "data",
"id": "T383"
},
{
"offsets": [
[
41163,
41208
]
],
"text": "['All our methods achieve real-time performance']",
"type": "own_claim",
"id": "T384"
},
{
"offsets": [
[
41216,
41357
]
],
"text": "['efficient pose-space interpolation of low-complexity linear elastic forces and modal reduction of either linear or semi-nonlinear StVK forces']",
"type": "own_claim",
"id": "T385"
},
{
"offsets": [
[
41476,
41517
]
],
"text": "['our method has only a marginal extra cost']",
"type": "own_claim",
"id": "T386"
},
{
"offsets": [
[
41463,
41470
]
],
"text": "['Table 1']",
"type": "data",
"id": "T387"
},
{
"offsets": [
[
41359,
41444
]
],
"text": "['Comparing our method with the performance of single (pose-independent) elastic models']",
"type": "data",
"id": "T388"
},
{
"offsets": [
[
41526,
41595
]
],
"text": "['efficient polynomial interpolation of the dynamic morph target models']",
"type": "own_claim",
"id": "T389"
},
{
"offsets": [
[
41606,
41734
]
],
"text": "['precomputation of the force polynomial coefficients in Eqn. (7) can be significant in case of StVK models, but never prohibitive']",
"type": "own_claim",
"id": "T390"
},
{
"offsets": [
[
41829,
41926
]
],
"text": "['The quality of the output animation depends on the quality and number of the artist’s input poses']",
"type": "own_claim",
"id": "T391"
},
{
"offsets": [
[
41928,
42014
]
],
"text": "['Too few input poses can cause slight popping of the animation towards the input shapes']",
"type": "own_claim",
"id": "T392"
},
{
"offsets": [
[
42019,
42097
]
],
"text": "['the simulation of the pose-dependent elastic model will be stable nevertheless']",
"type": "own_claim",
"id": "T393"
},
{
"offsets": [
[
42110,
42175
]
],
"text": "['stable results can be achieved with little effort from the artist']",
"type": "own_claim",
"id": "T394"
},
{
"offsets": [
[
42180,
42282
]
],
"text": "['poses can then be added incrementally to areas of the posespace where the behavior is not satisfactory']",
"type": "own_claim",
"id": "T395"
},
{
"offsets": [
[
42284,
42376
]
],
"text": "['The effect of the number of poses on the run-time performance of the algorithm is very small']",
"type": "own_claim",
"id": "T396"
},
{
"offsets": [
[
42380,
42448
]
],
"text": "['it doesn’t affect the number of degrees of freedom in the simulation']",
"type": "own_claim",
"id": "T397"
},
{
"offsets": [
[
42454,
42497
]
],
"text": "['comparison 6 vs. 9 shoulder DMTs in Table 1']",
"type": "data",
"id": "T398"
},
{
"offsets": [
[
42501,
42596
]
],
"text": "['Only the cost of the RBF interpolation in Eq. (2) is affected linearly in the number of poses m']",
"type": "own_claim",
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A13 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A13_M04_Efficient_Synthesis_of_Physically_Valid_Human_Motion_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/882262.882286
To appear in the ACM SIGGRAPH 2003 conference proceedings
<Title>Efficient Synthesis of Physically Valid Human Motion</Title>
Anthony C. Fang Department of Computer Science Brown University ∗
<Abstract>Optimization is a promising way to generate new animations from a minimal amount of input data. Physically based optimization techniques, however, are difficult to scale to complex animated characters, in part because evaluating and differentiating physical quantities becomes prohibitively slow. Traditional approaches often require optimizing or constraining parameters involving joint torques; obtaining first derivatives for these parameters is generally an O(D 2 ) process, where D is the number of degrees of freedom of the character. In this paper, we describe a set of objective functions and constraints that lead to linear time analytical first derivatives. The surprising finding is that this set includes constraints on physical validity, such as ground contact constraints. Considering only constraints and objective functions that lead to linear time first derivatives results in fast per-iteration computation times and an optimization problem that appears to scale well to more complex characters. We show that qualities such as squash-and-stretch that are expected from physically based optimization result from our approach. Our animation system is particularly useful for synthesizing highly dynamic motions, and we show examples of swinging and leaping motions for characters having from 7 to 22 degrees of freedom.</Abstract>
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation G.1.6 [Numerical Analysis]: Optimization—Constrained Optimizatio Keywords: animation, physically based animation
Providence
RI
[acf|nsp]@cs.brown.edu
<H1>1 Introduction</H1>
One appealing vision in animation is that the animator should be able to create and edit motion by defining and adjusting a small number of keyframes and constraints—and that the resulting motion should remain optimal in some way. An optimization approach to animation has proven useful for editing human motion capture data, refining a “sketched” version of an animation, and for creating entirely new motions for simple characters or short segments. Several challenges remain, however, to achieving fast, flexible, and realistic optimization of human motion. One challenge is incorporating physics into an interactive animation system. Despite impressive results obtained from physically based optimization,
Nancy S. Pollard
Figure 1: A single-flip dismount from a high-bar. (Top) Initial guess. (Middle) Flight duration is 0.6 seconds; Flip posture is tight and maximum height is below high-bar. (Bottom) Flight duration increased to 0.8 seconds. Flip posture is relaxed, and maximum height exceeds high-bar.
constraints and objective functions that require computing physical quantities such as momentum, force, and torque are typically viewed as slow, cumbersome, and difficult to control, especially for complex humanlike characters. As a result, physical validity is often sacrificed for performance. Physical validity is important, however, in situations such as those shown in Figure 1 . Kinematic optimization alone is unlikely to capture the coordination of different parts of the body that is required to perform this task, such as the preparatory back swing, the tuck, or the motion of the legs to drive the character upward that is shown in the bottom row of the figure. This paper presents an approach to physically based optimization that is efficient and appears to scale well to more complex characters. We use a standard problem formulation—iteratively adjust character motion to meet animator constraints and minimize an objective function. Our approach is based on restricting the definition of this optimization problem to constraints and objective functions that can be differentiated in time linear in the degrees of freedom of the character. The motivation for this approach is that solution techniques for nonlinear constrained optimization problems (e.g. SQP) typically require either analytical or numerical derivatives. Obtaining these derivatives is a computational bottleneck, and complex derivatives can lead to poor optimization performance and problems with local minima. Kinematic optimization [Gleicher 1997]
, which has been shown to be successful for complex characters, depends on constraints and objective functions for which first derivatives can be computed in linear time. We have found that constraints on physics that can be derived from the aggregate force and torque applied to the character can also be differentiated in linear time. This set includes most common constraints required for physically correct animation, such as conserving linear and angular momentum during flight, ensuring that ground contact forces can be explained by foot placement, constraining torque applied about an axis (e.g. the high bar in Figure 1 ), and limiting the coefficient of friction at any contact with the environment. Linear time derivatives for physics constraints do not result from direct differentiation of the equations of motion in either the Newton-Euler or the Lagrangian formulation; in either case, symbolic differentiation would result in a quadratic time algorithm. In this paper, we describe how the Newton-Euler equations of motion can be rewritten to allow first derivatives of aggregate forces and torques to be computed in linear time. We note that it is not possible to compute derivatives for torques at all of the characters joints in linear time. Intuitively, quadratic time is required because motion at any joint affects torque at all joints. As a result, typical objective functions such as minimizing the sum of squared joint torques are excluded from our restricted problem setup. Our results suggest, however, that physics constraints and a kinematic measure of smooth motion such as minimizing the sum of squared joint accelerations are sufficient to capture dynamic effects such as squashand-stretch and tucking for faster rotation, as shown in Figure 1 . While animator constraints such as key poses or an objective based on proximity to a reference motion can easily be incorporated into the system, no motion capture data is used in our examples, and user-supplied constraints are minimal (e.g., see Figure 7 ). The characteristics of the final motions fall out of the requirements of physical validity, a simple kinematic optimization function, and timing values selected for each phase of the motion.
1
To appear in the ACM SIGGRAPH 2003 conference proceedings
<H1>2 Background</H1>
Constrained optimization techniques were introduced to the graphics community by
Witkin and Kass [1988]
, who created a variety of animations involving a jumping Luxo lamp from simple descriptions including start pose, end pose, and a physically based objective function. Optimization approaches with physically based objective functions have proven difficult to extend to complex articulated characters, however, and much research has been focused on this problem. Cohen and his colleagues [Cohen 1992] [Liu et al. 1994]
introduced techniques to give the user more control, including an ability to focus on windows in time, and employed a hierarchical wavelet description to allow incremental changes to affect the motion at different time scales.
In his dissertation, Liu [1996]
also describes how symbolic differentiation of the equations of motion can be made efficient (although still quadratic time) by cleverly aggregating terms. Grzeszczuk, Terzopoulos, and Hinton [1998]
developed a neural network approximation of dynamics so that gradient search could be performed on this neural network, resulting in faster convergence to a solution.
The mix of animator control and physics present in
Witkin and Kass [1988] has been expanded upon in interactive techniques developed to control physical simulations of rigid bodies
[Popović et al. 2000], and a number of researchers have shown that the freefall portion of a dive can be efficiently optimized for a simplified character [Liu and Cohen 1994][Crawford 1998][Albro et al. 2000], as can motions such as weight lifting and pushups [Lo and Metaxas 1999].
Optimal control techniques, introduced to the graphics community by
Brotman and Netravali [1988], have been used with success by
Pandy and Anderson [2000]
for simulating human lower body motions such as optimal height jumping and walking. Running times were far from interactive, but show that optimization techniques can produce realistic motion for systems of human-level complexity. Preexisting motion data can simplify the optimization process.
Full scale human motion can be optimized when closely spaced keyframes are available
[Liu and Cohen 1995] or when only transitions between existing motion segments are required
[Rose et al. 1996]. Popović and Witkin [1999]
have shown that significant changes to motion capture data can be made by optimizing with a physically based objective function when the character is reduced to the degrees of freedom most important for the task. When physics does not dominate the motion, kinematic techniques can give the animator interactive control for motion editing (e.g.,
[Gleicher 1997] [Lee and Shin 1999] [Arikan and Forsyth 2002]). The idea of physically valid motion has appeared in both graphics and robotics. Dynamic filters have been developed for processing motion capture data for physical correctness
[Yamane and Nakamura 2000] [Dasgupta and Nakamura 1999] [Pollard and Reitsma 2001].
Physics constraints have been used to plan biped walking motions, exploiting the idea that dynamic equilibrium can be maintained by ensuring that the zero moment point (ZMP)—the point on the ground at which ground reaction moments about horizontal axes are zero—lies within the support polygon of the feet
[Vukobratović 1970] [Takanishi et al. 1985] [Nagasaka et al. 1999]. Similar ideas have also been developed in graphics by
[Ko and Badler 1996]
, who bend the torso of a character to reduce torques at the desired ZMP, and [van de Panne 1997] who ensure that reasonable forces are available to accelerate the center of mass without creating angular acceleration. Liu and Popović [2002]
show that some dynamic effects can be preserved by enforcing patterns of linear and angular momentum, which does not require computation of dynamic parameters such as contact forces and joint torques. We add to this body of work the insight that it is possible to incorporate constraints on physics as efficiently as constraints on kinematic parameters and an O(D) algorithm for computing first derivatives of a broad range of physics constraints for improved performance in a optimization context.
free parameters x in numerical optimizer x initial motion out h, C objectives x dh/dx constraints dC/dx view / edit problem desc. compute objective function and final motion derivatives
Figure 2: Optimizing motion synthesis or editing system. Parameter h is the objective function, and C are the constraint errors. This paper discusses efficient computation of the objective function, constraint errors, and their derivatives.
<H1>3 Constrained Optimization</H1>
Constrained optimization has been shown to be a very powerful approach for obtaining appealing dynamic motions from a minimal amount of input information. The user adjusts the problem description in the form of keyframes, constraints, and objectives; an optimizer computes an optimal animation given this problem description; and the process repeats until the user obtains a final animation ( Figure 2 ). We state the optimization problem solved at each stage in the following form: min x h(B(t)x) subject to c(t i ) = 0, i = 1..m, t i ∈ [t s ,t f ] where h is the optimization function; B(t) is a set of basis functions; x are the coefficients, the free parameters of the optimization; and c(t i ) are the constraints. We use cubic B-splines as basis functions and follow the standard approach of enforcing constraints at a fixed set of points in time (t i ). Enforcing physics constraints or minimizing a dynamic property such as sum squared joint torques requires an inverse dynamics computation at each time t i . Although the inverse dynamics computation is relatively expensive, many efficient algorithms exist, and the process is well known to require time linear in the number of degrees of freedom of the character. However, typical choices for the numerical optimizer in Figure 2 also require derivatives of the constraints and objective function. For example, the sequential quadratic programming algorithm used in
[Witkin and Kass 1988]
makes use of first derivatives of the constraints (the constraint Jacobian) and both first and second derivatives of the objective function (the Jacobian and the Hessian). This paper describes how a broad range of physics constraints can be expressed based on aggregate forces and torques applied to the character, and how expressing physics constraints in this way allows us to compute the constraint Jacobian in linear time (Section 4). Objective functions are compared for efficiency in Section 5. We used an objective function that enforces smooth motion, with a linear time Jacobian computation and a constant Hessian. With this objective function and our linear time algorithm for computing the constraint Jacobian, we are able to show that physically based optimization can be performed for a 22 degree of freedom character at interactive speeds.
2
To appear in the ACM SIGGRAPH 2003 conference proceedings
<H1>4 Efficient Physics Constraints</H1>
Constraints that enforce physical validity can be formulated as linear equality or inequality constraints on aggregate force. The aggregate force is a representation of all external forces and torques (excluding gravity) that would have to be applied to the character root to explain the character’s motion. We classify the physics constraints for the motions in our examples into the categories of flight, bar contact, and ground contact. Flight. One way of enforcing correct physics during flight is to ensure that the aggregate momentum of the body remains constant throughout the flight phase. Unfortunately, the constraint Jacobian that results from constraining momenta is denser than necessary as the control points that determine take-off affect all constraint equations governing the flight phase. A more elegant solution is to restrict illegal forces during flight. During flight, no forces, with the exception of gravity, may be derived from the environment. In our system, gravity is simulated by tion adding due an to acceleration gravity (See of Appendix −G to the B). root, When where gravity G is the is simulated accelerain this manner, the aggregate forces computed by the dynamics equations are the sum of forces acting on the body in excess of the gravitational forces. Let the aggregate force be denoted by f 0 . (In the spatial notation used here, f 0 contains both linear forces and torques.) The flight constraint is thus f 0 = 0. Bar contact. When the character is swinging on a high bar or monkey bars, the amount of torque that can be applied about the bar axis is constrained. Let aggregate force f 0 be represented as
1
f 0 = f f a b 0 0
where f a 0 is linear force and f b 0 is torque about the world origin. Aggregate force is translated to a constraint point c as follows:
2
f c = f f b c c a = f b 0 − c f a 0 0 × f a 0
where c 0 is the world vector from the base of the articulation to c.
The bar contact constraint can then be expressed as
3
− τ max < s bar · f b c < τ max
where is a projection τ max is the operation scalar torque that results limit, in s bar torque is the about bar axis, the and bar axis. s bar · f c b Ground contact. During ground contact, the feet can only push, not pull on the ground, contact forces should not require an unreasonable amount of friction, and the center of pressure must fall within the support polygon of the feet. These effects can be modeled with equations that constrain the linear and angular forces separately. We constrain the linear force using Coulomb’s contact model. Coulomb’s model dictates that the linear reaction force must fall within a friction cone oriented along the contact normal with angular half-width tan −1 μ , where μ is the coefficient of friction. The inequality constraint for the linear aggregate force is therefore
4
cos −1 N · a f 0 a < tan −1 μ
4
|f 0 |
where N is the unit contact normal. Equivalently, the constraint without the inverse trigonometric functions is:
5
N |f · a 0 f | a 0 > ( μ 1 2 + 1)
The magnitude of the normal force can be constrained as follows:
6
0 ≤ f a 0 · N ≤ K orce
6
f
Contact torques are constrained by geometrically confining the center of pressure to the support area. In the simplified case of a rectangular support area (or a linear support area in a twodimensional set up), the aggregate torques may be constrained directly as follows: Translate f 0 to the center c of the support as in Equation 2. Let T x and T y be orthogonal vectors spanning the rectangular support, and let δ x and δ y be the distances from c to the edge of the support along along T x and T y respectively. The torques about T x and T y may be constrained as:
7
− δ y (N · f 0 a ) < f b c · T x < δ y (N · f 0 a )
8
− δ x (N · f 0 a ) < f b c · T y < δ x (N · f a 0 ) The magnitude of the twist force is constrained as follows:
9
−K twist (N · f a 0 ) < f b c · N < K twist (N · f a 0 )
All physics constraints. Once all physics constraints have been expressed as constraints on aggregate force, computing derivatives on the physics constraints becomes a problem of differentiating aggregate force with respect to the free parameters of the problem. At any time t, character position q, velocity q, ̇ and acceleration q ̈ are known. The derivative of interest can be expressed in terms of q, q, ̇ and q ̈ using the chain rule:
10
∂ ∂ f x 0 = ∂ ∂ f q 0 ∂ ∂ q x + ∂ ∂ f q ̇ 0 ∂ ∂ q x ̇ + ∂ ∂ f q ̈ 0 ∂ ∂ q x ̈
where terms ∂ q/ ∂ x, ∂ q/ ̇ ∂ x, and ∂ q/ ̈ ∂ x are available trivially from the equations expressing joint degrees of freedom q as a function of B-spline coefficients x. The term ∂ f 0 / ∂ q, which we will refer to as the force Jacobian, is the most difficult term in this expression. The main point of the paragraphs below is to show how the force Jacobian can be computed efficiently. 1 We show that straightforward analytical computation of the force Jacobian would require time quadratic in the number of degrees of freedom of the character. However, if joint torques are not required, then this value and first derivatives for constraints based on this value can be computed in linear time. To our knowledge, our paper is the first to present a linear time algorithm for computing the force Jacobian for an articulated character or robot.
3
To appear in the ACM SIGGRAPH 2003 conference proceedings
p q i all parameters are affected by q i v p 0
Figure 3: The effect of parameter q i is propagated up the tree with velocities v and back down the tree with momentum terms p. Computing ∂ p 0 / ∂ q i requires O(D) time and results in an O(D 2 ) algorithm for computing the momentum Jacobian.
<H2>4.1 Notation</H2>
Our argument and implementation is constructed around a NewtonEuler formulation of inverse dynamics. We use spatial notation as in
Featherstone [1987]
for conciseness. Spatial notation involves 6-dimensional vectors, 6x6 coordinate transformations, and 6x6 inertia tensors. It combines linear and angular quantities such as force and torque or linear and angular velocity into single vectors, as shown in Equations 1 through 3. An overview of spatial notation is given in Appendix A, and terms are summarized here for reference. D Degrees of freedom of the articulated figure q i Scalar position of link i, from motion curves (DOF i) q ̇ i Scalar velocity of link i, from motion curves q ̈ i Scalar acceleration of link i, from motion curves X i j Spatial transform from frame i to frame j X 0 i Spatial transform from frame i to world frame s i Joint axis of link i (frame i) v i Local velocity of link i (frame i) v i Global velocity of link i (frame i) a i Local acceleration of link i (frame i) a i Global acceleration of link i (frame i) I i Spatial inertia of link i (frame i) p 0 Aggregate momentum of articulated figure (world frame) f 0 Aggregate force of articulated figure (world frame)
<H2>4.2 Linear Time Momentum Jacobian</H2>
Efficiently computing ∂ f 0 / ∂ q, the force Jacobian, requires efficiently computing ∂ p 0 / ∂ q, the momentum Jacobian, because aggregate force f 0 is the time derivative of aggregate momentum p 0 . We begin with a discussion of the momentum equations and present an argument that the momentum Jacobian can be computed in linear time. Section 4.3 extends this linear time result to the force
1 Final expressions for the force Jacobian and other terms of Equation 10 are summarized in Appendix B.
p * q i I * parameters from link i to the base are affected by q i p 0
Figure 4: The effect of rewriting the recursion is to limit the effect of q i to parameters collected at joints between i and 0. Terms required for the momentum Jacobian are accumulated in a single pass from leaf to base, and the momentum Jacobian can be computed in linear time.
Jacobian, the quantity required to compute derivatives of physics constraints. The usual way to compute aggregate momentum is to formulate the following recursion:
11
v i = X i i−1 v i−1 + s i q ̇ i
12
p i = X i+1 i p i+1 + I i v i
where p 0 is the desired result. Velocities v i are propagated from base to leaf, and momentum p i is propagated from leaf to base. Figure 3 shows this process. Parameter q i appears in the coordinate transforms X i i+1 and X i+1 i , and so every v j for j > i depends on q i , and every p j for j ≥ 0 depends on q i . Unrolling the recursion to collect terms for ∂ p 0 / ∂ q i requires O(D) time. There are D terms q i , and this approach will lead to an O(D 2 ) computation for the momentum Jacobian. There is no clever way to simplify the calculation by aggregating terms when it is presented in this form. We observe that rewriting the recursion solves this dilemma:
13
I ∗ i = X i+1 i I ∗ i+1 X i+1 i + I i
14
p i ∗ = X i i+1 p ∗ i+1 + I i ∗ v i
15
p 0 = p ∗ 0
The key thing to notice here is that p ∗ i is expressed as a function of v i , which is a local variable at link i. As a result, only propagation from leaf to base is required, and each parameter q j does not affect terms computed for joints j + 1 and beyond ( Figure 4 ). Also note that p ∗ i is in general not equal to p i if i = 0. A term superscripted with an asterix should be treated only as an intermediary quantity, unless its subscript is zero in which case it is the desired aggregate result. A linear time expression for the momentum Jacobian can be derived in a straightforward manner based on this form of the recursion. The results of this calculation are presented in Appendix B. Note that we are not simplifying or changing the outcome of the dynamics computation, only changing the order in which terms are computed. Aggregate momentum p 0 and the momentum Jacobian are exactly the same in both formulations. 4.3 Linear Time Force Jacobian In a traditional inverse dynamics formulation, accelerations and forces are expressed as the time derivatives of Equations 11 and 12:
17
a i = X i−1 i a i−1 + s i q ̈ i + v i ×s ˆ i q ̇ i f = X i+1 i f i+1 + I a + v ×I ˆ v
17
i i i i i i
4
To appear in the ACM SIGGRAPH 2003 conference proceedings
Timings for 500 computations of the aggregate force Jacobian of a serial chain 25 Direct differentiation Aggregate differentiation 20 15 Seconds 10 5 0 0 5 10 15 20 25 30 35 40 45 50 DOF
Figure 5: Timing of 500 computations of the Jacobian of the aggregate force by direct analytical differentiation and by our linear time analytical method.
where (Appendix the symbol A). As × ˆ with is the momentum, cross product this operator form results for spatial in an expresvectors sion for the force Jacobian that requires O(D 2 ) time to compute. For fast computation, we instead take the time derivative of Equation 14, which results in
18
f ∗ = X i i+1 f i+1 ∗ + v ×p ˆ ∗ + I ∗ a + ̇ I ∗ v
18
i i i i i i i
This equation has the properties we are looking for. Velocity v i and acceleration a i are local to link i, and terms are propagated from leaf to base only. Note that as with aggregate momentum, f ∗ i is in general different from the actual joint force f i if i = 0. Differentiating Equation 18 and accumulating the coefficients of derivative elements results in the simplified form as given in Appendix B. Each partial derivative of the aggregate force with respect to joint positions, velocities, and accelerations may be obtained in constant time, and hence the full Jacobian may be obtained in linear time. 4.4 Benchmarking Figure 5 shows timing results for computation of all partial derivatives of the aggregate force by the proposed method and by direct differentiation of the Newton-Euler equations of motion. Numerically the partial derivatives are identical. The articulated model is a serial chain ranging from 3 to 50 links. As expected, the proposed method is linear in the degrees of freedom, while direct differentiation shows quadratic growth. It is also observed that despite overheads in computing aggregate intermediate terms, the linear time method shows a computational advantage with as few as 5 degrees of freedom. 4.5 The Cost of a Linear Time Algorithm One obvious question is why has this technique of rewriting the recursion for fast computation not been explored in the robotics community? One possible reason is that there is a cost to this approach that may be higher for robotics applications than for graphics applications. In a standard Newton-Euler formulation, force parameter f i (Equation 17) contains all of the joint force information for joint i, in particular forces in the actuated directions of motion (joint torques). In robotics, this information must be computed because it corresponds to signals sent to the motors of the robot. It must in general also be part of optimization routines, because en- ergy consumption and joint torque limits are of particular concern when operating a robot, and none of the joints can be ignored. In contrast, we argue that for animation of human motion, many of the effects we expect to see in physically based optimization do not depend on joint torques. We believe that physical correctness and optimization functions enforcing smooth motion are sufficient to obtain many natural characteristics of human motion. If some torques (e.g. torques at the hip joints) are found to be important, it seems quite certain that many others (e.g. torques at the fingers) can be ignored for many motions. If a subset of K torques are required, it is straightforward to extend our approach to measure torques at these joints in O(KD) time.
<H1>5 Optimization Criteria</H1>
We now return to a discussion about the objective function, h(x). One traditional approach is to use the integral of the sum of squared joint torques to produce a motion that approximately minimizes energy expenditure:
19
h(x) = t f ∑ D τ i 2 (x,t) dt t=t s i=1
This function is expensive because computing its gradient requires O(D 2 ) work. Adopting this function would negate our effort in constructing efficient physics constraints. An objective function that we have found to work well is to minimize the integral of the sum of squared, weighted joint accelerations:
20
h(x) = t f ∑ D w q ̈ (x,t) 2 dt
20
i i t=t s i=1
where w i is aggregate mass subtended at joint i with respect to the effective root. For example, the weight for the left-knee during a left-legged support is the entire body mass minus the left lowerleg. Parameters q ̈ i do not include translational or rotational acceleration of the character root. Note that the analytical Hessian for this objective function is constant, symmetric, positive definite, and band-diagonal. Where a reference motion q R (t) is available, a simple objective function with low cost is to simply minimize the distance from the reference motion:
21
h(x) = t=t t f s (q(x,t) − q R (t)) 2 dt
This objective function is similar to the one used in
Gleicher [1997]. Other objective functions we have attempted include an integral of squared contact forces:
22
h(x) = t f f c 2 (t) dt t=t s
The Jacobian of this function is computable in linear time; our physics constraints are based upon it. Gaits generated using this function have a certain ‘tip-toe’ quality to them, as the function minimizes the amount of reaction force derived from the contacts. Minimizing contact jerk (the time derivative of force) can be achieved using forward differences:
23
m−1 h(x) = ∑ ( f c (t i ) − f c (t i+1 )) 2 i=1
5
To appear in the ACM SIGGRAPH 2003 conference proceedings
Figure 6: Samples of our results. Rows 1, 4, and 6 are initial motions. Details on the experiments are given in the text of Section 6.
6
To appear in the ACM SIGGRAPH 2003 conference proceedings
Swing Setup Information DOF 7 (5 rotational, 2 translational) 2D/3D 2D Number of variables 105 Implicit constraints Hand contact during swing Feet contact during landing Swing time 0.9s, Flight time 0.6s, 0.8s Explicit constraints Initial COM velocity zero Fixed final pose, final joint velocity zero Number of iterations 650 Time per iteration 0.04s Total time 0.43min
Figure 7: All setup information for the swing example.
<H1>6 Results</H1>
Optimal Motions. Figure 6 shows a sampling of our results. The first three rows show a dismount. From top to bottom: initial motion, results with a flight time of 0.6s, and results with a flight time of 0.8s. Note the looser tuck and the higher flight trajectory in the 0.8s motion. The initial motion (shown in the top row of Figure 6) appears very unstable at landing. The character would fall over. This effect is eliminated in the optimization by enforcing the physics constraints of ground contact. Details of the optimization setup are in Figure 7 . All timing information is for a 750 MHz Pentium 3 computer. Rows 4 and 5 of Figure 6 show initial and final motion for a monkey bars example. Details are in Figure 8 . Rows 6, 7, and 8 show initial and final results for a leaping character, with setup information in Figure 9 . In row 7, ground penetration constraints are enforced. In row 8, they are not; the character is leaping from peg to peg. No touch-up was done on the results. In particular, the geometry of the monkey bars and the pegs was not modeled. In these examples, notice the swinging of the legs and arms, as well as body roll, pitch, and yaw. All of these effects are obtained as a result of the optimization process. In these examples, the initial motion is rigid translation of the entire character. Our goal was to require a minimal amount of information from the animator. To set up these examples, we used 15-30 control points per degree of freedom. We found that a number of time slices (for constraint evaluation) equal to the number of control points produced good results and did not need to adjust this value for individual motions. Finer time slices would overly constrain the system, and sparser time slices allowed too much freedom for error. Each motion was set up using a constraint configuration file containing the information listed in the tables. In general, the initial motion was determined directly from constraints, with no additional user input, using linear interpolation between constrained poses. The exception was initial control points for the character root in the first example, which were set to create the overall body rotation required for the backflip. To automatically compute initial motion in a constrained pose, all joints are set at zero angle, the character is in a vertical posture, and the relevant end effector is placed at a user-specified point (e.g. hand at a specific point on the monkey bars). The vertical ”zero posture” had arms up for the bar swings, legs out for the monkey bars, and arms down for the ground motions. The high bar final pose was the only pose provided as a constraint in these examples. Timing. To empirically test the advantage of our method for fast derivative computation, we ran the peg example (bottom row of Figure 6) 5 times, each time with the identical setup except that a different technique was used to compute all required first derivatives. Figure 10 summarizes the results. The differentiation techniques tested were: • Our proach. method. Analytical gradient computation using our ap• Direct entiation method. of the equations Analytical of gradient motion. obtained by direct differ• NR1. ences. Numerical differentiation by ordinary forward differ• NR2. Numerical differentiation by central differences. • NR3. Richardson-extrapolation of order 6. Implementation Issues. Two implementation issues were especially important for achieving the results described in this paper. First, we note that if the basis functions have local influence, the vector and matrix quantities computed during optimization are very sparse. We use the publicly-available Lancelot optimization package
[Conn et al. 1992]
where sparsity is accounted for by groupseparability. Second, we outline the issue of rerooting. Implementing any inverse dynamics algorithm requires selecting a character root. An ability to move the effective root to different parts of the character is very convenient. For example, when there is a single point of constraint between the character and the environment, and that point has a known and fixed trajectory, it is convenient to place the
Monkey Bar Setup Information DOF 22 (19 rotational, 3 translational) 2D/3D 3D Number of variables 532 Implicit constraints Hand contact during support Support time 0.7s, Zero flight time Explicit constraints none Number of iterations 1330 Time per iteration 0.11s Total time 2.4min
Figure 8: All setup information for the monkey bar example.
Leap Setup Information DOF 22 (19 rotational, 3 translational) 2D/3D 3D Number of variables 532 Implicit constraints Foot contact during support Support time 0.35s, Flight time 0.4s Explicit constraints none Number of iterations 2213 Time per iteration 0.11s Total time 4.0min
Figure 9: All setup information for the leap example.
Technique Time per iteration Average % error Our method 0.11s 0 Direct method 0.62s 0 NR1 0.97s 0.10 NR2 1.92s 1.0e-04 NR3 5.73s 1.5e-06
Figure 10: Time required for one iteration of the peg example using a variety of differentiation techniques.
7
To appear in the ACM SIGGRAPH 2003 conference proceedings
character root at that point. In the swing example of Figure 1 , it may be convenient to root the character at the hands for the swing, at the center of mass for flight, and at the feet for landing. In a Newton-Euler inverse dynamics formulation, rerooting is typically done by changing parent / child relationships, which requires inverting joint angles and transforms at each joint and altering the flow of dynamic terms from leaves to root. Both of these changes complicate the problem description presented to the optimizer. The effective root can be relocated more easily, however, by leaving the actual root and the flow of the dynamics computation fixed and computing velocities and accelerations at the root to maintain the desired constraint. Details are given in Appendix C.
<H1>7 Discussion</H1>
This paper contributes to physically based optimization by defining and exploring a restricted class of optimization problems where physics constraints are included and first derivatives of constraints and objective functions can be computed in linear time. The fact that first derivatives can be computed in linear time instead of quadratic time suggests that our problem is simpler than previous physically based approaches and similar in complexity to very successful kinematic approaches such as minimizing distance to a reference motion. We suspect that our solution landscape will be smoother than previous physically based optimization approaches, making it feasible to handle more complex characters. When the optimization does not converge, we can usually trace it back to the problem setup. Sometimes it is due to overconstrained equations (setup error). But often it is due to overly restrictive parameters, such as friction coefficients, joint limits, poor selection of timings, etc. At present, timings are set by the user and their values need to be reasonable (e.g., the character cannot leap too far in too short a time). Any optimization technique that makes use of local derivatives has potential problems with local minima. Our experience, however, was that as long as an expected motion sequence could be thought of as motion about some neutral position, then when the character was started in that neutral position there was no problem descending toward the expected minimum. We were able to create a jumping Luxo and highly dynamic human motions with good success. For less dynamic activities, our system would require additional input; physics constraints plus smooth motion would not in general produce the desired results. An extreme example is “stand for 5 seconds.” Given this problem definition, our system would identify a static pose near the initial guess where the projection of the center of mass is in support area. Additional information would be required to fill in the details of the standing motion. For activities where joint torque limits are important, this torque information must be taken into account to produce good results. An extreme example of this situation is the passive swing of a multilink chain. Minimizing accelerations while maintaining physics constraints would produce a result that was valid for the body as a whole but would require non-zero torques at the joints—no whipping motion would be seen. Minimizing sum squared torques would produce the desired results. (Of course, truly passive motion can be created much more easily using forward dynamic simulation.) More commonly, a limited set of torques or energy terms may be important. For example, the peg running motion appears very athletic because it would require high torques at the knee and hip joints. When physical parameters at certain joints are identified as important, our method can be extended to provide and differentiate these parameters for any K joints with running times of O(KD), reaching the expected bound of O(D 2 ) when all joint torques are required. An interesting research problem is to determine automatically when torques at a given joint should be considered. Running on flat ground shows a combination of difficulties. To make this motion appear more natural, we would need to consider proper timing for the running stride, a more accurate foot model, torques at some of the joints, and perhaps also aspects of style that are not driven by physics or energy. Complexity in the number of degrees of freedom of the character is not the only concern in physically based optimization. The number of free parameters of the optimization problem also grows linearly with total time allotted for the animation. We have not yet attempted any long motion sequences, but we note that Liu, Gortler, and Cohen [Liu et al. 1994] have shown that time complexity can be effectively managed in an optimization context, in part because the influence of any one parameter is localized in time. It is interesting to compare our approach to that of
Liu and Popović [2002]. Their paper describes the power of patterns (e.g., momentum patterns) in creating desirable animation effects, and their approach could be adapted easily to obtain linear time performance by rewriting the momentum equations as described in Section 4.2 of this paper. The idea of dynamic patterns is an exciting one. However, relying on momentum patterns without computing interaction forces between the character and the environment may result in problems with certain types of physics constraints (e.g., keeping forces within a friction cone) when the initial motion is not favorable. In the present paper, we show that it is possible to optimize motion with physics constraints in an efficient manner, so that reasonable friction conditions, for example, can be easily enforced. We believe the combination of correct physics and knowledge of natural dynamic patterns of human motion such as momentum or movement of the center of pressure in the roll of the foot on the ground could be very powerful. Finally, we would like to emphasize that the main advantage of our approach may be as part of a more complete animation system. Our vision is that the ability to enforce physics constraints efficiently should be just one of the tools available to the animator. Details of the desired motion could be fleshed out using motion capture data, procedural techniques, keyframes, and/or objective functions appropriate to the specific task. We have shown that physics constraints can be enforced in an efficient manner. Incorporating physics constraints into traditionally kinematic animation approaches is one direction of future work.
<H1>Acknowledgments</H1>
We would like to thank Chris Atkeson for discussions and comments on the presentation of this paper. Thanks also to Jessica Hodgins and John Hughes for many helpful suggestions during the course of this project. This work was supported in part by NSF CAREER award CCR-0093072.
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Appendix A: Overview of Spatial Notation
For rotational joints, joint axis s i is represented as follows:
24
i
24
s i = r i × α α i
where α i is the axis of rotation and r i is the point about which the joint rotates. Both terms are expressed in the body i local frame, and the superscript on s i indicates that the spatial vector is expressed in body i local frame coordinates. For prismatic (translational) joints,
25
s = α 0
25
i i
where α i is the axis of translation. We represent multiple degree of freedom joints as sequences of single degree of freedom joints, connected by massless and inertialess bodies. Spatial velocity and acceleration are represented as:
26
v i = q ̇ i s i , a i = q ̈ i s i
where q ̇ and q ̈ are the scalar velocity and acceleration of degree of freedom i—angular velocity and acceleration for rotational joints and linear velocity and acceleration for prismatic joints. Spatial force also combines linear and angular quantities:
27
f i = f f a b i i
where f a i is linear force and f b i is torque. Spatial transform X i j takes spatial quantities from frame i to frame j:
28
X i j = r i j × R i j R i j R 0 i j
where R i j is the 3x3 matrix rotating vectors from frame i to frame j, and r i j is the position of frame j expressed in frame i. Spatial inertia represents both body mass and rotational inertia:
29
I i = I i − −m m i c i i c × i × c i × m i m c i i × = I i − −m m i i c c i i c i m m i c i i
where c i is the vector to the body i center of mass in frame i, c i is that same vector expressed as a cross product matrix, m i is the mass of body i, and I i is the rotational inertia of body i about its center of mass, expressed in frame i. lows: The spatial cross product × ˆ is expressed in matrix form as fol-
30
z = z z b a × ˆ = z z b a × × z a 0 × = z z b a z 0 a
The spatial transpose is
31
z S = z z a b = (z b ) T (z a ) T
where superscript S indicates a spatial transpose and superscript T indicates an ordinary 3-vector transpose.
9
To appear in the ACM SIGGRAPH 2003 conference proceedings
Appendix B: First Derivative Expressions
Conventions. For clarity, we assume a serial chain composed of of L single degree of freedom links numbered 1, 2, ..., L. The subscript zero is reserved for the quantities representing the entire multibody. Where superscripted with an asterix (e.g., I ∗ i ) the quantity represents aggregated information accumulated from L to i. Newton-Euler equations of motion. The equations of motion of a serial multibody chain are compactly expressed in recursive form as follows:
32
v i = X i i−1 v i−1 + s i q ̇ i
34
a i = X i−1 i a i−1 + s i q ̈ i + v i ×s ˆ i q ̇ i p i = X i+1 i p i+1 + I i v i
35
f i = X i+1 i f i+1 + I i a i + v i ×I ˆ i v i
where the second and fourth equations are the time derivatives of the first and third equations respectively. For a multibody rooted at its base joint, the following end condition for simulating gravity is
36
used: a 0 = −G , v 0 = 0 , p L+1 = 0 , f L+1 = 0
The Newton-Euler equations propagate quantities in two directions. To compute aggregate quantities and their derivatives efficiently, rewrite the dynamics equations as follows:
Aggregate equations — momentum. Compute:
37
I ∗ i = X i+1 I ∗ i+1 X i+1 i + I i
37
i
38
p i ∗ = X i+1 i p i+1 ∗ + I i ∗ s i q ̇ i
where body inertias and momenta propagate from the leaf to the base. I ∗ 0 and p ∗ 0 are the aggregate inertia and momentum of the entire body, and p ∗ 0 is equal to p 0 computed from the previous NewtonEuler recursive equations. However, p ∗ i is in general not equal to p i where i = 0 and should only be used as an intermediary quantity in computing the aggregates.
Aggregate equations — force. Compute:
40
I ̇ i ∗ = X i i+1 ̇ I ∗ i+1 X i i+1 + ( s i I i ∗ − I i ∗ s i ) q ̇ i f ∗ i = X i i+1 f i+1 ∗ + s i q ̇ i ×p ˆ ∗ i + I ∗ i s i q ̈ i + ̇ I ∗ i s i q ̇ i
where f ∗ 0 is the aggregate force applied to the entire body, equal to f 0 computed previously. As before, f ∗ i is in general not equal to f i where i = 0. First derivatives – Aggregate equations. The Jacobian may be constructed in time linear in the number of degrees of freedom as follows. All partial derivatives are expressed in frame i. 2
41
∂ ∂ I q 0 ∗ = ( s i I i ∗ − I i ∗ s i )
42
∂ ∂ p q ̇ ∗ 0 = I ∗ i s i
43
i ∂ ∂ p q 0 ∗ = ∂ ∂ I q 0 ∗ v i−1 i + s ×p ˆ ∗
43
i i i i
2 When implementing these expressions, we found it essential to compare the numerical values of these derivatives to the identical derivatives obtained from an alternative technique such as numerical differentiation.
∂ ∂ f q ̈ 0 ∗ i = ∂ ∂ p q ̇ ∗ 0 i (44) ∂ ∂ f q ̇ ∗ 0 i = ∂ ∂ p q ∗ 0 i + ̇ I i ∗ s i + v i × ˆ ∂ ∂ p q ̇ 0 ∗ i (45) ∂ ∂ f q ∗ 0 i = s i ×f ˆ ∗ i + ∂ ∂ I q ∗ 0 i a i−1 i + v i i−1 ×(s ˆ i ×p ˆ i ∗ + ∂ ∂ I q ∗ 0 i v i−1 i )
46
+ ( s i ̇ I ∗ i − ̇ I ∗ i s i )v i−1 i
where
47
v i i−1 = (X i−1 i v i−1 )
48
a i i−1 = (X i i−1 a i−1 )
Appendix C: Changing the Effective Root
Suppose we wish to place the effective root of the character at the point on body i that is located at point r in body i coordinates. We wish this point to have linear velocity b r,des and linear acceleration b ̇ r,des , expressed in the world coordinate frame. The current velocity of body i in the body i frame is v i .
49
v i = v v b a i i
The velocity of point r on body i is computed in the body i frame as
50
v r = v b i − v r a i × v a i and transformed to the world frame as follows:
51
v r = X i 0 v r = v v b a r r
In Equation 51, v b r is the linear velocity of the effective root expressed in world coordinates. This velocity should be b r,des . To obtain the correct velocity at the effective root, simply add the desired correction (b r,des − v b r ) to the reference frame velocity:
52
0 v 0 = b r,des − v r b
The adjustment to a 0 is derived using similar reasoning. When these changes are made, the actual character root can remain at the pelvis, for example, while the effective root is moved from hand to pelvis to foot or other bodies as needed. The effective root can even be set to the center of mass to obtain correct ballistic motion during flight. Derivatives of all equations with respect to parameters describing the motion can be computed in O(D) time.
10
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"text": "['optimal height jumping']",
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[
9111,
9118
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],
"text": "['walking']",
"type": "data",
"id": "T93"
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{
"offsets": [
[
9120,
9159
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],
"text": "['Running times were far from interactive']",
"type": "background_claim",
"id": "T94"
},
{
"offsets": [
[
9165,
9265
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],
"text": "['show that optimization techniques can produce realistic motion for systems of human-level complexity']",
"type": "background_claim",
"id": "T95"
},
{
"offsets": [
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9267,
9328
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],
"text": "['Preexisting motion data can simplify the optimization process']",
"type": "background_claim",
"id": "T96"
},
{
"offsets": [
[
9331,
9371
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],
"text": "['Full scale human motion can be optimized']",
"type": "background_claim",
"id": "T97"
},
{
"offsets": [
[
9418,
9436
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],
"text": "['Liu and Cohen 1995']",
"type": "data",
"id": "T98"
},
{
"offsets": [
[
9511,
9527
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],
"text": "['Rose et al. 1996']",
"type": "data",
"id": "T100"
},
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"offsets": [
[
9573,
9684
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],
"text": "['significant changes to motion capture data can be made by optimizing with a physically based objective function']",
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"id": "T101"
},
{
"offsets": [
[
9690,
9768
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],
"text": "['the character is reduced to the degrees of freedom most important for the task']",
"type": "data",
"id": "T102"
},
{
"offsets": [
[
9377,
9415
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],
"text": "['closely spaced keyframes are available']",
"type": "data",
"id": "T99"
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{
"offsets": [
[
9446,
9508
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],
"text": "['only transitions between existing motion segments are required']",
"type": "data",
"id": "T103"
},
{
"offsets": [
[
9530,
9555
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],
"text": "['Popović and Witkin [1999]']",
"type": "data",
"id": "T104"
},
{
"offsets": [
[
9775,
9811
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],
"text": "['physics does not dominate the motion']",
"type": "data",
"id": "T105"
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{
"offsets": [
[
9813,
9894
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],
"text": "['kinematic techniques can give the animator interactive control for motion editing']",
"type": "background_claim",
"id": "T106"
},
{
"offsets": [
[
9904,
9917
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],
"text": "['Gleicher 1997']",
"type": "data",
"id": "T107"
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{
"offsets": [
[
9920,
9937
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],
"text": "['Lee and Shin 1999']",
"type": "data",
"id": "T108"
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{
"offsets": [
[
9940,
9963
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],
"text": "['Arikan and Forsyth 2002']",
"type": "data",
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},
{
"offsets": [
[
9967,
10045
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],
"text": "['The idea of physically valid motion has appeared in both graphics and robotics']",
"type": "background_claim",
"id": "T110"
},
{
"offsets": [
[
10047,
10142
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],
"text": "['Dynamic filters have been developed for processing motion capture data for physical correctness']",
"type": "background_claim",
"id": "T111"
},
{
"offsets": [
[
10145,
10169
]
],
"text": "['Yamane and Nakamura 2000']",
"type": "data",
"id": "T112"
},
{
"offsets": [
[
10172,
10198
]
],
"text": "['Dasgupta and Nakamura 1999']",
"type": "data",
"id": "T113"
},
{
"offsets": [
[
10201,
10225
]
],
"text": "['Pollard and Reitsma 2001']",
"type": "data",
"id": "T114"
},
{
"offsets": [
[
10229,
10396
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],
"text": "['Physics constraints have been used to plan biped walking motions, exploiting the idea that dynamic equilibrium can be maintained by ensuring that the zero moment point']",
"type": "background_claim",
"id": "T115"
},
{
"offsets": [
[
10491,
10534
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],
"text": "['lies within the support polygon of the feet']",
"type": "background_claim",
"id": "T116"
},
{
"offsets": [
[
10537,
10554
]
],
"text": "['Vukobratović 1970']",
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"id": "T117"
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{
"offsets": [
[
10557,
10578
]
],
"text": "['Takanishi et al. 1985']",
"type": "data",
"id": "T118"
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{
"offsets": [
[
10581,
10601
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],
"text": "['Nagasaka et al. 1999']",
"type": "data",
"id": "T119"
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{
"offsets": [
[
10604,
10654
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],
"text": "['Similar ideas have also been developed in graphics']",
"type": "background_claim",
"id": "T120"
},
{
"offsets": [
[
10660,
10678
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],
"text": "['Ko and Badler 1996']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
10682,
10757
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],
"text": "['who bend the torso of a character to reduce torques at the desired ZMP, and']",
"type": "background_claim",
"id": "T122"
},
{
"offsets": [
[
10759,
10776
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],
"text": "['van de Panne 1997']",
"type": "data",
"id": "T123"
},
{
"offsets": [
[
10778,
10896
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],
"text": "['who ensure that reasonable forces are available to accelerate the center of mass without creating angular acceleration']",
"type": "background_claim",
"id": "T124"
},
{
"offsets": [
[
10898,
10920
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],
"text": "['Liu and Popović [2002]']",
"type": "data",
"id": "T125"
},
{
"offsets": [
[
10932,
11080
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],
"text": "['some dynamic effects can be preserved by enforcing patterns of linear and angular momentum, which does not require computation of dynamic parameters']",
"type": "background_claim",
"id": "T126"
},
{
"offsets": [
[
11168,
11419
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],
"text": "['it is possible to incorporate constraints on physics as efficiently as constraints on kinematic parameters and an O(D) algorithm for computing first derivatives of a broad range of physics constraints for improved performance in a optimization context']",
"type": "own_claim",
"id": "T127"
},
{
"offsets": [
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11950,
12103
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],
"text": "['Constrained optimization has been shown to be a very powerful approach for obtaining appealing dynamic motions from a minimal amount of input information']",
"type": "background_claim",
"id": "T128"
},
{
"offsets": [
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12105,
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],
"text": "['The user adjusts the problem description in the form of keyframes, constraints, and objectives']",
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"id": "T129"
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12201,
12274
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],
"text": "['an optimizer computes an optimal animation given this problem description']",
"type": "background_claim",
"id": "T130"
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12280,
12340
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],
"text": "['the process repeats until the user obtains a final animation']",
"type": "background_claim",
"id": "T131"
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{
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[
12343,
12351
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],
"text": "['Figure 2']",
"type": "data",
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},
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"offsets": [
[
12811,
12965
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"text": "['Enforcing physics constraints or minimizing a dynamic property such as sum squared joint torques requires an inverse dynamics computation at each time t i']",
"type": "background_claim",
"id": "T133"
},
{
"offsets": [
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12977,
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],
"text": "['the inverse dynamics computation is relatively expensive']",
"type": "background_claim",
"id": "T134"
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{
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13035,
13066
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],
"text": "['many efficient algorithms exist']",
"type": "background_claim",
"id": "T135"
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{
"offsets": [
[
13072,
13173
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],
"text": "['the process is well known to require time linear in the number of degrees of freedom of the character']",
"type": "background_claim",
"id": "T136"
},
{
"offsets": [
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13184,
13306
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],
"text": "['typical choices for the numerical optimizer in Figure 2 also require derivatives of the constraints and objective function']",
"type": "background_claim",
"id": "T137"
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{
"offsets": [
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13321,
13367
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],
"text": "['the sequential quadratic programming algorithm']",
"type": "background_claim",
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13378,
13398
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"text": "['Witkin and Kass 1988']",
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13401,
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"text": "['makes use of first derivatives of the constraints']",
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13477,
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"text": "['and both first and second derivatives of the objective function']",
"type": "background_claim",
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"offsets": [
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13598,
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],
"text": "['a broad range of physics constraints can be expressed based on aggregate forces and torques applied to the character, and how expressing physics constraints in this way allows us to compute the constraint Jacobian in linear time']",
"type": "own_claim",
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13828,
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"text": "['Section 4']",
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"text": "['With this objective function and our linear time algorithm for computing the constraint Jacobian, we are able to show that physically based optimization can be performed for a 22 degree of freedom character at interactive speeds']",
"type": "own_claim",
"id": "T144"
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14402,
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"text": "['Constraints that enforce physical validity can be formulated as linear equality or inequality constraints on aggregate force']",
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"id": "T145"
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14850,
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],
"text": "['One way of enforcing correct physics during flight is to ensure that the aggregate momentum of the body remains constant throughout the flight phase']",
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15015,
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],
"text": "['the constraint Jacobian that results from constraining momenta is denser than necessary']",
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"id": "T147"
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"offsets": [
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15106,
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],
"text": "['the control points that determine take-off affect all constraint equations governing the flight phase']",
"type": "own_claim",
"id": "T148"
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15209,
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"text": "['A more elegant solution is to restrict illegal forces during flight']",
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"id": "T149"
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15278,
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"text": "['During flight, no forces, with the exception of gravity, may be derived from the environment']",
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"offsets": [
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"text": "['the character is swinging on a high bar or monkey bars']",
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"text": "['the amount of torque that can be applied about the bar axis is constrained']",
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"text": "['During ground contact, the feet can only push, not pull on the ground']",
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"text": "['contact forces should not require an unreasonable amount of friction']",
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"text": "['the center of pressure must fall within the support polygon of the feet']",
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16947,
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"text": "['These effects can be modeled with equations that constrain the linear and angular forces separately']",
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"offsets": [
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17109,
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"text": "['Coulomb’s model dictates that the linear reaction force must fall within a friction cone oriented along the contact normal with angular half-width tan −1 μ']",
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17307,
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"text": "['The inequality constraint for the linear aggregate force is']",
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17408,
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],
"text": "['cos −1 N · a f 0 a < tan −1 μ']",
"type": "own_claim",
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17481,
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"text": "['|f 0 |']",
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"text": "['Aggregate force is translated to a constraint point c as follows']",
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"text": "['f c = f f b c c a = f b 0 − c f a 0 0 × f a 0']",
"type": "own_claim",
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16425,
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],
"text": "['The bar contact constraint can']",
"type": "own_claim",
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16461,
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"text": "['be expressed as']",
"type": "own_claim",
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16508,
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"text": "['− τ max < s bar · f b c < τ max']",
"type": "own_claim",
"id": "T165"
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{
"offsets": [
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17555,
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],
"text": "['the constraint without the inverse trigonometric functions is']",
"type": "own_claim",
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17649,
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"text": "['N |f · a 0 f | a 0 > ( μ 1 2 + 1)']",
"type": "own_claim",
"id": "T167"
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"offsets": [
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17703,
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"text": "['The magnitude of the normal force can be constrained as follows']",
"type": "own_claim",
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17799,
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"text": "['0 ≤ f a 0 · N ≤ K orce']",
"type": "own_claim",
"id": "T169"
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"offsets": [
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17862,
17863
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],
"text": "['f']",
"type": "own_claim",
"id": "T170"
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"offsets": [
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17881,
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],
"text": "['Contact torques are constrained by geometrically confining the center of pressure to the support area']",
"type": "own_claim",
"id": "T171"
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18927,
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],
"text": "['computing derivatives on the physics constraints becomes a problem of differentiating aggregate force with respect to the free parameters of the problem']",
"type": "own_claim",
"id": "T172"
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"offsets": [
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18848,
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],
"text": "['all physics constraints have been expressed as constraints on aggregate force']",
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},
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"offsets": [
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19081,
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],
"text": "['At any time t, character position q, velocity q, ̇ and acceleration q ̈ are known']",
"type": "own_claim",
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"offsets": [
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19635,
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],
"text": "['is the most difficult term in this expression']",
"type": "own_claim",
"id": "T175"
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"offsets": [
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19567,
19587
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],
"text": "['The term ∂ f 0 / ∂ q']",
"type": "own_claim",
"id": "T176"
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[
19799,
19941
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],
"text": "['straightforward analytical computation of the force Jacobian would require time quadratic in the number of degrees of freedom of the character']",
"type": "own_claim",
"id": "T177"
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"offsets": [
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19955,
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"text": "['joint torques are not required']",
"type": "data",
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{
"offsets": [
[
19992,
20091
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],
"text": "['this value and first derivatives for constraints based on this value can be computed in linear time']",
"type": "own_claim",
"id": "T179"
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{
"offsets": [
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20111,
20239
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],
"text": "['our paper is the first to present a linear time algorithm for computing the force Jacobian for an articulated character or robot']",
"type": "own_claim",
"id": "T180"
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"offsets": [
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20796,
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"text": "['We use spatial notation as in']",
"type": "own_claim",
"id": "T181"
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{
"offsets": [
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20848,
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],
"text": "['for conciseness']",
"type": "own_claim",
"id": "T182"
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{
"offsets": [
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20827,
20846
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],
"text": "['Featherstone [1987]']",
"type": "data",
"id": "T183"
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"offsets": [
[
21069,
21088
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],
"text": "['into single vectors']",
"type": "own_claim",
"id": "T184"
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{
"offsets": [
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20971,
21012
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"text": "['It combines linear and angular quantities']",
"type": "own_claim",
"id": "T185"
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{
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21102,
21123
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],
"text": "['Equations 1 through 3']",
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"offsets": [
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21978,
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"text": "['Efficiently computing ∂ f 0 / ∂ q, the force Jacobian, requires efficiently computing ∂ p 0 / ∂ q, the momentum Jacobian']",
"type": "own_claim",
"id": "T187"
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{
"offsets": [
[
22108,
22176
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],
"text": "['aggregate force f 0 is the time derivative of aggregate momentum p 0']",
"type": "own_claim",
"id": "T188"
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{
"offsets": [
[
22261,
22313
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],
"text": "['the momentum Jacobian can be computed in linear time']",
"type": "own_claim",
"id": "T189"
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"offsets": [
[
22981,
23064
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],
"text": "['The usual way to compute aggregate momentum is to formulate the following recursion']",
"type": "background_claim",
"id": "T190"
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{
"offsets": [
[
23098,
23130
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],
"text": "['v i = X i i−1 v i−1 + s i q ̇ i']",
"type": "background_claim",
"id": "T191"
},
{
"offsets": [
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23172,
23201
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],
"text": "['p i = X i+1 i p i+1 + I i v i']",
"type": "background_claim",
"id": "T192"
},
{
"offsets": [
[
23380,
23450
]
],
"text": "['Parameter q i appears in the coordinate transforms X i i+1 and X i+1 i']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
23460,
23538
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],
"text": "['every v j for j > i depends on q i , and every p j for j ≥ 0 depends on q i']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
23541,
23618
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],
"text": "['Unrolling the recursion to collect terms for ∂ p 0 / ∂ q i requires O(D) time']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
23725,
23796
]
],
"text": "['There is no clever way to simplify the calculation by aggregating terms']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
23802,
23830
]
],
"text": "['it is presented in this form']",
"type": "data",
"id": "T197"
},
{
"offsets": [
[
23648,
23723
]
],
"text": "['this approach will lead to an O(D 2 ) computation for the momentum Jacobian']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
23848,
23891
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],
"text": "['rewriting the recursion solves this dilemma']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
[
23925,
23962
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],
"text": "['I ∗ i = X i+1 i I ∗ i+1 X i+1 i + I i']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
24004,
24039
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],
"text": "['p i ∗ = X i i+1 p ∗ i+1 + I i ∗ v i']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
24081,
24092
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],
"text": "['p 0 = p ∗ 0']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
24239,
24285
]
],
"text": "['only propagation from leaf to base is required']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
24291,
24368
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],
"text": "['each parameter q j does not affect terms computed for joints j + 1 and beyond']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
24147,
24224
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],
"text": "['p ∗ i is expressed as a function of v i , which is a local variable at link i']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
24371,
24379
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],
"text": "['Figure 4']",
"type": "data",
"id": "T206"
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{
"offsets": [
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24398,
24443
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],
"text": "['p ∗ i is in general not equal to p i if i = 0']",
"type": "own_claim",
"id": "T207"
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{
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24445,
24532
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],
"text": "['A term superscripted with an asterix should be treated only as an intermediary quantity']",
"type": "own_claim",
"id": "T208"
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{
"offsets": [
[
24541,
24562
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],
"text": "['its subscript is zero']",
"type": "data",
"id": "T209"
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{
"offsets": [
[
24577,
24611
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],
"text": "['it is the desired aggregate result']",
"type": "own_claim",
"id": "T210"
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{
"offsets": [
[
24613,
24742
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],
"text": "['A linear time expression for the momentum Jacobian can be derived in a straightforward manner based on this form of the recursion']",
"type": "own_claim",
"id": "T211"
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{
"offsets": [
[
24815,
24942
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],
"text": "['we are not simplifying or changing the outcome of the dynamics computation, only changing the order in which terms are computed']",
"type": "own_claim",
"id": "T212"
},
{
"offsets": [
[
24944,
25034
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],
"text": "['Aggregate momentum p 0 and the momentum Jacobian are exactly the same in both formulations']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
[
25067,
25199
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],
"text": "['In a traditional inverse dynamics formulation, accelerations and forces are expressed as the time derivatives of Equations 11 and 12']",
"type": "background_claim",
"id": "T214"
},
{
"offsets": [
[
25233,
25320
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],
"text": "['a i = X i−1 i a i−1 + s i q ̈ i + v i ×s ˆ i q ̇ i f = X i+1 i f i+1 + I a + v ×I ˆ v']",
"type": "background_claim",
"id": "T215"
},
{
"offsets": [
[
26045,
26135
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],
"text": "['For fast computation, we instead take the time derivative of Equation 14, which results in']",
"type": "own_claim",
"id": "T217"
},
{
"offsets": [
[
26168,
26219
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],
"text": "['f ∗ = X i i+1 f i+1 ∗ + v ×p ˆ ∗ + I ∗ a + ̇ I ∗ v']",
"type": "own_claim",
"id": "T216"
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{
"offsets": [
[
26292,
26343
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],
"text": "['This equation has the properties we are looking for']",
"type": "own_claim",
"id": "T218"
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{
"offsets": [
[
26487,
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],
"text": "['f ∗ i is in general different from the actual joint force f i if i = 0']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
26404,
26447
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],
"text": "['terms are propagated from leaf to base only']",
"type": "own_claim",
"id": "T220"
},
{
"offsets": [
[
26559,
26674
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],
"text": "['Differentiating Equation 18 and accumulating the coefficients of derivative elements results in the simplified form']",
"type": "own_claim",
"id": "T221"
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{
"offsets": [
[
26687,
26697
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],
"text": "['Appendix B']",
"type": "data",
"id": "T222"
},
{
"offsets": [
[
26699,
26841
]
],
"text": "['Each partial derivative of the aggregate force with respect to joint positions, velocities, and accelerations may be obtained in constant time']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
26853,
26901
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],
"text": "['the full Jacobian may be obtained in linear time']",
"type": "own_claim",
"id": "T224"
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{
"offsets": [
[
27110,
27159
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],
"text": "['Numerically the partial derivatives are identical']",
"type": "own_claim",
"id": "T225"
},
{
"offsets": [
[
27161,
27227
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],
"text": "['The articulated model is a serial chain ranging from 3 to 50 links']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
27242,
27297
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],
"text": "['the proposed method is linear in the degrees of freedom']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
27305,
27350
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],
"text": "['direct differentiation shows quadratic growth']",
"type": "own_claim",
"id": "T228"
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{
"offsets": [
[
27438,
27528
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],
"text": "['the linear time method shows a computational advantage with as few as 5 degrees of freedom']",
"type": "own_claim",
"id": "T229"
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{
"offsets": [
[
27385,
27436
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],
"text": "['overheads in computing aggregate intermediate terms']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
27738,
27846
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],
"text": "['there is a cost to this approach that may be higher for robotics applications than for graphics applications']",
"type": "own_claim",
"id": "T231"
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{
"offsets": [
[
28054,
28100
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],
"text": "['In robotics, this information must be computed']",
"type": "own_claim",
"id": "T232"
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{
"offsets": [
[
28109,
28166
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],
"text": "['it corresponds to signals sent to the motors of the robot']",
"type": "own_claim",
"id": "T233"
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{
"offsets": [
[
27848,
27907
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],
"text": "['In a standard Newton-Euler formulation, force parameter f i']",
"type": "background_claim",
"id": "T234"
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{
"offsets": [
[
27922,
27977
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],
"text": "['contains all of the joint force information for joint i']",
"type": "background_claim",
"id": "T235"
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{
"offsets": [
[
27909,
27920
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],
"text": "['Equation 17']",
"type": "data",
"id": "T236"
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{
"offsets": [
[
28168,
28224
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],
"text": "['It must in general also be part of optimization routines']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
28234,
28327
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],
"text": "['en- ergy consumption and joint torque limits are of particular concern when operating a robot']",
"type": "own_claim",
"id": "T238"
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{
"offsets": [
[
28333,
28366
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],
"text": "['none of the joints can be ignored']",
"type": "own_claim",
"id": "T239"
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{
"offsets": [
[
28426,
28526
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],
"text": "['many of the effects we expect to see in physically based optimization do not depend on joint torques']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
28399,
28424
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],
"text": "['animation of human motion']",
"type": "data",
"id": "T241"
},
{
"offsets": [
[
28544,
28685
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],
"text": "['physical correctness and optimization functions enforcing smooth motion are sufficient to obtain many natural characteristics of human motion']",
"type": "own_claim",
"id": "T242"
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{
"offsets": [
[
28690,
28702
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],
"text": "['some torques']",
"type": "data",
"id": "T243"
},
{
"offsets": [
[
28736,
28761
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],
"text": "['are found to be important']",
"type": "data",
"id": "T244"
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{
"offsets": [
[
28791,
28802
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],
"text": "['many others']",
"type": "own_claim",
"id": "T245"
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{
"offsets": [
[
28833,
28864
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"text": "['can be ignored for many motions']",
"type": "own_claim",
"id": "T246"
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{
"offsets": [
[
28869,
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],
"text": "['a subset of K torques are required']",
"type": "data",
"id": "T247"
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{
"offsets": [
[
28905,
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],
"text": "['it is straightforward to extend our approach to measure torques at these joints in O(KD) time']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
[
29129,
29280
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],
"text": "['One traditional approach is to use the integral of the sum of squared joint torques to produce a motion that approximately minimizes energy expenditure']",
"type": "background_claim",
"id": "T249"
},
{
"offsets": [
[
29314,
29353
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],
"text": "['h(x) = t f ∑ D τ i 2 (x,t) dt t=t s i=1']",
"type": "background_claim",
"id": "T250"
},
{
"offsets": [
[
29371,
29397
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],
"text": "['This function is expensive']",
"type": "background_claim",
"id": "T251"
},
{
"offsets": [
[
29406,
29450
]
],
"text": "['computing its gradient requires O(D 2 ) work']",
"type": "background_claim",
"id": "T252"
},
{
"offsets": [
[
29452,
29544
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],
"text": "['Adopting this function would negate our effort in constructing efficient physics constraints']",
"type": "own_claim",
"id": "T253"
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{
"offsets": [
[
29546,
29679
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],
"text": "['An objective function that we have found to work well is to minimize the integral of the sum of squared, weighted joint accelerations']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
29713,
29745
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],
"text": "['h(x) = t f ∑ D w q ̈ (x,t) 2 dt']",
"type": "own_claim",
"id": "T255"
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{
"offsets": [
[
29787,
29800
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],
"text": "['i i t=t s i=1']",
"type": "own_claim",
"id": "T256"
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{
"offsets": [
[
30130,
30241
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],
"text": "['the analytical Hessian for this objective function is constant, symmetric, positive definite, and band-diagonal']",
"type": "own_claim",
"id": "T257"
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{
"offsets": [
[
30290,
30392
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],
"text": "['a simple objective function with low cost is to simply minimize the distance from the reference motion']",
"type": "own_claim",
"id": "T258"
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{
"offsets": [
[
30249,
30288
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],
"text": "['a reference motion q R (t) is available']",
"type": "data",
"id": "T259"
},
{
"offsets": [
[
30426,
30466
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],
"text": "['h(x) = t=t t f s (q(x,t) − q R (t)) 2 dt']",
"type": "own_claim",
"id": "T260"
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{
"offsets": [
[
30484,
30537
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],
"text": "['This objective function is similar to the one used in']",
"type": "own_claim",
"id": "T261"
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{
"offsets": [
[
30539,
30554
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],
"text": "['Gleicher [1997]']",
"type": "data",
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{
"offsets": [
[
30556,
30645
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],
"text": "['Other objective functions we have attempted include an integral of squared contact forces']",
"type": "own_claim",
"id": "T263"
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{
"offsets": [
[
30679,
30708
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],
"text": "['h(x) = t f f c 2 (t) dt t=t s']",
"type": "own_claim",
"id": "T264"
},
{
"offsets": [
[
30726,
30784
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],
"text": "['The Jacobian of this function is computable in linear time']",
"type": "own_claim",
"id": "T265"
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{
"offsets": [
[
30829,
30905
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],
"text": "['Gaits generated using this function have a certain ‘tip-toe’ quality to them']",
"type": "own_claim",
"id": "T266"
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{
"offsets": [
[
30910,
30987
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],
"text": "['the function minimizes the amount of reaction force derived from the contacts']",
"type": "own_claim",
"id": "T267"
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{
"offsets": [
[
30989,
31012
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],
"text": "['Minimizing contact jerk']",
"type": "own_claim",
"id": "T268"
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{
"offsets": [
[
31044,
31085
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],
"text": "['can be achieved using forward differences']",
"type": "own_claim",
"id": "T269"
},
{
"offsets": [
[
31119,
31166
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],
"text": "['m−1 h(x) = ∑ ( f c (t i ) − f c (t i+1 )) 2 i=1']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
32349,
32381
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],
"text": "['appears very unstable at landing']",
"type": "own_claim",
"id": "T271"
},
{
"offsets": [
[
32295,
32313
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],
"text": "['The initial motion']",
"type": "own_claim",
"id": "T272"
},
{
"offsets": [
[
32328,
32347
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],
"text": "['top row of Figure 6']",
"type": "data",
"id": "T273"
},
{
"offsets": [
[
32383,
32412
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],
"text": "['The character would fall over']",
"type": "own_claim",
"id": "T274"
},
{
"offsets": [
[
32414,
32514
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],
"text": "['This effect is eliminated in the optimization by enforcing the physics constraints of ground contact']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
32557,
32565
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],
"text": "['Figure 7']",
"type": "data",
"id": "T276"
},
{
"offsets": [
[
33180,
33253
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],
"text": "['All of these effects are obtained as a result of the optimization process']",
"type": "own_claim",
"id": "T277"
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{
"offsets": [
[
33506,
33529
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],
"text": "['a number of time slices']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
33558,
33678
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],
"text": "['equal to the number of control points produced good results and did not need to adjust this value for individual motions']",
"type": "own_claim",
"id": "T279"
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{
"offsets": [
[
33680,
33731
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],
"text": "['Finer time slices would overly constrain the system']",
"type": "own_claim",
"id": "T280"
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{
"offsets": [
[
33737,
33791
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],
"text": "['sparser time slices allowed too much freedom for error']",
"type": "own_claim",
"id": "T281"
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{
"offsets": [
[
35395,
35496
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],
"text": "['Two implementation issues were especially important for achieving the results described in this paper']",
"type": "own_claim",
"id": "T282"
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{
"offsets": [
[
35521,
35561
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],
"text": "['the basis functions have local influence']",
"type": "data",
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{
"offsets": [
[
35563,
35640
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],
"text": "['the vector and matrix quantities computed during optimization are very sparse']",
"type": "own_claim",
"id": "T284"
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{
"offsets": [
[
35642,
35701
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],
"text": "['We use the publicly-available Lancelot optimization package']",
"type": "own_claim",
"id": "T285"
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"offsets": [
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35723,
35775
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],
"text": "['where sparsity is accounted for by groupseparability']",
"type": "own_claim",
"id": "T286"
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{
"offsets": [
[
35704,
35720
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"text": "['Conn et al. 1992']",
"type": "data",
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},
{
"offsets": [
[
35820,
35899
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],
"text": "['Implementing any inverse dynamics algorithm requires selecting a character root']",
"type": "own_claim",
"id": "T288"
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{
"offsets": [
[
35901,
35993
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],
"text": "['An ability to move the effective root to different parts of the character is very convenient']",
"type": "own_claim",
"id": "T289"
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{
"offsets": [
[
36013,
36092
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],
"text": "['there is a single point of constraint between the character and the environment']",
"type": "data",
"id": "T290"
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"offsets": [
[
36098,
36141
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],
"text": "['that point has a known and fixed trajectory']",
"type": "data",
"id": "T291"
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{
"offsets": [
[
36143,
36172
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],
"text": "['it is convenient to place the']",
"type": "own_claim",
"id": "T292"
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{
"offsets": [
[
37284,
37312
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],
"text": "['character root at that point']",
"type": "own_claim",
"id": "T293"
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{
"offsets": [
[
37349,
37481
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],
"text": "['it may be convenient to root the character at the hands for the swing, at the center of mass for flight, and at the feet for landing']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
37321,
37346
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],
"text": "['swing example of Figure 1']",
"type": "data",
"id": "T295"
},
{
"offsets": [
[
37483,
37726
]
],
"text": "['In a Newton-Euler inverse dynamics formulation, rerooting is typically done by changing parent / child relationships, which requires inverting joint angles and transforms at each joint and altering the flow of dynamic terms from leaves to root']",
"type": "background_claim",
"id": "T296"
},
{
"offsets": [
[
37728,
37811
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],
"text": "['Both of these changes complicate the problem description presented to the optimizer']",
"type": "background_claim",
"id": "T297"
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{
"offsets": [
[
37813,
37860
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],
"text": "['The effective root can be relocated more easily']",
"type": "own_claim",
"id": "T298"
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{
"offsets": [
[
37871,
38034
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],
"text": "['by leaving the actual root and the flow of the dynamics computation fixed and computing velocities and accelerations at the root to maintain the desired constraint']",
"type": "own_claim",
"id": "T299"
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{
"offsets": [
[
38057,
38067
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],
"text": "['Appendix C']",
"type": "data",
"id": "T300"
},
{
"offsets": [
[
38121,
38377
]
],
"text": "['This paper contributes to physically based optimization by defining and exploring a restricted class of optimization problems where physics constraints are included and first derivatives of constraints and objective functions can be computed in linear time']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
38379,
38612
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],
"text": "['The fact that first derivatives can be computed in linear time instead of quadratic time suggests that our problem is simpler than previous physically based approaches and similar in complexity to very successful kinematic approaches']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
38680,
38828
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],
"text": "['our solution landscape will be smoother than previous physically based optimization approaches, making it feasible to handle more complex characters']",
"type": "own_claim",
"id": "T303"
},
{
"offsets": [
[
38835,
38869
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],
"text": "['the optimization does not converge']",
"type": "data",
"id": "T304"
},
{
"offsets": [
[
38871,
38920
]
],
"text": "['we can usually trace it back to the problem setup']",
"type": "own_claim",
"id": "T305"
},
{
"offsets": [
[
38922,
38970
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],
"text": "['Sometimes it is due to overconstrained equations']",
"type": "own_claim",
"id": "T306"
},
{
"offsets": [
[
38990,
39038
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],
"text": "['often it is due to overly restrictive parameters']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
39048,
39069
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],
"text": "['friction coefficients']",
"type": "data",
"id": "T308"
},
{
"offsets": [
[
39071,
39083
]
],
"text": "['joint limits']",
"type": "data",
"id": "T309"
},
{
"offsets": [
[
39085,
39110
]
],
"text": "['poor selection of timings']",
"type": "data",
"id": "T310"
},
{
"offsets": [
[
39117,
39195
]
],
"text": "['At present, timings are set by the user and their values need to be reasonable']",
"type": "own_claim",
"id": "T311"
},
{
"offsets": [
[
39259,
39362
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],
"text": "['Any optimization technique that makes use of local derivatives has potential problems with local minima']",
"type": "own_claim",
"id": "T312"
},
{
"offsets": [
[
39409,
39494
]
],
"text": "['an expected motion sequence could be thought of as motion about some neutral position']",
"type": "data",
"id": "T313"
},
{
"offsets": [
[
39506,
39556
]
],
"text": "['the character was started in that neutral position']",
"type": "data",
"id": "T314"
},
{
"offsets": [
[
39557,
39616
]
],
"text": "['there was no problem descending toward the expected minimum']",
"type": "own_claim",
"id": "T315"
},
{
"offsets": [
[
39618,
39706
]
],
"text": "['We were able to create a jumping Luxo and highly dynamic human motions with good success']",
"type": "own_claim",
"id": "T316"
},
{
"offsets": [
[
39712,
39735
]
],
"text": "['less dynamic activities']",
"type": "data",
"id": "T317"
},
{
"offsets": [
[
39737,
39778
]
],
"text": "['our system would require additional input']",
"type": "own_claim",
"id": "T318"
},
{
"offsets": [
[
39780,
39867
]
],
"text": "['physics constraints plus smooth motion would not in general produce the desired results']",
"type": "own_claim",
"id": "T319"
},
{
"offsets": [
[
39892,
39911
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],
"text": "['stand for 5 seconds']",
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"text": "['It is interesting to compare our approach to that']",
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"text": "['their approach could be adapted easily to obtain linear time performance by rewriting the momentum equations']",
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<Title>A Data-driven Approach for Real-Time Clothes Simulation</Title>
Frederic Cordier
Nadia Magnenat-Thalmann
MIRALab, University of Geneva, 24 rue du General Dufour, CH-1211 Geneva, Switzerland {cordier, thalmann}@miralab.unige.ch
<Abstract>A data-driven approach for real-time processing of clothes, particularly suitable for simulating dresses worn by virtual characters, is proposed. It starts, prior to realtime simulation, by analyzing cloth behavior in relation to the underlying skeleton movement from a pre-simulated sequence of the cloth obtained using any high quality offline simulators. The idea is to use this analysis to find an optimal combination of physics-based simulation and geometric approximation of the simulator; potentially colliding regions are defined on the cloth such that they will hold true for the skeleton movement that closely matches that of pre-simulated sequence. At runtime, using these analyses, our simulation process provides both visually pleasing results and performance, as long as the motion of the character remains sufficiently close to the original sequence used for the pre-computation. The key contributions of this paper are (1) efficient collision handling that prunes out potentially colliding objects by using the off-line simulation sequence as examples; (2) data-driven fix-up process for the coarse mesh simulation that deduces the gross behavior of the cloth; and (3) geometric approximation of the fine mesh deformation, responsible for details in the shape of the cloth such as wrinkles.</Abstract>
<H1>1. Introduction</H1>
The problem of simulating the behavior of clothes is one subject the graphics community has been grappling with since almost two decades ago
[19] [21]. Relatively little emphasis has been placed on the separate problem of how to achieve real-time performance in simulating cloth. A number of strategies have been suggested, such as using simplifying assumptions for the physics model and/or collision detection [ 7 ] [12]. A recent work by James et al. [10]
suggests a different approach by adopting a data-driven method. These techniques do not suffice, however, when simulating fully dressed virtual characters in real-time, leaving the topic unexplored. We present a data-driven method for simulating clothes worn by 3D characters in real-time. To effectively optimize the physics-based deformation, which is the bottleneck of the simulation, we use a coarse representation of the cloth mesh to drive the gross behavior in simulation. We consider that the gross cloth behavior is driven mainly by two separable contributions: the skeleton-driven movement of the character and the mechanical properties of the cloth. This consideration was partly inspired by the hybrid real-time simulation method proposed in Cordier et al. [ 5 ], where a hybrid deformation method is used to combine dynamic surfaces with Skeleton-Driven Deformation (SDD). Unlike that method, however, our method exhibits significantly more efficient and realistic behavior. This effect is achieved by focusing on the analysis of cloth movements in relation to its associated skin surface, and adopting a learning strategy. The idea is to use the analysis of the presimulated sequence to identify the region largely explained by joint movement and to replace the physics based simulation with geometric methods wherever possible. In our approach, the key ingredients of the new technique are associated with different facets of cloth simulation: First, our novel collision detection prunes out unnecessary collision tests by tightly localizing potentially colliding regions through the analysis of the cloth movement in relation to the skeleton. Second, we use the pre-simulated sequence to approximate the dynamic behavior of the coarse mesh geometrically wherever possible. Finally, fine details such as wrinkles are also simulated in a data-driven manner, by using the pre-simulated cloth sequence as examples. Subsequently, real-time animation of fully dressed human could be generated, which would be suitable for applications such as games where visual plausibility is more important than accuracy. The remainder of this paper is organized as follows: We begin by reviewing previous approaches in Section 2. Section 3 gives an overview, followed by the description of our method for SDD in the Section 4. The next two sections are dedicated to the simulation of gross behavior and the generation of wrinkles, respectively. After demonstrating results and performance in Section 7, we conclude with discussion and future work in Section 8.
<H1>2. Previous Work</H1>
<H2>2.1. Numerical solvers</H2>
Probably the most common technique for simulating the physical properties of clothes is the particle system. Simulation process is broken down into calculating the internal forces and solving the system of Partial Differential Equations (PDE). The latter point has attracted much interest in the field of real-time applications, since it requires high computation power. The explicit Euler method
[ 2 ] has been one of the first numerical solvers. Unfortunately, this method is notorious for its instability when using large time steps and stiff equations.
Several improvements have been proposed to reduce instability, such as the Verlet integration
[11] and the explicit Euler combined with inverse dynamics
[17] [20]. Unfortunately, the simulation quality is sacrificed in favor of computation speed, due to the approximations employed in these models. The implicit Euler method presented by Baraff et al.
[ 2 ] performs the computation not by using the derivative at the current time, but the predicted derivative at the next time step. Unlike explicit Euler integration, the implicit Euler method offers higher stability while using large time-steps and clothes with stiff mechanical properties. A major drawback of this numerical solver, however, is the computation of a large linear system, More recently, researchers worked on saving the computation time of the linear system solver. Desbrun et al
[ 7 ]
proposed solving the linear system with a precomputed inverse matrix. Kang et al.
[12] proposed further optimization with a direct update formula for the positions and velocities of the cloth vertices. As indicated by the authors, these methods are not intended to provide a physically-correct cloth animation. Our approach to that problem is a data-driven mass-spring system: the simulation is corrected with a set of functions built from the pre-simulated animation. By doing so, we bring the deformation of the mass-spring system closer to the original cloth behavior. Another approach to fast garment deformations is the hybrid approach. They aim for a neat combination of physically based deformation and geometric deformation. Cordier et al.
[ 5 ]
proposed to segment the cloth into pieces and simulate these by different algorithms, depending on how they lie on the body surface and whether they adhere to it or flow over it. Others have noted that wrinkle deformation is geometric in nature and therefore can be computed with a geometric method.
Wrinkles can be generated either by tessellating the cloth mesh
[12] or rendering details on texture using bump mapping
[9]. The main difficulty is defining a fold function that can simulate all kinds of wrinkle patterns. Moreover, determining the location and shape of wrinkles is left to CG artists. One of our contributions is a geometric wrinkling method that is “trained” by using a pre-simulated cloth sequence, rather than relying on users.
<H2>2.2. Collision handling</H2>
Collision detection is usually one of the bottlenecks in real-time animation. The problem is particularly acute in the case of clothes because these objects are highly deformable. Several algorithms have been proposed to process robustly collisions in cloth simulation [21] [22]
without reaching real-time performance.
Some other methods exploit graphics hardware to compute collisions on bump maps
[20]; others use implicit surfaces to check collisions on the body
[18], or voxel trees, which partition the space hierarchically
[14]. Using frame coherency to reduce computation cost has been explored by Zhang et al
[23]. In this work, we propose a data-driven collision detection method; we use the pre-simulated sequence to localize the collision checks to neighboring cloth regions that have high probability to collide.
<H2>2.3. Data-driven approaches</H2>
The idea of building an interpolator from examples or pre-simulated data has proven to be a valuable tool in a variety of areas of CG, e.g. for modeling a variety of human body shapes and for motion synthesis. The basic idea is to build an interpolation space filled with a set of pairs of input parameters and the targeted graphical objects. Cloth animation depends on a high number of parameters and therefore a data-driven approach is difficult to adapt. Very recently, James et al.
[10]
resented such an approach, where physics-based deformation and collision detection are both handled in a unified framework. By blending of pre-computed orbits rather than using a mass-spring system, previous unseen results could be achieved, such as garments with stiff mechanical properties in real-time. However, they show little degrees of freedom (DoF) to the clothes under simulation; Instead of resorting to a data-driven approach for the entire simulation, we seek a neat combination of a data-driven approach with the mass-spring system. Unlike previous works, our simulator allows a much higher degree of interaction, as it is often needed in animating clothes on moving characters.
The history of research on real-time cloth is relatively recent. Researchers have concentrated mainly on two aspects of real-time cloth animation: simulating the physical properties of garments and collision handling.
<H1>3. Overview of our approach</H1>
The primary focus of this paper is the development of a fast cloth simulator for real-time applications. Dynamic simulation of complex deformable models, however, can easily involve thousands of degrees of freedom. For example, a physics-based simulator would require several minutes to compute one frame of a cloth model worn by a character. Simulating large models directly would therefore be computationally impractical. In what follows, we present the chosen optimization strategies as well as the workflow of the method.
<H2>3.1. Optimization strategies</H2>
Figure 1 summarizes our strategies for optimizing the cloth simulation. Our simulator is based on two levels of deformation: the first deduces the gross cloth behavior by working on a coarse mesh with a physics-based approach whereas the second generates wrinkles on a fine mesh with a geometric method. The coarse mesh is generated by simplifying the original cloth mesh through segmentation. The reason for this choice is to lower the computation time; geometric methods are in general much faster than physically-based ones [9]. Simulation stages Strategies to reduce the computation time Simulating the Simplified mass-spring system gross movements Implicit Euler integrator on the coarse mesh Post-correction to maintain the cloth behaviour closer to the pre-simulated one Collision detection Each vertex is enclosed into a collision hull on the coarse mesh that is rigidly attached to the skeleton Extra collision detection is also computed between the floating regions and the skeleton joints Modelling the cloth Geometric deformation with a linear function details on the fine which coefficients are defined by linear mesh regression on the pre-simulated animation Next, our simulator assumes that the clothes are worn on 3D characters, and that the character movement is the only external force acting upon the cloth. When observing the behavior of garment worn by a character, there are considerable correlations between the body motion and the movement of the garment. These correlations are especially clear for some clothes like tight shirts and trousers. In our method we take advantage of these relationships to reduce the computation load on the mass-spring system and collision detection. We first construct the cloth-to-joint relation by analyzing a presimulated sequence of the cloth to be animated. We then reduce the number of vertices to be physically simulated by identifying the garment regions in which the shape follows that of the underlying skin. The cloth-to-joint relation enables us also to optimize collision detection by restricting the collision check to a small area around each vertex of the coarse mesh. Finally, we use the cloth shape of a pre-simulated cloth sequence to correct the physicsbased simulation of the coarse mesh in order to match the original cloth behavior more closely.
Figure 1. The chosen strategies to save computation time.
<H2>3.2. Workflow</H2>
An overview of the workflow is given in Figure 2 . The pre-processing stage involves generating the coarse mesh, computing the cloth-to-joint relation, and constructing the collision hulls and the interpolation functions for data-driven coarse mesh deformation and wrinkle animation. Global movements Cloth details (Section 5) (Section 6) Pre-simulated Cloth Animation Sequence Pre-processing Cloth-to-joint Pre-processing relation of the Coarse Coarse Mesh of Pre-processing the Fine Mesh Collision Hulls Mesh Coefficients of the Structure wrinkle interpolator Simulation Detection Collision Data-driven Spring System MassWrinkle Shape Skeleton Coarse Mesh Interpolator Runtime Animation Geometry Fine Mesh Geometry The run-time simulator includes deformation of the coarse mesh using the simplified mass-spring system; a post-correction on the position and velocity of the masspoints is processed in order to approach the pre-simulated cloth behavior. Collisions are handled by collision hulls the position of which is computed by our SDD. The final mesh is then obtained using the winkle shape interpolator and the computed geometry of the coarse mesh.
Data Process
Figure 2. Workflow of our approach.
<H1>4. Improving the skeleton driven deformation</H1>
The skeleton-driven deformation (SDD), a classical method for the basic skin deformation is perhaps the most widely used technique in 3D character animation. This method works first by assigning a set of joints with weights to each vertex in the character. The location of a vertex is then calculated by a weighted combination of the transformation of the influencing joints. Although developing a new SDD method is not our main goal, the way the skin deforms is important in our framework since natural looking cloth shape also requires natural skin shape. There are two requirements which the method should fulfill for this particular use: first, it must overcome the undesirable effect of vertex collapse as shown in Figure 3(a) . Second, the method must provide an easy way to compute the local coordinate system for each skin vertex. This is necessary as we want to compute the deformation of the cloth surface in relation to the skin surface. We found that the classical SDD can be greatly improved by replacing the linear combination of the matrices by the matrix operator defined by Alexa
[ 1 ]. The combination of i matrices Mi with their blending weight w i is given by: ∑ w i log( M i ) M SSD = ⊕ w i ⋅ M i = e i i In the remainder of this paper, we denote the function that returns the SDD matrix of the vertex P by SDD P (), SDD P () taking the joint angles of the skeleton as input. The SDD position of P is simply given by M SDD .X P,Dress , X P,Dress being the position of P at initial character pose (see
[15] for further details on SDD). Note that the operator is not continuous. It is not defined for a rotation of 2π radians between the matrices to be blended. In practice, such case is rare; in general, the largest angle range does not exceed π radians.
Figure 3. Twisting of 2⁄3 π radians: (a) the classical SDD, (b) the modified SDD in our approach and (c) its corresponding coordinate systems.
<H1>5. Simulation of the gross behavior</H1>
Due to the computational expenses of solving the full numerical system of the physics-based deformation, we seek simplifications by constructing a coarse mesh representation of the garment. The coarse mesh is used to deduce the gross behavior of the cloth in a data-driven manner, based on the input pre-simulated sequence. A number of optimization strategies are adopted: The two following sections describe a preprocessing that constructs and segments a coarse mesh representation into different region types. We then describe in the next two sections the spring-mass system and collision handling of the coarse mesh at each frame of the simulation. Also described is the runtime process.
<H2>5.1. Construction of the coarse mesh ∈</H2>
We begin by constructing a coarse representation of the given cloth model that will drive the gross behavior of the simulated garment. It consists of two following steps: (1) The cloth surface is partitioned into a set of patches as shown in Figure 4(a) and (b). (2) A coarse mesh representation is obtained by combining a set of vertices in a patch into a single mass point located at the center. The generation of a patch starts by finding a vertex that has not yet been attributed to a patch that is already generated. The patch is then grown by adding neighboring vertices one after the other. To select a new vertex into the current patch, we evaluate each neighboring vertex that has not been already assigned to a patch, using a penalty function. To enforce the regularity of coarse mesh, which is one condition for obtaining efficient deformation with the mass-spring system
[21], we consider two following components. • Minimizing the "shape factor": Square Root (Surface Area)/Contour Length. The objective is to obtain "well-shaped patches", patches that have a circular shape. • Obtaining patches of equivalent surface area. This component gives a cost that increases with the surface area of the patch. By modifying the significance of this component, we can easily control the number of vertices to be simulated with the physically-based deformation (see Figure 4 ). The vertex with the lowest cost is selected. When the lowest cost exceeds a threshold, the construction of the patch is completed. We proceed until no vertices can be found to start a new patch. Deciding a good granularity in the coarse mesh is hand-tuned, so that a neat compromise between the simulation quality and the computation load is found. We have found that best simulations are obtained when patch area covers one or two cloth wrinkles. Note that that each patch is associated with a vertex on the coarse mesh. We denote the vector position of a vertex P as XP, and the vector position of its neighbors as X N R 3n (n: number of neighbors of P).
Figure 4. Two examples of segmentation with (a) patches of large (260 cm2), and (b) small surface area (95 cm2). (c) is the coarse mesh corresponding to the segmentation (b). Different colors are randomly assigned to the patches.
<H2>5.2. Identifying cloth-to-joint relations and region types</H2>
Next we carry out cloth-to-skin (or body) attachment through skin fitting, by which the skinning data on the cloth mesh are approximated in such a way that the skinning-driven cloth shape best fits the simulated cloth shape throughout the whole pre-simulated sequence. The basic idea is to use the pre-simulated results as examples and find the error-minimizing skin data through optimization. An optimization approach, such as the one presented by Mohr et al
[15], could be adopted here. In our case, however, our SDD method is non-linear and therefore the linear regression as adopted by Mohr et al is not beneficial. Function minimization techniques such as Powell’s method
[16] can deal with non-linear functions. Performance is slightly slower, but only pre-processing performance is affected and not runtime performance. The fitting results for a dress model are shown in Figure 5 . Notably, the floating regions (colored in red in Figure 5(d) ) are attached to the root of the character, as shown in Figure 5(b) ; this is contributable to the fact that these regions are large in volume and they rarely collide with limbs during the walk motion. The residual values of the fitting provide useful information on how the garments behave in relation to the body. Intuitively, floating garments such as a skirt, cloth patches may collide with several joints; collisions need to be computed on these regions. On the other hand, the local movements of some cloth patches (like underwear) are negligible and these patches can be considered as being attached rigidly to the skeleton. In our approach, three regions are identified from the residual values of the skin fitting process ( Figure 5(d) ): those that potentially interact with several joints, those that are loosely attached to the skeleton and those that are rigidly attached to the skeleton. The threshold values are chosen in a way that the coarse mesh deformation remains sufficiently close to the pre-simulated sequence. For example, a false assignment of loose region into tight region would produce elongated deformations instead of slipping garment over the skin, and therefore generate an overly deformed coarse mesh, which is beyond the training data of the wrinkle generator. Similarly, a false assignment of region 3 into region 2 would result in the garment crossing the legs. In practice, values of 0.5 cm and 4.0 cm are used to identify tight regions and floating regions, respectively. The deformation of tight regions is directly computed with the SDD (line 2 and 6 on Figure 6 ). The use of SDD for these regions makes it possible to reduce the number of mass points even further. The pseudo code of the simulation loop is: 1 for each vertex P ∈ Coarse_Mesh do 2 M SDD ← SDD P (Skeleton_Joint_Angles) 3 if P ∉ Tight_Region 4 X P ← Mass_Spring_system(time_step) 5 else 6 X P ← M SDD .X D,Dress //Skeleton Driven Deformation High residual values indicate much less dependency on a specific body region of the cloth movement. Therefore, an additional collision check is required to handle the interaction of the clothes with the whole body skeleton. A list of potentially colliding body patches is defined by selecting those that approach within a certain distance of the floating regions during the pre-simulated cloth sequence. Apart from the position, our SDD computes the local transformation matrix of the vertices, the simulator to be optimized at least for the two following points: limiting collision checks to a small area around the vertices, and the geometric wrinkling which is processed in the SDD local coordinate system.
Residual Value (cm)
Identification of the garment regions Garments rigidly attached to the skeleton through SDD Garments for which the collision detection is made with collision hulls only Garments for which the collision detection is made with collision hulls and a predefined list of potentially colliding joints (d)
Figure 5. (a) and (b) influence of the joints on the dress shown in color, (c) quality of the fitting of the SDD data, (d) the three regions computed on an analysis of the residual values.
Figure 6. The coarse mesh is computed with both mass-spring system and SDD.
<H2>5.3. Data-driven post-correction of the coarse mesh</H2>
At each frame of the simulation, we compute the coarse mesh by a mass-spring system with the implicit Euler numerical solver
[ 2 ]. The simulation run on the coarse mesh hardly reproduces the gross movement of the original cloth because the initial mesh has been significantly simplified (from 4000 to a few dozen vertices) and the topology has been modified. Moreover, unlike the simulator used for the pre-simulated cloth sequence, the simplified mass-spring model does not accurately simulate the bending and shearing properties of the fabrics
[21]. We approach the problem by modifying the behavior of the mass-spring system through a fix-up process (similar to
[14]) where the position and velocity of the coarse mesh vertices are modified in order to maintain the cloth shape as close as possible to the original one ( Figure 7 ). X N,Pre-computed X P,Pre-computed Neighbor's positions Vertex positions Frame 1 ... Frame n ... Ideally, the local shape (e.g. position of the vertices in relation to their neighbors) should be a blend of those of the pre-simulated animation. This is achieved by constructing a set of functions of local shape deformation. Post-correction is accomplished with a function that evaluates the "ideal" position of the vertex given the position of its neighbors connected by the edges. For each vertex, we construct an interpolating function F Post by using a set of (X N,Pre-simulated , X P,Pre-simulated ) pairs extracted from each frame of the pre-simulated sequence, where X N ∈R 3 (n: number of neighbors of P) denotes the position of the neighbors and X N ∈R 3n the position of the vertex in question. All these positions are described in the SDD coordinate system of P. The evaluation of the "ideal" position is made with the inverse distance weighting on the pre-simulated frames. Given a position of neighbors X N,Input as input, the interpolation computes the corresponding X P by a weighted summation of the X P,pre-simulated values, each weight being computed from the Euclidian distance between X N,Input and all the X N,Presimulated values. The computation cost of this interpolator grows as the number of pre-simulated frames increases. We wish to keep the computation cost constant regardless of the duration of the pre-simulated sequence. A common solution is to construct a lookup table filled with values pre-simulated by the interpolator on grid sampling. In order to reduce the memory usage of the lookup table, the dimension of XN,Pre-simulated was reduced prior to the construction of the interpolator, by Principal Component Analysis [16]. The first three principal components, which describe 95 % of the average variability of the data, are used. The positions of the vertices are corrected after every simulation loop. The velocity is updated as well. Its new value is set to the sum of the original velocity and the velocity due to the modification of the vertex position (line 11 on Figure 9 ). 7 X’ P ← X P 8 for each vertex P ∈ Coarse_Mesh do 9 if P ∉ Tight_Region 10 X P ← F Post (X’ P , M SDD ) 11 V P ← V P + (X’ P – X P ) / time_step
t 1 t 2 t 3 t 4 t 1 t 3 t 2 t 4 t 1 t 2 t 3 t 4 Simulation before postSimulation after postPre-computed frames correction correction
Figure 7. Post-correction of the mass-spring system.
Dimension reduction by PCA Construction of the inverse distance weighting interpolator X P =F DD (X N ) Values Lookup table of F Post generated according to a regular grid X N,1 , X N,2 , X N,3 ... X P,1 , X P,2 , X P,3 ...
Figure 8. Construction of the lookup table for the datadriven post-correction.
Figure 9. Post-correction of the mass-spring system.
<H2>5.4. Collision hulls</H2>
To prune unnecessary collision tests, we pre-compute what we term “collision hulls” that exploit the skin-tocloth relation obtained from the pre-simulated sequence. These are built once at the beginning of the simulation (prior to the runtime simulation) after the SDD has been computed on the coarse mesh, using the pre-simulated sequence. At each pre-simulated frame, we calculate the difference between the SDD motion model and the presimulated cloth model in the local coordinate system of the SDD. After a sweep, we get a set of points that cover the path a patch takes during the simulation. The smallest convex hull that contains all these points is generated for every patch using the “Quickhull” algorithm presented by Barber et al [ 3 ]. Given enough variation and range of character motion, we expect these hulls to cover the allowable positions of corresponding cloth patches during the runtime simulation. By using collision hulls, collision tests are restricted to a small area around the patch; the overall computation can be significantly reduced in comparison to classical collision detection methods in which collisions are computed between the whole skin and cloth surface. Note that the collision hulls are generated for loose and floating garment regions only. The collision hulls of tight regions are small enough to be approximated by a single point. Figure 10(d) shows the convex hulls computed for the dress model. Collision handling at runtime consists of correcting the position of coarse mesh vertices after every simulation step so that they remain inside their respective hulls. The algorithm is summarized as follows: 12 for every vertex P ∈ Coarse_Mesh do 13 if P ∉ Tight_Region 14 if X p ∉ Collion_Hull(M SDD ) 15 Collision_Response with Collion_Hull Thus, collision detection returns to computing if the particle is in its associated collision hull (line 14 on Figure 11 ); the Gilbert-Johnson-Keerthi algorithm [ 8 ]
is ideally suited to this task. We used constrained dynamics
[22] to handle the collision response (i.e. modification of position and velocity in response to collision detection) at line 15. Collision detection is also computed between floating regions and skeleton joints as follows: 16 for each edge e ∈ Coarse_Mesh do 17 if e ∈ Floating_Region 18 if Collision(e, body_segment) 19 Collision_Response with body_segment
(b) (c)
Figure 10. Computation of the collision hull for each cloth patch; (a): residual values of SDD attachment fitting, (b): computation of local cloth displacement, (c): convex hulls covering all displacements, (d): resulting collision hulls.
Figure 11. Collision handling using collision hulls (loose and floating regions).
Figure 12. Collision handling on floating regions.
<H2>5.5. Runtime computations</H2>
The real-time computation of global cloth movements is obtained with a mass-spring system together with the collision response and post-correction described above. The runtime computation of the coarse mesh is obtained in the following order: • Mass-spring computation ( Figure 6 ) • Post-correction ( Figure 9 ) • Collision response on hulls ( Figure 11 ) • Collision response on floating garments ( Figure 12 ) Note that the collision response on floating garments comes last to ensure collision avoidance between cloth and body.
<H1>6. Generating garment details</H1>
So far we have shown the first part of our simulation, that is, the coarse level simulation. We continue now to describe the second part of the simulation, by which detailed cloth shape such as wrinkles or folds are depicted. Again, the main challenge here is obtaining the highest possible realism while maintaining acceptable computation load, in order to meet the real-time requirements. As recognized in earlier works [9] [13]
, wrinkles can be efficiently animated with a geometric method as they are geometric in nature. Unlike previous methods, however, our wrinkling function is not hand-drawn, nor geometrically approximated, but rather trained from on the analysis of the pre-simulated sequence. In this work, we choose to represent the wrinkle displacement in the local coordinate system used for SDD. This makes our wrinkle parameterization invariant of all joints of higher hierarchy than the currently influencing joint. Several techniques exist for shape interpolation using examples, such as Radial Basis Functions or parametric interpolation. We have used linear interpolation in which coefficients are defined by multi-linear regression on the pre-simulated animation, since it provides satisfactory results at a very low computation cost. For every vertex x in a patch, the interpolator function takes the associated mass point in the coarse mesh, and its neighbors as input. To calculate the position of x from the input, the wrinkle interpolator interpolates the positions of the coarse mesh points, weighted by coefficients determined the regression model of the following form: Error! Objects cannot be created from editing field codes. The values α, α P and α N are the interpolation coefficients. They are defined by multi-linear regression on a set of pairs (positions of coarse mesh vertices, fine mesh vertices) extracted from the pre-simulated cloth sequence. X P and X N are respectively the position of the vertex x and its neighbors; they are all expressed in the SDD coordinate system of x. Despite its simplicity, linear interpolation works fairly well provided a sufficient number of pre-simulated frames for the multi-linear regression. A condition of a good working interpolator is that the input (i.e. position of the coarse mesh vertices) should be within the range of the pre-simulated data. In other word, the wrinkle interpolator can only work for the input range for which it has been trained. This condition is maintained thank to the data-driven post-correction (see Section 5.3). This also keeps the smoothness of the boundaries between patches. Figure 14 illustrates the deformation of the wrinkles.
Coarse Mesh Frame
N 1
Main control point Neighbors Patch of a fine mesh
Figure 13. Shape of the patch with respect to the positions of the control point P and its neighbors N1, N2, and N3.
Figure 14. The wrinkling interpolator in action: wrinkles in (b) and (d) are generated geometrically with (a) and (c) as input.
<H1>7. Results and Discussion</H1>
We measure and validate the proposed real-time cloth simulation method along three criteria: the variety of clothes to be simulated, the computation speed and the range of body motion in the pre-simulated cloth sequence. Pre-simulated sequences obtained by the cloth simulator of Volino et al
[21] were used in our preprocessing.
<H2>7.1. Variety of clothes</H2>
We used our framework to different types of clothes, as shown on the demonstration video. • The “evening” dress ( Figure 14 ) is chosen to demonstrate our wrinkle interpolator on large garment regions. • The “cocktail” dress ( Figure 18 ) is a relatively complex model; the bottom is composed of two layers of tissues and has folds made of large number of vertices, inducing many self collisions. • The “Jeans” outfit is a good example of a model where the SDD based geometric approximation can reduce the number of mass points substantially by simulating only a few regions that contribute significantly to the dynamic behavior. Our simulator behaves fairly well on a wide variety of clothes, including those with highly stiff mechanical properties. Figure 18 show the pre-processing and runtime simulation results for the cocktail dress. Moreover, performance will increase due to the fact that the smallest number of triangles will be processed for the real-time rendering. However, the method may introduce flaws in simulation for some tight clothes, due to the approximate handling of collision detection. For some body movements, the skin surface may slightly intersect the cloth surface. Similarly, the same problem may arise for self-collisions on clothes. The deletion of the skin triangles covered by the garment surface can partially correct this drawback. Note that the cloth simulation is also restricted to clothes worn on bodies. While offering high computation speed, the cloth simulator cannot handle some cloth movements such as those appearing during dressing or undressing. More generally, the clothes are unable to interact with objects other than those that have been taken into consideration during the pre-processing phase. The list of objects that can potentially interact with clothes and the way these objects interact are defined at the preprocessing stage and cannot be changed during the realtime simulation. Finding a method to update the list of possible interacting objects automatically could be a subject for future research.
<H2>7.2. Variety of clothes</H2>
Figure 15 summarizes the performance of our simulator on a 1 GHz Windows PC. Evening Jeans Cocktail Dress Outfit Dress Number of faces 2992 2131 1331 Pre-processing time (min.) 8 8 7 Number of vertices on the rough mesh 110 97 82 Time performance of coarse mesh (fps) 31 63 74 Time performance of the fine mesh (fps) 207 322 588 Overall Performance (fps) 26 51 63
Figure 15. Computation speed.
The pre-processing of all the cloth models took less than 10 minutes. All examples run in real-time at approximately 25 to 50 frames per second (fps), with the coarse mesh deformation process taking about 75 % of the total CPU time. As expected, the duration of the pre- simulated sequence is not a factor of the runtime computation speed. In practice, the performance lowers down at a low rate as the complexity of the collision hulls increases, which tends to be governed by the number of pre-simulated frames (see Section 5.3).
<H2>7.3. Variety of clothes</H2>
As expected, the quality of the simulation depends on the number and variety of examples – the pre-simulated sequence in our case. To show that the simulator faithfully recreates the cloth movement used for training, we compared the real-time simulation with the presimulated one in the first video. The character walks at a normal pace without any fast movements. In the second video, different body movements from those of the training were supplied as input to our realtime simulator and the results are compared with the ones generated with a high quality simulator. To measure the simulation quality, we compared our simulation results with the pre-simulated sequence, using a deformation metric. It measures the still shape and movement by the sum of edge length difference and the mass velocity difference over the cloth mesh. The Figure 16 shows the importance of the variability of the body motion in the pre-simulated sequence. The best quality is achieved when the range of the body motion in the presimulated sequence is approximately 30 % larger than the one used in the real-time simulation. Our simulator works well for interpolation (i.e. joint angles within the range of those of the pre-simulated sequence) but often fails for extrapolation. The main reason for this limitation is collision detection, which does not allow the clothes to have different locations on the body from those calculated in the pre-simulated sequence; this makes the clothes being attached rigidly to the skeleton. The Figure 17 shows the effect of using motion of different durations (expressed in number of frames) with same joint angle ranges. With less than 70 pre-simulated frames, the real-time simulation loses its quality.
100 cm in error 10 Simulation 6.34 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Joint angle range of the pre-simulated sequence Ratio of Joint angle range of the real-time sequence
Figure 16. Estimation of the error when reducing the range of body motion in the pre-simulated sequence.
cm 100 in error Simulation 6.34 10 0 20 40 60 80 100 Number of pre-simulated frames
Figure 17. Estimation of the error when reducing the number of pre-simulated frames.
<H1>8. Conclusion</H1>
The recent advent of cloth simulation techniques has matured enough to produce highly realistic cloth movements on animated characters. However, real-time simulation has been largely unexplored until now. This paper presents the first report of a practical and efficient method for handling real-time simulation almost automatically. We used our framework to produce visually pleasing motion of a wide range of clothes. Both the mass-spring system and collision detection have been rewritten to take advantage of the pre-simulated sequence of the clothes to be animated. Consequently, our cloth simulator is able to construct a model for real-time animation without user intervention and can deal with different types of clothes from tight to floating with low computation consumption. There are many interesting avenues for future work. First, the approach could be extended to simulating other physics-based models such as hair and fluid. We also believe that the work on collision hulls is promising. The current mesh model of collision hulls could be replaced by implicit surfaces or voxel maps. Therefore, for a cloth vertex, it could be possible to compute several collisions hulls in relation to different objects in the scene and to compute their intersection for real-time collision detection. By doing so, it may be possible to process collisions on a higher number of objects while maintaining low computation cost. We also believe that the precision of the collision detection could be improved by replacing the convex shape by a surface to follows more closely the trajectories of the vertices.
<H1>9. Acknowledgements</H1>
This work is supported by the Swiss National Science Foundation (SNSF). Thanks to Marlène Arévalo and Christiane Luible for their design work to the garment models illustrating this paper.
<H1>10. References</H1>
[1] Alexa, M., "Linear Combination of Transformations", SIGGRAPH 2002 Conference Proceedings, Annual Conference Series, ACM Press, Vol. 21(3), pp. 380-387, 2002. - [2] Baraff D., Witkin, A., "Large steps in cloth simulation", ACM Transactions on Graphics, Proceedings of ACM SIGGRAPH, ACM Press, pp. 43-54, 1998. [3] Barber C. B., Dobkin D.P., Huhdanpaa H.T., "The Quickhull Algorithm for Convex Hulls", ACM Transactions on Mathematical Software, ACM Press, Vol. 22(4), pp. 469-483, 1996. [4] Burden R. L., Faires J. D., "Numerical Analysis, Fifth Edition", PWS Publishing, ISBN 0-534-93219-3, 1993. [5] Cordier F., Magnenat-Thalmann N., "Real-time Animation of Dressed Virtual Humans", Eurographics, Blackwell publishers, Vol. 21(3), pp 327-336, 2002. [possible DOI] [6] Choi K.-J., Ko H.-S., "Stable but Responsive Cloth", ACM Transactions on Graphics, Proceedings of ACM SIGGRAPH 2002, ACM Press, Vol. 21(3), pp. 604-611, 2002. [7] Desbrun M., Schröder P., Barr A. H. "Interactive Animation of Structured Deformable Objects", In Graphics [possible DOI] [alternative DOI] Interface'99 proceedings, Morgan Kaufmann, pp. 1-8, 1999. [possible DOI] [8] Gilbert E. G., Johnson D.W., Keerthi S. S., "A fast procedure for computing the distance between complex objects in three-dimensional space", IEEE Journal of Robotics and Automation, IEEE Press, Vol. 4(2), pp. 193-203, 1988. [9] Hadap S., Bangarter E., Volino P., Magnenat-Thalmann N., "Animating Wrinkles on Clothes", IEEE Visualization '99, San Francisco, USA, IEEE Press, pp. 175-182, 1999. [10] James D. L., Fatahalian K., "Precomputing Interactive Dynamic Deformable Scenes", ACM Transactions on Graphics, ACM Press, Vol. 22(3), pp. 165-172, 2003. [11] Kacic-Alesic Z., Nordenstam M., Bullock D., "A practical dynamics system", ACM SIGGRAPH/Eurographics Symposium on Computer Animation, ACM Press, pp. 7-16, 2003. [possible DOI] [alternative DOI] [12] Kang Y.-M., Cho H.-G., "Bilayered Approximate Integration for Rapid and Plausible Animation of Virtual Cloth with Realistic Wrinkles", Computer Animation 2002, Switzerland, IEEE Press, pp. 203-214, 2002. [13] Kang Y.-M., Choi J.-H., Cho H.-G., Lee D.-H., "An efficient animation of wrinkled cloth with approximate implicit integration", The Visual Computer Journal, Spinger-Verlag, Vol. 17(3), pp. 147-157, 2001. [14] Meyer M., Debunne G., Desbrun, M., Barr A. H., "Interactive Animation of Cloth-like Objects in Virtual Reality", Journal of Visualization and Computer Animation, John Wiley & Sons, Vol. 12(1), pp. 112, 2001. [15] Mohr A., and Gleicher M., "Building Efficient, Accurate Character Skins from Examples", ACM Transactions on Graphics, ACM Press, Vol. 22(3), pp. 165-172, 2003. [16] Press W. H., Flannery B. P., Teukolsky S. A., and Vetterling, W. T., “Numerical Recipes in C, The art of scientific computing”, Cambridge University Press, pp. 412-420, 1988. [17] Provot, X., "Deformation Constraints in a Mass-Spring Model to Describe Rigid Cloth Behavior", Graphics Interface'95 proceedings, A K Peters, pp 147-154, 1995. - [18] Rudomin, I., Meln, M., "Multi-Layer Garments Using Hybrid Models", Visual 2000, Springer, pp. 118, 2000. - [19] Terzopoulos D., Fleischer K., "Deformable Models", The Visual Computer, Springler-Verlag, Vol. 4(6), pp.306-331, 1988. [20] Vassilev T., Spanlang B., "Fast Cloth Animation on Walking Avatars", Eurographics, Blackwell Publishers, Vol. 20(3), pp. 260-267, 2001. - [21] Baraff D., Witkin A. P., Kass M., "Untangling cloth", ACM Transaction on Graphics, ACM Press, Vol. 22(3), pp. 862-870, 2003. - [22] Bridson R., Fedkiw R., Anderson J., "Robust treatment of collisions, contact and friction for cloth animation", ACM Transactions on Graphics, vol. 21(3), ACM Press, pp. 594-603, 2003.
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"text": "['the simulation is corrected with a set of functions built from the pre-simulated animation']",
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"text": "['Another approach to fast garment deformations is the hybrid approach']",
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"text": "['22']",
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"text": "['Figure 3(a)']",
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"text": "['the classical SDD can be greatly improved by replacing the linear combination of the matrices by the matrix operator defined by Alexa']",
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"text": "['It is not defined for a rotation of 2π radians between the matrices to be blended']",
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"text": "['the largest angle range does not exceed π radians']",
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"text": "['Figure 4(a)']",
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"text": "['(b)']",
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"text": "['A coarse mesh representation is obtained by combining a set of vertices in a patch into a single mass point located at the center']",
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"text": "['the regularity of coarse mesh, which is one condition for obtaining efficient deformation with the mass-spring system']",
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"text": "['21']",
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"text": "['The objective is to obtain \"well-shaped patches\", patches that have a circular shape']",
"type": "own_claim",
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"text": "['By modifying the significance of this component, we can easily control the number of vertices to be simulated with the physically-based deformation']",
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"text": "['Figure 4']",
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"text": "['the construction of the patch is completed']",
"type": "own_claim",
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"text": "['the lowest cost exceeds a threshold']",
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"text": "['Deciding a good granularity in the coarse mesh is hand-tuned']",
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"text": "['a neat compromise between the simulation quality and the computation load is found']",
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"text": "['best simulations are obtained']",
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"text": "['patch area covers one or two cloth wrinkles']",
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"text": "['each patch is associated with a vertex on the coarse mesh']",
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"text": "['The basic idea is to use the pre-simulated results as examples and find the error-minimizing skin data through optimization']",
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"text": "['An optimization approach']",
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"text": "['could be adopted here']",
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"text": "['the one presented by Mohr et al']",
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"text": "['15']",
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"text": "['our SDD method is non-linear']",
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"text": "['the linear regression']",
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"text": "['is not beneficial']",
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"text": "['adopted by Mohr et al']",
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"text": "['In our case']",
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"text": "['Function minimization techniques']",
"type": "own_claim",
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"text": "['can deal with non-linear functions']",
"type": "own_claim",
"id": "T191"
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"offsets": [
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"text": "['Powell’s method']",
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"text": "['16']",
"type": "data",
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"text": "['Performance is slightly slower']",
"type": "own_claim",
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"text": "['only pre-processing performance is affected and not runtime performance']",
"type": "own_claim",
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"text": "['this is contributable to the fact that these regions are large in volume and they rarely collide with limbs during the walk motion']",
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"text": "['The residual values of the fitting provide useful information on how the garments behave in relation to the body']",
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"text": "['floating garments']",
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"text": "['may collide with several joints']",
"type": "own_claim",
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"text": "['collisions need to be computed on these regions']",
"type": "own_claim",
"id": "T200"
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"offsets": [
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"text": "['a skirt']",
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],
"text": "['cloth patches']",
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"text": "['the local movements of some cloth patches']",
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"text": "['are negligible']",
"type": "own_claim",
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"text": "['these patches can be considered as being attached rigidly to the skeleton']",
"type": "own_claim",
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"text": "['underwear']",
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"offsets": [
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"text": "['three regions are identified from the residual values of the skin fitting process']",
"type": "own_claim",
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"text": "['Figure 5(d)']",
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"text": "['The threshold values are chosen in a way that the coarse mesh deformation remains sufficiently close to the pre-simulated sequence']",
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"id": "T209"
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"offsets": [
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"text": "['a false assignment of loose region into tight region would produce elongated deformations instead of slipping garment over the skin']",
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"text": "['generate an overly deformed coarse mesh']",
"type": "own_claim",
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"text": "['which is beyond the training data of the wrinkle generator']",
"type": "own_claim",
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"text": "['values of 0.5 cm and 4.0 cm are used to identify tight regions and floating regions']",
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],
"text": "['In practice']",
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"text": "['The deformation of tight regions is directly computed with the SDD']",
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"offsets": [
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"text": "['Figure 6']",
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"text": "['The use of SDD for these regions makes it possible to reduce the number of mass points even further']",
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"offsets": [
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"text": "['The pseudo code of the simulation loop is: 1 for each vertex P ∈ Coarse_Mesh do 2 M SDD ← SDD P']",
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"text": "['3']",
"type": "own_claim",
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"text": "['P ∉ Tight_Region']",
"type": "data",
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"offsets": [
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"text": "['4 X P ← Mass_Spring_system(time_step)']",
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"text": "['5 else 6 X P ← M SDD .X D,Dress //Skeleton Driven Deformation']",
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"offsets": [
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"text": "['High residual values indicate much less dependency on a specific body region of the cloth movement']",
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"text": "['an additional collision check is required to handle the interaction of the clothes with the whole body skeleton']",
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"text": "['A list of potentially colliding body patches is defined by selecting those that approach within a certain distance of the floating regions during the pre-simulated cloth sequence']",
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"text": "['we compute the coarse mesh by a mass-spring system with the implicit Euler numerical solver']",
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"text": "['The simulation run on the coarse mesh hardly reproduces the gross movement of the original cloth']",
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"text": "['the initial mesh has been significantly simplified']",
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"text": "['the topology has been modified']",
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"text": "['the simplified mass-spring model does not accurately simulate the bending and shearing properties of the fabrics']",
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"text": "['21']",
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"offsets": [
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"text": "['We approach the problem by modifying the behavior of the mass-spring system through a fix-up process (similar to']",
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"text": "['14']",
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"text": "['where the position and velocity of the coarse mesh vertices are modified in order to maintain the cloth shape as close as possible to the original one']",
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"text": "['Figure 7']",
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"text": "['the local shape']",
"type": "own_claim",
"id": "T239"
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"text": "['should be a blend of those of the pre-simulated animation']",
"type": "own_claim",
"id": "T240"
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"text": "['position of the vertices in relation to their neighbors']",
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"text": "['This is achieved by constructing a set of functions of local shape deformation']",
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"text": "['Post-correction is accomplished with a function that evaluates the \"ideal\" position of the vertex']",
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"text": "['the position of its neighbors connected by the edges']",
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"offsets": [
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"text": "['The computation cost of this interpolator grows']",
"type": "own_claim",
"id": "T245"
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"offsets": [
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26882,
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"text": "['the number of pre-simulated frames increases']",
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"offsets": [
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"text": "['A common solution is to construct a lookup table filled with values pre-simulated']",
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"offsets": [
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"text": "['the dimension of XN,Pre-simulated was reduced prior to the construction of the interpolator, by Principal Component Analysis']",
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"text": "['16']",
"type": "data",
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"text": "['Its new value is set to the sum of the original velocity and the velocity']",
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"text": "['the modification of the vertex position']",
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"text": "['Figure 9']",
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"text": "['The smallest convex hull']",
"type": "own_claim",
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"text": "['is generated for every patch using the “Quickhull” algorithm presented by Barber et al']",
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"text": "['3']",
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"text": "['we expect these hulls to cover the allowable positions of corresponding cloth patches during the runtime simulation']",
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"text": "['enough variation and range of character motion']",
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"text": "['the overall computation can be significantly reduced in comparison to classical collision detection methods']",
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"id": "T258"
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"text": "['the collision hulls are generated for loose and floating garment regions only']",
"type": "own_claim",
"id": "T259"
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{
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"text": "['The collision hulls of tight regions are small enough to be approximated by a single point']",
"type": "own_claim",
"id": "T260"
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{
"offsets": [
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"text": "['the Gilbert-Johnson-Keerthi algorithm']",
"type": "own_claim",
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"text": "['8']",
"type": "data",
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"text": "['is ideally suited to this task']",
"type": "own_claim",
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"text": "['We used constrained dynamics']",
"type": "own_claim",
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"text": "['to handle the collision response']",
"type": "own_claim",
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"text": "['22']",
"type": "data",
"id": "T266"
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"offsets": [
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"text": "['the main challenge here is obtaining the highest possible realism']",
"type": "own_claim",
"id": "T267"
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{
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"text": "['As recognized in earlier works']",
"type": "own_claim",
"id": "T268"
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{
"offsets": [
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"text": "['wrinkles can be efficiently animated with a geometric method']",
"type": "own_claim",
"id": "T269"
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"text": "['9']",
"type": "data",
"id": "T270"
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"offsets": [
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"text": "['13']",
"type": "data",
"id": "T271"
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{
"offsets": [
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"text": "['they are geometric in nature']",
"type": "background_claim",
"id": "T272"
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{
"offsets": [
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32747,
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],
"text": "['our wrinkling function is not hand-drawn, nor geometrically approximated']",
"type": "own_claim",
"id": "T273"
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{
"offsets": [
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],
"text": "['previous methods']",
"type": "own_claim",
"id": "T274"
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{
"offsets": [
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32890
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],
"text": "['trained from on the analysis of the pre-simulated sequence']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
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],
"text": "['This makes our wrinkle parameterization invariant of all joints of higher hierarchy than the currently influencing joint']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
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32997
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],
"text": "['we choose to represent the wrinkle displacement in the local coordinate system used for SDD']",
"type": "data",
"id": "T277"
},
{
"offsets": [
[
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33184
]
],
"text": "['Several techniques exist for shape interpolation using examples']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
33194,
33216
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],
"text": "['Radial Basis Functions']",
"type": "data",
"id": "T279"
},
{
"offsets": [
[
33220,
33244
]
],
"text": "['parametric interpolation']",
"type": "data",
"id": "T280"
},
{
"offsets": [
[
33289,
33371
]
],
"text": "['coefficients are defined by multi-linear regression on the pre-simulated animation']",
"type": "own_claim",
"id": "T281"
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{
"offsets": [
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33379,
33442
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],
"text": "['it provides satisfactory results at a very low computation cost']",
"type": "data",
"id": "T282"
},
{
"offsets": [
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33794,
33844
]
],
"text": "['Objects cannot be created from editing field codes']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
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33846,
33906
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],
"text": "['The values α, α P and α N are the interpolation coefficients']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
34075,
34150
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],
"text": "['X P and X N are respectively the position of the vertex x and its neighbors']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
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34234,
34272
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],
"text": "['linear interpolation works fairly well']",
"type": "own_claim",
"id": "T286"
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http://dx.doi.org/10.1145/1187112.1187281
<Title>A Semi-Lagrangian Contouring Method for Fluid Simulation</Title>
ADAM W. BARGTEIL
TOLGA G. GOKTEKIN
JAMES F. O’BRIEN
JOHN A. STRAIN University of California
Berkeley
<Abstract>In this article, we present a semi-Lagrangian surface tracking method for use with fluid simulations. Our method maintains an explicit polygonal mesh that defines the surface, and an octree data structure that provides both a spatial index for the mesh and a means for efficiently approximating the signed distance to the surface. At each timestep, a new surface is constructed by extracting the zero set of an advected signed-distance function. Semi-Lagrangian backward path tracing is used to advect the signed-distance function. One of the primary advantages of this formulation is that it enables tracking of surface characteristics, such as color or texture coordinates, at negligible additional cost. We include several examples demonstrating that the method can be effectively used as part of a fluid simulation to animate complex and interesting fluid behaviors.</Abstract>
Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation; I.6.8 [Simulation and Modeling]: Types of Simulation—Animation General Terms: Algorithms Additional Key Words and Phrases: Natural phenomena, physically based animation, computational fluid dynamics, surface tracking, level-set methods, semi-Lagrangian contouring
<H1>1. INTRODUCTION</H1>
The fundamental problem of tracking a surface as it is advected by some velocity field arises frequently in applications such as surface reconstruction, image segmentation, and fluid simulation. Unfortunately, the na ̈ ive approach of simply advecting the vertices of a polygonal mesh, or other explicit representation of the surface, quickly encounters problems such as tangling and self-intersection. Instead, a family of methods, known as level-set methods, has been developed for surface tracking. These methods represent the surface implicitly as the zero set of a scalar field defined over the problem domain.
The methods are widely used, and the texts by
Sethian [1999] and Osher and Fedkiw [2003], and
Osher and Sethian’s [1988] seminal article, provide an excellent introduction to the topic. One of the key issues that distinguishes various level-set and similar approaches is the representation of the scalar field, which must capture whatever surface properties are important to a given application. In this article we present a surface tracking method that explicitly represents the surface as a set of polygons. However, rather than attempting to advect these polygons forward with the flow, we update the surface in time with an implicit representation: an advected signed-distance function, ψ, whose
This work was supported in part by California MICRO 04-066 and 05-044, and by generous support from Apple Computer, Alias, Pixar Animation Studios, Intel Corporation, Sony Computer Entertainment America, and the Alfred P. Sloan Foundation. Adam Bargteil was supported by a Siebel Scholarship. Author’s address: Computer Science Department, University of California, Berkeley, 537 Soda Hall, Berkeley, CA 94720; email: {adamb,goktekin,job}@cs.berkeley.edu , [email protected] . Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 1515 Broadway, New York, NY 10036 USA, fax: +1 (212) 869-0481, or [email protected] . c 2006 ACM 0730-0301/06/0100-0019 $5.00
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006, Pages 19–38.
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• A. W. Bargteil et al.
zero set defines the surface. A new polygonal surface is generated by contouring or extracting the zero set of ψ. The value of ψ at a point x, at current time t, is obtained by first tracing backward through the flow field to find the previous location x at time t − t, and then returning the signed distance of x from the previous surface. Using adaptive octree data structures, we can efficiently and reliably construct the new surface and corresponding signed-distance function. The theoretical framework for this method comes from a series of articles by Strain [1999b, 1999c, 1999a, 2000, 2001] that described and analyzed a method for contour tracking in two dimensions. While the semi-Lagrangian procedure for backward advection does not change significantly when going from twoto three-dimensional problems, significant surface tracking issues arise when moving to three dimensions. This article discusses these issues, as well as the general method, and demonstrates how semi-Lagrangian surface contouring can be useful for animating the complex and interesting behavior of fluids. One of the primary advantages of this method is that it enables tracking surface characteristics, such as color or texture coordinates. These properties can be easily stored directly on the polygonal mesh and efficiently mapped onto the new surface during semi-Lagrangian advection. The explicit surface representation also facilitates other common operations, such as rendering, while reconstruction from a scalar function allows operations that rely on an implicit representation. Finally, the method produces detailed, well-defined surfaces that are suitable for realistic animation and that do not jitter or exhibit other undesirable behaviors.
<H1>2. BACKGROUND</H1>
Our method pulls together solutions to a number of well-studied problems to arrive at a method for tracking surfaces. In this section we will first discuss other surface tracking methods and then discuss related work and the mathematical foundation for several of the individual components of our method.
<H2>2.1 Previous Surface Tracking Methods</H2>
Because surface tracking arises in a variety of contexts, the topic has received a significant amount of attention.
Even in the limited context of fluid animation, there has been a great deal of excellent work on simulating fluids with free surfaces, including
Foster and Metaxas [1996], Foster and Fedkiw [2001], Enright et al. [2002b], Carlson et al. [2002, 2004],
Losasso et al. [2004], Goktekin et al. [2004], Hong and Kim [2005], Wang et al. [2005], Guendelman et al. [2005], and
Zhu and Bridson [2005]. The methods available for tracking free surfaces of liquids can be roughly sorted into four categories: level-set methods, particle-based methods, particle level-set methods, and semi-Lagrangian contouring. 2.1.1 Level-Set Methods. Many of the most successful solutions to the surface tracking problem are based on level-set methods, which were originally introduced by
Osher and Sethian [1988]. A complete review of level-set methods is beyond the scope of this article, and we recommend the excellent surveys by
Sethian [1999] and Osher and Fedkiw [2003]. Level-set methods represent a surface as the zero set of a scalar function which is updated over time by solving a partial differential equation, known as the level-set equation. This equation relates change of the scalar function to an underlying velocity field. By using this implicit representation, level-set methods avoid dealing with complex topological changes. However, the scalar function is defined and maintained in the embedding three-dimensional space, rather than just on the two-dimensional surface. In practice, scalar function values need only be accurately maintained very near the surface, resulting in a cost that is roughly linear in the complexity of the surface. One difficulty with level-set methods is that they generally require very high-order conservation-law solvers, though fast semi-Lagrangian methods have been shown to work in some cases
[Strain 1999b; Enright et al. 2005]. The most significant drawback to using level-set methods to track liquid surfaces is their tendency to lose volume in underresolved, high-curvature regions. See Enright et al. [2002a]
for an excellent discussion of the reasons for this volume loss. Bærentzen and Christensen [2002]
built a sculpting system using a level-set surface representation which could be manipulated by a user with a variety of sculpting tools. Like us, they used adaptive grid structures to store the scalar field. However, they used a two-level structure rather than a full octree. They also used semi-Lagrangian methods to update their level-set function. However, when evaluating the distance function after the semi-Lagrangian path tracing, they interpolated distance values stored on a regular grid, while our explicit surface representation allows us to compute exact distances near the surface. Sussman and Puckett [2000]
coupled volume-of-fluid and level-set methods to model droplet dynamics in ink-jet devices. Volume-of-fluid
[Hirt and Nichols 1981]
techniques represent the surface by storing, in each voxel, a volume fraction—the proportion of the voxel filled with liquid. Any cell whose fraction is not one or zero contains surface. Unfortunately, this representation does not admit accurate curvature estimates, which are essential to surface tension computations. However, accurate curvature estimates are easily computed from level-set representations. Thus, the authors combined volume-of-fluid and level-set representations to model surface tension in ink droplets. Some volume-of-fluid methods build an explicit surface representation from the volume fractions stored in each voxel. The key difference between our method and volume-of-fluid methods is that we never compute volume fractions. Instead, our explicit representation is generated by contouring an advected signed-distance function. 2.1.2 Particle-Based Methods. A number of researchers [Terzopoulos et al. 1989; Desbrun and Gascuel 1995; Foster and Metaxas 1996; Desbrun and Cani 1996; Cani and Desbrun 1997; Stora et al. 1999; M uller ̈ et al. 2003, 2004; Premo ze et al. 2003; Zhu and Bridson 2005; Pauly et al. 2005] have used particles to track surfaces. In many of these methods, the simulation elements are particles, which are already being tracked throughout the volume of the deforming liquid or solid. The surface can then be implicitly defined as the boundary between where the particles are and where they aren’t. The particles can be visualized directly, or can be used to define an implicit representation using blobbies or moving least-squares methods. Premo ze et al. [2003]
went a step further and used particle positions and velocities to guide a level-set solution. Mueller et al. [2004] and Pauly et al. [2005]
used special particles, called surfels, to represent the surface. Surfels store a surface normal as well as position and there are generally many more surfels than simulation particles. The principal drawback of these methods is that generating high-quality time-coherent surfaces can be difficult: directly visualizing the particles is insufficient for high-quality animations, methods which convert the particles to some other representation on a per-frame basis often lack temporal coherence, and methods which must run sequentially through the frames or run during the simulation are often quite costly. Additional difficulties arise when trying to ensure a good sampling of the surface. 2.1.3 Particle Level-Set Methods. To address the volume loss of level-set methods, Enright and his colleagues [2002a, 2002b, 2005] built on the work of
Foster and Fedkiw [2001]
to develop particle level-set methods. These methods track the characteristics of the fluid flow with Lagrangian particles, which are then used to fix the level-set solution, essentially increasing the effective resolution of the method. Recently, these methods have been extended to work with octrees
[Enright et al. 2005; Losasso et al. 2004]
, allowing for very high-resolution surface tracking. These methods represent the current state of the art on tracking liquid surfaces for animation, but do have some drawbacks. In particular, the published particle correction rules choose a single particle to provide the signed-distance value. Since there is no guarantee that the same particle will be chosen at subsequent timesteps, the method is extremely susceptible to high-frequency temporally incoherent perturbations of the surface. The artifacts are most noticeable when the surface thins out below the grid resolution and particles happen to be near some of the sample points, but not others. Also, the method has a large number of parameters and rules, such as the number of particles per cell and the reseeding strategy, which need to be decided, often in an application-specific way. Finally, the method tends to produce very smooth surfaces with very little detail, which is desirable in some, but not all, applications. Despite these drawbacks, the particle level-set methods have been very successful and represent a significant step forward in the area of surface tracking for liquid simulations. 2.1.4 Semi-Lagrangian Contouring. Recently, Strain [1999b, 1999c, 1999a, 2000, 2001] has written a series of articles building a theoretical framework culminating in the formulation of surface tracking as a contouring problem. He demonstrated his semi-Lagrangian contouring method on a variety of two-dimensional examples. Our method is based on the method presented by
Strain [2001]
, but with variations and extensions to deal with problems that arise in three-dimensional computer animation. While our method bears a number of similarities to level-set methods and takes advantage of many techniques developed for those methods, we are not directly solving the level-set equation. By formulating surface tracking as a contouring problem, we avoid many of the issues that complicate level-set methods. In particular, we do not have the same volume loss issues which prompted the particle levelset methods: while we do not explicitly conserve volume, our semi-Lagrangian path tracing tends to conserve volume in the same way as the Lagrangian particles in the particle level-set method.
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
Semi-Lagrangian Contouring Method for Fluid Simulation •
21
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
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• A. W. Bargteil et al.
<H2>2.2 Implicit Representations</H2>
The octree structure we use to build and index the polygonal mesh is quite similar to adaptively sampled distance fields [Frisken et al. 2000]. These structures adaptively sample distance fields according to local detail and store samples in a spatial hierarchy. The key difference between adaptively sampled distance fields and our surface representation is that we store a polygon mesh in addition to distance samples. This polygon mesh is used for exact evaluation of the distance function near the surface. Additionally, our splitting criterion is different from that presented by
Frisken et al. [2000]. An alternative structure for storing narrow-band level-set functions is the dynamic tubular grid of
Nielsen and Museth [2006]. This structure can be combined with run-length encoding schemes [Houston et al. 2006]
, providing extremely compact, high-resolution representations of level-set functions. While the asymptotic times for their structure match ours, they are able to exploit cache coherence to provide extremely fast run times for most level-set operations. Integrating the methods presented here with this data structure is a promising area for future work.
<H2>2.3 Contouring</H2>
As our title suggests, we formulate surface tracking as a contouring problem. The contouring problem has been well studied in computer graphics and a number of approaches have been suggested.
The oldest and most widely used is marching cubes, which was first presented by
Wyvill et al. [1986], and later named and popularized by
Lorensen and Cline [1987]. Marching cubes suffers from a tendency to create ill-shaped triangles. This problem is fixed to some degree by dual contouring [Ju et al. 2002]
, which also provides adaptive contouring and an elegant means of preserving sharp boundaries. Dual contouring depends on normal estimates at edge crossings and is very sensitive to inaccuracies in these normal estimates. Unfortunately, in our method we do not have accurate normal information until after the contouring step, when we have the triangle mesh. More recently, Boissonnat and Oudot [2003]
presented a contouring technique which uses Delaunay triangulation methods to generate provably good triangulations. However, this method appears to be prohibitively expensive for something which must run at every timestep. Yet another alternative is marching triangles
[Hilton et al. 1996]
, which takes a surface-based rather than volume-based approach to contouring. Marching triangles requires significantly less computation time and fewer triangles, and produces higher-quality triangles than marching cubes. Unfortunately, marching triangles is not guaranteed to produce closed, manifold meshes in the presence of sharp or thin features.
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
Semi-Lagrangian Contouring Method for Fluid Simulation •
23
<H2>2.4 Semi-Lagrangian Methods</H2>
Semi-Lagrangian methods have been widely used in computer graphics since they were introduced by
Stam [1999]
to solve the nonlinear advection term of the Navier-Stokes equations. These methods provide the foundation for our surface tracking method. Consequently, we briefly discuss the mathematical foundation of semi-Lagrangian methods. Our discussion follows that of
Strain [1999b]. Consider the simplest linear hyperbolic PDE
1
φ t + v(x, t) · ∇φ = 0,
where φ is a scalar field and v(x, t) is a velocity function. Equation (1) passively advects φ through the velocity field v. Semi-Lagrangian methods are based on the observation that Equation (1) propagates φ values along characteristic curves x = s(t) defined by
2
s(t) = v(s(t), t), s(0) = x 0 .
Thus we can find φ values at any time t by finding the characteristic curve passing through (x, t), following it backward to some previous point (x 0 , t 0 ) where the value of φ is known, and setting φ(x, t) = φ(x 0 , t 0 ). This observation forms the basis of the backward characteristic or CIR scheme developed by Courant, Isaacson, and Rees [1952]
, which is the simplest semi-Lagrangian scheme. Given φ at time t n , CIR approximates φ(x, t n+1 ) at any point x at time t n+1 = t n + t by evaluating the previous speed v(x, t n ), approximating the backward characteristic through x by a straight line
3
s(t) ≈ x − (t n+1 − t)v(x, t n ),
and interpolating φ at time t n to the point
4
s(t n ) ≈ x − ( t)v(x, t n ).
Then φ(x, t n+1 ) is set equal to the interpolated value, φ(s(t n ), t n ). For linear PDEs, such as Equation (1), the Lax-Richtmyer equivalence theorem
[LeVeque 1990]
guarantees that CIR will converge to the exact solution as t, x → 0 if it is stable and consistent. The stability properties of the CIR scheme are excellent. Each new value φ(x, t n+1 ) is a single interpolated value of φ at time t n , so unconditional stability is guaranteed in any norm where the interpolation does not increase norms. For example, CIR with linear interpolation is unconditionally stable in the 2-norm. In general, semi-Lagrangian schemes satisfy the CFL condition by shifting the stencil, rather than restricting the timestep. Thus information propagates over long distances in one timestep. Consistency (loosely speaking, the local accuracy of the method), however, is conditional. The global error of CIR is
5
( x) 2 O + O( t),
5
t
due to the O(( x) 2 ) error in linear interpolation accumulated over O(1/( t)) timesteps, plus the O( t) error due to freezing F and approximating the characteristics by straight lines. Thus CIR is consistent to O( t) if a condition t ≥ O( x) is satisfied, contrary to the usual hyperbolic condition t ≤ C x. This condition is extremely convenient, because t = O( x) balances time and space resolution in this first-order accurate scheme.
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
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Fig. 1.
An overview of our method. At each timestep we begin with an explicit surface representation, from which we can build a signed-distance function (a) and a velocity field given by the fluid simulator (b). We then define a field function, the zero set of which will be our new surface. To get the value of the field function at the green point (c), we trace backward through the flow field to find the yellow point (b), which is the image of the green point at the previous timestep. We then evaluate the signed distance of the yellow point to the previous surface (a). We can evaluate this field function at every point in the domain and extract the zero set (c).
For nonlinear PDEs, CIR still converges when the solution is smooth. But nonsmooth shock solutions of conservation laws move at the wrong speed because CIR is not in conservative form. Since level-set solutions have no shocks, CIR is a natural scheme for moving interfaces.
<H1>3. METHOD OVERVIEW</H1>
The surface tracking problem can be phrased as: given a surface representation and a velocity field at time t, build a representation of the surface at time t + t. We begin with a triangle mesh and an octree annotated with signed-distance field samples. We could try to advect the mesh points through the flow field, but would quickly encounter significant topological difficulties. Instead, we avoid topological issues by updating the surface using an implicit representation. The implicit representation is then used to construct a new mesh at the current timestep. More specifically, we define a scalar-valued function which relates the surface at the current timestep to the surface at the previous timestep. Next, we extract the zero set of this function using a contouring algorithm. Finally, a new signed-distance field is computed through a process known as redistancing (see Figure 1 ).
<H1>4. EXPLICIT REPRESENTATION</H1>
One of the key differences between our method and other surface tracking methods is that we build an explicit representation of the surface at every timestep. This explicit representation is a closed, manifold triangle mesh, which is stored as an array of vertices and an array of triangles. The vertices are shared between triangles, allowing for easy computation of smooth vertex normals and other common mesh operations. The distance tree (see Section 6) provides a spatial index for the mesh. The explicit representation provides our method with several advantages. First, it allows us to compute exact signeddistance values near the mesh. Second, it allows us to store properties on mesh vertices, rather than at points near the mesh. Finally, it allows us to take advantage of the many tools and algorithms which have been developed in computer graphics for manipulating and rendering triangle meshes.
<H1>5. IMPLICIT REPRESENTATION</H1>
To avoid the topological difficulties of directly updating an explicit surface representation, we update the surface in time through an implicit representation (see Figure 2 ). We define a scalar-valued field function, ψ(x), which relates the surface at the current timestep to the surface at the previous timestep.
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Fig. 2.
An example of how our implicit representation accommodates merging surfaces. The red grid points trace back through the velocity field to points inside the surface. The green grid points trace back to points outside the surface. When the contouring algorithm runs, it will look for zero crossings only between positive and negative (green and red) grid points and create a surface that does not pass between two grid points of the same color. Thus, without even explicitly determining that a topological change has occurred, the method handles the change.
The surface at the current timestep will be the zero set of this function,
6
S n = {x : ψ(x) = 0}.
For a point x at the current timestep, the function, ψ, first uses backward path tracing, a semiLagrangian integration technique, to find the point x at the previous timestep which flows to x. It then returns the distance from x to the surface, S n−1 , at the previous timestep. If we denote the backward path tracing as b(x) : R 3 → R 3 and let φ n (x) be the signed distance from x to the surface S n ,
7
ψ n (x) = φ n−1 (b(x)) = φ n−1 (x ).
Essentially, we are advecting the signed-distance function through the velocity field given by the fluid simulator. In solving this advection term, our method differs from the simple CIR scheme discussed earlier in two ways. First, instead of the simple linear backward path tracing, we use a second-order Runge-Kutta scheme (also known as the midpoint method with an Euler predictor)
8
t
9
x n−1/2 = x (t n−1/2 ) = x n − v(x n , t n ), 2 x n−1 = x (t n−1 ) = x n − ( t)v(x n−1/2 , t n ),
where v(x, t) is the velocity function. It is important to note that, while this method traces back through the velocity field with second-order accuracy, the velocity field is frozen over the course of the timestep, leading to first-order accuracy in time. The second difference is that, when evaluating φ at points near the surface, we do not interpolate values stored on a grid. Instead, we compute exact distance values. These changes only improve the accuracy (consistency) of our method and do not affect the unconditional stability. To compute the exact distance from a point x , we compute the distances d i to all the nearby triangles. The distance to the surface is min i d i . Schneider and Eberly [2002]
detailed a method for computing the distance from a point to a triangle. This operation is relatively expensive, but many triangles can be pruned, especially when x is very close to the surface, by using standard bounding-box techniques and our octree data structure (see Section 6). Signing the distance values turns out to be somewhat difficult near sharp corners. Let y and n(y) denote the closest point on the surface to x and its normal, respectively. When y lies strictly inside a
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Fig. 3. The figure shows a part of the surface passing through a grid cell. The cell’s vertices have been annotated with signeddistance values. Linear interpolation of these values incorrectly chooses the red point as the zero crossing along the bottom edge. The green point is the actual zero crossing, which will be found with our exact evaluation.
triangle then the sign can be easily computed as
10
s = (x − y) · n(y),
where n(y) is the normal of the triangle containing y. However, if the nearest point in the mesh lies on more than one triangle (i.e., on an edge or vertex of the mesh), the triangles do not always agree on the sign. These situations can be resolved by computing an angle-weighted pseudonormal for each edge and vertex of the mesh and using these pseudonormals to determine the sign when the nearest point is on an edge or vertex of the mesh. Bærentzen and Aanæs [2002] provided a proof that this procedure results in accurate signing (in exact arithmetic). The ability to compute exact distances is one of the chief advantages of having an explicit surface representation. Interpolation can produce substantial errors (see Figure 3 ) which are compounded over time. In fact, this interpolation error is one of the most significant drawbacks to semi-Lagrangian methods in general. When used for velocity advection, interpolation produces such significant smoothing that researchers have proposed a number of methods to add detail back to the flow [Fedkiw et al. 2001] or avoid semi-Lagrangian advection altogether
[Zhu and Bridson 2005]. In this work, we are able to leverage the advantages of semi-Lagrangian advection, without incurring the interpolation error that would otherwise undesireably smooth surface detail.
<H1>6. THE DISTANCE TREE</H1>
Our implementation makes heavy use of a structure we call the distance tree. The distance tree is a balanced octree subdivision of the spatial domain. The octree vertices are annotated with signeddistance values and each cell of the octree contains a list of the triangles with which it intersects. The distance tree serves three purposes: (1) It provides a fast spatial index for the mesh so that nearby triangles can be found quickly. (2) It provides a fast, approximate signed-distance function, which is sufficient when evaluating the signed distance far from the surface. (3) It guides the contouring algorithm, quickly identifying cells which have vertices of different sign and, thus, contain triangles.
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Fig. 4. A two-dimensional distance tree. Distance samples are stored at the octree vertices and triangle lists are stored in cells which intersect the surface. This distance tree could be generated using our implementation of Criterion (13), which considers ψ only at cell centers.
<H2>6.1 Approximating the Signed-Distance Function</H2>
When computing the signed distance from a point x to a surface, S, we first find the smallest octree cell, C, containing x . If C is at the finest level of the octree, then x may be near the surface and all the triangles in the up to 27 cells in the concentric triple 1 of C are considered when computing the minimum distance to the surface. By storing the nearest distance seen so far and using standard bounding-box techniques, many of these triangles can be pruned before computing distances, especially when x is very near the surface. If the computed distance is less than C’s edge length, then the distance is guaranteed to be exact. Otherwise, the computed distance is a very good estimate but may be slightly larger than the actual distance. Contrariwise, if C is not at the finest level of the octree or if there are no triangles in the concentric triple of C, then x is not near the surface and we do not require an exact distance. An approximation with the correct sign is sufficient. In this case, we use trilinear interpolation of the distance values stored at the vertices of C.
<H2>6.2 General Splitting Criterion</H2>
We make use of two different methods for building distance trees in this work. Most often, we wish to build a distance tree to resolve the zero set of our field function ψ. However, it is also useful to build a distance tree from an existing triangle mesh. Our octrees are always built in a top-down manner where each cell is split based on some variation of the following splitting criterion:
11
Split any cell whose edge length exceeds its minimum distance to the surface.
Splitting ends when the tree reaches a predetermined maximum depth. Criterion (11) results in a three-color octree, as described by Samet [1990]
, where each cell of the octree has one of three types: interior, exterior, and boundary (see Figure 4 ). In general, Criterion (11) builds octrees with several useful properties: —Adjacent cells differ in size by no more than a factor of 2, producing a smooth mesh and simplifying procedures such as neighbor finding and triangulation of the vertices. —A cell’s size is proportional to its distance to the surface. —If φ is the signed distance to the surface at vertices and we extend φ into each cell by trilinear interpolation, then, because cells vary in size, φ will be discontinuous. However, the jumps in φ decrease in size in cells near the surface because of the triangle inequality. Thus the interpolated φ is nearly continuous near the surface. —Cells coarsen very rapidly away from the surface: if there are N childless cells touching the surface, then the entire tree contains only O(N log N ) cells. Hence the surface is resolved accurately at minimal cost.
1 If cell C = {x : x − c ∞ ≤ r} has center c and edge length 2r then its concentric triple T is given by T = {x : x − c ∞ ≤ 3r}.
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<H2>6.3 Building a Distance Tree to Resolve</H2>
ψ
When building a new octree at the beginning of each timestep, we are essentially trying to resolve our approximation
12
ψ n+1 (x) = φ n (x − ( t)v(x n−1/2 , t n ))
to the signed-distance function φ n+1 (x). The octree is built recursively from the root cell C 0 by the following splitting criterion:
13
Split every cell where |ψ n+1 | is larger than the edge length.
Thus we apply Criterion (13) as if ψ n+1 were a distance function. Redistancing every timestep keeps
14
ψ n+1 = φ n + ( t)v · ∇φ n + O( t) = φ n + O( t)
within O( t) of the signed-distance function φ n . Thus in the limit, t = O( x) → 0, Criterion (13) reduces to (11), yielding the properties noted above. In practice, we use the value of φ at the cell’s center to determine whether we should split the cell. To deal with the fact that ψ n+1 is not a distance function and that the value at the cell’s center may not be the minimum over the cell, we multiply the edge length by some constant before doing the comparison. We have found that 1/3 works well in practice—always dividing near the surface, without spuriously dividing too many cells. Notice that we can vary this constant to achieve high-resolution bands of varying width around the surface.
<H2>6.4 Building a Distance Tree from a Triangle Mesh</H2>
When building an octree from a triangle mesh (either in initialization, or after some geometric operation has been applied to the triangle mesh) we use the following splitting criterion:
15
Split every cell whose concentric triple intersects the surface.
This test is efficiently implemented using
Green and Hatch’s [1995]
cube/triangle intersection test. Notice that we need not check every cell in the concentric triple of C individually, but can just increase the size of C. In practice we have found it sufficient to increase the cell size by a factor of 2, rather than 3, but such trees may not satisfy all the properties listed above.
<H1>7. CONTOURING</H1>
Once we have resolved ψ on our distance tree, we need to create an explicit representation of our surface at the new timestep. Creating this explicit representation amounts to extracting the zero set of ψ and is an instance of the contouring problem, which has been well studied in computer graphics. For its simplicity, robustness, and speed, we choose to use a marching-cubes method in our implementation. Our implementation is based on Bloomenthal’s [1994]. Our cubes are the leaf cells in the distance tree which have vertices of differing sign. We divide each cube into six tetrahedra to simplify the implementation. Additionally, when finding the zero crossing along any edge (which will eventually be a vertex in the triangle mesh), we use a secant method to speed up convergence and evaluate our full composite field function, including exact evaluation of the previous signed-distance function. Consequently, the vertices of our polygon mesh are guaranteed to lie on the implicit surface (within an tolerance). In fact, each vertex in our polygon mesh can be mapped to some point on some triangle in the mesh at the previous timestep. We take advantage of this fact when advecting surface properties. The marching-cubes algorithm works well for our purposes because each triangle generated by marching cubes sits strictly inside a single cell of the distance tree, making the distance tree an especially effective spatial index. Furthermore, we use the distance tree we have already built to guide the marching cubes, avoiding the need to build a second structure to determine the topology of the new mesh. Near the surface, our distance tree is refined to the maximum level and looks like a uniform grid. Consequently, we need not worry about patching the marching-cubes solution. Our choice of contouring algorithm does result in some limitations. In addition to creating poorly shaped triangles, marching cubes is nonadaptive. That is, the sampling is as dense in flat regions as in regions of high curvature. Unfortunately, the nonadaptive nature of marching cubes limits the resolution we can achieve in high-curvature areas, but is necessary to ensure compatibility. To address this lack of resolution in high-curvature areas, Strain [2001]
split line segments whose centers were far from the surface, yielding arbitrarily high accuracy. Unfortunately, this splitting technique is not easily extended to three dimensions as splitting a triangle either creates an incompatible triangulation or produces even more poorly shaped triangles. It is also very difficult to guarantee that we will still have a manifold when the inserted vertices are moved to the surface. Alternatively, several adaptive contouring methods
[Shu et al. 1995; Shekhar et al. 1996; Poston et al. 1998]
seek to use adaptive grids and regain compatibility through various crack-patching techniques. Such methods could easily be used here and we plan to explore adaptive methods in future work. Although we did not find it necessary, after the contouring step the mesh can be processed in any way that preserves the closed-manifold invariant. This optional processing might include smoothing the surface, improving the shape of the triangles, or any other operation that returns a closed manifold. A new distance tree can then be built from this modified mesh using Criterion (15). A new distance must be built only if the mesh is modified. By taking advantage of the details of our method, we can very efficiently achieve limited smoothing in two ways. First, we can define a second composite function to be the combination of path tracing backward in time followed by the evaluation of a high-order polynomial interpolant of the distances at the vertices of the octree. This function is quite similar to the functions used in semi-Lagrangian level-set methods [Strain 1999b; Enright et al. 2005]. When marching cubes encounters an edge whose vertices have different signs, we find a point which evaluates to zero for each composite function. The final mesh vertex is an average of these two points. By constraining the mesh vertex to be on the edge of the marching-cubes grid, we still guarantee a consistent, closed, manifold triangulation. While this smoothing technique may be quite useful in some applications, we did not use this method for any of the results in this article. Second, repeatedly using the same grid for contouring can produce grid artifacts. For example, a sphere of fluid falling under gravity will develop creases along the coordinate axes. Such artifacts are a form of aliasing and can be reduced by jittering the grid each timestep. Most of the examples in this article used grids which were slightly larger than the simulation domain. These grids were then randomly perturbed so that grids at adjacent timesteps were slightly offset from one another. This jittering limits the reusability of our octrees, but since we build new octrees every timestep, this limitation is not significant.
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<H1>8. REDISTANCING</H1>
After the triangle mesh at the current timestep has been extracted, we must assign true distance values to the vertices of our octree. This problem, referred to as redistancing, has been well studied by the level-set community and a number of methods have been suggested. Strain [1999a]
suggested redistancing by performing an exact evaluation at every vertex of the octree. This method is relatively efficient since the tree coarsens rapidly away from the surface and works well in two dimensions. However, in three dimensions, we have found it to be prohibitively expensive and unnecessary. Instead, we perform exact evaluation at all vertices of the cells that contain triangles, but then run a fast marching method [Sethian 1996; Losasso et al. 2004]
over the remaining vertices. In our method, there may be some parts of the domain where the octree was refined but did not result in any triangles, such as when the surface becomes thinner than the resolution of the tree. Consequently, our octree, unlike those used by Losasso et al. [2004]
, does not necessarily coarsen away from the surface. To address this problem, we coarsen parts of the tree which have been refined but did not generate surface. We do this coarsening in two steps. First, we propagate the triangle lists up the tree so that the triangle list of a cell is the union of the triangle lists of a cell’s descendants. Second, we remove all the children of any cell whose concentric triple does not contain any triangles. Our redistancing method comprises three steps: —coarsen the octree; —compute exact distances at vertices of cells which contain triangles; —run a fast marching method over the remaining vertices.
<H1>9. TRACKING SURFACE PROPERTIES</H1>
One of the primary advantages of our method is the ability to track surface properties, such as color, texture coordinates, or even simulation variables, accurately at negligible additional cost. As pointed out earlier, every vertex in a polygon mesh corresponds to some point on some triangle in the previous mesh. Thus, semi-Lagrangian advection provides a mapping between surfaces at adjacent timesteps. If vertex v in the current mesh maps to point p in the old mesh and some surface property was stored at p, this property can be copied to v. In this way we can track surface properties on the actual surface as we build the surface, so we do not incur any significant additional cost. Previous methods, such as the one proposed by
Rassmussen et al. [2004]
, have been limited to tracking properties in the volume near the surface and interpolating them to the surface. Such methods incur significant cost, introduce substantial smoothing, and blur properties between nearby surfaces. In many applications there is no value actually stored at p. Instead, the properties are stored at the vertices of the triangle containing p. In these cases the problem is slightly more involved. In many cases it is sufficient to use barycentric interpolation to compute a value at p and copy this interpolated value to v. However, for some applications this interpolation can produce unwanted smoothing. A simple alternative is to set the value at pto the value stored at the vertex nearest p. Unfortunately, this approach may introduce unwanted aliasing. Essentially, we are having trouble because we are resampling the surface at every timestep. However, if we know something about the property we are tracking, we may be able to “clean up” the blurred signal. For example, in our examples with checkerboard textures, we tracked reference coordinates which were passed to a simple function to determine color. Since we know that the tracked value should always be a point on the initial surface we could find the point on the initial mesh which was closest to the value the tracking method supplied. In this way, we ensured that, at every timestep, every vertex in the mesh mapped back to some point on the initial surface. Once we had this mapping we could copy any property stored on the initial surface, whether it be the reference coordinates, texture coordinates, or color values. Thus, if image textures were preferred over procedural textures, texture coordinates could be copied instead of reference coordinates. There are still plenty of open problems in the area of texturing liquid surfaces. In particular, it is difficult to deal with large discontinuities in surface properties, which occur when two surfaces merge, or a surface splits. Creating detail where a surface stretches is also an open problem.
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Fig. 5. These images shows an invisible tank being filled as multicolored balls of fluid fall into it. The resulting surface contains complex geometric details which retain the different colors of the balls. The left image was rendered with a matte shader, while the right image was rendered with a colored glass shader.
Fig. 6. Two balls of viscoelastic fluid are thrown at each other and merge.
<H1>10. RESULTS AND DISCUSSION</H1>
We have tested this surface tracking method coupled with a fluid simulation on several examples such as the ones shown in Figures 5 and 6. We also tested it in the spiraling analytical test field from
Enright et al. [2002a]. Figure 7 shows two objects being advected in this divergence-free velocity field to a midpoint after which the field reverses. The sphere was restored to a nearly identical shape (see Figure 8 ), while the bunny exhibited a small amount of smoothing. The surface of the bunny was textured by a spot-generating reaction-diffusion system that ran on the surface as the object was being advected
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Fig. 7. This figure shows the behavior generated when two different surfaces are passed through an analytical flow field that stretches and distorts them. The first three images show the object flowing along the field, the last three show the behavior when the distorted object then flows back along the reverse field. The bunny is textured using a reaction-diffusion system that is running on the surface during the sequence.
Fig. 8. This figure shows the error in the final frame of the sphere example in Figure 7 . The color maps to the error as a percentage of the sphere’s radius, with blue points slightly inside and yellow points slightly outside.
Fig. 9.
In this figure we show the result generated when we continue to distort and stretch an object past the point where it thins out and tears.
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Fig. 10.
This sequence shows a thick viscoelastic fluid sliding off of a shelf. A checkerboard texture is mapped onto the surface. It is interesting to notice that the corners of the checkerboard texture stay sharp, despite the significant deformation.
[Turk 1991; Witkin and Kass 1991]. At each timestep, the morphogens were advected along with the surface and then allowed to react. In Figure 9 we show the result of running the sphere through the flow field for several revolutions to highlight the behavior generated when the surface thins below the resolution of the octree’s finest level. All of our fluid examples used a standard regular-grid Eulerian fluid simulator with the elasticity model of
Goktekin et al. [2004]. The fluid simulator and the surface tracking module were only very loosely coupled: the fluid simulator provided the surface tracker with a velocity function and, in turn, the surface tracker provided the simulator with the signed-distance function. Because our fluid simulator has a regular grid its resolution is notably coarser than the surface tracker, which uses an octree.
The idea of using different resolutions for the fluid and surface is not new;
Foster and Fedkiw [2001] used different timesteps for their fluid and surface calculations and
Goktekin et al. [2004]
found that increasing the spatial resolution of the surface tracking grid dramatically reduced volume loss. As noted by Losasso et al. [2004]
, using different spatial resolutions can produce artifacts. For example, pieces of surface could appear connected when the simulator thinks they are disconnected and vice versa. Additionally, surface features may be maintained when a more detailed fluid simulator would smooth them away. In general, we found the increased surface resolution to be worth these artifacts. Ideally we would use a multiresolution fluid simulation, like the octree method of Losasso et al. [2004]. We plan to incorporate a multiresolution fluid simulator as part of our future work. For most of our examples the surface tracking module took roughly 1 min/timestep at an effective resolution of 512 3 . The fluid simulation also required about 1 min/timestep. Both the fluid simulator and the surface tracking module took 11 timesteps per frame. Thus it took about 2 days to simulate 10 s of animation, with roughly half the time spent solving for the velocity field and half the time updating the surface. It is important to note that, given a perfect semi-Lagrangian path tracer, the method could take arbitrarily large timesteps. Decoupling the timesteps of the fluid simulator and surface tracker, so that the surface tracker runs only once per frame, is an interesting area of future work. In Figure 10 we show the behavior when a thick viscoelastic fluid is allowed to flow off a shelf into a basin. This surface is textured by advecting reference coordinates along with the flow and applying a procedural checkerboard texture. Figure 11 shows beginning and ending frames using both an offthe-shelf procedural shader, which includes a displacement map, and a reaction-diffusion system. The motion of the spots on the surface occurs both from the motion of the surface and from the reactiondiffusion system seeking equilibrium on the moving surface. Figures 12 and 13 show two streams of liquid that are being sprayed toward each other. As the streams oscillate from side-to-side, they collide and produce a thin, web-like surface between them. The motion of the two streams causes this thin surface to form a spiral shape as the streams separate. Similar effects can be seen in real-world footage. All of our images were rendered with the open-source renderer Pixie
[Arikan 2005]. Since we generated a polygonal mesh for each frame, we could take advantage of standard rendering techniques,
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Fig. 11.
This figure shows the beginning and ending frames of an animation similar to that shown in Figure 10 . The left images were rendered with an off-the-shelf procedural texture which includes a displacement map, while the images on the right were generated with a reaction-diffusion texture.
Fig. 12. These images show closeups of the fluid web created between two intersecting sprays. The left image is rendered realistically, the right is rendered with a matte shader where the color has been advected with the flow.
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Fig. 13. This sequence shows a realistic rendering of two liquid sprays. As the sprays move from side to side, they periodically intersect and create a web-like spiral pattern.
allowing for very fast rendering times; most of our renderings took less than 3 min/frame. Many of our examples were rendered with a matte shader so that the surface detail can be seen. A number of our examples were also rendered with a glass shader (using water’s index of refraction) for comparison to previous methods and real fluids, and to demonstrate how the method can be used to generate realistic results. Our colored and textured examples illustrate how easily a variety of properties may be attached to the surface. In practice, we believe that advected properties could be used effectively with standard shading techniques to generate a wide range of interesting effects.
<H1>11. CONCLUSIONS</H1>
Semi-Lagrangian contouring offers an elegant and effective means for surface tracking and has a number of advantages over competing methods. First, we have an explicit representation. In addition to enabling exact evaluation, this explicit representation also allows us to leverage 30 years of computer graphics technology which has been optimized for polygonal meshes. Rendering, texture mapping, and a variety of other applications are all very straightforward. Second, we have an implicit representation. This implicit representation allows us to update the surface without explicitly addressing any of the difficult topological issues which plague other approaches. Third, semi-Lagrangian advection gives us a mapping between surfaces at adjacent timesteps. This mapping allows us to accurately track surface properties on the actual surface at negligible complexity and cost. Fourth, our method does not have any ad hoc rules or parameters to tune. In fact, the only parameters to our system are the upper and lower corners of the domain, the maximum depth of the octree (a resolution parameter), and some resolution tolerances. Finally, and most importantly, we are able to produce detailed, flicker-free animations of complex fluid motions.
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<H1>ACKNOWLEDGMENTS</H1>
We thank the other members of the Berkeley Graphics Group for their helpful criticism and comments. In particular, we thank Ravi Kolluri, Okan Arikan, and Bryan Feldman for many fruitful discussions. We also thank the anonymous reviewers for their many helpful comments and suggestions.
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Received September 2005; accepted December 2005
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
Semi-Lagrangian Contouring Method for Fluid Simulation •
37
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
38
• A. W. Bargteil et al.
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006.
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"text": "['By using this implicit representation, level-set methods avoid dealing with complex topological changes']",
"type": "background_claim",
"id": "T86"
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"offsets": [
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"text": "['Level-set methods represent a surface as the zero set of a scalar function which is updated over time by solving a partial differential equation']",
"type": "background_claim",
"id": "T87"
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{
"offsets": [
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8288,
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],
"text": "['level-set equation']",
"type": "data",
"id": "T88"
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"offsets": [
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8308,
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"text": "['This equation relates change of the scalar function to an underlying velocity field']",
"type": "background_claim",
"id": "T89"
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"text": "['the scalar function is defined and maintained in the embedding three-dimensional space']",
"type": "background_claim",
"id": "T90"
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8644,
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"text": "['In practice, scalar function values need only be accurately maintained very near the surface']",
"type": "background_claim",
"id": "T91"
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"offsets": [
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8738,
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],
"text": "['resulting in a cost that is roughly linear in the complexity of the surface']",
"type": "background_claim",
"id": "T92"
},
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"offsets": [
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8815,
8924
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],
"text": "['One difficulty with level-set methods is that they generally require very high-order conservation-law solvers']",
"type": "background_claim",
"id": "T93"
},
{
"offsets": [
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8933,
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],
"text": "['fast semi-Lagrangian methods have been shown to work in some cases']",
"type": "background_claim",
"id": "T94"
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{
"offsets": [
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9002,
9014
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],
"text": "['Strain 1999b']",
"type": "data",
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{
"offsets": [
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9016,
9035
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],
"text": "['Enright et al. 2005']",
"type": "data",
"id": "T96"
},
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"offsets": [
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9038,
9193
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],
"text": "['The most significant drawback to using level-set methods to track liquid surfaces is their tendency to lose volume in underresolved, high-curvature regions']",
"type": "background_claim",
"id": "T97"
},
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"offsets": [
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9227,
9286
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],
"text": "['an excellent discussion of the reasons for this volume loss']",
"type": "background_claim",
"id": "T98"
},
{
"offsets": [
[
9199,
9213
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],
"text": "['Enright et al.']",
"type": "data",
"id": "T99"
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"offsets": [
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9215,
9220
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],
"text": "['2002a']",
"type": "data",
"id": "T100"
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"offsets": [
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9460,
9529
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"text": "['Like us, they used adaptive grid structures to store the scalar field']",
"type": "own_claim",
"id": "T101"
},
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"offsets": [
[
9540,
9597
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],
"text": "['they used a two-level structure rather than a full octree']",
"type": "background_claim",
"id": "T102"
},
{
"offsets": [
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9599,
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],
"text": "['They also used semi-Lagrangian methods to update their level-set function']",
"type": "background_claim",
"id": "T103"
},
{
"offsets": [
[
9322,
9458
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],
"text": "['built a sculpting system using a level-set surface representation which could be manipulated by a user with a variety of sculpting tools']",
"type": "background_claim",
"id": "T104"
},
{
"offsets": [
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9288,
9313
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],
"text": "['Bærentzen and Christensen']",
"type": "data",
"id": "T105"
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"offsets": [
[
9315,
9319
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],
"text": "['2002']",
"type": "data",
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"offsets": [
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9761,
9819
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"text": "['they interpolated distance values stored on a regular grid']",
"type": "background_claim",
"id": "T107"
},
{
"offsets": [
[
9827,
9916
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],
"text": "['our explicit surface representation allows us to compute exact distances near the surface']",
"type": "own_claim",
"id": "T108"
},
{
"offsets": [
[
9688,
9759
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],
"text": "['evaluating the distance function after the semi-Lagrangian path tracing']",
"type": "data",
"id": "T109"
},
{
"offsets": [
[
9946,
10036
]
],
"text": "['coupled volume-of-fluid and level-set methods to model droplet dynamics in ink-jet devices']",
"type": "background_claim",
"id": "T110"
},
{
"offsets": [
[
9918,
9937
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],
"text": "['Sussman and Puckett']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
9939,
9943
]
],
"text": "['2000']",
"type": "data",
"id": "T112"
},
{
"offsets": [
[
10080,
10157
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],
"text": "['techniques represent the surface by storing, in each voxel, a volume fraction']",
"type": "background_claim",
"id": "T113"
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{
"offsets": [
[
10038,
10053
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],
"text": "['Volume-of-fluid']",
"type": "background_claim",
"id": "T114"
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{
"offsets": [
[
10056,
10077
]
],
"text": "['Hirt and Nichols 1981']",
"type": "data",
"id": "T115"
},
{
"offsets": [
[
10158,
10204
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],
"text": "['the proportion of the voxel filled with liquid']",
"type": "data",
"id": "T116"
},
{
"offsets": [
[
10206,
10265
]
],
"text": "['Any cell whose fraction is not one or zero contains surface']",
"type": "background_claim",
"id": "T117"
},
{
"offsets": [
[
10267,
10398
]
],
"text": "['Unfortunately, this representation does not admit accurate curvature estimates, which are essential to surface tension computations']",
"type": "background_claim",
"id": "T118"
},
{
"offsets": [
[
10409,
10488
]
],
"text": "['accurate curvature estimates are easily computed from level-set representations']",
"type": "background_claim",
"id": "T119"
},
{
"offsets": [
[
10496,
10603
]
],
"text": "['the authors combined volume-of-fluid and level-set representations to model surface tension in ink droplets']",
"type": "background_claim",
"id": "T120"
},
{
"offsets": [
[
10605,
10721
]
],
"text": "['Some volume-of-fluid methods build an explicit surface representation from the volume fractions stored in each voxel']",
"type": "background_claim",
"id": "T121"
},
{
"offsets": [
[
10723,
10830
]
],
"text": "['The key difference between our method and volume-of-fluid methods is that we never compute volume fractions']",
"type": "own_claim",
"id": "T122"
},
{
"offsets": [
[
10841,
10932
]
],
"text": "['our explicit representation is generated by contouring an advected signed-distance function']",
"type": "own_claim",
"id": "T123"
},
{
"offsets": [
[
10964,
10987
]
],
"text": "['A number of researchers']",
"type": "background_claim",
"id": "T124"
},
{
"offsets": [
[
11223,
11260
]
],
"text": "['have used particles to track surfaces']",
"type": "background_claim",
"id": "T125"
},
{
"offsets": [
[
10989,
11012
]
],
"text": "['Terzopoulos et al. 1989']",
"type": "data",
"id": "T127"
},
{
"offsets": [
[
11014,
11038
]
],
"text": "['Desbrun and Gascuel 1995']",
"type": "data",
"id": "T128"
},
{
"offsets": [
[
11040,
11063
]
],
"text": "['Foster and Metaxas 1996']",
"type": "data",
"id": "T129"
},
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"offsets": [
[
11065,
11086
]
],
"text": "['Desbrun and Cani 1996']",
"type": "data",
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},
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"offsets": [
[
11088,
11109
]
],
"text": "['Cani and Desbrun 1997']",
"type": "data",
"id": "T131"
},
{
"offsets": [
[
11111,
11128
]
],
"text": "['Stora et al. 1999']",
"type": "data",
"id": "T132"
},
{
"offsets": [
[
11130,
11152
]
],
"text": "['M uller ̈ et al. 2003']",
"type": "data",
"id": "T133"
},
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"offsets": [
[
11154,
11158
]
],
"text": "['2004']",
"type": "data",
"id": "T134"
},
{
"offsets": [
[
11160,
11180
]
],
"text": "['Premo ze et al. 2003']",
"type": "data",
"id": "T126"
},
{
"offsets": [
[
11182,
11202
]
],
"text": "['Zhu and Bridson 2005']",
"type": "data",
"id": "T135"
},
{
"offsets": [
[
11204,
11221
]
],
"text": "['Pauly et al. 2005']",
"type": "data",
"id": "T136"
},
{
"offsets": [
[
11262,
11413
]
],
"text": "['In many of these methods, the simulation elements are particles, which are already being tracked throughout the volume of the deforming liquid or solid']",
"type": "background_claim",
"id": "T137"
},
{
"offsets": [
[
11719,
11787
]
],
"text": "['used particle positions and velocities to guide a level-set solution']",
"type": "background_claim",
"id": "T138"
},
{
"offsets": [
[
11671,
11686
]
],
"text": "['Premo ze et al.']",
"type": "data",
"id": "T139"
},
{
"offsets": [
[
11688,
11692
]
],
"text": "['2003']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
11789,
11803
]
],
"text": "['Mueller et al.']",
"type": "data",
"id": "T141"
},
{
"offsets": [
[
11805,
11809
]
],
"text": "['2004']",
"type": "data",
"id": "T142"
},
{
"offsets": [
[
11815,
11827
]
],
"text": "['Pauly et al.']",
"type": "data",
"id": "T143"
},
{
"offsets": [
[
11836,
11858
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],
"text": "['used special particles']",
"type": "background_claim",
"id": "T144"
},
{
"offsets": [
[
11876,
11900
]
],
"text": "['to represent the surface']",
"type": "background_claim",
"id": "T145"
},
{
"offsets": [
[
11829,
11833
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],
"text": "['2005']",
"type": "data",
"id": "T146"
},
{
"offsets": [
[
11867,
11874
]
],
"text": "['surfels']",
"type": "data",
"id": "T147"
},
{
"offsets": [
[
11902,
11952
]
],
"text": "['Surfels store a surface normal as well as position']",
"type": "background_claim",
"id": "T148"
},
{
"offsets": [
[
11957,
12020
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],
"text": "['there are generally many more surfels than simulation particles']",
"type": "background_claim",
"id": "T149"
},
{
"offsets": [
[
12022,
12133
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],
"text": "['The principal drawback of these methods is that generating high-quality time-coherent surfaces can be difficult']",
"type": "background_claim",
"id": "T150"
},
{
"offsets": [
[
12135,
12213
]
],
"text": "['directly visualizing the particles is insufficient for high-quality animations']",
"type": "data",
"id": "T151"
},
{
"offsets": [
[
12215,
12330
]
],
"text": "['methods which convert the particles to some other representation on a per-frame basis often lack temporal coherence']",
"type": "data",
"id": "T152"
},
{
"offsets": [
[
12336,
12442
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],
"text": "['methods which must run sequentially through the frames or run during the simulation are often quite costly']",
"type": "data",
"id": "T153"
},
{
"offsets": [
[
12444,
12526
]
],
"text": "['Additional difficulties arise when trying to ensure a good sampling of the surface']",
"type": "background_claim",
"id": "T154"
},
{
"offsets": [
[
12562,
12609
]
],
"text": "['To address the volume loss of level-set methods']",
"type": "background_claim",
"id": "T155"
},
{
"offsets": [
[
12659,
12679
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],
"text": "['built on the work of']",
"type": "background_claim",
"id": "T156"
},
{
"offsets": [
[
12707,
12744
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],
"text": "['to develop particle level-set methods']",
"type": "background_claim",
"id": "T157"
},
{
"offsets": [
[
12611,
12637
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],
"text": "['Enright and his colleagues']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
12653,
12657
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],
"text": "['2005']",
"type": "data",
"id": "T159"
},
{
"offsets": [
[
12646,
12651
]
],
"text": "['2002b']",
"type": "data",
"id": "T160"
},
{
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12639,
12644
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],
"text": "['2002a']",
"type": "data",
"id": "T161"
},
{
"offsets": [
[
12681,
12698
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],
"text": "['Foster and Fedkiw']",
"type": "data",
"id": "T162"
},
{
"offsets": [
[
12700,
12704
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],
"text": "['2001']",
"type": "data",
"id": "T163"
},
{
"offsets": [
[
12746,
12880
]
],
"text": "['These methods track the characteristics of the fluid flow with Lagrangian particles, which are then used to fix the level-set solution']",
"type": "background_claim",
"id": "T164"
},
{
"offsets": [
[
12882,
12943
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],
"text": "['essentially increasing the effective resolution of the method']",
"type": "background_claim",
"id": "T165"
},
{
"offsets": [
[
12955,
13008
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],
"text": "['these methods have been extended to work with octrees']",
"type": "background_claim",
"id": "T166"
},
{
"offsets": [
[
13055,
13105
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],
"text": "['allowing for very high-resolution surface tracking']",
"type": "background_claim",
"id": "T167"
},
{
"offsets": [
[
13011,
13030
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],
"text": "['Enright et al. 2005']",
"type": "data",
"id": "T168"
},
{
"offsets": [
[
13032,
13051
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],
"text": "['Losasso et al. 2004']",
"type": "data",
"id": "T169"
},
{
"offsets": [
[
13107,
13201
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],
"text": "['These methods represent the current state of the art on tracking liquid surfaces for animation']",
"type": "background_claim",
"id": "T170"
},
{
"offsets": [
[
13207,
13229
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],
"text": "['do have some drawbacks']",
"type": "background_claim",
"id": "T171"
},
{
"offsets": [
[
13440,
13544
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],
"text": "['the method is extremely susceptible to high-frequency temporally incoherent perturbations of the surface']",
"type": "background_claim",
"id": "T172"
},
{
"offsets": [
[
13355,
13438
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],
"text": "['there is no guarantee that the same particle will be chosen at subsequent timesteps']",
"type": "data",
"id": "T173"
},
{
"offsets": [
[
13546,
13706
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],
"text": "['The artifacts are most noticeable when the surface thins out below the grid resolution and particles happen to be near some of the sample points, but not others']",
"type": "background_claim",
"id": "T174"
},
{
"offsets": [
[
13714,
13767
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],
"text": "['the method has a large number of parameters and rules']",
"type": "background_claim",
"id": "T175"
},
{
"offsets": [
[
13838,
13900
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],
"text": "['which need to be decided, often in an application-specific way']",
"type": "background_claim",
"id": "T176"
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{
"offsets": [
[
13777,
13809
]
],
"text": "['the number of particles per cell']",
"type": "data",
"id": "T177"
},
{
"offsets": [
[
13814,
13836
]
],
"text": "['the reseeding strategy']",
"type": "data",
"id": "T178"
},
{
"offsets": [
[
14026,
14038
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],
"text": "['applications']",
"type": "background_claim",
"id": "T179"
},
{
"offsets": [
[
13911,
14011
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],
"text": "['the method tends to produce very smooth surfaces with very little detail, which is desirable in some']",
"type": "background_claim",
"id": "T180"
},
{
"offsets": [
[
14065,
14217
]
],
"text": "['the particle level-set methods have been very successful and represent a significant step forward in the area of surface tracking for liquid simulations']",
"type": "background_claim",
"id": "T181"
},
{
"offsets": [
[
14304,
14444
]
],
"text": "['has written a series of articles building a theoretical framework culminating in the formulation of surface tracking as a contouring problem']",
"type": "background_claim",
"id": "T182"
},
{
"offsets": [
[
14263,
14269
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],
"text": "['Strain']",
"type": "data",
"id": "T183"
},
{
"offsets": [
[
14446,
14540
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],
"text": "['He demonstrated his semi-Lagrangian contouring method on a variety of two-dimensional examples']",
"type": "background_claim",
"id": "T184"
},
{
"offsets": [
[
14271,
14276
]
],
"text": "['1999b']",
"type": "data",
"id": "T185"
},
{
"offsets": [
[
14278,
14283
]
],
"text": "['1999c']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
14285,
14290
]
],
"text": "['1999a']",
"type": "data",
"id": "T187"
},
{
"offsets": [
[
14292,
14296
]
],
"text": "['2000']",
"type": "data",
"id": "T188"
},
{
"offsets": [
[
14298,
14302
]
],
"text": "['2001']",
"type": "data",
"id": "T189"
},
{
"offsets": [
[
14542,
14588
]
],
"text": "['Our method is based on the method presented by']",
"type": "own_claim",
"id": "T190"
},
{
"offsets": [
[
14610,
14713
]
],
"text": "['with variations and extensions to deal with problems that arise in three-dimensional computer animation']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
14590,
14596
]
],
"text": "['Strain']",
"type": "data",
"id": "T192"
},
{
"offsets": [
[
14598,
14602
]
],
"text": "['2001']",
"type": "data",
"id": "T193"
},
{
"offsets": [
[
14721,
14783
]
],
"text": "['our method bears a number of similarities to level-set methods']",
"type": "own_claim",
"id": "T194"
},
{
"offsets": [
[
14788,
14850
]
],
"text": "['takes advantage of many techniques developed for those methods']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
[
14852,
14902
]
],
"text": "['we are not directly solving the level-set equation']",
"type": "own_claim",
"id": "T196"
},
{
"offsets": [
[
14904,
15022
]
],
"text": "['By formulating surface tracking as a contouring problem, we avoid many of the issues that complicate level-set methods']",
"type": "own_claim",
"id": "T197"
},
{
"offsets": [
[
15039,
15126
]
],
"text": "['we do not have the same volume loss issues which prompted the particle levelset methods']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
15134,
15170
]
],
"text": "['we do not explicitly conserve volume']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
[
15172,
15306
]
],
"text": "['our semi-Lagrangian path tracing tends to conserve volume in the same way as the Lagrangian particles in the particle level-set method']",
"type": "own_claim",
"id": "T200"
},
{
"offsets": [
[
15651,
15771
]
],
"text": "['The octree structure we use to build and index the polygonal mesh is quite similar to adaptively sampled distance fields']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
15773,
15792
]
],
"text": "['Frisken et al. 2000']",
"type": "data",
"id": "T202"
},
{
"offsets": [
[
15914,
16070
]
],
"text": "['The key difference between adaptively sampled distance fields and our surface representation is that we store a polygon mesh in addition to distance samples']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
15795,
15912
]
],
"text": "['These structures adaptively sample distance fields according to local detail and store samples in a spatial hierarchy']",
"type": "background_claim",
"id": "T204"
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16162,
16235
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"text": "['Additionally, our splitting criterion is different from that presented by']",
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16237,
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"text": "['Frisken et al.']",
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16253,
16257
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"text": "['2000']",
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16260,
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"text": "['An alternative structure for storing narrow-band level-set functions is the dynamic tubular grid of']",
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16361,
16379
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"text": "['Nielsen and Museth']",
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16381,
16385
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"text": "['2006']",
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16388,
16451
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"text": "['This structure can be combined with run-length encoding schemes']",
"type": "background_claim",
"id": "T211"
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{
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16453,
16472
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"text": "['Houston et al. 2006']",
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"offsets": [
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16476,
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"text": "['providing extremely compact, high-resolution representations of level-set functions']",
"type": "background_claim",
"id": "T213"
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{
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16620,
16726
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"text": "['they are able to exploit cache coherence to provide extremely fast run times for most level-set operations']",
"type": "background_claim",
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"text": "['the asymptotic times for their structure match ours']",
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"offsets": [
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16728,
16827
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"text": "['Integrating the methods presented here with this data structure is a promising area for future work']",
"type": "own_claim",
"id": "T216"
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{
"offsets": [
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16914,
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"text": "['we formulate surface tracking as a contouring problem']",
"type": "own_claim",
"id": "T217"
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16969,
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"text": "['The contouring problem has been well studied in computer graphics']",
"type": "background_claim",
"id": "T218"
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17039,
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"text": "['a number of approaches have been suggested']",
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17141,
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"text": "['was first presented by']",
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17187,
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"text": "['and later named and popularized by']",
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17165,
17178
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],
"text": "['Wyvill et al.']",
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17180,
17184
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"text": "['1986']",
"type": "data",
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17223,
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"text": "['Lorensen and Cline']",
"type": "data",
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"offsets": [
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17084,
17133
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],
"text": "['The oldest and most widely used is marching cubes']",
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"offsets": [
[
17243,
17247
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],
"text": "['1987']",
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17250,
17319
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"text": "['Marching cubes suffers from a tendency to create ill-shaped triangles']",
"type": "background_claim",
"id": "T227"
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{
"offsets": [
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17321,
17376
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],
"text": "['This problem is fixed to some degree by dual contouring']",
"type": "background_claim",
"id": "T228"
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{
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17396,
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"text": "['which also provides adaptive contouring and an elegant means of preserving sharp boundaries']",
"type": "background_claim",
"id": "T229"
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{
"offsets": [
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17378,
17392
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"text": "['Ju et al. 2002']",
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"offsets": [
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17489,
17614
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"text": "['Dual contouring depends on normal estimates at edge crossings and is very sensitive to inaccuracies in these normal estimates']",
"type": "background_claim",
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17616,
17719
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"text": "['Unfortunately, in our method we do not have accurate normal information until after the contouring step']",
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17726,
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"text": "['we have the triangle mesh']",
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"offsets": [
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17797,
17912
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],
"text": "['presented a contouring technique which uses Delaunay triangulation methods to generate provably good triangulations']",
"type": "background_claim",
"id": "T234"
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{
"offsets": [
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17768,
17788
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"text": "['Boissonnat and Oudot']",
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"offsets": [
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17790,
17794
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"text": "['2003']",
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17923,
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"text": "['this method appears to be prohibitively expensive for something which must run at every timestep']",
"type": "background_claim",
"id": "T237"
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{
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18021,
18066
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],
"text": "['Yet another alternative is marching triangles']",
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"id": "T238"
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{
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18091,
18166
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],
"text": "['which takes a surface-based rather than volume-based approach to contouring']",
"type": "background_claim",
"id": "T239"
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{
"offsets": [
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18069,
18087
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"text": "['Hilton et al. 1996']",
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18168,
18310
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"text": "['Marching triangles requires significantly less computation time and fewer triangles, and produces higher-quality triangles than marching cubes']",
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"id": "T241"
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18327,
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],
"text": "['marching triangles is not guaranteed to produce closed, manifold meshes in the presence of sharp or thin features']",
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},
{
"offsets": [
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18667,
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],
"text": "['Semi-Lagrangian methods have been widely used in computer graphics']",
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"id": "T243"
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{
"offsets": [
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18778,
18846
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"text": "['to solve the nonlinear advection term of the Navier-Stokes equations']",
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{
"offsets": [
[
18771,
18775
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"text": "['1999']",
"type": "data",
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18765,
18769
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"text": "['Stam']",
"type": "data",
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18848,
18916
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"text": "['These methods provide the foundation for our surface tracking method']",
"type": "own_claim",
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19007,
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"text": "['Our discussion follows that of']",
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19039,
19045
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],
"text": "['Strain']",
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19047,
19052
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"text": "['1999b']",
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"offsets": [
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19761,
19851
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"text": "['This observation forms the basis of the backward characteristic or CIR scheme developed by']",
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"id": "T251"
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{
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19852,
19879
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"text": "['Courant, Isaacson, and Rees']",
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[
19881,
19885
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"text": "['1952']",
"type": "data",
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19889,
19933
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"text": "['which is the simplest semi-Lagrangian scheme']",
"type": "background_claim",
"id": "T254"
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20447,
20458
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"text": "['linear PDEs']",
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20468,
20480
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"text": "['Equation (1)']",
"type": "data",
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20482,
20519
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"text": "['the Lax-Richtmyer equivalence theorem']",
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20537,
20592
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"text": "['guarantees that CIR will converge to the exact solution']",
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"id": "T258"
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{
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20522,
20534
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"text": "['LeVeque 1990']",
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20596,
20604
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"text": "['t, x → 0']",
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20608,
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"text": "['it is stable and consistent']",
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20637,
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"text": "['The stability properties of the CIR scheme are excellent']",
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20776,
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"text": "['unconditional stability is guaranteed in any norm where the interpolation does not increase norms']",
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20695,
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"text": "['Each new value φ(x, t n+1 ) is a single interpolated value of φ at time t n']",
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20888,
20957
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"text": "['CIR with linear interpolation is unconditionally stable in the 2-norm']",
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20971,
21082
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"text": "['semi-Lagrangian schemes satisfy the CFL condition by shifting the stencil, rather than restricting the timestep']",
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21089,
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"text": "['information propagates over long distances in one timestep']",
"type": "own_claim",
"id": "T267"
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21584,
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"text": "['CIR is consistent to O( t)']",
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21614,
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"text": "['a condition t ≥ O( x) is satisfied']",
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21702,
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"text": "['This condition is extremely convenient']",
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21750,
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"text": "['t = O( x) balances time and space resolution in this first-order accurate scheme']",
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22707,
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"text": "['CIR still converges when the solution is smooth']",
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22691,
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22760,
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"text": "['nonsmooth shock solutions of conservation laws move at the wrong speed']",
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22839,
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"text": "['CIR is not in conservative form']",
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22878,
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"text": "['level-set solutions have no shocks']",
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22914,
22959
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"text": "['CIR is a natural scheme for moving interfaces']",
"type": "own_claim",
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{
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23349,
23409
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"text": "['would quickly encounter significant topological difficulties']",
"type": "own_claim",
"id": "T278"
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{
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23298,
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"text": "['advect the mesh points through the flow field']",
"type": "data",
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23420,
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"text": "['we avoid topological issues by updating the surface using an implicit representation']",
"type": "own_claim",
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{
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23990,
24147
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],
"text": "['One of the key differences between our method and other surface tracking methods is that we build an explicit representation of the surface at every timestep']",
"type": "own_claim",
"id": "T281"
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24487,
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"text": "['The explicit representation provides our method with several advantages']",
"type": "own_claim",
"id": "T282"
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{
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24567,
24632
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],
"text": "['it allows us to compute exact signeddistance values near the mesh']",
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"id": "T283"
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24642,
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"text": "['it allows us to store properties on mesh vertices']",
"type": "own_claim",
"id": "T284"
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24739,
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],
"text": "['it allows us to take advantage of the many tools and algorithms which have been developed in computer graphics for manipulating and rendering triangle meshes']",
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26794,
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"text": "['In solving this advection term, our method differs from the simple CIR scheme discussed earlier in two ways']",
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26962,
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"text": "['we use a second-order Runge-Kutta scheme']",
"type": "own_claim",
"id": "T287"
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{
"offsets": [
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27018,
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"text": "['the midpoint method with an Euler predictor']",
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27586,
27631
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"text": "['we do not interpolate values stored on a grid']",
"type": "own_claim",
"id": "T289"
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{
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27545,
27584
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"text": "['evaluating φ at points near the surface']",
"type": "data",
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27642,
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"text": "['we compute exact distance values']",
"type": "own_claim",
"id": "T291"
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{
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27730,
27789
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"text": "['of our method and do not affect the unconditional stability']",
"type": "own_claim",
"id": "T292"
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{
"offsets": [
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27676,
27715
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],
"text": "['These changes only improve the accuracy']",
"type": "own_claim",
"id": "T293"
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{
"offsets": [
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27968,
28039
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],
"text": "['detailed a method for computing the distance from a point to a triangle']",
"type": "background_claim",
"id": "T294"
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{
"offsets": [
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27939,
27959
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],
"text": "['Schneider and Eberly']",
"type": "data",
"id": "T295"
},
{
"offsets": [
[
27961,
27965
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],
"text": "['2002']",
"type": "data",
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{
"offsets": [
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28085,
28234
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],
"text": "['many triangles can be pruned, especially when x is very close to the surface, by using standard bounding-box techniques and our octree data structure']",
"type": "background_claim",
"id": "T297"
},
{
"offsets": [
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28041,
28079
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],
"text": "['This operation is relatively expensive']",
"type": "background_claim",
"id": "T298"
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{
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28240,
28249
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],
"text": "['Section 6']",
"type": "data",
"id": "T299"
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{
"offsets": [
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28252,
28333
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],
"text": "['Signing the distance values turns out to be somewhat difficult near sharp corners']",
"type": "own_claim",
"id": "T300"
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{
"offsets": [
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29252,
29297
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],
"text": "['the triangles do not always agree on the sign']",
"type": "own_claim",
"id": "T301"
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{
"offsets": [
[
29149,
29209
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],
"text": "['the nearest point in the mesh lies on more than one triangle']",
"type": "data",
"id": "T302"
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{
"offsets": [
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29217,
29249
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],
"text": "['on an edge or vertex of the mesh']",
"type": "data",
"id": "T303"
},
{
"offsets": [
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29552,
29616
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],
"text": "['provided a proof that this procedure results in accurate signing']",
"type": "background_claim",
"id": "T305"
},
{
"offsets": [
[
29299,
29523
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],
"text": "['These situations can be resolved by computing an angle-weighted pseudonormal for each edge and vertex of the mesh and using these pseudonormals to determine the sign when the nearest point is on an edge or vertex of the mesh']",
"type": "own_claim",
"id": "T304"
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{
"offsets": [
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29525,
29544
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],
"text": "['Bærentzen and Aanæs']",
"type": "data",
"id": "T306"
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{
"offsets": [
[
29546,
29550
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],
"text": "['2002']",
"type": "data",
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"offsets": [
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29640,
29754
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"text": "['The ability to compute exact distances is one of the chief advantages of having an explicit surface representation']",
"type": "own_claim",
"id": "T308"
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{
"offsets": [
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29756,
29800
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],
"text": "['Interpolation can produce substantial errors']",
"type": "own_claim",
"id": "T309"
},
{
"offsets": [
[
29817,
29847
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],
"text": "['which are compounded over time']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
29806,
29814
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],
"text": "['Figure 3']",
"type": "data",
"id": "T311"
},
{
"offsets": [
[
29858,
29950
]
],
"text": "['this interpolation error is one of the most significant drawbacks to semi-Lagrangian methods']",
"type": "background_claim",
"id": "T312"
},
{
"offsets": [
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29997,
30128
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],
"text": "['interpolation produces such significant smoothing that researchers have proposed a number of methods to add detail back to the flow']",
"type": "background_claim",
"id": "T313"
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{
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30150,
30195
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"text": "['or avoid semi-Lagrangian advection altogether']",
"type": "background_claim",
"id": "T314"
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{
"offsets": [
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29968,
29995
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"text": "['used for velocity advection']",
"type": "data",
"id": "T315"
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{
"offsets": [
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30130,
30148
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],
"text": "['Fedkiw et al. 2001']",
"type": "data",
"id": "T316"
},
{
"offsets": [
[
30198,
30218
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],
"text": "['Zhu and Bridson 2005']",
"type": "data",
"id": "T317"
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{
"offsets": [
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30235,
30401
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],
"text": "['we are able to leverage the advantages of semi-Lagrangian advection, without incurring the interpolation error that would otherwise undesireably smooth surface detail']",
"type": "own_claim",
"id": "T318"
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{
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30463,
30538
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],
"text": "['Our implementation makes heavy use of a structure we call the distance tree']",
"type": "own_claim",
"id": "T319"
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{
"offsets": [
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30540,
30612
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],
"text": "['The distance tree is a balanced octree subdivision of the spatial domain']",
"type": "data",
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30762,
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"text": "['The distance tree serves three purposes']",
"type": "own_claim",
"id": "T321"
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{
"offsets": [
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30807,
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"text": "['It provides a fast spatial index for the mesh so that nearby triangles can be found quickly']",
"type": "own_claim",
"id": "T322"
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{
"offsets": [
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30904,
31038
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],
"text": "['It provides a fast, approximate signed-distance function, which is sufficient when evaluating the signed distance far from the surface']",
"type": "own_claim",
"id": "T323"
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31155,
31172
]
],
"text": "['contain triangles']",
"type": "own_claim",
"id": "T324"
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{
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[
31044,
31143
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],
"text": "['It guides the contouring algorithm, quickly identifying cells which have vertices of different sign']",
"type": "own_claim",
"id": "T326"
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{
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[
32324,
32362
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],
"text": "['the distance is guaranteed to be exact']",
"type": "own_claim",
"id": "T325"
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{
"offsets": [
[
32267,
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],
"text": "['the computed distance is less than C’s edge length']",
"type": "data",
"id": "T327"
},
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"offsets": [
[
32364,
32472
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],
"text": "['Otherwise, the computed distance is a very good estimate but may be slightly larger than the actual distance']",
"type": "own_claim",
"id": "T328"
},
{
"offsets": [
[
32629,
32664
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],
"text": "['we do not require an exact distance']",
"type": "own_claim",
"id": "T329"
},
{
"offsets": [
[
32599,
32624
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],
"text": "['x is not near the surface']",
"type": "own_claim",
"id": "T330"
},
{
"offsets": [
[
32491,
32533
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],
"text": "['C is not at the finest level of the octree']",
"type": "data",
"id": "T331"
},
{
"offsets": [
[
32540,
32592
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],
"text": "['there are no triangles in the concentric triple of C']",
"type": "data",
"id": "T332"
},
{
"offsets": [
[
32666,
32718
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],
"text": "['An approximation with the correct sign is sufficient']",
"type": "own_claim",
"id": "T333"
},
{
"offsets": [
[
33078,
33151
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],
"text": "['it is also useful to build a distance tree from an existing triangle mesh']",
"type": "own_claim",
"id": "T334"
},
{
"offsets": [
[
32975,
33067
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],
"text": "['Most often, we wish to build a distance tree to resolve the zero set of our field function ψ']",
"type": "own_claim",
"id": "T335"
},
{
"offsets": [
[
33153,
33288
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],
"text": "['Our octrees are always built in a top-down manner where each cell is split based on some variation of the following splitting criterion']",
"type": "own_claim",
"id": "T336"
},
{
"offsets": [
[
33328,
33404
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],
"text": "['Split any cell whose edge length exceeds its minimum distance to the surface']",
"type": "data",
"id": "T337"
},
{
"offsets": [
[
33313,
33315
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],
"text": "['11']",
"type": "data",
"id": "T338"
},
{
"offsets": [
[
33510,
33541
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],
"text": "['results in a three-color octree']",
"type": "background_claim",
"id": "T339"
},
{
"offsets": [
[
33506,
33508
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],
"text": "['11']",
"type": "background_claim",
"id": "T340"
},
{
"offsets": [
[
33574,
33660
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],
"text": "['where each cell of the octree has one of three types: interior, exterior, and boundary']",
"type": "background_claim",
"id": "T341"
},
{
"offsets": [
[
33559,
33564
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],
"text": "['Samet']",
"type": "data",
"id": "T342"
},
{
"offsets": [
[
33566,
33570
]
],
"text": "['1990']",
"type": "data",
"id": "T343"
},
{
"offsets": [
[
33666,
33674
]
],
"text": "['Figure 4']",
"type": "data",
"id": "T344"
},
{
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[
33690,
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],
"text": "['Criterion']",
"type": "background_claim",
"id": "T345"
},
{
"offsets": [
[
33705,
33750
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],
"text": "['builds octrees with several useful properties']",
"type": "background_claim",
"id": "T346"
},
{
"offsets": [
[
33701,
33703
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],
"text": "['11']",
"type": "background_claim",
"id": "T347"
},
{
"offsets": [
[
33753,
33812
]
],
"text": "['Adjacent cells differ in size by no more than a factor of 2']",
"type": "background_claim",
"id": "T348"
},
{
"offsets": [
[
33495,
33504
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],
"text": "['Criterion']",
"type": "background_claim",
"id": "T349"
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{
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[
33814,
33864
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],
"text": "['producing a smooth mesh and simplifying procedures']",
"type": "background_claim",
"id": "T350"
},
{
"offsets": [
[
33873,
33889
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],
"text": "['neighbor finding']",
"type": "data",
"id": "T351"
},
{
"offsets": [
[
33894,
33923
]
],
"text": "['triangulation of the vertices']",
"type": "data",
"id": "T352"
},
{
"offsets": [
[
33926,
33986
]
],
"text": "['A cell’s size is proportional to its distance to the surface']",
"type": "background_claim",
"id": "T353"
},
{
"offsets": [
[
34137,
34160
]
],
"text": "['φ will be discontinuous']",
"type": "background_claim",
"id": "T354"
},
{
"offsets": [
[
33992,
34043
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],
"text": "['φ is the signed distance to the surface at vertices']",
"type": "data",
"id": "T355"
},
{
"offsets": [
[
34048,
34101
]
],
"text": "['we extend φ into each cell by trilinear interpolation']",
"type": "data",
"id": "T356"
},
{
"offsets": [
[
34117,
34135
]
],
"text": "['cells vary in size']",
"type": "data",
"id": "T357"
},
{
"offsets": [
[
34171,
34228
]
],
"text": "['the jumps in φ decrease in size in cells near the surface']",
"type": "background_claim",
"id": "T358"
},
{
"offsets": [
[
34240,
34263
]
],
"text": "['the triangle inequality']",
"type": "data",
"id": "T359"
},
{
"offsets": [
[
34270,
34326
]
],
"text": "['the interpolated φ is nearly continuous near the surface']",
"type": "background_claim",
"id": "T360"
},
{
"offsets": [
[
34382,
34430
]
],
"text": "['there are N childless cells touching the surface']",
"type": "data",
"id": "T361"
},
{
"offsets": [
[
34437,
34484
]
],
"text": "['the entire tree contains only O(N log N ) cells']",
"type": "background_claim",
"id": "T362"
},
{
"offsets": [
[
34329,
34377
]
],
"text": "['Cells coarsen very rapidly away from the surface']",
"type": "background_claim",
"id": "T363"
},
{
"offsets": [
[
34492,
34542
]
],
"text": "['the surface is resolved accurately at minimal cost']",
"type": "background_claim",
"id": "T364"
},
{
"offsets": [
[
34961,
35015
]
],
"text": "['we are essentially trying to resolve our approximation']",
"type": "own_claim",
"id": "T365"
},
{
"offsets": [
[
35054,
35097
]
],
"text": "['ψ n+1 (x) = φ n (x − ( t)v(x n−1/2 , t n ))']",
"type": "own_claim",
"id": "T366"
},
{
"offsets": [
[
35039,
35041
]
],
"text": "['12']",
"type": "own_claim",
"id": "T367"
},
{
"offsets": [
[
35119,
35160
]
],
"text": "['to the signed-distance function φ n+1 (x)']",
"type": "own_claim",
"id": "T368"
},
{
"offsets": [
[
34904,
34959
]
],
"text": "['building a new octree at the beginning of each timestep']",
"type": "data",
"id": "T369"
},
{
"offsets": [
[
37355,
37445
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],
"text": "['is an instance of the contouring problem, which has been well studied in computer graphics']",
"type": "background_claim",
"id": "T370"
},
{
"offsets": [
[
37273,
37310
]
],
"text": "['Creating this explicit representation']",
"type": "background_claim",
"id": "T371"
},
{
"offsets": [
[
37554,
37584
]
],
"text": "['Our implementation is based on']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
37585,
37598
]
],
"text": "['Bloomenthal’s']",
"type": "data",
"id": "T373"
},
{
"offsets": [
[
37600,
37604
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],
"text": "['1994']",
"type": "data",
"id": "T374"
},
{
"offsets": [
[
38356,
38412
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],
"text": "['The marching-cubes algorithm works well for our purposes']",
"type": "own_claim",
"id": "T375"
},
{
"offsets": [
[
38421,
38518
]
],
"text": "['each triangle generated by marching cubes sits strictly inside a single cell of the distance tree']",
"type": "data",
"id": "T376"
},
{
"offsets": [
[
38520,
38582
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],
"text": "['making the distance tree an especially effective spatial index']",
"type": "own_claim",
"id": "T377"
},
{
"offsets": [
[
38175,
38288
]
],
"text": "['each vertex in our polygon mesh can be mapped to some point on some triangle in the mesh at the previous timestep']",
"type": "own_claim",
"id": "T378"
},
{
"offsets": [
[
38597,
38760
]
],
"text": "['we use the distance tree we have already built to guide the marching cubes, avoiding the need to build a second structure to determine the topology of the new mesh']",
"type": "own_claim",
"id": "T379"
},
{
"offsets": [
[
38875,
38935
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],
"text": "['we need not worry about patching the marching-cubes solution']",
"type": "own_claim",
"id": "T380"
},
{
"offsets": [
[
38937,
39003
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],
"text": "['Our choice of contouring algorithm does result in some limitations']",
"type": "own_claim",
"id": "T381"
},
{
"offsets": [
[
39005,
39083
]
],
"text": "['In addition to creating poorly shaped triangles, marching cubes is nonadaptive']",
"type": "own_claim",
"id": "T382"
},
{
"offsets": [
[
39094,
39166
]
],
"text": "['the sampling is as dense in flat regions as in regions of high curvature']",
"type": "data",
"id": "T383"
},
{
"offsets": [
[
39183,
39326
]
],
"text": "['the nonadaptive nature of marching cubes limits the resolution we can achieve in high-curvature areas, but is necessary to ensure compatibility']",
"type": "own_claim",
"id": "T384"
},
{
"offsets": [
[
39328,
39386
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],
"text": "['To address this lack of resolution in high-curvature areas']",
"type": "background_claim",
"id": "T385"
},
{
"offsets": [
[
39403,
39498
]
],
"text": "['split line segments whose centers were far from the surface, yielding arbitrarily high accuracy']",
"type": "background_claim",
"id": "T386"
},
{
"offsets": [
[
39388,
39394
]
],
"text": "['Strain']",
"type": "data",
"id": "T387"
},
{
"offsets": [
[
39396,
39400
]
],
"text": "['2001']",
"type": "data",
"id": "T388"
},
{
"offsets": [
[
39515,
39582
]
],
"text": "['this splitting technique is not easily extended to three dimensions']",
"type": "own_claim",
"id": "T389"
},
{
"offsets": [
[
39586,
39697
]
],
"text": "['splitting a triangle either creates an incompatible triangulation or produces even more poorly shaped triangles']",
"type": "data",
"id": "T390"
},
{
"offsets": [
[
39699,
39824
]
],
"text": "['It is also very difficult to guarantee that we will still have a manifold when the inserted vertices are moved to the surface']",
"type": "own_claim",
"id": "T391"
},
{
"offsets": [
[
39841,
39876
]
],
"text": "['several adaptive contouring methods']",
"type": "background_claim",
"id": "T392"
},
{
"offsets": [
[
39938,
40031
]
],
"text": "['seek to use adaptive grids and regain compatibility through various crack-patching techniques']",
"type": "background_claim",
"id": "T393"
},
{
"offsets": [
[
39879,
39894
]
],
"text": "['Shu et al. 1995']",
"type": "data",
"id": "T394"
},
{
"offsets": [
[
39896,
39915
]
],
"text": "['Shekhar et al. 1996']",
"type": "data",
"id": "T395"
},
{
"offsets": [
[
39917,
39935
]
],
"text": "['Poston et al. 1998']",
"type": "data",
"id": "T396"
},
{
"offsets": [
[
40033,
40071
]
],
"text": "['Such methods could easily be used here']",
"type": "own_claim",
"id": "T397"
},
{
"offsets": [
[
40076,
40126
]
],
"text": "['we plan to explore adaptive methods in future work']",
"type": "own_claim",
"id": "T398"
},
{
"offsets": [
[
40905,
40994
]
],
"text": "['This function is quite similar to the functions used in semi-Lagrangian level-set methods']",
"type": "background_claim",
"id": "T400"
},
{
"offsets": [
[
40996,
41008
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],
"text": "['Strain 1999b']",
"type": "data",
"id": "T401"
},
{
"offsets": [
[
41010,
41029
]
],
"text": "['Enright et al. 2005']",
"type": "data",
"id": "T402"
},
{
"offsets": [
[
41383,
41448
]
],
"text": "['this smoothing technique may be quite useful in some applications']",
"type": "own_claim",
"id": "T403"
},
{
"offsets": [
[
41450,
41515
]
],
"text": "['we did not use this method for any of the results in this article']",
"type": "own_claim",
"id": "T404"
},
{
"offsets": [
[
42114,
42148
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],
"text": "['this limitation is not significant']",
"type": "own_claim",
"id": "T399"
},
{
"offsets": [
[
42013,
42065
]
],
"text": "['This jittering limits the reusability of our octrees']",
"type": "own_claim",
"id": "T405"
},
{
"offsets": [
[
42077,
42112
]
],
"text": "['we build new octrees every timestep']",
"type": "data",
"id": "T406"
},
{
"offsets": [
[
42595,
42607
]
],
"text": "['This problem']",
"type": "background_claim",
"id": "T407"
},
{
"offsets": [
[
42638,
42730
]
],
"text": "['has been well studied by the level-set community and a number of methods have been suggested']",
"type": "background_claim",
"id": "T408"
},
{
"offsets": [
[
42624,
42636
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],
"text": "['redistancing']",
"type": "data",
"id": "T409"
},
{
"offsets": [
[
42748,
42834
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],
"text": "['suggested redistancing by performing an exact evaluation at every vertex of the octree']",
"type": "background_claim",
"id": "T410"
},
{
"offsets": [
[
42836,
42871
]
],
"text": "['This method is relatively efficient']",
"type": "background_claim",
"id": "T411"
},
{
"offsets": [
[
42732,
42738
]
],
"text": "['Strain']",
"type": "data",
"id": "T412"
},
{
"offsets": [
[
42740,
42745
]
],
"text": "['1999a']",
"type": "data",
"id": "T413"
},
{
"offsets": [
[
42878,
42958
]
],
"text": "['the tree coarsens rapidly away from the surface and works well in two dimensions']",
"type": "data",
"id": "T414"
},
{
"offsets": [
[
42969,
43052
]
],
"text": "['in three dimensions, we have found it to be prohibitively expensive and unnecessary']",
"type": "own_claim",
"id": "T415"
},
{
"offsets": [
[
43453,
43463
]
],
"text": "['our octree']",
"type": "own_claim",
"id": "T416"
},
{
"offsets": [
[
43510,
43560
]
],
"text": "['does not necessarily coarsen away from the surface']",
"type": "own_claim",
"id": "T417"
},
{
"offsets": [
[
43562,
43668
]
],
"text": "['To address this problem, we coarsen parts of the tree which have been refined but did not generate surface']",
"type": "own_claim",
"id": "T418"
},
{
"offsets": [
[
43956,
44001
]
],
"text": "['Our redistancing method comprises three steps']",
"type": "own_claim",
"id": "T419"
},
{
"offsets": [
[
44004,
44022
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],
"text": "['coarsen the octree']",
"type": "data",
"id": "T420"
},
{
"offsets": [
[
44025,
44093
]
],
"text": "['compute exact distances at vertices of cells which contain triangles']",
"type": "data",
"id": "T421"
},
{
"offsets": [
[
44096,
44150
]
],
"text": "['run a fast marching method over the remaining vertices']",
"type": "data",
"id": "T422"
},
{
"offsets": [
[
44222,
44308
]
],
"text": "['One of the primary advantages of our method is the ability to track surface properties']",
"type": "own_claim",
"id": "T423"
},
{
"offsets": [
[
44376,
44416
]
],
"text": "['accurately at negligible additional cost']",
"type": "own_claim",
"id": "T424"
},
{
"offsets": [
[
44318,
44323
]
],
"text": "['color']",
"type": "data",
"id": "T425"
},
{
"offsets": [
[
44325,
44344
]
],
"text": "['texture coordinates']",
"type": "data",
"id": "T426"
},
{
"offsets": [
[
44354,
44374
]
],
"text": "['simulation variables']",
"type": "data",
"id": "T427"
},
{
"offsets": [
[
44544,
44627
]
],
"text": "['semi-Lagrangian advection provides a mapping between surfaces at adjacent timesteps']",
"type": "own_claim",
"id": "T428"
},
{
"offsets": [
[
44442,
44536
]
],
"text": "['every vertex in a polygon mesh corresponds to some point on some triangle in the previous mesh']",
"type": "own_claim",
"id": "T429"
},
{
"offsets": [
[
44736,
44768
]
],
"text": "['this property can be copied to v']",
"type": "own_claim",
"id": "T430"
},
{
"offsets": [
[
44632,
44692
]
],
"text": "['vertex v in the current mesh maps to point p in the old mesh']",
"type": "data",
"id": "T431"
},
{
"offsets": [
[
44697,
44734
]
],
"text": "['some surface property was stored at p']",
"type": "data",
"id": "T432"
},
{
"offsets": [
[
44782,
44859
]
],
"text": "['we can track surface properties on the actual surface as we build the surface']",
"type": "own_claim",
"id": "T433"
},
{
"offsets": [
[
44864,
44911
]
],
"text": "['we do not incur any significant additional cost']",
"type": "own_claim",
"id": "T434"
},
{
"offsets": [
[
44913,
44929
]
],
"text": "['Previous methods']",
"type": "background_claim",
"id": "T435"
},
{
"offsets": [
[
44987,
45096
]
],
"text": "['have been limited to tracking properties in the volume near the surface and interpolating them to the surface']",
"type": "background_claim",
"id": "T436"
},
{
"offsets": [
[
44960,
44977
]
],
"text": "['Rassmussen et al.']",
"type": "data",
"id": "T437"
},
{
"offsets": [
[
44979,
44983
]
],
"text": "['2004']",
"type": "data",
"id": "T438"
},
{
"offsets": [
[
45098,
45211
]
],
"text": "['Such methods incur significant cost, introduce substantial smoothing, and blur properties between nearby surfaces']",
"type": "background_claim",
"id": "T439"
},
{
"offsets": [
[
45783,
45860
]
],
"text": "['we are having trouble because we are resampling the surface at every timestep']",
"type": "own_claim",
"id": "T440"
},
{
"offsets": [
[
45723,
45768
]
],
"text": "['this approach may introduce unwanted aliasing']",
"type": "own_claim",
"id": "T441"
},
{
"offsets": [
[
45213,
45272
]
],
"text": "['In many applications there is no value actually stored at p']",
"type": "own_claim",
"id": "T442"
},
{
"offsets": [
[
45283,
45353
]
],
"text": "['the properties are stored at the vertices of the triangle containing p']",
"type": "own_claim",
"id": "T443"
},
{
"offsets": [
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"text": "['Similar effects can be seen in real-world footage']",
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A04 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A04_S05_Bar-net_driven_skinning_for_character_animation_CORPUS__v4__onlyOriginalMarkups__v5_onlyAnalysis.xml_CITATION_PURPOSE_M_v1.xml">
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10.1002/cav.211
COMPUTER ANIMATION AND VIRTUAL WORLDS Comp. Anim. Virtual Worlds 2007; 18: 437–446 Published online 6 July 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/cav.211 *******************************************************************************************************************
<Title>Bar-net driven skinning for character animation</Title>
By Jian J. Zhang
Xiaosong Yang
Yunfeng Zhao
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*Correspondence to: X. Yang, The Media School, Bournemouth University, United Kingdom. E-mail: [email protected]
<Abstract>In this paper we present a physically motivated technique for the deformation of animated characters, called the bar-net driven skinning. We use a bar-network (bar-net) as a deforming mechanism. This technique can be used similarly to a conventional skinning tool, but can also make a skin surface behave in a physically plausible manner due to the inherent physical properties of the network. A bar-net is a structure commonly used in structural engineering. Its shape depends on the structural and material properties and the forces acting upon it. Computing the rest shape of an arbitrary bar-net is a time-consuming non-linear problem. In order to speed up the computation and also for such a bar-net to be used intuitively to help computer animation, we have defined a set of properties that a desirable bar-net should satisfy. This allows a bar-net shape finding problem to be solved using linear equations. We adopt a two-layer structure for the representation of a skin surface, including a coarse mesh and a fine mesh. To deform a skin surface, we couple a bar-net to its coarse mesh, which in turn deforms the fine mesh when the coupled bar-net is deformed. The fine surface mesh can be of different forms, including Nurbs, subdivision surfaces and polygons. Copyright # 2007 John Wiley & Sons, Ltd. Received: 14 May 2007; Accepted: 15 May 2007</Abstract>
Keywords: 15 May 2007 Keywords: character animation; skin deformation; physically based animation; barnetwork form finding
<H1>Introduction</H1>
Skin deformation resulting from the movement of characters, such as humans and animals, is one of the most interesting and challenging topics in computer animation. The modelling and deformation of such characters are inevitably complicated and timeconsuming, because of their structural complexity. While realism is important, other factors such as intuitiveness, ease of interaction and computational cost are also of great importance in animation production. Often a compromise among these factors has to be reached. There are two general categories of methods in animation practice: simulation and authoring. Simulation
1–3
refers to the use of a mathematical model to automatically recreate the physical reality on computers. A simulation method lets the animator easily create certain effect which could be otherwise almost an impossible mission with manual manipulations. However, the disadvantage is also obvious. Although the simulation techniques provide some parameters for the animator to control the animation, the connection between these parameters and the result is often implicit. It is usually difficult for an artist to understand the exact physical meaning of these parameters and connect them to the final outcome. Second, such methods are often computationally expensive. The authoring methods refer to those that the animator can use to manipulate the modelling or deformation directly. The animator is able to see the result immediately and has a full control over the deformed shape of the character in question. Both types of methods were around for a long time. However, animators turn to favour those tools that they feel they have a control and can evaluate the results directly. In this paper we present a new skinning technique for the deformation of computer-animated characters. A key advantage is that it combines the strengths of both prevalent categories discussed above. This technique is based on a physically inspired deformation model from structural engineering, known as the bar-networks (bar-nets) and can therefore deform realistically based on the physical properties leading to physically plausible outcomes. Meanwhile, instead of letting the mechanical model taking its full course, the animator is able to operate it as a physically based authoring tool in the same way as other conventional deformation tools. We call this technique the bar-net driven deformations. In animation practice, either for the film industry or games design, it is quite often for each character model to have two layers of mesh, a rough mesh (low resolution) and a fine mesh (high resolution). The high-resolution (high-res) mesh may take various forms, such as Nurbs, subdivision surfaces and polygon meshes. The detailed skin shapes including skin deformations, wrinkle, squama and feather are created on this layer. Because the mesh is very dense involving fine detail, it is inefficient to animate directly on this layer. The low-resolution (low-res) mesh thus works as an efficient intermediate layer for the modelling and deformation of the fine skin layer. Our technique adopts this two-layered strategy. The animator creates almost all skin deformation effects on the low-res layer. In order to take advantage of the physics, we couple an aforementioned mechanical bar-net with the low-res mesh in areas where deformations are expected to occur. This low-res layer gets deformed physically and in turn deforms the high-res mesh of the character’s skin model. In order for a mechanical network to be useful in skin deformation for animated characters, we devise a set of properties for the network to satisfy. These properties allow the behaviours of skin and anatomic tissues, e.g. muscle groups, to be mimicked intuitively in computer animation and to be computed rapidly. What needs pointing out is although there is a similarity between a bar-net and a mass–spring model, they have substantially different behaviours. A mass–spring model will not be able to satisfy the properties we define here. This paper is organized as follows. The next section provides a brief survey of the previous related work. The following section outlines the procedure of creating an animated character model with the coupled bar-nets. The section ‘Bar-net Definition and Form Finding’ discusses the properties an ideal bar-net should have together with a form finding method. Using a human arm model as an example, the section ‘Deformation for Character Animation’ describes how our deformation technique can be used to deform skin surfaces. The last section concludes our work.
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Copyright # 2007 John Wiley & Sons, Ltd.
J. J. ZHANG, X. YANG AND Y. ZHAO
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<H1>Background and Related Work</H1>
Most of the techniques on character deformation can be roughly categorized into two groups: authoring and simulation methods, although the boundary between them does not always seem clear. What’s follows is a brief review of some relevant previous work. The technique of Free Form Deformations (FFDs) first introduced by Sederberg and Parry
4
remains popular and has been adopted by many animation software packages due to its simplicity and modelling speed. FFDs were later extended by several other researchers. 5–8 All these techniques are purely geometric in nature and make no attempt to simulate the physical properties or behaviours of a character. Based on the FFDs, two very popular deformation tools were developed in Maya, the Lattice and Wrap deformers. An intuitive attempt to deform a character was involving a skeleton into skin deformation. This approach has a long history and it treats the skin as a shell that moves by an explicit function of the skeleton. Vertices of the skin are deformed by a weighted combination of the joint transformations of the character’s skeleton. 9–13 Collectively, such methods are known as the smooth skinning. They are easy to understand and intuitive to use. A tedious part is the proper assignment of the weights. In production, the weights are painted by the animator, and thus the animator has full control over the outcomes. The smooth skinning approach suffers from some notorious drawbacks, called the candy wrapper effect or collapsing elbow effect, due to its lack of consideration of volume preservation for the soft tissues. The example-based methods were developed as an alternative in order to overcome this kind of problems
14–17
and have had some success. With this method, which is called Blend Shape in animation production, the animator can control the exact appearance of the character. In facial animation, for example, the animator often needs to dictate how a facial model deforms to achieve different expressions. On the downside, however, a large number of models have to be made in the pose space and stored for shape interpolation. This is an expensive process. The drive for realism in computer graphics has lead to some new modelling and deformation techniques. A group of techniques that have gained increasing popularity in the computer animation of characters are those based on characters anatomy. These models attempt to mimic their real life counterparts by reproducing their anatomical structures. These anatomy-based skinning methods differ on the complexity of the models and their behaviours of the underlying anatomical structures.
Some use simple muscle shapes, such as abstract muscle operators,
18 meatballs,
19 some employ detailed models. 20–23
The obvious advantage of this group of methods is its ability in achieving detailed visual quality during animation. One of the difficulties of these techniques,
24–28
however, is that they are indirect to use, as one has to model the anatomical structures before its appearance arrives. Achieving a particular look of the skin requires the determination of the shape, number and the layout of the muscles underneath. Until the skin mesh envelops the underlying structure, it is very hard to anticipate how the character looks like from the outside. To retain the advantage of the anatomy-based technique without losing intuitiveness, recent research has looked into the issue of estimating the muscles from the skin shape. 29–30
This new technique has had a degree of success. The current limitations are that they could only use simple muscle shapes, which are sufficient in obtaining detailed deformations.
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Copyright # 2007 John Wiley & Sons, Ltd.
438
Comp. Anim. Virtual Worlds 2007; 18: 437–446 DOI: 10.1002/cav
BAR-NET DRIVEN SKINNING
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<H1>The Proposed Model</H1>
Anatomy based multi-layered models have significantly improved the realism of the modelling of complex living creatures. Character animation based on the deformation of underlying anatomical structures, such as muscles or fat, is a very complicated process. Issues like mechanical forces, material properties and collision among anatomic structures all need to be properly addressed. The computational cost is inevitably excessive. Such computational costs place severe restrictions on many applications. Our bar-net driven skinning method endeavours to take advantage of the anatomy-based approach, the smooth skinning approach and the physically based approach. Figure 1 shows a simple flowchart of this method. It follows the current animation workflow, except that a bar-net is coupled with the low-res skin layer. When the low-res mesh (or a part of it) is coupled with a bar-net, the couple mesh is called the control mesh in this paper. Bar-nets deform according to both the external forces it is subject to and the stiffness properties of the network. By controlling these two factors, the animator can easily create the various skin deformation effects including muscle bulge, wrinkles and creases easily. Our system consists of three layers: The skeleton layer controls the movement of the character and determines the basic shape of its body. The control mesh is bound to the character skeleton in the same way as the traditional smooth skinning method. The bar-nets work like a deformer (a term used in many animation packages, e.g. Maya) to change the shape of the skin surface. It is compatible with all the other deformation tools incorporated in current animation software. They can accumulatively deform the skin shape in a certain order which can be easily changed by the animator on the fly. The fine mesh, either in the form of Nurbs, subdivision surfaces or polygons, is deformed by the control mesh using the wrapping deformation method 8 which is available with many commercial animation packages. Further detailed deformations including wrinkle can also be added on either by manipulating the skin surface directly or by coupling a bar-net with the fine mesh using the same mechanism. A bar-net connects n s points, P i , in three-dimensional space with straight-line segments, called bars. These points on the net are known as nodes. The nodes can be either fixed or free. Fixed nodes will not have their positions changed regardless of whether they are subjected to external forces. Free nodes can be moved to balance the acting forces on the net. Each bar connects two nodes. These bars can be stretched and squashed resulting from the positioning of the end nodes, but they cannot be bent. The network described above is in fact a graph with links connecting pairs of nodes. A matrix C s , called the branch–node matrix can be formed, which represents in a tabular form the graph of the network. Assuming that there are n free nodes and n f fixed nodes, the branch–node matrix can be further subdivided into two sub-matrices, C and C f , by grouping the free-node columns and fixed-node columns of the original matrix, respectively. These matrices are used in computing the rest shape of a bar-net. A deformable part of the low-res mesh of a character can be considered as a bar-net. This analogy establishes a natural link between a mechanical bar-net and a surface patch. If a bar-net is coupled with a surface, the surface can be made to behave like a piece of elastic material. Thus many numerical methods developed in structural and mechanical engineering for the manipulation of structures and networks can be applied to control the deformation of the surfaces. Deforming the bar-net deforms the coupled surface, hence the name bar-net driven deformation. Bar-nets can have any arbitrary topology. They are not restricted to a quadrilateral topology unlike most curved surface patches. Quadrilateral patches are the easiest to control and there have been many algorithms developed to implement them. But methods for controlling the deformation of a non-quadrilateral surface patch analytically remain an interesting research topic. Such a problem could be resolved by coupling a general mechanical bar-net with the control points of a surface patch of the same topology.
Architecture of Bar-Net Driven Skinning
Figure 1. Workflow of bar-net driven skinning.
Bar-Networks
P i 2 R 3 i 1⁄4 1; . . . ; n s
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A Bar-Net as a Control Mesh
<H1>Bar-Net Def|nition and Form Finding</H1>
The principle idea of the proposed bar-net driven deformation technique is to regard the deformable area of an animated creature as a network, which deforms under an acting force. The final shape of the surface represents the rest shape of the network and is the result of the balance of all external and internal forces. To make it useful and intuitive for the animator, we define a number of properties essential for such a bar-net to satisfy: The network deformation is determined only by the topology, the stiffness parameter and the loading (force) of the network, but is independent of the geometry. One does not need to worry about the shape of the network itself. This because we use a bar-net only as a control mechanism. Changing the stiffness with other parameters unchanged has an influence on the whole network. This is in line with the physical property of human tissues and therefore makes physical sense. The x, y, z components of the external loads applied to the free nodes (non-fixed nodes) have independent influences on the deformation. The x component of the displacement is only determined by the x component of the applied force, and similarly for the y and z components. So when the animator wishes to finetune the effects on the x, y or z direction separately, the surface will deform as expected. The deformation of the free nodes satisfies the superposition principle. In other words, if one free node is subject to the influence of a number of forces simultaneously, the general deformation applied to the node is the same as the sum of all the deformations generated by applying these loads independently. The benefit from this property is that several muscles, bones, fat tissues can affect the skin deformation simultaneously through summing up of their individual forces. Network form finding is always a numerically complicated problem in mechanical and structural engineering. Various numerical methods exist. As far as most mechanical networks are concerned, the relationship between the equilibrium state and the acting forces is non-linear. Shape change cannot be trivially related to the magnitude and direction of the external forces. Often numerical algorithms are deployed to determine the rest form of a network, which is inevitably time-consuming and not very useful for animation production. In our case, the effect of stiffness of a network can be approximated by the quantity of force-length ratios of all the bars. Some researchers call this quantity the force density. Using this stiffness parameter, we found the force density method 31 satisfies the above-defined properties. y An added advantage is that it is also able to solve the
y In the interest of space, in this paper we do not give the detailed proofs that the force-density form finding method satisfies our specified properties.
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form finding problem with a set of linear equations. The mathematical details are given in the appendix. In the following, we will discuss how such bar-nets can be used to assist the deformation of skin surfaces of deformable characters
<H1>Deformation for Character Animation</H1>
Using a bar-net together with the force density form finding method, the prevailing advantages of this technique are its speed of computation and intuitiveness in shape control. Coupling bar-nets with a skin surface makes it ‘mechanically deformable’. Skin deformation can happen around the joints where its surface bends and also in places where the underlying anatomic structure, such as muscles, pushes and pulling the skin surface. Using above defined bar-net properties, deformations are achieved by applying virtual forces to the appropriate free nodes of the control mesh. We use the force density parameter (equivalent to stiffness) and external forces to control the deformation. Multiple bars can be grouped together to simulate the effect of muscle groups. The user can manipulate the force on each node to tune the deformation interactively. One can also change the force densities to make the network firmer or softer. Figure 2 shows an example of using a bar-net to solve the collapsing elbow problem and obtain a fat bulge effect when the forearm bends towards the upper arm. Forces are applied to only eight nodes of the bar-net. In this example, the force densities are kept unchanged. The gradually changed forces on the control mesh are bound to the elbow rotation angle, which produce both the bulge effect and compensate for the volume loss that the traditional smooth skinning method suffers. Deformed muscles always change the shape of the skin surface. Using the model of a human arm, we illustrate how to generate the muscle effect with bar-nets. There are approximately 50 muscles in a human upper limb, most of which are large and complex. Muscles usually act in groups, some muscles act to move the joint, some to support the movement by avoiding unwanted secondary movements. The combination of these actions causes the muscles, hence the arm, to deform. In the animated arm model, we are only concerned with the muscles producing major influence on the skin. The deformation of the forearm is complex but relatively unnoticeable. Therefore, in this case, only the deformations caused by biceps brachii and triceps brachii are generated. When the arm flexes, the biceps brachii contracts and bulges. At the same time, the triceps brachii relaxes to allow this action. The opposite occurs during the extension of the forearm. The biceps brachii and triceps brachii are positioned on opposite sides of the upper arm. Accordingly only the nodes lying around the central line of the two muscles are set to free, all other nodes are fixed. All the bars in the network are initially assigned a uniform force density. Because of the tendon of the biceps brachii, flexing the arm deforms the biceps brachii in all three directions (x, y and z): it is shortened along the arm due to its contraction and it bulges in the other two directions to maintain its volume. To simulate the force of the muscles, we apply some simple loads to the midpoints of the network as shown in Figure 3(b) . These loads deform the surface to form a natural muscle bulge, as shown in Figure 3(c) . This example demonstrates that the animator can easily shape the characters using the virtual forces as user-handles. There are 2891 vertices and 2816 faces in the subdivision model. While in the bar-net, there are 12 free nodes which are the only necessary resources involved in the form finding and it involves little computation cost. Local deformations can be similarly achieved by changing the force densities. For example, reducing the force density of the network increases the size of the bulge effect as shown in Figure 3(d) . The human shoulder is a typical area where notorious skin deformities occur using a traditional skinning method. Figure 4 shows a few snapshots from our bar-net driven skinning technique. Most computer-animated characters are complex both geometrically and topologically. The use of quadrilateral meshes to model the geometry of such characters is frequently inadequate. Computer-animated characters come in different shapes, e.g. in a form of a human, an animal or a completely imaginary figure. Branches, holes, non-manifolds and irregularities are possible geometric features of their body forms. Satisfying our designed properties, the network is capable of handling any connectivity (topology). Figure 5 shows a bar-net bound to various topologies, including meshes with holes and branches. In practice, an animated character can be initially modelled by sketching its basic shape roughly. This rough model is coupled with a bar-net to act as the control mesh of the character. The fine skin surface can be represented in various surface forms. Our implementation includes three major surface modelling forms: Nurbs, subdivision surfaces and polygons. Once the control mesh is deformed, it can deform the fine surface model using the Wrap deformer available in many animation packages. Bar-net driven skinning is applicable also to the modelling of wrinkle, where the bar-net is bound to the fine mesh rather than the rough mesh in order to obtain a detailed look. Figure 6 shows an example of the bar-net working on wrinkle deformation.
Interactive Deformation
Figure 2. Interactive manipulation of the forces on eight free nodes to produce a specific deformation effect (Left: traditional smooth skinning; middle: volume compensated by bar-net driven skinning; right: control mesh and free nodes).
Mimicking Muscle Deformation
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Figure 3. A deformed arm model.
Figure 4. Deforming muscles on the shoulder area.
The Deformation of Skin Surface With an ArbitraryT opology and Representative Form
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Figure 5. Bar-network deforming a mesh with an arbitrary topology, including holes, branches and non-manifolds.
<H1>Conclusions and Future Work</H1>
Character deformation in computer animation has attracted a great deal of research effort over the last two decades. The earlier models, despite being cheap, had difficulties in creating realistic character deformations. With the quest for realism, more physically based and CPU intensive computation models have emerged, notably the multi-layered anatomy-based approach. However, in addition to the computational cost, it is undesirable to require the animator to model many muscles before the skin shape is developed. In this paper, we propose a physically motivated deformation authoring technique, called the bar-net driven skinning. Its main strength lies in the combination of speed, intuitiveness and good realism. Our technique can achieve similar results to those of the anatomy-based techniques, but in an interactive manner. Bar-nets reach their rest shape when the acted forces equilibrate. Changing the forces and/or stiffness leads to a change of their shape. Coupling a part of surface mesh with bar-nets allows the surface deformation to be controlled by manipulating the networks and can take advantage of the physical behaviour inherent to the network. In order to allow deformations to be produced quickly and intuitively, we have devised a set of properties that an ‘ideal’ bar-net should satisfy, which make intuitive shape control and fast computation possible. To deform the skin surface of a character, we couple a bar-net with a low-res mesh, called the control mesh, which links with the skin surface. This makes the skin mechanically deformable and achieves realistic deformation outcomes. We provide two types of user-handles associated with a bar-net, the virtual forces applied to the free nodes of a network and the force density values. They can be used individually as an interactive modelling tool or collectively to mimic the muscle forces from a muscle group. We have implemented this technique into prototype program in a form of a plug-in for the Autodesk Maya software ( Figure 7 ). It provides the animator with a new deformer which can be used both as a modelling and an animation tool. The animator can interactively change the fix–free status of each node, define and manipulate the forces on each free node, tune the force densities for selected bars. On the downside, the tools developed so far are still relatively primitive. The user needs to understand the basic principles of the bar-net properties before the technique can be used efficiently. To remedy this problem we are currently designing higher-level tools with an interactive user interface, which will hide this complexity from the user.
Figure 6. Local deformation modelling.
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Figure 7. Bar-net driven skinning implemented as a Maya plug-in.
<H1>ACKNOWLEDGEMENTS</H1>
This research is funded by the British Arts and Humanities Research Council grant B/RG/AN5263/APN12727 and Great Western Research Fellowship. We are grateful to Autodesk for their donation of the Maya software licences. Appendix: The Force Density Method For each bar branch, we define an impulse function k(j) to evaluate the connectivity between branch j and node k in the net: 1 k is an end note of branch j kðjÞ 1⁄4 0 otherwise
The usual branch–node matrix C s is defined by: 8 < þ1 for iðjÞ 1⁄4 1 C s ðj; iÞ 1⁄4 À1 for kðjÞ 1⁄4 1 ( 1 ) : 0 otherwise Here i is one extremity and k is the other extremity of branch j. The equilibrium shape of the net structure is reached when all the forces applied at each node sum up to zero. This equilibrium requirement can be written as a linear equation system in the following matrix form: 31 C t QC x þ C t QC f x f 1⁄4 p x C t QC ỹ þ C t QC f ỹ f 1⁄4 p y ( 2 ) C t QCz þ C t QC f z f 1⁄4 p z where Q is the diagonal matrix built from the force densities, ðQÞ ij 1⁄4 q 0 i otherwise if i 1⁄4 j and force density q i of bar i is defined as the ratio of the force to the length of the bar, x f ; ỹ f ;z f are the vectors of the x, y, z co-ordinates of the fixed nodes and are considered as input data, while x; ỹ;z are the vectors of the x, y, z co-ordinates of the moving nodes (free nodes) and are unknowns in the equation. p x ; p y ; p z are the external load vectors. By setting D 1⁄4 C t QC and D f 1⁄4 C t QC f , Equation ( 2 ) becomes:
D x 1⁄4 p x À D f x f D ỹ 1⁄4 p y À D f ỹ f ( 3 ) Dz 1⁄4 p z À D f z f
Thus at the equilibrium state the coordinates of the free nodes are determined by: x 1⁄4 D À1 ð p x À D f x f Þ ỹ 1⁄4 D À1 ð p y À D f ỹ f Þ ( 4 ) z 1⁄4 D À1 ð p z À D f z f Þ Therefore, with the given interconnection, the force density vector, the load and the co-ordinates of the fixed nodes, and the positioning of the free nodes are determined by the equilibrium of the loads. It is clear from Equation ( 4 ) that any state of equilibrium of a general network structure can be obtained by the solution of one system of linear equations, which is computationally inexpensive.
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<H1>References</H1>
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Authors’ biographies:
Jian J. Zhang is Professor of Computer Graphics at the National Centre for Computer Animation and Director of Computer Animation Research Centre, Bournemouth Media School, Bournemouth University. His research interests include computer graphics, computer animation, physically based simulation, geometric modelling, medical simulation and visualisation. Xiaosong Yang is a Research Fellow in the National Centre for Computer Animation, Bournemouth Media School, Bournemouth University, United Kingdom. He received his bachelor (1993) and master degree (1996) in Computer Science from Zhejiang University (P. R. China), Ph.D. (2000) in Computing Mechanics from Dalian University of Technology (P. R. China). He worked as PostDoc (2000–2002) in the Department of Computer Science and Technology of Tsinghua University for 2 years, and Research Assistant (2001–2002) at the ‘Virtual Reality, Visualization and Imaging Research Centre’ of Chinese University of Hong Kong. His research interests include 3D modelling, animation, real-time rendering, virtual reality, virtual surgery simulation and computer-aided design. Yunfeng Zhao did his research on computer graphics in Bournemouth University. He was then a Research Fellow in Leeds University and contributed greatly to the success of the VITAL project. He became a Partner and Senior Engineer in Icona Solutions Ltd which was set up to explore the innovative technology developed in VITAL project. His research interests include computer graphics and computer simulation, especially in the area of computer-simulated deformation.
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"text": "['Our technique adopts this two-layered strategy']",
"type": "own_claim",
"id": "T23"
},
{
"offsets": [
[
5825,
5902
]
],
"text": "['The animator creates almost all skin deformation effects on the low-res layer']",
"type": "background_claim",
"id": "T24"
},
{
"offsets": [
[
5423,
5447
]
],
"text": "['The detailed skin shapes']",
"type": "background_claim",
"id": "T25"
},
{
"offsets": [
[
5505,
5530
]
],
"text": "['are created on this layer']",
"type": "background_claim",
"id": "T26"
},
{
"offsets": [
[
5458,
5475
]
],
"text": "['skin deformations']",
"type": "data",
"id": "T27"
},
{
"offsets": [
[
5477,
5484
]
],
"text": "['wrinkle']",
"type": "data",
"id": "T28"
},
{
"offsets": [
[
5486,
5492
]
],
"text": "['squama']",
"type": "data",
"id": "T29"
},
{
"offsets": [
[
5497,
5504
]
],
"text": "['feather']",
"type": "data",
"id": "T30"
},
{
"offsets": [
[
6330,
6396
]
],
"text": "['These properties allow the behaviours of skin and anatomic tissues']",
"type": "own_claim",
"id": "T31"
},
{
"offsets": [
[
6418,
6493
]
],
"text": "['to be mimicked intuitively in computer animation and to be computed rapidly']",
"type": "own_claim",
"id": "T32"
},
{
"offsets": [
[
6403,
6416
]
],
"text": "['muscle groups']",
"type": "data",
"id": "T33"
},
{
"offsets": [
[
6531,
6594
]
],
"text": "['there is a similarity between a bar-net and a mass–spring model']",
"type": "own_claim",
"id": "T34"
},
{
"offsets": [
[
6596,
6640
]
],
"text": "['they have substantially different behaviours']",
"type": "own_claim",
"id": "T35"
},
{
"offsets": [
[
6642,
6719
]
],
"text": "['A mass–spring model will not be able to satisfy the properties we define here']",
"type": "own_claim",
"id": "T36"
},
{
"offsets": [
[
7724,
7814
]
],
"text": "['Most of the techniques on character deformation can be roughly categorized into two groups']",
"type": "background_claim",
"id": "T37"
},
{
"offsets": [
[
11887,
12006
]
],
"text": "['Anatomy based multi-layered models have significantly improved the realism of the modelling of complex living creatures']",
"type": "background_claim",
"id": "T38"
},
{
"offsets": [
[
12008,
12088
]
],
"text": "['Character animation based on the deformation of underlying anatomical structures']",
"type": "background_claim",
"id": "T39"
},
{
"offsets": [
[
12114,
12143
]
],
"text": "['is a very complicated process']",
"type": "background_claim",
"id": "T40"
},
{
"offsets": [
[
12098,
12105
]
],
"text": "['muscles']",
"type": "data",
"id": "T41"
},
{
"offsets": [
[
12109,
12112
]
],
"text": "['fat']",
"type": "data",
"id": "T42"
},
{
"offsets": [
[
12236,
12269
]
],
"text": "['all need to be properly addressed']",
"type": "background_claim",
"id": "T43"
},
{
"offsets": [
[
12145,
12151
]
],
"text": "['Issues']",
"type": "background_claim",
"id": "T44"
},
{
"offsets": [
[
12157,
12174
]
],
"text": "['mechanical forces']",
"type": "data",
"id": "T45"
},
{
"offsets": [
[
12176,
12195
]
],
"text": "['material properties']",
"type": "data",
"id": "T46"
},
{
"offsets": [
[
12200,
12235
]
],
"text": "['collision among anatomic structures']",
"type": "data",
"id": "T47"
},
{
"offsets": [
[
12271,
12317
]
],
"text": "['The computational cost is inevitably excessive']",
"type": "background_claim",
"id": "T48"
},
{
"offsets": [
[
12319,
12390
]
],
"text": "['Such computational costs place severe restrictions on many applications']",
"type": "background_claim",
"id": "T49"
},
{
"offsets": [
[
12392,
12549
]
],
"text": "['Our bar-net driven skinning method endeavours to take advantage of the anatomy-based approach, the smooth skinning approach and the physically based approach']",
"type": "own_claim",
"id": "T50"
},
{
"offsets": [
[
12551,
12559
]
],
"text": "['Figure 1']",
"type": "data",
"id": "T51"
},
{
"offsets": [
[
12601,
12642
]
],
"text": "['It follows the current animation workflow']",
"type": "own_claim",
"id": "T52"
},
{
"offsets": [
[
12656,
12704
]
],
"text": "['a bar-net is coupled with the low-res skin layer']",
"type": "own_claim",
"id": "T53"
},
{
"offsets": [
[
12831,
12945
]
],
"text": "['Bar-nets deform according to both the external forces it is subject to and the stiffness properties of the network']",
"type": "own_claim",
"id": "T54"
},
{
"offsets": [
[
12947,
13048
]
],
"text": "['By controlling these two factors, the animator can easily create the various skin deformation effects']",
"type": "own_claim",
"id": "T55"
},
{
"offsets": [
[
13059,
13071
]
],
"text": "['muscle bulge']",
"type": "data",
"id": "T56"
},
{
"offsets": [
[
13139,
13239
]
],
"text": "['The skeleton layer controls the movement of the character and determines the basic shape of its body']",
"type": "own_claim",
"id": "T57"
},
{
"offsets": [
[
13241,
13350
]
],
"text": "['The control mesh is bound to the character skeleton in the same way as the traditional smooth skinning method']",
"type": "own_claim",
"id": "T58"
},
{
"offsets": [
[
13352,
13385
]
],
"text": "['The bar-nets work like a deformer']",
"type": "own_claim",
"id": "T59"
},
{
"offsets": [
[
13438,
13477
]
],
"text": "['to change the shape of the skin surface']",
"type": "own_claim",
"id": "T60"
},
{
"offsets": [
[
13479,
13575
]
],
"text": "['It is compatible with all the other deformation tools incorporated in current animation software']",
"type": "own_claim",
"id": "T61"
},
{
"offsets": [
[
14096,
14187
]
],
"text": "['A bar-net connects n s points, P i , in three-dimensional space with straight-line segments']",
"type": "own_claim",
"id": "T62"
},
{
"offsets": [
[
14246,
14283
]
],
"text": "['The nodes can be either fixed or free']",
"type": "own_claim",
"id": "T63"
},
{
"offsets": [
[
14285,
14394
]
],
"text": "['Fixed nodes will not have their positions changed regardless of whether they are subjected to external forces']",
"type": "own_claim",
"id": "T64"
},
{
"offsets": [
[
14396,
14459
]
],
"text": "['Free nodes can be moved to balance the acting forces on the net']",
"type": "own_claim",
"id": "T65"
},
{
"offsets": [
[
14584,
14602
]
],
"text": "['they cannot be ben']",
"type": "own_claim",
"id": "T66"
},
{
"offsets": [
[
14490,
14578
]
],
"text": "['These bars can be stretched and squashed resulting from the positioning of the end nodes']",
"type": "own_claim",
"id": "T67"
},
{
"offsets": [
[
14644,
14688
]
],
"text": "['a graph with links connecting pairs of nodes']",
"type": "own_claim",
"id": "T68"
},
{
"offsets": [
[
14605,
14635
]
],
"text": "['The network described above is']",
"type": "own_claim",
"id": "T69"
},
{
"offsets": [
[
14735,
14809
]
],
"text": "['can be formed, which represents in a tabular form the graph of the network']",
"type": "own_claim",
"id": "T70"
},
{
"offsets": [
[
14690,
14702
]
],
"text": "['A matrix C s']",
"type": "own_claim",
"id": "T71"
},
{
"offsets": [
[
14825,
14867
]
],
"text": "['there are n free nodes and n f fixed nodes']",
"type": "data",
"id": "T72"
},
{
"offsets": [
[
14869,
14939
]
],
"text": "['the branch–node matrix can be further subdivided into two sub-matrices']",
"type": "own_claim",
"id": "T73"
},
{
"offsets": [
[
14953,
15032
]
],
"text": "['by grouping the free-node columns and fixed-node columns of the original matrix']",
"type": "own_claim",
"id": "T74"
},
{
"offsets": [
[
15114,
15197
]
],
"text": "['A deformable part of the low-res mesh of a character can be considered as a bar-net']",
"type": "own_claim",
"id": "T75"
},
{
"offsets": [
[
15199,
15287
]
],
"text": "['This analogy establishes a natural link between a mechanical bar-net and a surface patch']",
"type": "own_claim",
"id": "T76"
},
{
"offsets": [
[
3147,
3203
]
],
"text": "['Often a compromise among these factors has to be reached']",
"type": "background_claim",
"id": "T77"
},
{
"offsets": [
[
3205,
3270
]
],
"text": "['There are two general categories of methods in animation practice']",
"type": "background_claim",
"id": "T78"
},
{
"offsets": [
[
3272,
3282
]
],
"text": "['simulation']",
"type": "background_claim",
"id": "T79"
},
{
"offsets": [
[
3287,
3296
]
],
"text": "['authoring']",
"type": "background_claim",
"id": "T80"
},
{
"offsets": [
[
3310,
3313
]
],
"text": "['1–3']",
"type": "data",
"id": "T81"
},
{
"offsets": [
[
3298,
3308
]
],
"text": "['Simulation']",
"type": "background_claim",
"id": "T82"
},
{
"offsets": [
[
3315,
3416
]
],
"text": "['refers to the use of a mathematical model to automatically recreate the physical reality on computers']",
"type": "background_claim",
"id": "T83"
},
{
"offsets": [
[
3418,
3564
]
],
"text": "['A simulation method lets the animator easily create certain effect which could be otherwise almost an impossible mission with manual manipulations']",
"type": "background_claim",
"id": "T84"
},
{
"offsets": [
[
3575,
3607
]
],
"text": "['the disadvantage is also obvious']",
"type": "background_claim",
"id": "T85"
},
{
"offsets": [
[
3618,
3709
]
],
"text": "['the simulation techniques provide some parameters for the animator to control the animation']",
"type": "background_claim",
"id": "T86"
},
{
"offsets": [
[
3711,
3783
]
],
"text": "['the connection between these parameters and the result is often implicit']",
"type": "background_claim",
"id": "T87"
},
{
"offsets": [
[
3785,
3921
]
],
"text": "['It is usually difficult for an artist to understand the exact physical meaning of these parameters and connect them to the final outcome']",
"type": "background_claim",
"id": "T88"
},
{
"offsets": [
[
3931,
3979
]
],
"text": "['such methods are often computationally expensive']",
"type": "background_claim",
"id": "T89"
},
{
"offsets": [
[
3981,
4095
]
],
"text": "['The authoring methods refer to those that the animator can use to manipulate the modelling or deformation directly']",
"type": "background_claim",
"id": "T90"
},
{
"offsets": [
[
4097,
4147
]
],
"text": "['The animator is able to see the result immediately']",
"type": "background_claim",
"id": "T91"
},
{
"offsets": [
[
4152,
4223
]
],
"text": "['has a full control over the deformed shape of the character in question']",
"type": "background_claim",
"id": "T92"
},
{
"offsets": [
[
4225,
4274
]
],
"text": "['Both types of methods were around for a long time']",
"type": "background_claim",
"id": "T93"
},
{
"offsets": [
[
4285,
4394
]
],
"text": "['animators turn to favour those tools that they feel they have a control and can evaluate the results directly']",
"type": "background_claim",
"id": "T94"
},
{
"offsets": [
[
4499,
4593
]
],
"text": "['A key advantage is that it combines the strengths of both prevalent categories discussed above']",
"type": "own_claim",
"id": "T95"
},
{
"offsets": [
[
4595,
4689
]
],
"text": "['This technique is based on a physically inspired deformation model from structural engineering']",
"type": "own_claim",
"id": "T96"
},
{
"offsets": [
[
4732,
4735
]
],
"text": "['can']",
"type": "own_claim",
"id": "T97"
},
{
"offsets": [
[
4746,
4840
]
],
"text": "['deform realistically based on the physical properties leading to physically plausible outcomes']",
"type": "own_claim",
"id": "T98"
},
{
"offsets": [
[
4917,
5044
]
],
"text": "['the animator is able to operate it as a physically based authoring tool in the same way as other conventional deformation tools']",
"type": "own_claim",
"id": "T99"
},
{
"offsets": [
[
5102,
5123
]
],
"text": "['In animation practice']",
"type": "background_claim",
"id": "T100"
},
{
"offsets": [
[
5171,
5240
]
],
"text": "['it is quite often for each character model to have two layers of mesh']",
"type": "background_claim",
"id": "T101"
},
{
"offsets": [
[
5136,
5153
]
],
"text": "['the film industry']",
"type": "data",
"id": "T102"
},
{
"offsets": [
[
5157,
5169
]
],
"text": "['games design']",
"type": "data",
"id": "T103"
},
{
"offsets": [
[
5242,
5254
]
],
"text": "['a rough mesh']",
"type": "data",
"id": "T104"
},
{
"offsets": [
[
5276,
5287
]
],
"text": "['a fine mesh']",
"type": "data",
"id": "T105"
},
{
"offsets": [
[
7816,
7825
]
],
"text": "['authoring']",
"type": "background_claim",
"id": "T106"
},
{
"offsets": [
[
7830,
7848
]
],
"text": "['simulation methods']",
"type": "background_claim",
"id": "T107"
},
{
"offsets": [
[
7859,
7911
]
],
"text": "['the boundary between them does not always seem clear']",
"type": "background_claim",
"id": "T108"
},
{
"offsets": [
[
7978,
8017
]
],
"text": "['The technique of Free Form Deformations']",
"type": "background_claim",
"id": "T109"
},
{
"offsets": [
[
8069,
8141
]
],
"text": "['remains popular and has been adopted by many animation software packages']",
"type": "background_claim",
"id": "T111"
},
{
"offsets": [
[
8149,
8163
]
],
"text": "['its simplicity']",
"type": "data",
"id": "T110"
},
{
"offsets": [
[
8168,
8183
]
],
"text": "['modelling speed']",
"type": "data",
"id": "T112"
},
{
"offsets": [
[
8185,
8238
]
],
"text": "['FFDs were later extended by several other researchers']",
"type": "background_claim",
"id": "T113"
},
{
"offsets": [
[
8066,
8067
]
],
"text": "['4']",
"type": "data",
"id": "T114"
},
{
"offsets": [
[
4410,
4497
]
],
"text": "['we present a new skinning technique for the deformation of computer-animated characters']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
8240,
8243
]
],
"text": "['5–8']",
"type": "data",
"id": "T116"
},
{
"offsets": [
[
8244,
8295
]
],
"text": "['All these techniques are purely geometric in nature']",
"type": "background_claim",
"id": "T117"
},
{
"offsets": [
[
8300,
8380
]
],
"text": "['make no attempt to simulate the physical properties or behaviours of a character']",
"type": "background_claim",
"id": "T118"
},
{
"offsets": [
[
8382,
8458
]
],
"text": "['Based on the FFDs, two very popular deformation tools were developed in Maya']",
"type": "background_claim",
"id": "T119"
},
{
"offsets": [
[
8460,
8471
]
],
"text": "['the Lattice']",
"type": "data",
"id": "T120"
},
{
"offsets": [
[
8476,
8490
]
],
"text": "['Wrap deformers']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
8492,
8581
]
],
"text": "['An intuitive attempt to deform a character was involving a skeleton into skin deformation']",
"type": "background_claim",
"id": "T122"
},
{
"offsets": [
[
8583,
8615
]
],
"text": "['This approach has a long history']",
"type": "background_claim",
"id": "T123"
},
{
"offsets": [
[
8620,
8700
]
],
"text": "['it treats the skin as a shell that moves by an explicit function of the skeleton']",
"type": "background_claim",
"id": "T124"
},
{
"offsets": [
[
8702,
8818
]
],
"text": "['Vertices of the skin are deformed by a weighted combination of the joint transformations of the character’s skeleton']",
"type": "background_claim",
"id": "T125"
},
{
"offsets": [
[
8820,
8824
]
],
"text": "['9–13']",
"type": "data",
"id": "T126"
},
{
"offsets": [
[
8886,
8934
]
],
"text": "['They are easy to understand and intuitive to use']",
"type": "background_claim",
"id": "T127"
},
{
"offsets": [
[
8936,
8990
]
],
"text": "['A tedious part is the proper assignment of the weights']",
"type": "background_claim",
"id": "T128"
},
{
"offsets": [
[
8992,
9046
]
],
"text": "['In production, the weights are painted by the animator']",
"type": "background_claim",
"id": "T129"
},
{
"offsets": [
[
9057,
9104
]
],
"text": "['the animator has full control over the outcomes']",
"type": "background_claim",
"id": "T130"
},
{
"offsets": [
[
9106,
9172
]
],
"text": "['The smooth skinning approach suffers from some notorious drawbacks']",
"type": "background_claim",
"id": "T131"
},
{
"offsets": [
[
9181,
9205
]
],
"text": "['the candy wrapper effect']",
"type": "data",
"id": "T132"
},
{
"offsets": [
[
9209,
9232
]
],
"text": "['collapsing elbow effect']",
"type": "data",
"id": "T133"
},
{
"offsets": [
[
9241,
9310
]
],
"text": "['its lack of consideration of volume preservation for the soft tissues']",
"type": "data",
"id": "T134"
},
{
"offsets": [
[
9312,
9413
]
],
"text": "['The example-based methods were developed as an alternative in order to overcome this kind of problems']",
"type": "background_claim",
"id": "T135"
},
{
"offsets": [
[
9426,
9447
]
],
"text": "['have had some success']",
"type": "background_claim",
"id": "T136"
},
{
"offsets": [
[
9415,
9420
]
],
"text": "['14–17']",
"type": "data",
"id": "T137"
},
{
"offsets": [
[
9520,
9582
]
],
"text": "['the animator can control the exact appearance of the character']",
"type": "background_claim",
"id": "T138"
},
{
"offsets": [
[
9449,
9465
]
],
"text": "['With this method']",
"type": "background_claim",
"id": "T139"
},
{
"offsets": [
[
9584,
9603
]
],
"text": "['In facial animation']",
"type": "background_claim",
"id": "T140"
},
{
"offsets": [
[
9618,
9713
]
],
"text": "['the animator often needs to dictate how a facial model deforms to achieve different expressions']",
"type": "background_claim",
"id": "T141"
},
{
"offsets": [
[
9741,
9834
]
],
"text": "['a large number of models have to be made in the pose space and stored for shape interpolation']",
"type": "background_claim",
"id": "T142"
},
{
"offsets": [
[
9836,
9864
]
],
"text": "['This is an expensive process']",
"type": "background_claim",
"id": "T143"
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{
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9866,
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],
"text": "['The drive for realism in computer graphics has lead to some new modelling and deformation techniques']",
"type": "background_claim",
"id": "T144"
},
{
"offsets": [
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9968,
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"text": "['A group of techniques that have gained increasing popularity in the computer animation of characters are those based on characters anatomy']",
"type": "background_claim",
"id": "T145"
},
{
"offsets": [
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10108,
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"text": "['These models attempt to mimic their real life counterparts by reproducing their anatomical structures']",
"type": "background_claim",
"id": "T146"
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{
"offsets": [
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10211,
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"text": "['These anatomy-based skinning methods differ on the complexity of the models and their behaviours of the underlying anatomical structures']",
"type": "background_claim",
"id": "T147"
},
{
"offsets": [
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10351,
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],
"text": "['Some use simple muscle shapes']",
"type": "background_claim",
"id": "T148"
},
{
"offsets": [
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10390,
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],
"text": "['abstract muscle operators']",
"type": "data",
"id": "T149"
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{
"offsets": [
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10418,
10420
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],
"text": "['18']",
"type": "data",
"id": "T150"
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{
"offsets": [
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10421,
10430
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],
"text": "['meatballs']",
"type": "data",
"id": "T151"
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{
"offsets": [
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10433,
10435
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],
"text": "['19']",
"type": "data",
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{
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],
"text": "['some employ detailed models']",
"type": "background_claim",
"id": "T153"
},
{
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10465,
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],
"text": "['20–23']",
"type": "data",
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{
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],
"text": "['The obvious advantage of this group of methods is its ability in achieving detailed visual quality during animation']",
"type": "background_claim",
"id": "T155"
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{
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"text": "['One of the difficulties of these techniques']",
"type": "background_claim",
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"text": "['is that they are indirect to use']",
"type": "background_claim",
"id": "T157"
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{
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10635,
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"text": "['24–28']",
"type": "data",
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10688,
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"text": "['one has to model the anatomical structures before its appearance arrives']",
"type": "data",
"id": "T159"
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{
"offsets": [
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10762,
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"text": "['Achieving a particular look of the skin requires the determination of the shape, number and the layout of the muscles underneath']",
"type": "background_claim",
"id": "T160"
},
{
"offsets": [
[
10892,
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"text": "['Until the skin mesh envelops the underlying structure, it is very hard to anticipate how the character looks like from the outside']",
"type": "background_claim",
"id": "T161"
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{
"offsets": [
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"text": "['recent research has looked into the issue of estimating the muscles from the skin shape']",
"type": "background_claim",
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"text": "['29–30']",
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"text": "['This new technique has had a degree of success']",
"type": "background_claim",
"id": "T164"
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11253,
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"text": "['The current limitations are that they could only use simple muscle shapes']",
"type": "background_claim",
"id": "T165"
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{
"offsets": [
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11328,
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"text": "['which are sufficient in obtaining detailed deformations']",
"type": "background_claim",
"id": "T166"
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{
"offsets": [
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15292,
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"text": "['a bar-net is coupled with a surface']",
"type": "data",
"id": "T167"
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{
"offsets": [
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15329,
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],
"text": "['the surface can be made to behave like a piece of elastic material']",
"type": "own_claim",
"id": "T168"
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{
"offsets": [
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15402,
15581
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"text": "['many numerical methods developed in structural and mechanical engineering for the manipulation of structures and networks can be applied to control the deformation of the surfaces']",
"type": "own_claim",
"id": "T169"
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{
"offsets": [
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15583,
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"text": "['Deforming the bar-net deforms the coupled surface']",
"type": "own_claim",
"id": "T170"
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{
"offsets": [
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15677,
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"text": "['Bar-nets can have any arbitrary topology']",
"type": "own_claim",
"id": "T171"
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{
"offsets": [
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15771,
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"text": "['unlike most curved surface patches']",
"type": "background_claim",
"id": "T172"
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{
"offsets": [
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15719,
15770
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"text": "['They are not restricted to a quadrilateral topology']",
"type": "own_claim",
"id": "T173"
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{
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15807,
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"text": "['Quadrilateral patches are the easiest to control']",
"type": "background_claim",
"id": "T174"
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{
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15860,
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"text": "['there have been many algorithms developed to implement them']",
"type": "background_claim",
"id": "T175"
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{
"offsets": [
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15925,
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"text": "['methods for controlling the deformation of a non-quadrilateral surface patch analytically remain an interesting research topic']",
"type": "background_claim",
"id": "T176"
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{
"offsets": [
[
16053,
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"text": "['Such a problem could be resolved by coupling a general mechanical bar-net with the control points of a surface patch of the same topology']",
"type": "background_claim",
"id": "T177"
},
{
"offsets": [
[
16939,
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"text": "['The principle idea of the proposed bar-net driven deformation technique is to regard the deformable area of an animated creature as a network, which deforms under an acting force']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
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17545,
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"text": "['One does not need to worry about the shape of the network itself']",
"type": "own_claim",
"id": "T179"
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{
"offsets": [
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17624,
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"text": "['we use a bar-net only as a control mechanism']",
"type": "data",
"id": "T180"
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{
"offsets": [
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17670,
17762
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],
"text": "['Changing the stiffness with other parameters unchanged has an influence on the whole network']",
"type": "own_claim",
"id": "T181"
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{
"offsets": [
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17764,
17823
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"text": "['This is in line with the physical property of human tissues']",
"type": "own_claim",
"id": "T182"
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{
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17838,
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],
"text": "['makes physical sense']",
"type": "own_claim",
"id": "T183"
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{
"offsets": [
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18143,
18224
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],
"text": "['the animator wishes to finetune the effects on the x, y or z direction separately']",
"type": "data",
"id": "T184"
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{
"offsets": [
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18226,
18261
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"text": "['the surface will deform as expected']",
"type": "own_claim",
"id": "T185"
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{
"offsets": [
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18263,
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"text": "['The deformation of the free nodes satisfies the superposition principle']",
"type": "own_claim",
"id": "T186"
},
{
"offsets": [
[
18355,
18433
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],
"text": "['one free node is subject to the influence of a number of forces simultaneously']",
"type": "data",
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"offsets": [
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18435,
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"text": "['the general deformation applied to the node is the same as the sum of all the deformations generated by applying these loads independently']",
"type": "own_claim",
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{
"offsets": [
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18575,
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"text": "['The benefit from this property is that several muscles, bones, fat tissues can affect the skin deformation simultaneously through summing up of their individual forces']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
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18744,
18849
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"text": "['Network form finding is always a numerically complicated problem in mechanical and structural engineering']",
"type": "background_claim",
"id": "T190"
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{
"offsets": [
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18851,
18882
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"text": "['Various numerical methods exist']",
"type": "background_claim",
"id": "T191"
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{
"offsets": [
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18894,
18932
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"text": "['most mechanical networks are concerned']",
"type": "data",
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"offsets": [
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18934,
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"text": "['the relationship between the equilibrium state and the acting forces is non-linear']",
"type": "background_claim",
"id": "T193"
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{
"offsets": [
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19112
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"text": "['Shape change cannot be trivially related to the magnitude and direction of the external forces']",
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"offsets": [
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19114,
19193
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"text": "['Often numerical algorithms are deployed to determine the rest form of a network']",
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{
"offsets": [
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19229
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"text": "['is inevitably time-consuming']",
"type": "background_claim",
"id": "T196"
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{
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19234,
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"text": "['not very useful for animation production']",
"type": "background_claim",
"id": "T197"
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{
"offsets": [
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19289,
19400
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"text": "['the effect of stiffness of a network can be approximated by the quantity of force-length ratios of all the bars']",
"type": "own_claim",
"id": "T198"
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{
"offsets": [
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19276,
19287
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"text": "['In our case']",
"type": "data",
"id": "T199"
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"offsets": [
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19523,
19525
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"text": "['31']",
"type": "data",
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"offsets": [
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19526,
19564
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"text": "['satisfies the above-defined properties']",
"type": "own_claim",
"id": "T201"
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{
"offsets": [
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19457,
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"text": "['Using this stiffness parameter, we found the force density method']",
"type": "own_claim",
"id": "T202"
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{
"offsets": [
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20240,
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"text": "['form finding problem with a set of linear equations']",
"type": "own_claim",
"id": "T203"
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{
"offsets": [
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19568,
19623
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"text": "['An added advantage is that it is also able to solve the']",
"type": "own_claim",
"id": "T204"
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{
"offsets": [
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20383,
20476
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],
"text": "['such bar-nets can be used to assist the deformation of skin surfaces of deformable characters']",
"type": "own_claim",
"id": "T205"
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{
"offsets": [
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20621,
20728
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"text": "['the prevailing advantages of this technique are its speed of computation and intuitiveness in shape control']",
"type": "own_claim",
"id": "T206"
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{
"offsets": [
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20730,
20801
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"text": "['Coupling bar-nets with a skin surface makes it ‘mechanically deformable']",
"type": "own_claim",
"id": "T207"
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{
"offsets": [
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20988,
21130
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"text": "['Using above defined bar-net properties, deformations are achieved by applying virtual forces to the appropriate free nodes of the control mesh']",
"type": "own_claim",
"id": "T209"
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{
"offsets": [
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21755,
21841
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],
"text": "['The gradually changed forces on the control mesh are bound to the elbow rotation angle']",
"type": "data",
"id": "T208"
},
{
"offsets": [
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21843,
21965
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"text": "['which produce both the bulge effect and compensate for the volume loss that the traditional smooth skinning method suffers']",
"type": "own_claim",
"id": "T210"
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{
"offsets": [
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21513,
21641
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],
"text": "['using a bar-net to solve the collapsing elbow problem and obtain a fat bulge effect when the forearm bends towards the upper arm']",
"type": "own_claim",
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{
"offsets": [
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21484,
21492
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],
"text": "['Figure 2']",
"type": "data",
"id": "T212"
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{
"offsets": [
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21967,
22027
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"text": "['Deformed muscles always change the shape of the skin surface']",
"type": "background_claim",
"id": "T213"
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{
"offsets": [
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22541,
22610
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"text": "['The deformation of the forearm is complex but relatively unnoticeable']",
"type": "own_claim",
"id": "T214"
},
{
"offsets": [
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22637,
22717
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],
"text": "['only the deformations caused by biceps brachii and triceps brachii are generated']",
"type": "own_claim",
"id": "T215"
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{
"offsets": [
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23560,
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"text": "['These loads deform the surface to form a natural muscle bulge']",
"type": "own_claim",
"id": "T216"
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{
"offsets": [
[
23635,
23646
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"text": "['Figure 3(c)']",
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23680,
23765
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"text": "['the animator can easily shape the characters using the virtual forces as user-handles']",
"type": "own_claim",
"id": "T218"
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{
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23950,
23985
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"text": "['it involves little computation cost']",
"type": "own_claim",
"id": "T219"
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{
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23854,
23945
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"text": "['there are 12 free nodes which are the only necessary resources involved in the form finding']",
"type": "own_claim",
"id": "T220"
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{
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23838,
23852
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"text": "['in the bar-net']",
"type": "data",
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{
"offsets": [
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23987,
24063
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"text": "['Local deformations can be similarly achieved by changing the force densities']",
"type": "own_claim",
"id": "T222"
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{
"offsets": [
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24078,
24158
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"text": "['reducing the force density of the network increases the size of the bulge effect']",
"type": "own_claim",
"id": "T223"
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{
"offsets": [
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24171,
24182
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"text": "['Figure 3(d)']",
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{
"offsets": [
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24185,
24296
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"text": "['The human shoulder is a typical area where notorious skin deformities occur using a traditional skinning method']",
"type": "background_claim",
"id": "T225"
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{
"offsets": [
[
24373,
24455
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"text": "['Most computer-animated characters are complex both geometrically and topologically']",
"type": "background_claim",
"id": "T226"
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{
"offsets": [
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24457,
24554
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"text": "['The use of quadrilateral meshes to model the geometry of such characters is frequently inadequate']",
"type": "background_claim",
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24821,
24872
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"text": "['the network is capable of handling any connectivity']",
"type": "own_claim",
"id": "T228"
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{
"offsets": [
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25168,
25233
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"text": "['The fine skin surface can be represented in various surface forms']",
"type": "own_claim",
"id": "T229"
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{
"offsets": [
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25235,
25298
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],
"text": "['Our implementation includes three major surface modelling forms']",
"type": "own_claim",
"id": "T230"
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{
"offsets": [
[
25300,
25305
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],
"text": "['Nurbs']",
"type": "data",
"id": "T231"
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{
"offsets": [
[
25307,
25327
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],
"text": "['subdivision surfaces']",
"type": "data",
"id": "T232"
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{
"offsets": [
[
25332,
25340
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],
"text": "['polygons']",
"type": "data",
"id": "T233"
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{
"offsets": [
[
25347,
25375
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],
"text": "['the control mesh is deformed']",
"type": "data",
"id": "T234"
},
{
"offsets": [
[
25377,
25474
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],
"text": "['it can deform the fine surface model using the Wrap deformer available in many animation packages']",
"type": "own_claim",
"id": "T235"
},
{
"offsets": [
[
25476,
25546
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],
"text": "['Bar-net driven skinning is applicable also to the modelling of wrinkle']",
"type": "own_claim",
"id": "T236"
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{
"offsets": [
[
25655,
25663
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"text": "['Figure 6']",
"type": "data",
"id": "T237"
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{
"offsets": [
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27440,
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"text": "['Character deformation in computer animation has attracted a great deal of research effort over the last two decades']",
"type": "background_claim",
"id": "T238"
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{
"offsets": [
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27598,
27659
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"text": "['had difficulties in creating realistic character deformations']",
"type": "background_claim",
"id": "T239"
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{
"offsets": [
[
27557,
27575
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],
"text": "['The earlier models']",
"type": "background_claim",
"id": "T240"
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{
"offsets": [
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27689,
27760
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],
"text": "['more physically based and CPU intensive computation models have emerged']",
"type": "background_claim",
"id": "T241"
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{
"offsets": [
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27770,
27810
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"text": "['the multi-layered anatomy-based approach']",
"type": "data",
"id": "T242"
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{
"offsets": [
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27860,
27958
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],
"text": "['it is undesirable to require the animator to model many muscles before the skin shape is developed']",
"type": "background_claim",
"id": "T243"
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{
"offsets": [
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27975,
28040
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"text": "['we propose a physically motivated deformation authoring technique']",
"type": "own_claim",
"id": "T244"
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{
"offsets": [
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28078,
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"text": "['Its main strength lies in the combination of speed, intuitiveness and good realism']",
"type": "own_claim",
"id": "T245"
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{
"offsets": [
[
28162,
28244
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"text": "['Our technique can achieve similar results to those of the anatomy-based techniques']",
"type": "own_claim",
"id": "T246"
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{
"offsets": [
[
28250,
28274
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"text": "['in an interactive manner']",
"type": "own_claim",
"id": "T247"
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{
"offsets": [
[
28682,
28822
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],
"text": "['we have devised a set of properties that an ‘ideal’ bar-net should satisfy, which make intuitive shape control and fast computation possible']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
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28541,
28609
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"text": "['can take advantage of the physical behaviour inherent to the network']",
"type": "own_claim",
"id": "T249"
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{
"offsets": [
[
28414,
28536
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],
"text": "['Coupling a part of surface mesh with bar-nets allows the surface deformation to be controlled by manipulating the networks']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
29016,
29055
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],
"text": "['achieves realistic deformation outcomes']",
"type": "own_claim",
"id": "T251"
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{
"offsets": [
[
28968,
29011
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],
"text": "['This makes the skin mechanically deformable']",
"type": "own_claim",
"id": "T252"
},
{
"offsets": [
[
29209,
29334
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],
"text": "['They can be used individually as an interactive modelling tool or collectively to mimic the muscle forces from a muscle group']",
"type": "own_claim",
"id": "T253"
},
{
"offsets": [
[
29462,
29566
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],
"text": "['It provides the animator with a new deformer which can be used both as a modelling and an animation tool']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
29753,
29810
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],
"text": "['the tools developed so far are still relatively primitive']",
"type": "own_claim",
"id": "T255"
},
{
"offsets": [
[
29812,
29932
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],
"text": "['The user needs to understand the basic principles of the bar-net properties before the technique can be used efficiently']",
"type": "own_claim",
"id": "T256"
},
{
"offsets": [
[
29957,
30037
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],
"text": "['we are currently designing higher-level tools with an interactive user interface']",
"type": "own_claim",
"id": "T257"
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A40 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A40_C10_Modelling_Effects_of_Wind_Fields_in_Cloth_Animations_CITATION_PURPOSE_M_v1.xml">
4fef03a6a1d1004158274410456824728d0af841568d973768d12f42e6ba9700
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http://dx.doi.org/10.1007/bf01782240
<Title>Modelling Effects of Wind Fields in Cloth Animations</Title>
M. Keckeisen, S. Kimmerle, B. Thomaszewski, and M. Wacker WSI/GRIS University of Tübingen Sand 14, D-72076 Tübingen, Germany {keckeisen, kimmerle, wacker}@gris.uni-tuebingen.de [email protected]
<Abstract>In this paper we show how to incorporate effects of wind fields in cloth animations. We discuss two different approaches to model force fields describing air motion and show how these models can be augmented to exhibit interaction with deformable thin objects such as textiles. The first model is based on the Navier-Stokes equations, while the second method extends simple particle tracing methods by the effect of lee. In each case, we present a method for simulating the interaction of cloth movements with the wind field. Both methods have been integrated in an existing cloth simulation system, and we compare their respective advantages and disadvantages. </Abstract>
Keywords: Wind Simulation, Cloth Animation, Physically Based Modelling, Computer Graphics.
1.
WSCG
2004
February
2-6
2003
Plzen
Czech Republic.
<H1>INTRODUCTION</H1>
While the simulation of wind is an area of vast interest in aerodynamic engineering, computational fluid dynamics (CFD), and animation/visualisation of fluids in computer graphics, it has been a rather abandoned subject in simulation of deformable objects such as cloth simulation. This is mostly due to the fact that conventional CFD applications require enormous computational power. However, aerodynamic effects are obviously capable of enhancing the realism of an animated scene and thus are an important part of a cloth simulation system. For example, air resistance is a vital component which cannot be neglected if realistic animation is desired [ BTH + 03 ]. In this work we discuss two different approaches to
model force fields describing air motion and show how these forces can be applied to generate aerodynamic effects on textiles. The first method is based on modelling air flows with the Navier-Stokes equations. With this model, the effect of wind fields on smoke has already been investigated
[FSJ01], and in this paper we extend this approach to wind effects on textiles. However, the solution of the sophisticated NavierStokes equations is not desired and not necessary in all situations where wind effects shall be integrated into cloth simulations. For this case, we present a much simpler model based on tracing wind particles that move along a global force field. During the simulation, the wind field is evaluated for each wind particle at its current position. The so determined flow field is used to compute the wind force which is added to the forces that act on the textiles. By detecting collisions between the wind particles and objects in the scene, we are able to simulate the important effect of lee even with this straightforward method. Of course, both approaches have different characteristics and aim at different applications for wind simulation. In this work, we compare the models’ advantages and disadvantages and show practical results of the described methods. The rest of this paper is organized as follows. After giving an overview of related work in section 2, we describe the aerodynamic force model applied in this work in section 3. In section 4 we discuss the two wind field models, namely the one based on the Navier-Stokes equations in 4.1, and the one based on tracing wind particles in 4.2. Finally, we show some practical results of the implemented algorithms in section 5, before we conclude the paper in section 6.
2.
<H1>PREVIOUS WORK</H1>
Models for fluid dynamics can be essentially subdivided into two categories. Simple models which are commonly used in most computer graphics applications describe the wind flow by predefined flow functions. Here, global functions are defined to model the velocity of wind. Either special flow primitives can be combined
[ WH91 ; LDG96 ; Li00 ] or visually pleasing functions introducing random turbulence [ SF92 ; SF93 ; Sta97 ] are taken into account to model even complex wind scenes. Many models use this method to move objects in the wind field through the scene [ Ree83 ; Sim90 ; WH91; BLM95 ]. In addition, physically based fluid dynamics solving equations of motions with particle methods were presented recently
[ IK03 ]. However, fixed flow functions lack interaction with the user or objects in the scene. Hence, with increasing computer power, computer graphics concentrates on physically more accurate simulations. In many fields the Navier-Stokes equations are the standard mathematical formulation to model fluid dynamics. A vast literature exists on how to solve these equations numerically. CFD are applied in this field for engineering tasks with a high degree of quality requirements. Unfortunately, it is quite difficult to apply these algorithms in computer graphics due to enormous calculation times. Hence, faster fluid solvers were investigated for computer graphics applications. Kajiya et al. [ KvH84 ], Yaeger et al. [ YU86 ], and Gamito et al. [ GLG95 ] worked on fluid dynamics solvers in two dimensions, and many improvements and variants followed [ CdVLHM97 ; KM90 ]. Foster and Metaxas [ FM96 ; FM97 ]
, and Griebel et al. [ GDN98 ]
presented a solver for the fully three dimensional Navier-Stokes equations.
Due to explicit integration methods very small step sizes had to be used.
To enable faster simulations, a solution with an unconditionally stable solver was introduced by Stam
[Sta99 ] and further extended in
[ FSJ01 ; Sta01 ; Sta03 ]. Modelling interaction of fluids with solid objects has been investigated by Takahashi et al. [ TFK + 03 ] and Génevaux et al.
[ GHD03 ]. Recently, Wei et al. [ WZF + 03 ] presented an interesting approach to simulate lightweight objects like soap bubbles and feathers in a wind flow using a Lattice Boltzmann Model extended with a subgrid model. For interaction of highly deformable objects and especially cloth-like objects only few models have been investigated. Simple models consist in the calculation of lift and drag forces from the surrounding velocity field [SF92; Pro95; KCC + 00; KCCL01]. More complex interaction models calculate the wind force by a panel method [LDG96; Li00] introducing local vortices. In this work, we show how recent results in fluid dynamics for computer graphics can be exploited to simulate interaction of wind flows with textiles. Moreover, we extend the more straightforward approach of global wind field functions by the effect of lee.
3.
<H1>AERODYNAMICS</H1>
To incorporate wind effects in a physically based animation we have to apply additional external forces in the dynamical model of the deformable objects. Hence, given a wind flow represented by a velocity field in the scene we calculate the forces which are exerted on the simulated objects. In this section, we briefly describe the model we use to compute the effective aerodynamic forces such as wind force and air resistance, mainly following
[SF92]. The wind force acting on objects in an air stream is decomposed into two components: the lift force F L and the drag force F D (see figure 1 ).
cloth wind direction u F L V rel q F D ^ n
Figure 1:
The decomposition of wind forces (side
view).
The direction of the drag force F D is diametral to the relative velocity v rel = v object − u, where v object is the object’s velocity and u is the velocity field of the wind. Note that in the case of a windless situation, i.e. u = 0, we still have air resistance for moving objects. Since two-dimensional objects do not exhibit an inside and outside, the unit normal n i of the i-th face of the object mesh (cf. figure 1 ) is replaced by
3.1
n i = n − i n i if n i . · v i,rel > 0 ,
The drag force per face is then given by F i,D = 2 1 C D ρ|v i,rel | 2 A · ( n i · v i,rel ) · (− v i,rel ) ,
where C D is the specific air resistance coefficient, ρ the density of air, A is the area of the corresponding face, and v i,rel the unit relative velocity vector of the face. The direction of the lift force, which is perpendicular to v i,rel and lies in the plane spanned by v i,rel and n i , is given by u i = ( n i × v i,rel ) × v i,rel .
Then the lift force is calculated as F i,L = 1 2 C L ρ|v i,rel | 2 A cos θ · u i , where C L is the lift force coefficient, and θ is the angle between v i,rel and the actual face.
4.
<H1>WIND FIELD MODELS</H1>
In this section we describe two different wind field models and show how they can be used to model wind effects on textiles. The first model is based on the work of Stam
[Sta97] and calculates the numerical solution of the Navier-Stokes equation with a semi-Lagrangian approach. This model is extended to interaction of the wind flow with textiles. The second model employs precomputed wind flows and particle tracing methods. This approach is much easier to implement and can be added to existing simulation modules without additional computational cost. In section 4.2, we show how to produce realistic effects of wind on textiles including lee effects. 4.1 The Navier-Stokes equations The Navier-Stokes equations describe a precise mathematical model for fluid flows. The numerical algorithms used in CFD to solve these equations are designed for physical accuracy for engineering applications and are expensive in computation. But in our case where this precision is not necessary simplifications can be made which greatly reduce the computation costs as described by Stam [Sta03]. Since the arising wind velocities are clearly below the speed of sound, compressibility effects are negligible, and the wind is modelled as an incompressible constant density fluid. This notably simplifies the numerical approximation, and the incompressible Navier-Stokes equations can be written in a compact vector notation as
4.1
∇ · u = 0,
4.2
∂u = −(u · ∇)u − 1 ∇p + ν∇ 2 u + f . ∂t ρ
Here, u describes the (three-dimensional) velocity field, ν is the kinematic viscosity of the fluid, ρ its density, p the pressure in the wind field, and f accounts for external forces. The first equation states that the velocity field should be incompressible while the second one describes the evolution of a velocity field over time. The first term on the right hand side reflects the change of velocity due to advection, while the second expression accounts for any external force f and acceleration caused by the local pressure gradient ∇p and by viscous drag depending on ν. In the following we briefly describe the numerical solution of the Navier-Stokes equations (4.1) and (4.2) by Stam
[Sta97]. To solve these equations numerically they first have to be discretised. For this, the computational domain is diced up into equally sized cubes forming a grid as described in section 4.2.2, and sample values of velocity and pressure are defined at the cell centres. Foster and Metaxas [FM96] use a finite difference approximation for the discretisation of the operators in equation (4.2). Then they update the cell’s velocities according to the divergence value computed in each direction, respectively, using an explicit integration scheme. Since time steps in explicit computations usually need to be very small, we follow Stam
[Sta99]
who proposes an implicit integration scheme, which allows stable simulations with large time steps. While the linear terms in equation (4.2) are straightforward to solve implicitly, the term −(u · ∇)u is nonlinear and deserves special attention. Here, a different approach based on the method of characteristics is used to solve the advection equation. Equation (4.2) does not provide a divergent-free velocity field. Therefore, the divergence of each cell in the grid has to be projected to zero using the Helmholtz-Hodge decomposition
[Sta03]. The major advantage of Navier-Stokes based approaches consists in the fact that the evolution of the wind flow over time is calculated. It enables us to model global effects like convection and diffusion on a physical basis. We present a model to exploit these wind models for calculating the interaction of deformable objects with the air flow by a boundary condition method. As already stated by Stam [Sta03] “a velocity field of its own isn’t really visually interesting until it starts moving objects [...]”. That means in particular that all objects in the scene interact with the fluid present in it, i.e. in our case clothes with the wind. On the one hand the wind deforms the objects which on the other hand change the wind flow. To describe the above situation by a physical model we require the Neumann boundary condition ∂u =0 ∂n to be satisfied for the wind flow u at any boundary point of an object with normal n. Rigid objects like walls will influence the fluid field but will not be affected by fluid forces themselves. Deformable objects like cloth are supposed to both experience fluid forces and itself influence the fluid flow. This in fact is a major difficulty. Consider a point p b on the boundary of a deformable object in the scene. Let u(p b ) be the corresponding wind velocity at that point and n be its normal. On the one hand, we want the Neumann boundary condition u(p b ) · n = 0 to be satisfied. On the other hand, the wind velocity orthogonal to the object’s surface is just what causes the aerodynamic forces. Without further remedial action setting the boundary according to the Neumann condition would mean that the fluid will not exert forces on the objects. Here we propose a method which meets both requirements. For every deformable object the velocity value of the surrounding wind field for every vertex of the representing mesh is tracked. In the solver’s cycle, the boundary conditions are then set prior to any other operation: For every object in the scene each triangle of its representing mesh is registered in the fluid grid, which means that the cell of the wind field occupied by the object is marked as occupied. The wind velocity at the vertex positions of the object is recorded. Additionally, the normals of these vertices are stored. Then, the aerodynamic forces as described in section 3 are calculated. Finally, for every marked cell in the scene the previously stored normals are averaged in one space cell which are used to update the velocity at the cell to satisfy the Neumann boundary condition. Thus, the boundary conditions are met and yet aerodynamic forces are obtained. A different issue is how to deal with the inside of (rigid) objects. The method to set boundary conditions as described above does not account for the interior of objects. Thus, a nonzero velocity could be mistakenly assigned to cells lying inside an object. To avoid this situation, the path of the wind flow is checked for object intersection, whereby the collision detection of the cloth simulation system provides a simple method to deal with this issue [MKE03]. 4.2 Particle Tracing on Wind Fields Here we combine the idea of creating wind fields by predefined flow primitives with particle tracing in given flow fields. To define a wind scene we first built up the air flow by simple primitives such as parallel directed wind fields, vortices, etc. We then use a par- ticle tracing method in the defined wind field to determine the effect of lee by detecting windless areas. This method is very easy to implement and yields very plausible and nicely looking results. A simple approach to generate complex air flows is to define a wind field by mathematical functions which assign to each point in space a unique velocity value. As Wejchert et al. [WH91] have shown, this already enables an animator to design even complex wind fields: Assuming an irrotational (∇ × u = 0), inviscid, and incompressible (∇ · u = 0) fluid, the NavierStokes equations which describe the mechanics of any fluid (see section 4.1) can be simplified to give the Laplace equation
4.1.1 The solution of the Navier-Stokes equations
4.1.2 Wind effects on clothes
4.2.1 Global Wind Fields
4.3
∇ · u = ∇∇φ = ∇ 2 φ = 0,
where φ is the potential of the given wind field. Thus, the velocity field u is given by The linearity of equation (4.3) enables an animator to combine basic flows which satisfy equation (4.3) as he likes and thus to obtain complex fluid flows. Some primitives common to fluid simulations are depicted in figure 2 . One drawback of this model is that it cannot handle objects exhibiting complex boundaries. The approach to model solid objects in the scene taken by Wejchert et al. consists in placing a wind source using a mirror principle in order to extinguish the air flow at the boundary of the object. While this works for simple objects this approach is not feasible at all with deformable objects like textiles. Another more serious drawback of this model for our application consists in the lack of interaction with objects. The wind flow defined by the primitives will not react on objects in the scene which means for example that tissues in the lee of other objects will be affected by the wind flow as well. However, this method can be combined with the aerodynamic model described in section 3 to give nice and fast results as will be shown in section 5. To solve the described problems we propose a model which combines the simple global wind flow techniques with a particle tracing method. Here, particles are moved along the wind field to determine the effect of objects in the scene. This model divides the scene into parallelepiped cells.
There are two common approaches to discretising the continuous velocity field defined in space: one can either choose the midpoint of a cell
[Sta99] or its six faces
[FM96] to define the field. As usual, values between the defining points of the grid are interpolated using trilinear functions. The basic idea of the particle tracing method is to trace wind particles through a field w = i w i defined by linear superposition of wind sources corresponding to flow primitives with respective velocity fields w i . The field w does not account for lee effects caused by objects in the flow. Therefore we compute the wind field u containing these effects as follows. In our model every wind source is also a particle source: These particles form an uncoupled particle system which can be considered as a wind gust. The wind particles are emitted into the velocity field w i of the corresponding wind source which is defined on a grid. The specific emission intervals and amounts depend on the properties of the flow sources. In every time step each particle in a wind gust moves along its velocity field w i defined by the corresponding wind source. Notice that the movement of the particles in a wind gust is only affected by the wind source they belong to. The global superposition of all wind sources has no effect on these particles. To calculate the wind particles’ positions we used the explicit Euler integration scheme. For a wind particle at position p t and time t this results in a path s(p t , p t+∆t ), where p t+∆t denotes the position after time step ∆t according to p t+∆t = p t + w i (p t , ∆t) . As a particle moves along its path in space, all grid cells colliding with the path are updated with the velocity of the associated wind source with respect to the position of the particle. The particle might cross several grid cells on its way during a single time step. If this is the case, the path of the particle has to be subdivided into parts not exceeding the size of a grid cell. This path is then tested for collisions with the objects in the scene. The velocity field u is then computed as for each grid cell separately, where w i are all those wind sources whose particles have reached the cell. If a collision is detected at position p col the normal of the colliding object n obj (p col ) is determined and the velocity of the particle is set to w i (p col , t+∆t) = w i (p col , t)−(n obj ·w i (p col , t))·n obj This assures that the velocity component of the resulting field u is orthogonal to the collision object’s surface at p col is zero, i.e. u(p col , t + ∆t) · n obj = 0 , and thus no flow propagates through the object. The wind force effective on objects in the scene is then computed from the velocity field u. Since u is determined using the wind particles, every point p that could not be reached by any wind particle will hold zero velocity even if w may hold a nonzero velocity. Thus, this method solves the problems described in section 4.2.1. Note that the somewhat tempting simplification of tagging each cell to either have wind in it or not is not valid. Imagine the simple scene in which there are two directional wind sources with opposite wind directions. Let them further have equal velocity magnitude and no distance attenuation. If we now place a solid object in between these two sources a rather undesired effect would occur using this simplification: on both sides of the solid object all cells would be tagged as having wind. But evaluating the wind field at every cell we would obtain a zero velocity. This is due to the extinguishing effect of the superposition of the two wind sources. Therefore, it is crucial for the particles to have the associated velocity of their wind source and not just the velocity resulting from the global superposition of all wind sources. 4.3 Comparison In this section we comment on the different models described in this paper. For physically accurate simulations based on the common method in fluid dynamics the model introduced by Stam produces realistic effects which global wind field models can never achieve. It produces nice swirls and vortices derived from dynamical characteristics of the fluid. However implementing the fluid solver is quite complex and using a high grid resolution is computationally expensive. Hence, the global wind field model is better suited for an easy to implement tool which is easy to adapt to specific situations. Particle systems are very common in the simulation engines and most functionality can be adapted to integrate the proposed wind model. Even with this straightforward approach, nice, realistic looking results can be achieved which is illustrated in the next section.
u = ∇φ .
Directional Vortex Point
Figure 2: Flow primitives
4.2.2 Particle Tracing
u = w i
.
5.
<H1>RESULTS</H1>
In this section we present some practical results of the previously described methods. We implemented the wind models described in sections 4.1 and 4.2 in a cloth animation system that employs a fast finite element method to simulate the drape of textiles with measured material properties
[EKS03]. For collision detection (between deformable objects, rigid objects, and wind particles) we use k-DOP hierarchies as described in
[MKE03]. Figure 3 shows a flag blowing in the wind. We used the particle tracing method described in section 4.2 to model the effects of a directional wind field on the flag. In figures 4 and 5 we show the ability of both the Navier-Stokes equations model (section 4.1) and the particle tracing method (section 4.2) to model the effect of lee. Two flags are exposed to a wind field, but the wind is blocked by a wall, so one of the flags is not affected by the wind. The images in figure 6 show a character wearing a shirt and standing in a wind stream coming from the front. Images (a) and (b) are snapshots from the beginning of the animation, images (c) and (d) show the result after the wind field has affected the clothes. To show the improved realism when simulating lee effects, we let the the wind act on all polygons of the shirt on the right (no lee effect). For the shirt on the left we used the Particle Tracing Method to simulate lee effects which, we think, gives more realistic results (see also the accompanying video).
6.
<H1>CONCLUSIONS</H1>
We presented two models for including advanced wind effects into cloth simulation systems. The first concentrates on physically accurate computations using a semi-Lagrangian approach to solve the Navier-Stokes equations, the second model incorporates a particle tracing method for global wind fields. As illustrated in the previous section both methods produce realistic looking results which are capable of enhancing the realism of computer animations. While the first model has a wider range of applications, the second one provides an easy method which still delivers realistic effects such as air resistance and lee. All methods described in this work should be easy to extend to three-dimensional deformable objects. All the methods apply the same except for simple changes. Since three-dimensional objects define an inner and outer part, the adaption of the face normals in equation (3.1) is not necessary. Moreover, in the wind field computation care has to be taken that no wind field is present in the object. Here, the same method as described for rigid objects (section 4.2.2) can be applied.
7.
<H1>Acknowledgements</H1>
This work was partially supported by the bmb+f research project Virtual Try-On (www.VirtualTryOn.de).
8.
<H1>References</H1>
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Figure 3: Flag blowing in the wind (particle tracing method).
Figure 4: Two flags blowing in the wind (Navier-Stokes equations). The wind is blocked by the wall, so the right flag is in the lee. Note that the simulation starts with both flags in an unfolded state.
Figure 5: Two flags blowing in the wind (particle tracing method). The wind is blocked by the wall, so the right flag is in the lee (compare figure 4 ).
(a) (b)
Figure 6: A character wearing a shirt and standing in a wind stream coming from the front. Images (a) and (b) are snapshots from the beginning of the animation, images (c) and (d) show the result after the wind field has affected the clothes. To show the improved realism when simulating lee effects, we let the the wind act on all polygons of the shirt on the right (no lee effect). For the shirt on the left we used the particle tracing method to simulate lee effects which gives more realistic results.
(c) (d)
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7070
]
],
"text": "['we show how recent results in fluid dynamics for computer graphics can be exploited to simulate interaction of wind flows with textiles']",
"type": "own_claim",
"id": "T88"
},
{
"offsets": [
[
7082,
7177
]
],
"text": "['we extend the more straightforward approach of global wind field functions by the effect of lee']",
"type": "own_claim",
"id": "T89"
},
{
"offsets": [
[
7242,
7394
]
],
"text": "['To incorporate wind effects in a physically based animation we have to apply additional external forces in the dynamical model of the deformable objects']",
"type": "own_claim",
"id": "T90"
},
{
"offsets": [
[
7403,
7532
]
],
"text": "['given a wind flow represented by a velocity field in the scene we calculate the forces which are exerted on the simulated objects']",
"type": "own_claim",
"id": "T91"
},
{
"offsets": [
[
7551,
7687
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],
"text": "['we briefly describe the model we use to compute the effective aerodynamic forces such as wind force and air resistance, mainly following']",
"type": "own_claim",
"id": "T92"
},
{
"offsets": [
[
7690,
7694
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],
"text": "['SF92']",
"type": "data",
"id": "T93"
},
{
"offsets": [
[
7697,
7780
]
],
"text": "['The wind force acting on objects in an air stream is decomposed into two components']",
"type": "own_claim",
"id": "T94"
},
{
"offsets": [
[
7782,
7800
]
],
"text": "['the lift force F L']",
"type": "data",
"id": "T95"
},
{
"offsets": [
[
7805,
7823
]
],
"text": "['the drag force F D']",
"type": "data",
"id": "T96"
},
{
"offsets": [
[
7829,
7837
]
],
"text": "['figure 1']",
"type": "data",
"id": "T97"
},
{
"offsets": [
[
8000,
8094
]
],
"text": "['The direction of the drag force F D is diametral to the relative velocity v rel = v object − u']",
"type": "own_claim",
"id": "T98"
},
{
"offsets": [
[
8236,
8283
]
],
"text": "['we still have air resistance for moving objects']",
"type": "own_claim",
"id": "T99"
},
{
"offsets": [
[
8202,
8222
]
],
"text": "['a windless situation']",
"type": "data",
"id": "T100"
},
{
"offsets": [
[
8353,
8408
]
],
"text": "['the unit normal n i of the i-th face of the object mesh']",
"type": "own_claim",
"id": "T101"
},
{
"offsets": [
[
8425,
8439
]
],
"text": "['is replaced by']",
"type": "own_claim",
"id": "T102"
},
{
"offsets": [
[
8473,
8495
]
],
"text": "['n i = n − i n i if n i']",
"type": "own_claim",
"id": "T103"
},
{
"offsets": [
[
8291,
8351
]
],
"text": "['two-dimensional objects do not exhibit an inside and outside']",
"type": "data",
"id": "T104"
},
{
"offsets": [
[
8498,
8514
]
],
"text": "['· v i,rel > 0']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
8414,
8422
]
],
"text": "['figure 1']",
"type": "data",
"id": "T106"
},
{
"offsets": [
[
8534,
8560
]
],
"text": "['The drag force per face is']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
[
8566,
8641
]
],
"text": "['given by F i,D = 2 1 C D ρ|v i,rel | 2 A · ( n i · v i,rel ) · (− v i,rel )']",
"type": "own_claim",
"id": "T108"
},
{
"offsets": [
[
8828,
8991
]
],
"text": "['The direction of the lift force, which is perpendicular to v i,rel and lies in the plane spanned by v i,rel and n i , is given by u i = ( n i × v i,rel ) × v i,rel']",
"type": "own_claim",
"id": "T109"
},
{
"offsets": [
[
9007,
9082
]
],
"text": "['the lift force is calculated as F i,L = 1 2 C L ρ|v i,rel | 2 A cos θ · u i']",
"type": "own_claim",
"id": "T110"
},
{
"offsets": [
[
9266,
9373
]
],
"text": "['we describe two different wind field models and show how they can be used to model wind effects on textiles']",
"type": "own_claim",
"id": "T111"
},
{
"offsets": [
[
9375,
9419
]
],
"text": "['The first model is based on the work of Stam']",
"type": "own_claim",
"id": "T112"
},
{
"offsets": [
[
9422,
9427
]
],
"text": "['Sta97']",
"type": "data",
"id": "T113"
},
{
"offsets": [
[
9429,
9528
]
],
"text": "['and calculates the numerical solution of the Navier-Stokes equation with a semi-Lagrangian approach']",
"type": "own_claim",
"id": "T114"
},
{
"offsets": [
[
9530,
9598
]
],
"text": "['This model is extended to interaction of the wind flow with textiles']",
"type": "own_claim",
"id": "T115"
},
{
"offsets": [
[
9600,
9676
]
],
"text": "['The second model employs precomputed wind flows and particle tracing methods']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
9678,
9805
]
],
"text": "['This approach is much easier to implement and can be added to existing simulation modules without additional computational cost']",
"type": "own_claim",
"id": "T117"
},
{
"offsets": [
[
10022,
10180
]
],
"text": "['The numerical algorithms used in CFD to solve these equations are designed for physical accuracy for engineering applications and are expensive in computation']",
"type": "background_claim",
"id": "T118"
},
{
"offsets": [
[
10236,
10327
]
],
"text": "['simplifications can be made which greatly reduce the computation costs as described by Stam']",
"type": "background_claim",
"id": "T119"
},
{
"offsets": [
[
10329,
10334
]
],
"text": "['Sta03']",
"type": "data",
"id": "T120"
},
{
"offsets": [
[
10204,
10235
]
],
"text": "['this precision is not necessary']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
10343,
10407
]
],
"text": "['the arising wind velocities are clearly below the speed of sound']",
"type": "background_claim",
"id": "T122"
},
{
"offsets": [
[
10409,
10447
]
],
"text": "['compressibility effects are negligible']",
"type": "background_claim",
"id": "T123"
},
{
"offsets": [
[
10453,
10517
]
],
"text": "['the wind is modelled as an incompressible constant density fluid']",
"type": "background_claim",
"id": "T124"
},
{
"offsets": [
[
10519,
10570
]
],
"text": "['This notably simplifies the numerical approximation']",
"type": "background_claim",
"id": "T125"
},
{
"offsets": [
[
10576,
10665
]
],
"text": "['the incompressible Navier-Stokes equations can be written in a compact vector notation as']",
"type": "background_claim",
"id": "T126"
},
{
"offsets": [
[
10699,
10708
]
],
"text": "['∇ · u = 0']",
"type": "background_claim",
"id": "T127"
},
{
"offsets": [
[
10752,
10786
]
],
"text": "['∂u = −(u · ∇)u − 1 ∇p + ν∇ 2 u + f']",
"type": "background_claim",
"id": "T128"
},
{
"offsets": [
[
10789,
10793
]
],
"text": "['∂t ρ']",
"type": "background_claim",
"id": "T129"
},
{
"offsets": [
[
10997,
11071
]
],
"text": "['The first equation states that the velocity field should be incompressible']",
"type": "background_claim",
"id": "T130"
},
{
"offsets": [
[
11078,
11146
]
],
"text": "['the second one describes the evolution of a velocity field over time']",
"type": "background_claim",
"id": "T131"
},
{
"offsets": [
[
11517,
11587
]
],
"text": "['To solve these equations numerically they first have to be discretised']",
"type": "background_claim",
"id": "T132"
},
{
"offsets": [
[
11783,
11801
]
],
"text": "['Foster and Metaxas']",
"type": "background_claim",
"id": "T133"
},
{
"offsets": [
[
11803,
11807
]
],
"text": "['FM96']",
"type": "data",
"id": "T134"
},
{
"offsets": [
[
11809,
11904
]
],
"text": "['use a finite difference approximation for the discretisation of the operators in equation (4.2)']",
"type": "background_claim",
"id": "T135"
},
{
"offsets": [
[
11906,
12057
]
],
"text": "['Then they update the cell’s velocities according to the divergence value computed in each direction, respectively, using an explicit integration scheme']",
"type": "background_claim",
"id": "T136"
},
{
"offsets": [
[
12065,
12130
]
],
"text": "['time steps in explicit computations usually need to be very small']",
"type": "background_claim",
"id": "T137"
},
{
"offsets": [
[
12157,
12255
]
],
"text": "['who proposes an implicit integration scheme, which allows stable simulations with large time steps']",
"type": "background_claim",
"id": "T138"
},
{
"offsets": [
[
12132,
12146
]
],
"text": "['we follow Stam']",
"type": "background_claim",
"id": "T139"
},
{
"offsets": [
[
12149,
12154
]
],
"text": "['Sta99']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
12263,
12337
]
],
"text": "['the linear terms in equation (4.2) are straightforward to solve implicitly']",
"type": "background_claim",
"id": "T141"
},
{
"offsets": [
[
12339,
12401
]
],
"text": "['the term −(u · ∇)u is nonlinear and deserves special attention']",
"type": "background_claim",
"id": "T142"
},
{
"offsets": [
[
12510,
12573
]
],
"text": "['Equation (4.2) does not provide a divergent-free velocity field']",
"type": "background_claim",
"id": "T143"
},
{
"offsets": [
[
12586,
12693
]
],
"text": "['the divergence of each cell in the grid has to be projected to zero using the Helmholtz-Hodge decomposition']",
"type": "background_claim",
"id": "T144"
},
{
"offsets": [
[
12696,
12701
]
],
"text": "['Sta03']",
"type": "data",
"id": "T145"
},
{
"offsets": [
[
12704,
12838
]
],
"text": "['The major advantage of Navier-Stokes based approaches consists in the fact that the evolution of the wind flow over time is calculated']",
"type": "background_claim",
"id": "T146"
},
{
"offsets": [
[
12840,
12927
]
],
"text": "['It enables us to model global effects like convection and diffusion on a physical basis']",
"type": "background_claim",
"id": "T147"
},
{
"offsets": [
[
12929,
13079
]
],
"text": "['We present a model to exploit these wind models for calculating the interaction of deformable objects with the air flow by a boundary condition method']",
"type": "own_claim",
"id": "T148"
},
{
"offsets": [
[
13116,
13208
]
],
"text": "['a velocity field of its own isn’t really visually interesting until it starts moving objects']",
"type": "background_claim",
"id": "T149"
},
{
"offsets": [
[
13108,
13113
]
],
"text": "['Sta03']",
"type": "data",
"id": "T150"
},
{
"offsets": [
[
13247,
13309
]
],
"text": "['all objects in the scene interact with the fluid present in it']",
"type": "background_claim",
"id": "T151"
},
{
"offsets": [
[
13442,
13629
]
],
"text": "['To describe the above situation by a physical model we require the Neumann boundary condition ∂u =0 ∂n to be satisfied for the wind flow u at any boundary point of an object with normal n']",
"type": "own_claim",
"id": "T152"
},
{
"offsets": [
[
13631,
13738
]
],
"text": "['Rigid objects like walls will influence the fluid field but will not be affected by fluid forces themselves']",
"type": "background_claim",
"id": "T153"
},
{
"offsets": [
[
13740,
13850
]
],
"text": "['Deformable objects like cloth are supposed to both experience fluid forces and itself influence the fluid flow']",
"type": "background_claim",
"id": "T154"
},
{
"offsets": [
[
13852,
13856
]
],
"text": "['This']",
"type": "background_claim",
"id": "T155"
},
{
"offsets": [
[
13865,
13886
]
],
"text": "['is a major difficulty']",
"type": "background_claim",
"id": "T156"
},
{
"offsets": [
[
14249,
14399
]
],
"text": "['Without further remedial action setting the boundary according to the Neumann condition would mean that the fluid will not exert forces on the objects']",
"type": "background_claim",
"id": "T157"
},
{
"offsets": [
[
14061,
14131
]
],
"text": "['we want the Neumann boundary condition u(p b ) · n = 0 to be satisfied']",
"type": "background_claim",
"id": "T158"
},
{
"offsets": [
[
14152,
14247
]
],
"text": "['the wind velocity orthogonal to the object’s surface is just what causes the aerodynamic forces']",
"type": "background_claim",
"id": "T159"
},
{
"offsets": [
[
14406,
14455
]
],
"text": "['we propose a method which meets both requirements']",
"type": "own_claim",
"id": "T160"
},
{
"offsets": [
[
15270,
15341
]
],
"text": "['the boundary conditions are met and yet aerodynamic forces are obtained']",
"type": "own_claim",
"id": "T161"
},
{
"offsets": [
[
15075,
15262
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],
"text": "['for every marked cell in the scene the previously stored normals are averaged in one space cell which are used to update the velocity at the cell to satisfy the Neumann boundary condition']",
"type": "own_claim",
"id": "T162"
},
{
"offsets": [
[
15343,
15410
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],
"text": "['A different issue is how to deal with the inside of (rigid) objects']",
"type": "own_claim",
"id": "T163"
},
{
"offsets": [
[
15412,
15513
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],
"text": "['The method to set boundary conditions as described above does not account for the interior of objects']",
"type": "own_claim",
"id": "T164"
},
{
"offsets": [
[
15521,
15600
]
],
"text": "['a nonzero velocity could be mistakenly assigned to cells lying inside an object']",
"type": "own_claim",
"id": "T165"
},
{
"offsets": [
[
15602,
15800
]
],
"text": "['To avoid this situation, the path of the wind flow is checked for object intersection, whereby the collision detection of the cloth simulation system provides a simple method to deal with this issue']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
15802,
15807
]
],
"text": "['MKE03']",
"type": "data",
"id": "T167"
},
{
"offsets": [
[
15851,
15967
]
],
"text": "['we combine the idea of creating wind fields by predefined flow primitives with particle tracing in given flow fields']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
16224,
16314
]
],
"text": "['This method is very easy to implement and yields very plausible and nicely looking results']",
"type": "own_claim",
"id": "T169"
},
{
"offsets": [
[
16316,
16475
]
],
"text": "['A simple approach to generate complex air flows is to define a wind field by mathematical functions which assign to each point in space a unique velocity value']",
"type": "background_claim",
"id": "T170"
},
{
"offsets": [
[
16515,
16582
]
],
"text": "['this already enables an animator to design even complex wind fields']",
"type": "background_claim",
"id": "T171"
},
{
"offsets": [
[
16497,
16501
]
],
"text": "['WH91']",
"type": "data",
"id": "T172"
},
{
"offsets": [
[
16593,
16668
]
],
"text": "['an irrotational (∇ × u = 0), inviscid, and incompressible (∇ · u = 0) fluid']",
"type": "data",
"id": "T173"
},
{
"offsets": [
[
16670,
16738
]
],
"text": "['the NavierStokes equations which describe the mechanics of any fluid']",
"type": "background_claim",
"id": "T174"
},
{
"offsets": [
[
16744,
16755
]
],
"text": "['section 4.1']",
"type": "data",
"id": "T175"
},
{
"offsets": [
[
16757,
16803
]
],
"text": "['can be simplified to give the Laplace equation']",
"type": "background_claim",
"id": "T176"
},
{
"offsets": [
[
16966,
16989
]
],
"text": "['∇ · u = ∇∇φ = ∇ 2 φ = 0']",
"type": "background_claim",
"id": "T177"
},
{
"offsets": [
[
17097,
17212
]
],
"text": "['The linearity of equation (4.3) enables an animator to combine basic flows which satisfy equation (4.3) as he likes']",
"type": "background_claim",
"id": "T178"
},
{
"offsets": [
[
17222,
17251
]
],
"text": "['to obtain complex fluid flows']",
"type": "background_claim",
"id": "T179"
},
{
"offsets": [
[
17324,
17413
]
],
"text": "['One drawback of this model is that it cannot handle objects exhibiting complex boundaries']",
"type": "background_claim",
"id": "T180"
},
{
"offsets": [
[
17415,
17613
]
],
"text": "['The approach to model solid objects in the scene taken by Wejchert et al. consists in placing a wind source using a mirror principle in order to extinguish the air flow at the boundary of the object']",
"type": "background_claim",
"id": "T181"
},
{
"offsets": [
[
17621,
17650
]
],
"text": "['this works for simple objects']",
"type": "background_claim",
"id": "T182"
},
{
"offsets": [
[
17651,
17725
]
],
"text": "['this approach is not feasible at all with deformable objects like textiles']",
"type": "background_claim",
"id": "T183"
},
{
"offsets": [
[
17727,
17839
]
],
"text": "['Another more serious drawback of this model for our application consists in the lack of interaction with objects']",
"type": "background_claim",
"id": "T184"
},
{
"offsets": [
[
17841,
17919
]
],
"text": "['The wind flow defined by the primitives will not react on objects in the scene']",
"type": "background_claim",
"id": "T185"
},
{
"offsets": [
[
17949,
18026
]
],
"text": "['tissues in the lee of other objects will be affected by the wind flow as well']",
"type": "background_claim",
"id": "T186"
},
{
"offsets": [
[
18037,
18144
]
],
"text": "['this method can be combined with the aerodynamic model described in section 3 to give nice and fast results']",
"type": "own_claim",
"id": "T187"
},
{
"offsets": [
[
18165,
18174
]
],
"text": "['section 5']",
"type": "data",
"id": "T188"
},
{
"offsets": [
[
18176,
18311
]
],
"text": "['To solve the described problems we propose a model which combines the simple global wind flow techniques with a particle tracing method']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
18466,
18560
]
],
"text": "['There are two common approaches to discretising the continuous velocity field defined in space']",
"type": "background_claim",
"id": "T190"
},
{
"offsets": [
[
18562,
18606
]
],
"text": "['one can either choose the midpoint of a cell']",
"type": "background_claim",
"id": "T191"
},
{
"offsets": [
[
18609,
18614
]
],
"text": "['Sta99']",
"type": "data",
"id": "T192"
},
{
"offsets": [
[
18616,
18632
]
],
"text": "['or its six faces']",
"type": "background_claim",
"id": "T193"
},
{
"offsets": [
[
18635,
18639
]
],
"text": "['FM96']",
"type": "data",
"id": "T194"
},
{
"offsets": [
[
18641,
18660
]
],
"text": "['to define the field']",
"type": "background_claim",
"id": "T195"
},
{
"offsets": [
[
18763,
18978
]
],
"text": "['The basic idea of the particle tracing method is to trace wind particles through a field w = i w i defined by linear superposition of wind sources corresponding to flow primitives with respective velocity fields w i']",
"type": "background_claim",
"id": "T196"
},
{
"offsets": [
[
18981,
19055
]
],
"text": "['The field w does not account for lee effects caused by objects in the flow']",
"type": "background_claim",
"id": "T197"
},
{
"offsets": [
[
19067,
19130
]
],
"text": "['we compute the wind field u containing these effects as follows']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
19132,
19188
]
],
"text": "['In our model every wind source is also a particle source']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
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19400,
19488
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"type": "own_claim",
"id": "T200"
},
{
"offsets": [
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19490,
19613
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"text": "['In every time step each particle in a wind gust moves along its velocity field w i defined by the corresponding wind source']",
"type": "own_claim",
"id": "T201"
},
{
"offsets": [
[
19627,
19722
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"text": "['the movement of the particles in a wind gust is only affected by the wind source they belong to']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
[
19724,
19801
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"text": "['The global superposition of all wind sources has no effect on these particles']",
"type": "own_claim",
"id": "T203"
},
{
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19940,
19979
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"text": "['this results in a path s(p t , p t+∆t )']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
19803,
19891
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"text": "['To calculate the wind particles’ positions we used the explicit Euler integration scheme']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
19897,
19939
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"text": "['a wind particle at position p t and time t']",
"type": "data",
"id": "T206"
},
{
"offsets": [
[
20124,
20267
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"text": "['all grid cells colliding with the path are updated with the velocity of the associated wind source with respect to the position of the particle']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
20082,
20122
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"text": "['a particle moves along its path in space']",
"type": "data",
"id": "T208"
},
{
"offsets": [
[
20269,
20349
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"text": "['The particle might cross several grid cells on its way during a single time step']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
20372,
20466
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"text": "['the path of the particle has to be subdivided into parts not exceeding the size of a grid cell']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
20690,
20731
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"text": "['a collision is detected at position p col']",
"type": "data",
"id": "T211"
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{
"offsets": [
[
20732,
20795
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"text": "['the normal of the colliding object n obj (p col ) is determined']",
"type": "own_claim",
"id": "T212"
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{
"offsets": [
[
20800,
20906
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"text": "['the velocity of the particle is set to w i (p col , t+∆t) = w i (p col , t)−(n obj ·w i (p col , t))·n obj']",
"type": "own_claim",
"id": "T213"
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{
"offsets": [
[
20907,
21037
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"text": "['This assures that the velocity component of the resulting field u is orthogonal to the collision object’s surface at p col is zero']",
"type": "own_claim",
"id": "T214"
},
{
"offsets": [
[
21085,
21122
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"text": "['no flow propagates through the object']",
"type": "own_claim",
"id": "T215"
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{
"offsets": [
[
21223,
21263
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"text": "['u is determined using the wind particles']",
"type": "own_claim",
"id": "T216"
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{
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[
21265,
21349
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"text": "['every point p that could not be reached by any wind particle will hold zero velocity']",
"type": "own_claim",
"id": "T217"
},
{
"offsets": [
[
21395,
21453
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],
"text": "['this method solves the problems described in section 4.2.1']",
"type": "own_claim",
"id": "T218"
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{
"offsets": [
[
21465,
21568
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"text": "['the somewhat tempting simplification of tagging each cell to either have wind in it or not is not valid']",
"type": "own_claim",
"id": "T219"
},
{
"offsets": [
[
21753,
21809
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"text": "['we now place a solid object in between these two sources']",
"type": "data",
"id": "T220"
},
{
"offsets": [
[
21810,
21873
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"text": "['a rather undesired effect would occur using this simplification']",
"type": "own_claim",
"id": "T221"
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{
"offsets": [
[
21875,
21949
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],
"text": "['on both sides of the solid object all cells would be tagged as having wind']",
"type": "own_claim",
"id": "T222"
},
{
"offsets": [
[
21955,
22026
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],
"text": "['evaluating the wind field at every cell we would obtain a zero velocity']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
22043,
22112
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],
"text": "['the extinguishing effect of the superposition of the two wind sources']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
22125,
22295
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],
"text": "['it is crucial for the particles to have the associated velocity of their wind source and not just the velocity resulting from the global superposition of all wind sources']",
"type": "own_claim",
"id": "T225"
},
{
"offsets": [
[
22388,
22573
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],
"text": "['For physically accurate simulations based on the common method in fluid dynamics the model introduced by Stam produces realistic effects which global wind field models can never achieve']",
"type": "background_claim",
"id": "T226"
},
{
"offsets": [
[
22575,
22663
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],
"text": "['It produces nice swirls and vortices derived from dynamical characteristics of the fluid']",
"type": "background_claim",
"id": "T227"
},
{
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[
22673,
22719
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],
"text": "['implementing the fluid solver is quite complex']",
"type": "background_claim",
"id": "T228"
},
{
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[
22724,
22781
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],
"text": "['using a high grid resolution is computationally expensive']",
"type": "background_claim",
"id": "T229"
},
{
"offsets": [
[
22790,
22910
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"text": "['the global wind field model is better suited for an easy to implement tool which is easy to adapt to specific situations']",
"type": "own_claim",
"id": "T230"
},
{
"offsets": [
[
22912,
22970
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"text": "['Particle systems are very common in the simulation engines']",
"type": "background_claim",
"id": "T231"
},
{
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[
22975,
23045
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],
"text": "['most functionality can be adapted to integrate the proposed wind model']",
"type": "own_claim",
"id": "T232"
},
{
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[
23052,
23135
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"text": "['with this straightforward approach, nice, realistic looking results can be achieved']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
23164,
23176
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"text": "['next section']",
"type": "data",
"id": "T234"
},
{
"offsets": [
[
23484,
23686
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],
"text": "['We implemented the wind models described in sections 4.1 and 4.2 in a cloth animation system that employs a fast finite element method to simulate the drape of textiles with measured material properties']",
"type": "own_claim",
"id": "T235"
},
{
"offsets": [
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23689,
23694
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"text": "['EKS03']",
"type": "data",
"id": "T236"
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{
"offsets": [
[
23697,
23720
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"text": "['For collision detection']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
23785,
23825
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"text": "['we use k-DOP hierarchies as described in']",
"type": "own_claim",
"id": "T238"
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{
"offsets": [
[
23828,
23833
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"text": "['MKE03']",
"type": "data",
"id": "T239"
},
{
"offsets": [
[
25265,
25377
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],
"text": "['both methods produce realistic looking results which are capable of enhancing the realism of computer animations']",
"type": "own_claim",
"id": "T240"
},
{
"offsets": [
[
25248,
25264
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],
"text": "['previous section']",
"type": "data",
"id": "T241"
},
{
"offsets": [
[
25385,
25434
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],
"text": "['the first model has a wider range of applications']",
"type": "own_claim",
"id": "T242"
},
{
"offsets": [
[
25436,
25544
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"text": "['the second one provides an easy method which still delivers realistic effects such as air resistance and lee']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
25546,
25645
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],
"text": "['All methods described in this work should be easy to extend to three-dimensional deformable objects']",
"type": "own_claim",
"id": "T244"
},
{
"offsets": [
[
25647,
25703
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],
"text": "['All the methods apply the same except for simple changes']",
"type": "own_claim",
"id": "T245"
},
{
"offsets": [
[
25711,
25767
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"text": "['three-dimensional objects define an inner and outer part']",
"type": "data",
"id": "T246"
},
{
"offsets": [
[
25769,
25836
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"text": "['the adaption of the face normals in equation (3.1) is not necessary']",
"type": "own_claim",
"id": "T247"
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{
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[
25848,
25942
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"text": "['in the wind field computation care has to be taken that no wind field is present in the object']",
"type": "own_claim",
"id": "T248"
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{
"offsets": [
[
25950,
25996
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"text": "['the same method as described for rigid objects']",
"type": "own_claim",
"id": "T249"
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{
"offsets": [
[
26013,
26027
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"text": "['can be applied']",
"type": "own_claim",
"id": "T250"
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25998,
26011
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"text": "['section 4.2.2']",
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A39 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A39_C09_Large_Steps_in_Cloth_Simulation_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/280814.280821
SIGGRAPH 98, Orlando, July 19–24
C OMPUTER G RAPHICS Proceedings, Annual Conference Series, 1998
<Title>Large Steps in Cloth Simulation</Title>
David Baraff Andrew Witkin Robotics Institute Carnegie Mellon University
<Abstract>The bottle-neck in most cloth simulation systems is that time steps must be small to avoid numerical instability. This paper describes a cloth simulation system that can stably take large time steps. The simulation system couples a new technique for enforcing constraints on individual cloth particles with an implicit integration method. The simulator models cloth as a triangular mesh, with internal cloth forces derived using a simple continuum formulation that supports modeling operations such as local anisotropic stretch or compression; a unified treatment of damping forces is included as well. The implicit integration method generates a large, unbanded sparse linear system at each time step which is solved using a modified conjugate gradient method that simultaneously enforces particles’ constraints. The constraints are always maintained exactly, independent of the number of conjugate gradient iterations, which is typically small. The resulting simulation system is significantly faster than previous accounts of cloth simulation systems in the literature.</Abstract>
Keywords: Cloth, simulation, constraints, implicit integration, physically-based modeling.
<H1>1 Introduction</H1>
Physically-based cloth animation has been a problem of interest to the graphics community for more than a decade. Early work by Terzopoulos et al. [ 17 ] and Terzopoulos and Fleischer [ 15 , 16 ]
on deformable models correctly characterized cloth simulation as a problem in deformable surfaces, and applied techniques from the mechanical engineering and finite element communities to the problem. Since then, other research groups (notably Carignan et al.
[ 4 ] and Volino et al. [ 20 , 21 ]
; Breen et al. [ 3 ]; and Eberhardt et al. [ 5 ]) have taken up the challenge of cloth. Although specific details vary (underlying representations, numerical solution methods, collision detection and constraint methods, etc.), there is a deep commonality amongst all the approaches: physically-based cloth simulation is formulated as a time-varying partial differential equation which, after discretization, is numerically solved as an ordinary differential equation
1
ẍ = M −1 − ∂E + F . ∂x
In this equation the vector x and diagonal matrix M represent the geometric state and mass distribution of the cloth, E—a scalar funcAuthor affiliation (September 1998): David Baraff, Andrew Witkin, Pixar Animation Studios, 1001 West Cutting Blvd., Richmond, CA 94804. Email: [email protected] , [email protected] . This is an electronic reprint. Permission is granted to copy part or all of this paper for noncommercial use provided that the title and this copyright notice appear. This electronic reprint is ©1998 by CMU. The original printed paper is ©1998 by the ACM.
tion of x—yields the cloth’s internal energy, and F (a function of x and x) describes other forces (air-drag, contact and constraint forces, internal damping, etc.) acting on the cloth. In this paper, we describe a cloth simulation system that is much faster than previously reported simulation systems. Our system’s faster performance begins with the choice of an implicit numerical integration method to solve equation (1). The reader should note that the use of implicit integration methods in cloth simulation is far from novel: initial work by Terzopoulos et al. [ 15 , 16 , 17 ] applied such methods to the problem. 1 Since this time though, research on cloth simulation has generally relied on explicit numerical integration (such as Euler’s method or Runge-Kutta methods) to advance the simulation, or, in the case of of energy minimization, analogous methods such as steepest-descent
[ 3 , 10 ]. This is unfortunate. Cloth strongly resists stretching motions while being comparatively permissive in allowing bending or shearing motions. This results in a “stiff” underlying differential equation of motion
[ 12 ]. Explicit methods are ill-suited to solving stiff equations because they require many small steps to stably advance the simulation forward in time. 2 In practice, the computational cost of an explicit method greatly limits the realizable resolution of the cloth. For some applications, the required spatial resolution—that is, the dimension n of the state vector x—can be quite low: a resolution of only a few hundred particles (or nodal points, depending on your formulation/terminology) can be sufficient when it comes to modeling flags or tablecloths. To animate clothing, which is our main concern, requires much higher spatial resolution to adequately represent realistic (or even semi-realistic) wrinkling and folding configurations. In this paper, we demonstrate that implicit methods for cloth overcome the performance limits inherent in explicit simulation methods. We describe a simulation system that uses a triangular mesh for cloth surfaces, eliminating topological restrictions of rectangular meshes, and a simple but versatile formulation of the internal cloth energy forces. (Unlike previous metric-tensor-based formulations
[ 15 , 16 , 17 , 4 ]
which model some deformation energies as quartic functions of positions, we model deformation energies only as quadratic functions with suitably large scaling. Quadratic energy models mesh well with implicit integration’s numerical properties.) We also introduce a simple, unified treatment of damping forces, a subject which has been largely ignored thus far. A key step in our simulation process is the solution of an O(n) × O(n) sparse linear system, which arises from the implicit integration method.
In this respect, our implementation differs greatly from the implementation by Terzopoulos et al.
[ 15 , 17 ], which for large simulations
1 Additional use of implicit methods in animation and dynamics work includes Kass and Miller [ 8 ], Terzopoulos and Qin
[ 18 ], and Tu
[ 19 ]. 2 Even worse, the number of time steps per frame tends to increase along with the problem size, for an explicit method. Cloth simulations of size n— meaning x ∈ IR O(n) —generally require O(n) explicit steps per unit simulated time. Because the cost of an explicit step is also O(n) (setting aside complications such as collision detection for now) explicit methods for cloth require time O(n 2 )—or worse.
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used an “alternating-direction” implicit (ADI) method [ 12 ]. An ADI method generates a series of tightly banded (and thus quickly solved) linear systems rather than one large sparse system. (The price, however, is that some of the forces in the system—notably between diagonally-adjacent and non-adjacent nodes involved in self-collisions—are treated explicitly, not implicitly.) The speed (and ease) with which our sparse linear systems can be robustly solved—even for systems involving 25,000 variables or more—has convinced us that there is no benefit to be gained from using an ADI method instead (even if ADI methods could be applied to irregular triangular meshes). Thus, regardless of simulation size, we treat all forces as part of the implicit formulation. Even for extremely stiff systems, numerical stability has not been an issue for our simulator.
<H2>1.1 Specific Contributions</H2>
Much of the performance of our system stems from the development of an implicit integration formulation that handles contact and geometric constraints in a direct fashion. Specifically, our simulator enforces constraints without introducing additional penalty terms in the energy function E or adding Lagrange-multiplier forces into the force F. (This sort of direct constraint treatment is trivial if equation (1) is integrated using explicit techniques, but is problematic for implicit methods.) Our formulation for directly imposing and maintaining constraints is harmonious with the use of an extremely fast iterative solution algorithm—a modified version of the conjugate gradient (CG) method—to solve the O(n) × O(n) linear system generated by the implicit integrator. Iterative methods do not in general solve linear systems exactly—they are run until the solution error drops below some tolerance threshold. A property of our approach, however, is that the constraints are maintained exactly, regardless of the number of iterations taken by the linear solver. Additionally, we introduce a simple method, tailored to cloth simulation, for dynamically adapting the size of time steps over the course of a simulation. The combination of implicit integration and direct constraint satisfaction is very powerful, because this approach almost always allows us to take large steps forward. In general, most of our simulations require on average from two to three time steps per frame of 30 Hz animation, even for (relatively) fast moving cloth. The large step sizes complement the fact that the CG solver requires relatively few iterations to converge. For example, in simulating a 6, 000 node system, the solver takes only 50–100 iterations to solve the 18, 000 × 18, 000 linear system formed at each step. Additionally, the running time of our simulator is remarkably insensitive to the cloth’s material properties (quite the opposite behavior of explicit methods). All of the above advantages translate directly into a fast running time.
For example, we demonstrate results similar to those in Breen et al.
[ 3 ] and Eberhardt et al.
[ 5 ] (draping of a 2,600 node cloth) with a running time just over 2 seconds per frame on an SGI Octane R10000 195 Mhz processor. Similarly, we show garments (shirts, pants, skirts) exhibiting complex wrinkling and folding behavior on both key-framed and motion-captured characters. Representative running times include a long skirt with 4,530 nodes (8,844 triangles) on a dancing character at a cost of 10 seconds per frame, and a shirt with 6,450 nodes (12,654 triangles) with a cost varying between 8 to 14 seconds per frame, depending on the underlying character’s motion.
<H2>1.2 Previous Work</H2>
Terzopoulos et al. [ 15 , 17 ]
discretized cloth as a rectangular mesh. Energy functions were derived using a continuum formulation. This work recognized the need for damping forces; however, only a simple viscous drag force −k x was used. The linear systems result- ing from the use of implicit integration techniques were solved, for small systems, by direct methods such as Choleski factorization, or using iterative techniques such as Gauss-Seidel relaxation or conjugate gradients. (For a square √ system of n nodes, the resulting linear system has bandwidth n. In this case, banded Choleski factorization
[ 6 ] requires time O(n 2 ).) As previously discussed, Terzopoulos et al. made use of an ADI method for larger cloth simulations. Following Terzopoulos et al.’s treatment of deformable surfaces, work by Carignan et al.
[ 4 ]
described a cloth simulation system using rectangular discretization and the same formulation as Terzopoulos et al. Explicit integration was used. Carignan et al. recognized the need for damping functions which do not penalize rigidbody motions of the cloth (as simple viscous damping does) and they added a force which damps cloth stretch and shear (but not bend). Later work by the same group includes Volino et al. [ 20 ]
, which focuses mainly on collision detection/response and uses a triangular mesh; no mention is made of damping forces. The system uses the midpoint method (an explicit method) to advance the simulation. Thus far, the accumulated work by this group (see Volino et al.
[ 21 ] for an overview) gives the only published results we know of for simulated garments on moving characters. Reported resolutions of the garments are approximately two thousand triangles per garment (roughly 1,000 nodal points)
[ 21 ]
with running times of several minutes per frame for each garment on an SGI R4400 150 Mhz processor. Breen et al. [ 3 ]
depart completely from continuum formulations of the energy function, and describe what they call a “particlebased” approach to the problem. By making use of real-world cloth material properties (the Kawabata measuring system) they produced highly realistic static images of draped rectangular cloth meshes with reported resolutions of up to 51 × 51 nodes. The focus of this work is on static poses for cloth, as opposed to animation: thus, their simulation process is best described as energy minimization, although methods analogous to explicit methods are used. Speed was of secondary concern in this work. Refinements by Eberhardt et al. [ 5 ]
—notably, the use of higher-order explicit integration methods and Maple-optimized code, as well as a dynamic, not static treatment of the problem—obtain similarly realistic results, while dropping the computational cost to approximately 20–30 minutes per frame on an SGI R8000 processor. No mention is made of damping terms. Provot [ 13 ]
focuses on improving the performance of explicit methods by a post-step modification of nodal positions. He iteratively adjusts nodal positions to eliminate unwanted stretch; the convergence properties of this method are unclear. A more comprehensive discussion on cloth research can be found in the survey paper by Ng and Grimsdale
[ 9 ].
<H1>2 Simulation Overview</H1>
In this section, we give a brief overview of our simulator’s architecture and introduce some notation. The next section derives the linear system used to step the simulator forward implicitly while section 4 describes the specifics of the internal forces and their derivatives that form the linear system. Section 5 describes how constraints are maintained (once established), with a discussion in section 6 on collision detection and constraint initialization. Section 7 describes our adaptive step-size control, and we conclude in section 8 with some simulation results.
<H2>2.1 Notation and Geometry</H2>
Our simulator models cloth as a triangular mesh of particles. Given a mesh of n particles, the position in world-space of the ith particle is x i ∈ IR 3 . The geometric state of all the particles is simply x ∈ IR 3n . The same component notation applies to forces: a force f ∈ IR 3n acting on the cloth exerts a force f i on the ith particle. Real-world cloth is cut from flat sheets of material and tends to resist deformations away from this initial flat state (creases and pleats not withstanding). We capture the rest state of cloth by assigning each particle an unchanging coordinate (u i , v i ) in the plane. 3 Section 4 makes use of these planar coordinates. Collisions between cloth and solid objects are handled by preventing cloth particles from interpenetrating solid objects. Our current implementation models solid objects as triangularly faced polyhedra. Each face has an associated thickness and an orientation; particles found to be sufficiently near a face, and on the wrong side, are deemed to have collided with that face, and become subject to a contact constraint. (If relative velocities are extremely high, this simple test may miss some collisions. In this case, analytically checking for intersection between previous and current positions can guarantee that no collisions are missed.) For cloth/cloth collisions, we detect both face-vertex collisions between cloth particles and triangles, as well as edge/edge collisions between portions of the cloth. As in the case of solids, close proximity or actual intersection of cloth with itself initiates contact handling.
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<H2>2.2 Energy and Forces</H2>
The most critical forces in the system are the internal cloth forces which impart much of the cloth’s characteristic behavior. Breen et al. [ 3 ]
describes the use of the Kawabata system of measurement for realistic determination of the in-plane shearing and out-of-plane bending forces in cloth. We call these two forces the shear and bend forces. We formulate the shear force on a per triangle basis, while the bend force is formulated on a per edge basis—between pairs of adjacent triangles. The strongest internal force—which we call the stretch force— resists in-plane stretching or compression, and is also formulated per triangle. Under normal conditions, cloth does not stretch appreciably under its own weight. This requires the stretch force to have a high coefficient of stiffness, and in fact, it is the stretch force that is most responsible for the stiffness of equation (1). A common practice in explicitly integrated cloth systems is to improve running time by decreasing the strength of the stretch force; however, this leads to “rubbery” or “bouncy” cloth. Our system uses a very stiff stretch force to combat this problem, without any detrimental effects on the run-time performance. While the shear and bend force stiffness coefficients depend on the material being simulated, the stretch coefficient is essentially the same (large) value for all simulations. (Of course, if stretchy cloth is specifically called for, the stretch coefficient can be made smaller.) Complementing the above three internal forces are three damping forces. In section 5, we formulate damping forces that subdue any oscillations having to do with, respectively, stretching, shearing, and bending motions of the cloth. The damping forces do not dissipate energy due to other modes of motion. Additional forces include air-drag, gravity, and user-generated generated mouse-forces (for interactive simulations). Cloth/cloth contacts generate strong repulsive linear-spring forces between cloth particles. Combining all forces into a net force vector f, the acceleration ẍ i of the ith particle is simply ẍ i = f i /m i , where m i is the ith particle’s mass. The mass m i is determined by summing one third the mass
3 In general, each particle has a unique (u, v) coordinate; however, to accommodate pieces of cloth that have been topologically seamed together (such as a sleeve), particles lying on the seam must have multiple (u, v) coordinates. For these particles, we let the (u, v) coordinate depend on which triangle we are currently examining. The (u, v) coordinates are useful for texturing.
of all triangles containing the ith particle. (A triangle’s mass is the product of the cloth’s density and the triangle’s fixed area in the uv coordinate system.) Defining the diagonal mass matrix M ∈ IR 3n×3n by diag(M) = (m 1 , m 1 , m 1 , m 2 , m 2 , m 2 , . . . , m n , m n , m n ), we can write simply that
2
ẍ = M −1 f(x, x).
<H2>2.3 Sparse Matrices</H2>
The use of an implicit integration method, described in the next section, generates large unbanded sparse linear systems. We solve these systems through a modified conjugate gradient (CG) iterative method, described in section 5. CG methods exploit sparsity quite easily, since they are based solely on matrix-vector multiplies, and require only rudimentary sparse storage techniques. The sparsity of the matrix generated by the implicit integrator is best represented in block-fashion: for a system with n particles, we deal with an n × n matrix, whose non-zero entries are represented as dense 3 × 3 matrices of scalars. The matrix is represented as an array of n rows; each row is a linked list of the non-zero elements of that row, to accommodate possible run-time changes in the sparsity pattern, due to cloth/cloth contact. The (dense) vectors that are multiplied against this matrix are stored simply as n element arrays of threecomponent vectors. The overall implementation of sparsity is completely straightforward.
<H2>2.4 Constraints</H2>
An individual particle’s position and velocity can be completely controlled in either one, two, or three dimensions. Particles can thus be attached to a fixed or moving point in space, or constrained to a fixed or moving surface or curve. Constraints are either user-defined (the time period that a constraint is active is user-controlled) or automatically generated, in the case of contact constraints between cloth and solids. During cloth/solid contacts, the particle may be attached to the surface, depending on the magnitudes of the frictional forces required; otherwise, the particle is constrained to remain on the surface, with sliding allowed. The mechanism for releasing a contact constraint, or switching between sliding or not sliding, is described in section 5. The constraint techniques we use on individual particles work just as well for collections of particles; thus, we could handle cloth/cloth intersections using the technique described in section 5, but the cost is potentially large. For that reason, we have chosen to deal with cloth/cloth contacts using penalty forces: whenever a particle is near a cloth triangle or is detected to have passed through a cloth triangle, we add a stiff spring with damping to pull the particle back to the correct side of the triangle. The implicit solver easily tolerates these stiff forces.
<H1>3 Implicit Integration</H1>
Given the known position x(t 0 ) and velocity x(t 0 ) of the system at time t 0 , our goal is to determine a new position x(t 0 + h) and velocity x(t 0 + h) at time t 0 + h. To compute the new state and velocity using an implicit technique, we must first transform equation (2) into a first-order differential equation. This is accomplished simply by defining the system’s velocity v as v = x and then writing
3
dt d x x = dt d v x = M −1 f(x, v v) .
To simplify notation, we will define x 0 = x(t 0 ) and v 0 = v(t 0 ). We also define x = x(t 0 + h) − x(t 0 ) and v = v(t 0 + h) − v(t 0 ).
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The explicit forward Euler method applied to equation (3) approximates x and v as x v = h M v −1 0 f 0
where the force f 0 is defined by f 0 = f(x 0 , v 0 ). As previously discussed, the step size h must be quite small to ensure stability when using this method. The implicit backward Euler method appears similar at first: x and v are approximated by
4
v x = h M −1 f(x 0 v + 0 + x, v v 0 + v) .
The difference in the two methods is that the forward method’s step is based solely on conditions at time t 0 while the backward method’s step is written in terms of conditions at the terminus of the step itself. 4 The forward method requires only an evaluation of the function f but the backward method requires that we solve for values of x and v that satisfy equation (4). Equation (4) is a nonlinear equation: rather than solve this equation exactly (which would require iteration) we apply a Taylor series expansion to f and make the firstorder approximation ∂f ∂f f(x 0 + x, v 0 + v) = f 0 + ∂x x + ∂v v. In this equation, the derivative ∂f/∂x is evaluated for the state (x 0 , v 0 ) and similarly for ∂f/∂v. Substituting this approximation into equation (4) yields the linear system
5
v x = h M −1 (f 0 + v 0 ∂x ∂f + x v + ∂v ∂f v) .
Taking the bottom row of equation (5) and substituting x = h(v 0 + v) yields v = hM −1 f 0 + ∂x ∂f h(v 0 + v) + ∂v ∂f v . Letting I denote the identity matrix, and regrouping, we obtain
6
I − hM −1 ∂v ∂f − h 2 M −1 ∂x ∂f v = hM −1 f 0 + h ∂x ∂f v 0
which we then solve for v. Given v, we trivially compute x = h(v 0 + v). Thus, the backward Euler step consists of evaluating f 0 , ∂f/∂x and ∂f/∂v; forming the system in equation (6); solving the system for v; and then updating x and v. We use the sparse data structures described in section 2.3 to store the linear system. The sparsity pattern of equation (6) is described in the next section, while solution techniques are deferred to section 5.
4 The method is called “backward” Euler because starting from the output state (x 0 + x, v 0 + v) and using a forward Euler step to run the system backward in time (i.e. taking the step −h(v(t 0 + h), f(x(t 0 + h), v(t 0 + h))) brings you back to (x 0 , v 0 ). What is the value in this? Forward Euler takes no notice of wildly changing derivatives, and proceeds forward quite blindly. Backward Euler, however, forces one to find an output state whose derivative at least points back to where you came from, imparting, essentially, an additional layer of consistency (or sanity-checking, if you will).
<H1>4 Forces</H1>
Cloth’s material behavior is customarily described in terms of a scalar potential energy function E(x); the force f arising from this energy is f = −∂E/∂x. Equation (6) requires both the vector f and the matrix ∂f/∂x. Expressing the energy E as a single monolithic function—encompassing all aspects of the cloth’s internal behavior—and then taking derivatives is impractical, from a bookkeeping point of view. A better approach is decompose E into a sum of sparse energy functions; that is, to write E(x) = α E α (x) where each E α depends on as few elements of x—as few particles—as possible. However, even decomposing E into sparse energy functions is not enough. Energy functions are an undesirable starting point because sensible damping functions cannot be derived from energy functions. Instead, we define internal behavior by formulating a vector condition C(x) which we want to be zero, and then defining the associated energy as k C(x) T C(x) where k is a stiffness constant. 2 In section 4.5, we show how sensible damping functions can be constructed based on this formulation. An added bonus is that starting from this vector-based energy description tends to result in a simpler, more compact, and more easily coded formulation for ∂f/∂x than proceeding from an energy function in which the structure of C has been lost.
<H2>4.1 Forces and Force Derivatives</H2>
Given a condition C(x) which we want to be zero, we associate an energy function E C with C by writing E C (x) = k 2 C(x) T C(x) where k is a stiffness constant of our choice. Assuming that C depends on only a few particle, C gives rise to a sparse force vector f. Recall from section 2.1 that we view the vector f in block form; each element f i is a vector in IR 3 . For each particle i that C depends on,
7
f i = − ∂E ∂x C i = −k ∂C(x) ∂x i C(x);
all the other elements of f are zero. Similarly, the derivative of f is also sparse. Defining the derivative matrix K = ∂f/∂x, the nonzero entries of K are K ij for all pairs of particles i and j that C depends on. Again, we treat K in block fashion: K ∈ IR 3n×3n , so an element K ij is a 3 × 3 matrix. From equation (7), we have
8
K ij = ∂x ∂f i j = −k ∂C(x) ∂x i ∂C(x) ∂x j T + ∂ ∂x 2 C(x) i ∂x j C(x) .
Additionally, since K ij is a second derivative—that is, K ij = ∂f i /∂x j = ∂ 2 E/∂x i ∂x j —we have K ij = K T ji so K is symmetric. Note that since C does not depend on v, the matrix ∂f/∂v is zero. We can now easily describe the internal forces acting on the cloth, by just writing condition functions. Forces and their derivatives are easily derived using equations (7) and (8).
<H2>4.2 Stretch Forces</H2>
Recall that every cloth particle has a changing position x i in world space, and a fixed plane coordinate (u i , v i ). Even though our cloth is modeled as a discrete set of points, grouped into triangles, it will be convenient to pretend momentarily that we have a single continuous function w(u, v) that maps from plane coordinates to world space. Stretch can be measured at any point in the cloth surface by examining the derivatives w u = ∂w/∂u and w v = ∂w/∂v at that point. The magnitude of w u describes the stretch or compression in the u direction; the material is unstretched wherever w u = 1. Stretch in the v direction is measured by w v . (Some previous continuum formulations have modeled stretch energy along an axis as essentially (w u T w u − 1) 2 , which is a quartic function of position
[ 15 , 16 , 17 , 4 ]. We find this to be needlessly stiff; worse, near the rest state, the force gradient—a quadratic function of position—is quite small, which partially negates the advantage implicit integration has in exploiting knowledge of the force gradient. A quadratic model for energy is, numerically, a better choice.) We apply this stretch/compression measure to a triangle as follows. Let us consider a triangle whose vertices are particles i, j and k. Define x 1 = x j − x i and x 2 = x k − x i . Also, let u 1 = u j − u i , while u 2 = u k − u i and similarly for v 1 and v 2 . We approximate w(u, v) as a linear function over each triangle; this is equivalent to saying that w u and w v are constant over each triangle. This lets us write x 1 = w u u 1 + w v v 1 and x 2 = w u u 2 + w v v 2 . Solving for w u and w v yields
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−1 (w u w v ) = ( x 1 x 2 ) v u 1 1 v u 2 2 .
Note that x 1 and x 2 vary during the simulation but the matrix in the above equation does not. We can treat w u and w v as functions of x, realizing that they depend only on x i , x j and x k and using equation (9) to obtain derivatives. The condition we use for the stretch energy is
10
C(x) = a w w u v (x) (x) − − b b v u
where a is the triangle’s area in uv coordinates. Usually, we set b u = b v = 1, though we need not always do so. In particular, if we want to slightly lengthen a garment (for example, a sleeve) in the u direction, we can increase b u , which causes w u to seek a larger value, and tends to induce wrinkles across the u direction. Likewise, we might decrease b v near the end of a sleeve, inducing a tight cuff, as on a sweatshirt. We have found the ability to control shrink/stretch anisotropically to be an indispensable modeling tool.
<H2>4.3 Shear and Bend Forces</H2>
Cloth likewise resists shearing in the plane. We can measure the extent to which cloth has sheared in a triangle by considering the inner product w u T w v . In its rest state, this product is zero. Since the stretch term prevents the magnitudes of w u and w v from changing overly much, we need not normalize. By the small angle approximation, the product w u T w v is a reasonable approximation to the shear angle. The condition for shearing is simply C(x) = aw u (x) T w v (x) with a the triangle’s area in the uv plane. We measure bend between pairs of adjacent triangles. The condition we write for the bend energy depends upon the four particles defining the two adjoining triangles. If we let n 1 and n 2 denote the unit normals of the two triangles and let e be a unit vector parallel to the common edge, the angle θ between the two faces is defined by the relations sin θ = (n 1 × n 2 ) · e and cos θ = n 1 · n 2 . We define a condition for bending by writing simply C(x) = θ which results in a force that counters bending. 5 The assumption that the stretch energy will keep the cloth from stretching much allows us to treat n 1 , 5 For reasonably equilateral triangles, as edge lengths decrease, the curvature represented by a particular angle θ between triangles increases. Since the square of the curvature—a a good measure of the bend energy in cloth— increases at the same rate that the triangle’s area decreases, the condition C should not be scaled by the triangles’ areas. See Breen et al. [ 3 ]
for a further discussion of relating curvature to bend angle. n 2 and e as having a constant length at each step of the simulation. This makes differentiating θ with respect to x a manageable task. Rectangular meshes make it simple to treat bending anisotropically. The uv coordinates associated with particles make this possible for triangular meshes as well. Given material for which bending in the u and v directions are weighted by stiffnesses k u and k v , we can emulate this anisotropy as follows. Let the edge between the triangles be between particles i and j, and define u = u i − u j and v = v i − v j . The stiffness weighting for this edge should simply be k u ( u) 2 + k v ( v) 2 . ( u) 2 + ( v) 2
<H2>4.4 Additional Forces</H2>
To the above forces we also add easily implemented forces such as gravity and air-drag (which is formulated on a per-triangle basis, and opposes velocities along the triangle’s normal direction). When the simulation is fast enough to interact with, we add user-controlled “mouse” forces. These forces and their gradients are easily derived.
<H2>4.5 Damping</H2>
The energies we have just described are functions of position only. Robust dynamic cloth simulation, however, is critically dependent on well-chosen damping forces that are a function of both position and velocity. For example, the strong stretch force must be accompanied by a suitably strong damping force if we are to prevent anomalous in-plane oscillations from arising between connected particles. However, this strong damping force must confine itself solely to damping in-plane stretching/compressing motions: stretch damping should not arise due to motions that are not causing stretch or compression. Terzopoulos et al.’s [ 16 , 17 ]
treatment of cloth used a simple viscous damping function which dissipated kinetic energy, independent of the type of motion. Carignan et al. [ 4 ]
improved upon this somewhat, borrowing a formulation due to Platt and Barr [ 11 ]; however, their damping function—a linear function of velocity—does not match the quartic energy functions of their continuum formulation. In this section we describe a general treatment for damping that is independent of the specific energy function being damped. It is tempting to formulate a damping function for an energy function E(x) by measuring the velocity of the energy, E ̇ = d E(x). dt This is an easy trap to fall into, but it gives nonsensical results. At an equilibrium point of E, the gradient ∂E/∂x vanishes. Since E ̇ = (∂E/∂x) T x, we find that E ̇ is zero when E is at its minimum, regardless of the system’s velocity x = v. In general, E ̇ is always too small near the system’s rest state. Clearly, basing the damping force on E ̇ is not what we want to do. We believe that the damping function should be defined not in terms of the energy E, but in terms of the condition C(x) we have been using to define energies. The force f arising from the energy acts only in the direction ∂C(x)/∂x, and so should the damping force. Additionally, the damping force should depend on the component of the system’s velocity in the ∂C(x)/∂x direction; in other words, the damping strength should depend on (∂C(x)/∂x) T x = C(x). ̇ Putting this together, we propose that the damping force d associated with a condition C have the form
11
d = −k d ∂C(x) ∂x C(x). ̇
∂C(x) This neatly parallels the fact that f = −k s ∂x C(x).
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Given the condition functions C we have defined in this section for stretch, bend and shear forces, we can now add accompanying damping forces by applying equation (11). As before, d i is nonzero only for those particles that C depends on, and ∂d/∂x has the same sparsity pattern as ∂f/∂x. Differentiating equation (11), we obtain
12
∂x ∂d i j = −k d ∂C(x) ∂x i ∂ C(x) ∂x ̇ j T + ∂ ∂x C(x) i ∂x j C(x) ̇ .
Note that ∂d/∂x is not a second derivative of some function as was the case in equation (8) so we cannot expect ∂d/∂x to be symmetrical. In equation (12), it is the term (∂C(x)/∂x i )(∂ C(x)/∂x ̇ j ) T which breaks the symmetry. Anticipating section 5.2, we find it expedient simply to leave this term out, thereby restoring symmetry. This simplification is clearly not physically justifiable, but we have not observed any ill effects from this omission. (Omitting all of equation (12), however, causes serious problems.) Finally, equation (6) requires the derivative ∂d/∂v. Since C(x) ̇ = (∂C(x)/∂x) T v, we have ∂ C(x) ̇ ∂ ∂C(x) T ∂C(x) = v = . ∂v ∂v ∂x ∂x Using this fact, we can write ∂v ∂d i j = −k d ∂C(x) ∂x i ∂ C(x) ∂v ̇ j T = −k d ∂C(x) ∂x i ∂C(x) ∂x j T . In this case, the result is symmetrical without dropping any terms.
<H1>5 Constraints</H1>
In this section, we describe how constraints are imposed on individual cloth particles. The constraints we discuss in this section are either automatically determined by the user (such as geometric attachment constraints on a particle) or are contact constraints (generated by the system) between a solid object and a particle. The techniques we describe in this section could be used for multi-particle constraints; however, constraints that share particle would need to be merged. Thus, a set of four-particle constraints (such as vertex/triangle or edge/edge contacts in the cloth) might merge to form a single constraint on arbitrarily many particles, which would be expensive to maintain. Because of this, we handle cloth/cloth contacts with strong springs (easily dealt with, given the simulator’s underlying implicit integration base) and “position alteration,” a technique described in section 6. At any given step of the simulation, a cloth particle is either completely unconstrained (though subject to forces), or the particle may be constrained in either one, two or three dimensions. Given the differential nature of our formulation, it is the particle’s acceleration, or equivalently, the change in the particle’s velocity, that is constrained. If the particle is constrained in all three dimensions, then we are explicitly setting the particle’s velocity (at the next step). If the constraint is in two or one dimensions, we are constraining the particle’s velocity along either two or one mutually orthogonal axes. Before describing our constraint method, we discuss several other possible enforcement mechanisms and explain why we chose not to use them.
Reduced Coordinates
An obvious and quite exact method for constraining a particle is to reduce the number of coordinates describing the particle’s position and velocity. A completely constrained particle would have no coordinates, while a particle with one dimension of constraint would have two coordinates. This is possible—but it complicates the system immensely. If we change the number of coordinates per particle, we alter the size of the derivative matrices in equation (6), as well as the sparsity pattern (this happens when a particle changes from having no coordinates to some coordinates, or vice versa). Given the transient nature of contact constraints between cloth and solids, this is most unappealing. The computation of the derivative matrices’ entries is also greatly complicated, because we must now introduce extra Jacobian matrices that relate a particle’s reduced coordinates to its motion in world-space. Finally, correct constraint-release behavior between cloth and solid objects is difficult to achieve using a reduced coordinate formulation. Considering all of this, we immediately rejected this method of constraints.
Penalty Methods
We could constrain particles through the use of strong energy functions—essentially, stiff springs that attempt to prevent illegal particle motions. Since our entire formulation is geared to handle stiffness, the usual objections to enforcing constraints with springs—very stiff equations—do not carry as much weight. We tried this for a time, and found it to be a not unreasonable constraint enforcement mechanism. However, penalty methods do not enforce constraints exactly, and they do add some additional stiffness to the system. Since the mechanism we describe enforces constraints exactly, and adds no extra stiffness, we turned away from penalty methods except in the case of cloth/cloth interactions.
Lagrange Multipliers
We could introduce additional constraint forces—that is, Lagrange multipliers—into our system to satisfy the constraints. This involves augmenting the linear system of equation (6) with extra variables (the multipliers) and extra equations (the constraint conditions). Unfortunately, this turns a positive definite system into an indefinite system, which means that iterative methods such as CG will need to square the system first, thereby doubling the running time and degrading the numerical conditionining of the linear system. Additionally, an iterative method will generally not enforce the constraints exactly without a large number of iterations. (A direct method for solving the augmented system would, however, avoid this problem.) Again, the constraint method we describe steps past these difficulties, so we turned away from using Lagrange multipliers.
<H2>5.1 Mass Modification</H2>
The idea behind our constraint enforcement mechanism is described quite simply, although the actual implementation is somewhat more complicated, to maximize performance. A dynamic simulation usually requires knowledge of the inverse mass of objects; for example, note the appearance of M −1 , and not M in equation (6). In the case of a single particle, we write x ̈ i = m 1 i f i to describe a particle’s acceleration. When inverse mass is used, it becomes trivial to enforce constraints by altering the mass. Suppose for example that we want to keep particle i’s velocity from changing. If we take 1/m i to be zero, we give the particle an infinite mass, making it ignore all forces exerted on it. Complete control over a particle’s acceleration is thus taken care of by storing a value of zero for the particle’s inverse mass. What if we wish to constrain the particle’s acceleration in only one or two dimensions? Although we normally think of a particle’s mass as a scalar, we need not always do so. Suppose we write ẍ i = 1/m 0 i 1/m 0 i 0 0 f i . Now ẍ i 0 0 0 must lie in the xy plane; no acceleration in the z direction is possible. Note than an unconstrained particle can be considered to have the 3 × 3 inverse mass matrix 1 I, with I the identity matrix. m i Of course, we are not restricted to coordinate-aligned constraints. More generally, given a unit vector p ∈ IR 3 , a particle is prevented from accelerating along p by using an inverse mass matrix 1 (I − m i pp T ); this follows from the fact that (I − pp T )p = 0. Similarly, given two mutually orthogonal unit vectors p and q, we prevent a particle from accelerating in either the p or q direction by using the inverse mass matrix 1 (I − pp T − qq T ). m i By allowing constrained particles to have these sorts of inverse masses, we can build constraints directly into equation (6). We will create a modified version W of M −1 ; W will be a block-diagonal matrix, with off-diagonal blocks being zero, and diagonal blocks defined as follows: let ndof(i) indicate the number of degrees of freedom particle i has, and let particle i’s prohibited directions be p i (if ndof(i) = 2) or p i and q i (if ndof(i) = 1) with p i and q i mutually orthogonal unit vectors. W’s diagonal blocks are W ii = m 1 i S i where
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S i = I 0 (I (I − − p p i i p p i i T T ) − q i q i T ) if if if ndof(i) ndof(i) ndof(i) ndof(i) = = = = 1 2 3 0.
We are not limited to constraining particles to have zero accelerations in certain directions; rather, we control exactly what the change in velocity is along the constrained directions. For every particle i, let z i be the change in velocity we wish to enforce in the particle’s constrained direction(s). (This implies we can choose any value of z i for a completely constrained particle, since all directions are constrained; an unconstrained particle must have z i = 0 since it has no constrained directions.) Using W and z, we rewrite equation (6) to directly enforce constraints. If we solve
14
I − hW ∂v ∂f − h 2 W ∂x ∂f v = hW f 0 + h ∂x ∂f v 0 + z
for v, we will obtain a v which is consistent with our constraints. Completely constrained particles will have v i = z i , while partially constrained particles will have a v i whose component in the constrained direction(s) is equal to z i .
<H2>5.2 Implementation</H2>
We initially implemented constraints using equation (14) and found that it worked exactly as advertised. For very small test systems, we solved equation (14) using a direct method (Gaussian elimination) without any problems. For larger systems, we planned to use the iterative, sparsity-exploiting CG method, which immediately presents us with a problem: equation (14) is not a symmetric linear system. (For that matter, neither is equation (6) unless all particles have the same mass.) CG methods, however, require symmetric matrices. 6 We could apply a CG method to the unsymmetric matrix of equation (14) by use of the “normal equations”; but this involves multiplying the matrix of equation (14) with its transpose which doubles the cost of each iteration while squaring the condition number of the system [ 14 ]
—a less than desirable plan. We decided that using a CG method to solve the unsymmetric problem was not acceptable. Note that without constraints, applying a CG method to equation (6) is not difficult, because we can transform this equation to 6 In fact, they work best on positive definite symmetric matrices. The matrices we ultimately hand to our CG method are positive definite. a symmetric (and positive definite) system by left-multiplying the entire equation by M: the system
15
M − h ∂v ∂f − h 2 ∂x ∂f v = h f 0 + h ∂x ∂f v 0
is symmetric and has the same solution v as equation (6). Unfortunately, we cannot apply the same transformation to equation (14), because W is singular—the filtering blocks in equation (13) are rank deficient—so we cannot multiply through by W −1 . The solution to the problem of asymmetry is to modify the CG method so that it can operate on equation (15), while procedurally applying the constraints inherent in the matrix W at each iteration. The modified method will need to know about the particles’ constraints and the vector z. Let us define the symmetric positive definite matrix A by
16
A = M − h ∂f − h 2 ∂f ∂v ∂x
and the vector b and residual vector r as ∂f b = h f 0 + h ∂x v 0 and r = A v − b. Given A, b, constraints on the particles, and z, our modified CG method will try to find v that satisfies two conditions: • For each particle i, the component of r i in the particle’s unconstrained direction(s) will be made equal to zero (assuming the method is run for sufficiently many iterations). • For each particle i, the component of v i in the particle’s constrained direction(s) will be exactly z i (no matter how many iterations are taken). Note that these two conditions imply that unconstrained particles have r i close to zero, while completely constrained particles have v i = z i . Thus in the case when no particles are constrained, our modified CG method should produce the same result as the regular CG method.
<H2>5.3 The Modified Conjugate Gradient Method</H2>
The CG method (technically, the preconditioned CG method) takes a symmetric positive semi-definite matrix A, a symmetric positive definite preconditioning matrix P of the same dimension as A, a vector b and iteratively solves A v = b. The iteration stops when b − A v is less than b where is a user-defined tolerance value. The preconditioning matrix P, which must be easily invertible, speeds convergence to the extent that P −1 approximates A. We wholeheartedly refer the reader to Shewchuk
[ 14 ] for information on the CG method. We derive our modified conjugate gradient method by observing that the effect of the matrix W in equation (14) is to filter out velocity changes in the constrained directions. Our idea then is to define an invariant— for all i, the component of v i in the constrained direction(s) of particle i is equal to z i —and then establish and maintain the invariant at each iteration, by defining a filtering procedure filter. The role of filter is to take a vector a and perform the same filtering operation (see equation (13)) as multiplying by W, but leaving out the scaling by 1/m i : procedure filter(a) for i = 1 to n a i = S i a i return a
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Using filter, we define the modified CG method modified-pcg as follows:
1 procedure modified-pcg 2 v = z 3 δ 0 = filter(b) T P filter(b) 4 r = filter(b − A v) 5 c = filter(P −1 r) 6 δ new = r T c 7 while δ new > 2 δ 0 8 q = filter(Ac) 9 α = δ new /(c T q) 10 v = v + αc 11 r = r − αq 12 s = P −1 r 13 δ old = δ new 14 δ new = r T s 15 c = filter(s + δ new c) δ old Line 2 of the procedure establishes our invariant. Lines 5 and 15 maintain the invariant by filtering c before adding it to v. The unmodified conjugate gradient method establishes a stopping criterion based on b T Pb. Since our constrained formulation ignores certain components of b, our stopping criterion should as well, so we add filtering to line 3. The vector r measures the solution error b − A v, and should not include error due to the constraints; hence we add filtering at lines 4 and 8. (Note that removing the calls to filter and changing line 2 to v = 0 yields the standard preconditioned conjugate gradient method.) We use a simple preconditioner P by making P be a diagonal matrix with P ii = 1/A ii so products involving P −1 are trivially computed. More elaborate preconditioners could be used, though we doubt there is a large speedup to be gained. Matrix-vector products with A are of course implemented in sparse matrix-vector fashion, using the data structures defined in section 2.3. Given modified-pcg, obvious questions are “does it work?” followed by “how does it compare with the unmodified CG method?” Proofs about CG methods are difficult in general; in practice, our method always converges, which answers the first question. Prior to implementing modified-pcg, we used a penalty method and applied the standard CG method to equation (15). When we began using procedure modified-pcg, we did not notice any substantial change in the number of iterations required by the method. Empirically, we conclude that the two methods have similar convergence behavior. Result in section 8 indicate that the running time is close to O(n 1.5 ), which is what unmodified CG would be expected to deliver on this sort of problem [ 14 ].
<H2>5.4 Determining the Constraint Forces</H2>
For contact constraints (between cloth and solid objects) we need to know what the actual force of constraint is, in order to determine when to terminate a constraint. Additionally, we need to know the constraint force actually exerted in order to model frictional forces properly. Fortunately, it is easy to add one more step to modified-pcg to determine the constraint force. When modified-pcg terminates, the residual error e = A v − b has the property that e i need not be close to zero if particle i is constrained. In fact, e i is exactly the extra constraint force that must have been supplied to enforce the constraint. Thus, we can compute constraint forces at the end of modified-pcg by performing one last matrixvector product to compute A v − b. (The vector r in modified-pcg is equal to filter(A v − b), so the extra matrix-vector product to compute e really is necessary.) The particles’ accelerations are inherently dependent on one another through the matrix A of equation (16). This means that the correct approach to determing constraint release is combinatoric, as in Baraff [ 2 ]. We reject this approach as impractical given the dimension of A. Instead, we allow contacts to release when the constraint force between a particle and a solid switches from a repulsive force to an attractive one. In practice, this has proven to work well. Friction presents a similar problem. When cloth contacts a solid, we lock the particle onto the surface, if the relative tangential velocity is low. We monitor the constraint force, and if the tangential force exceeds some fraction of the normal force, we allow the particle to slide on the surface. For high sliding velocities, we apply a dissipative tangential force, opposite the relative sliding direction, proportional to the normal force.
<H1>6 Collisions</H1>
Much has been written about collision detection for cloth; we have nothing substantial to add to the subject of collision detection per se. Cloth/cloth collisions are detected by checking pairs ( p, t) and (e 1 , e 2 ) for intersections, where p and t are a cloth particle and a cloth triangle respectively, and e 1 and e 2 are edges of cloth triangles. Given a previous known legal state of the cloth, we postulate a linear motion for the cloth particles to the current (possibly illegal) state and check for either particle/triangle or edge/edge crossings. To avoid O(n 2 ) comparisons, we use a coherency-based boundingbox approach
[ 1 ] to cull out the majority of pairs. When collisions between a cloth vertex and triangle, or two cloth edges are detected, we insert a strong damped spring force to push the cloth apart. A dissipative force tangent to the contact is also applied, countering any sliding motion. The force is not, strictly speaking, a frictional force: rather it is proportional to the slip velocity, so it is in actuality a damping force, although it reasonably emulates dynamic friction. Applying static friction forces to cloth contacts is far more difficult, and is a problem we have not solved yet. The forces, and their derivatives with respect to position and velocity, are of course included in equation (15). Our system detects collisions between cloth particles and solid objects by testing each individual cloth particle against the faces of each solid object. A solid object’s faces are grouped in a hierarchical bounding box tree, with the leaves of the tree being individual faces of the solid. The tree is created by a simple recursive splitting along coordinate axes. The maintenance of contacts and the application of friction forces was described in the previous section.
<H2>6.1 Constraint Initiation</H2>
Both cloth/cloth and cloth/solid collisions give rise to the same problem whenever two contacts form. For both types of collisions, our detection algorithm reports an intersection, and then takes action to remedy the situation: either by enforcing a constraint (cloth/solid collisions) or by adding a penalty force (cloth/cloth) collisions. However, since our simulator proceeds in discrete steps, collisions resulting in a reasonably substantial interpenetration depth can occur between one step and the next. Clearly, this situation needs to be remedied. For cloth/cloth collisions, this would not appear to be a problem: the spring forces that are added work to counter the colliding velocities and then push the cloth apart. For cloth/solid collisions, however, the situation is more complicated. If we simply enforce a constraint which causes the colliding cloth particle to have a velocity consistent with the solid object’s velocity, and continue to enforce that constraint, the cloth particle will continue to remain embedded somewhere below the solid object’s surface. This is unacceptable. One solution is to use Baumgarte stabilization [ 18 ]
, which schedules the particle’s acceleration so that the position and velocity error of the particle with respect to the surface decay asymptotically to zero. We experimented with this technique, but found it lacking. In particular, a fast rise to the surface was prone to noise and “jumpiness”; this could be eliminated, but at the cost of decreasing the step size. A slower rise to the surface caused visual artifacts. We tried a simpler solution: when intersections occurred, rather than wait for a scheduled constraint or a penalty force to eliminate the intersection, we simply altered the positions of the cloth particles, effecting an instantaneous (and discontinuous) change in position. While this would be problematic when using a multi-step differential equation solver which expects continuity (such as a RungeKutta method), it should not interfere with a one-step solver such as the backward Euler method. Unfortunately, simply changing particle positions produced disastrous results. The stretch energy term in a cloth system is extremely strong, and altering particle positions arbitrarily introduced excessively large deformation energies in an altered particle’s neighborhood. This resulted in visibly “jumpy” behavior of the cloth in localized regions.
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<H2>6.2 Position Alteration</H2>
Despite its initial failure, the ability to make arbitrary small changes in a particle’s position continued to attract our attention. The entire process of implicit integration can be considered to be a filtering process
[ 7 ], and we postulated that a mechanism for filtering energy changes caused by displacing particles might make position alteration a viable technique. We considered that perhaps some sort of extra implicit step could be used as a filter, but forming and solving an additional linear system at each step seemed too expensive. Happily, we can make use of the filtering effect of implicit integration without any extra work. Consider a particle that has collided with a solid object. The particle’s change in velocity at each step is under our control, using the constraint techniques described in section 5. Meanwhile, the particle’s position at the next step follows from equation (4): x i = h(v 0i + v i ) (recall that v 0i is the particle’s current velocity). The reason that changing positions after a step has been taken doesn’t work is because the particle’s neighbors receive no advance notification of the change in position: they are confronted with the alteration at the beginning of the next step. This presents an obvious solution: we simply modify the top row of equation (4) to
17
x i = h(v 0i + v i ) + y i
where y i is an arbitrary correction term of our choice, introduced solely to move a particle to a desired location during the backward Euler step. Having modified the top row of equation (4), we must follow this change through: using equation (17) and repeating the derivation of section 3 and the symmetric transform from section 5 yields the modified symmetric system
18
M − h ∂v ∂f − h 2 ∂x ∂f v = h f 0 + h ∂x ∂f v 0 + ∂x ∂f y .
This modification gives us complete control over both the position and velocity of a constrained particle in just one step, without any extra computational cost. We use this technique to bring particles quickly and stably to the surface of solid objects without creating visual artifacts or limiting the allowable step size. We can also add correction terms to particles involved in cloth/cloth collisions. Without a constraint on those particles’ velocities there is no guarantee that they will go exactly where we want in one step, but the ability to induce sizeable jumps in position without excessively stiff spring forces adds greatly to the stability of the simulation.
<H1>7 Adaptive Time Stepping</H1>
The methods introduced in all of the previous sections usually allow us to take sizeable steps forward, without loss of stability. Even so, there are still times when the step size must be reduced to avoid divergence. There are a large number of methods for altering the size of a time step, for both explicit and implicit integrators, but these methods tend to concentrate on the accuracy of the simulation, and not the stability. Our goal is animation, not engineering; thus visually pleasing results, meaning a numerically stable solution, rather than overall accuracy, is the deciding voice. The trick is to recognize instability before you see it on your screen—by then it’s too late. Stiffness, and thus any potential instability, arises almost completely from the strong stretch forces in the cloth. After each implicit step, we treat the resulting x as a proposed change in the cloth’s state, and examine the stretch terms (section 4.2) for each triangle in the newly proposed state. If any triangle undergoes a drastic change in its stretch (in either the u or v direction) we discard the proposed state, reduce the step size, and try again. Subtlety is not required: we find that an unstable step invariably results in stretch changes that are quite large, and are thus easily detected. Our simulation is run with a parameter that indicates the maximum allowable step size: this parameter is set by the user, and is always less than or equal to one frame. (Most of our simulations involving human motions use a step size of 0.02 seconds.) Whenever the simulator reduces the step size, after two successes with the reduced step size the simulator tries to increase the step size. If the simulator fails at the larger step size, it reduces the size again, and waits for a longer period of time before retrying to increase the step size. At its limit, the simulator will try increasing the step size every 40 steps; thus, if the user chooses too large a step, the simulator settles down to wasting only one out of every 40 steps in attempting too large a step. This method, though simple, has served us well.
<H1>8 Results</H1>
Table 1 gives a performance summary of assorted animations, shown in figures 1–6. Unaccounted overhead of the simulation (typically about 5%) includes tasks such as geometry transformations, memory allocation, etc. The clothes in figures 3–6 were modeled as discrete planar panels, and then topologically seamed. The simulator was used to relax the clothing from an initial deformed state, that got the clothes around the characters, to a well-fitting state on the characters. The b u and b v parameters (see equation (10)) were then made smaller in certain regions to produce cuffs and waistbands, or strategically increased to induce wrinkling behavior in other regions. We also ran the simulation in figure 1 with a range of stiffnesses for the bend term. Using the stiffness parameters in figure 1 as a reference, we ran the simulation with those bend stiffnesses multiplied by 0.1, 1.0, 10, 100 and 1,000 (for a total range of 10,000 in the stiffness). The variance in the running times was under 5%. We doubt that simulators based on explicit integration methods could make a similar claim. Finally, we tried to estimate our simulator’s performance as a function of n, the number of cloth particles. We ran the simulation in figure 1 with cloth resolutions of 500, 899, 2,602 (shown in figure 1 ) and 7,359 particles. The running times were, respectively, 0.23 seconds/frame, 0.46 seconds/frame, 2.23 seconds/frame, and 10.3 seconds/frame. This is slightly better than O(n 1.5 ) performance, which is in line with the convergence rates of the conjugate gradient method [ 14 ] for systems such as equation (18).
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figure
no. vertices/no. triangles
time/frame
cloth solid
(CPU sec.)
1
2,602/4,9442 322/640
2.23
2
2,602/4,9442 322/640
3.06
3
6,450/12,654 9,941/18,110
7.32
4
(shirt)
6,450/12,654 9,941/18,110
14.5
(pants)
8,757/17,352 9,941/18,110
38.5
5
(skirt)
2,153/4,020 7,630/14,008
3.68
(blouse)
5,108/10,016 7,630/14,008
16.7
6
(skirt)
4,530/8,844 7,630/14,008
10.2
(blouse)
5,188/10,194 7,630/14,008
16.6
figure no. vertices/no. triangles time/frame cloth solid (CPU sec.) 1 2,602/4,9442 322/640 2.23 2 2,602/4,9442 322/640 3.06 3 6,450/12,654 9,941/18,110 7.32 4 (shirt) 6,450/12,654 9,941/18,110 14.5 (pants) 8,757/17,352 9,941/18,110 38.5 5 (skirt) 2,153/4,020 7,630/14,008 3.68 (blouse) 5,108/10,016 7,630/14,008 16.7 6 (skirt) 4,530/8,844 7,630/14,008 10.2 (blouse) 5,188/10,194 7,630/14,008 16.6
Table 1: System performance for simulations in figures 1–6. Minimum and maximum time steps are in milliseconds of simulation time. Time/frame indicates actual CPU time for each frame, averaged over the simulation. Percentages of total running time are given for four tasks: EVAL— forming the linear system of equation (18); CG—solving equation (18); C/C—cloth/cloth collision detection; and C/S—cloth/solid collision detection.
<H1>9 Acknowledgments</H1>
This research was supported in part by an ONR Young Investigator award, an NSF CAREER award, and grants from the Intel Corporation. We thank Alias|Wavefront for supplying the models and motion capture data used in figures 5 and 6.
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C OMPUTER G RAPHICS Proceedings, Annual Conference Series, 1998
Figure 1 (top row): Cloth draping on cylinder; frames 8, 13 and 35. Figure 2 (second row): Sheet with two fixed particles; frames 10, 29 and 67. Figure 3 (third row): Shirt on twisting figure; frames 1, 24 and 46. Figure 4 (bottom row): Walking man; frames 30, 45 and 58.
53
SIGGRAPH 98, Orlando, July 19–24
C OMPUTER G RAPHICS Proceedings, Annual Conference Series, 1998
Figure 5 (top row): Dancer with short skirt; frames 110, 136 and 155. Figure 6 (middle row): frames 185, 215 and 236. Figure 7 (bottom row): Closeups from figures 4 and 6.
Dancer with long skirt;
54
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"text": "['Later work by the same group includes Volino et al.']",
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"text": "['20']",
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"text": "['gives the only published results we know of for simulated garments on moving characters']",
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"text": "['the accumulated work by this group']",
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"text": "['Breen et al.']",
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"text": "['3']",
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"text": "['describe what they call a “particlebased” approach to the problem']",
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"text": "['By making use of real-world cloth material properties']",
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"text": "['they produced highly realistic static images of draped rectangular cloth meshes with reported resolutions of up to 51 × 51 nodes']",
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"text": "['The focus of this work is on static poses for cloth']",
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"text": "['their simulation process is best described as energy minimization, although methods analogous to explicit methods are used']",
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"text": "['Speed was of secondary concern in this work']",
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"text": "['obtain similarly realistic results']",
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"text": "['dropping the computational cost to approximately 20–30 minutes per frame on an SGI R8000 processor']",
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"text": "['Refinements by Eberhardt et al.']",
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"text": "['5']",
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"text": "['focuses on improving the performance of explicit methods by a post-step modification of nodal positions']",
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"text": "['Provot']",
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"text": "['13']",
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"text": "['the convergence properties of this method are unclear']",
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"text": "['He iteratively adjusts nodal positions to eliminate unwanted stretch']",
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"text": "['A more comprehensive discussion on cloth research can be found in the survey paper by Ng and Grimsdale']",
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"text": "['9']",
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"text": "['this simple test may miss some collisions']",
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"text": "['relative velocities are extremely high']",
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"text": "['analytically checking for intersection between previous and current positions can guarantee that no collisions are missed']",
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"text": "['The most critical forces in the system are the internal cloth forces which impart much of the cloth’s characteristic behavior']",
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"text": "['describes the use of the Kawabata system of measurement for realistic determination of the in-plane shearing and out-of-plane bending forces in cloth']",
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"text": "['Breen et al.']",
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"text": "['3']",
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"text": "['cloth does not stretch appreciably under its own weight']",
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"text": "['normal conditions']",
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"text": "['This requires the stretch force to have a high coefficient of stiffness']",
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"text": "['it is the stretch force that is most responsible for the stiffness of equation (1)']",
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"text": "['A common practice in explicitly integrated cloth systems is to improve running time by decreasing the strength of the stretch force']",
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"text": "['this leads to “rubbery” or “bouncy” cloth']",
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"text": "['Our system uses a very stiff stretch force to combat this problem, without any detrimental effects on the run-time performance']",
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"text": "['the shear and bend force stiffness coefficients depend on the material being simulated']",
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"text": "['the stretch coefficient is essentially the same (large) value for all simulations']",
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"text": "['Complementing the above three internal forces are three damping forces']",
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"text": "['we formulate damping forces that subdue any oscillations having to do with']",
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"text": "['section 5']",
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"text": "['The damping forces do not dissipate energy due to other modes of motion']",
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"text": "['Additional forces include air-drag, gravity, and user-generated generated mouse-forces']",
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"text": "['Cloth/cloth contacts generate strong repulsive linear-spring forces between cloth particles']",
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"offsets": [
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"text": "['the acceleration ẍ i of the ith particle is simply ẍ i = f i /m i , where m i is the ith particle’s mass']",
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"text": "['Combining all forces into a net force vector f']",
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"offsets": [
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"text": "['to accommodate pieces of cloth that have been topologically seamed together (such as a sleeve), particles lying on the seam must have multiple (u, v) coordinates']",
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"text": "['each particle has a unique (u, v) coordinate']",
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"text": "['The (u, v) coordinates are useful for texturing']",
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"offsets": [
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19246,
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],
"text": "['Defining the diagonal mass matrix M ∈ IR 3n×3n by diag(M) = (m 1 , m 1 , m 1 , m 2 , m 2 , m 2 , . . . , m n , m n , m n )']",
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"text": "['ẍ = M −1 f(x, x)']",
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"id": "T172"
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"offsets": [
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19586,
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"text": "['generates large unbanded sparse linear systems']",
"type": "own_claim",
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"text": "['The use of an implicit integration method']",
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"text": "['next section']",
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"text": "['We solve these systems through a modified conjugate gradient (CG) iterative method']",
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"text": "['section 5']",
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"text": "['they are based solely on matrix-vector multiplies']",
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"text": "['require only rudimentary sparse storage techniques']",
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"text": "['The sparsity of the matrix generated by the implicit integrator is best represented in block-fashion']",
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"text": "['for a system with n particles, we deal with an n × n matrix, whose non-zero entries are represented as dense 3 × 3 matrices of scalars']",
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"text": "['The overall implementation of sparsity is completely straightforward']",
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"text": "['An individual particle’s position and velocity can be completely controlled in either one, two, or three dimensions']",
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"text": "['Particles can thus be attached to a fixed or moving point in space']",
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"text": "['constrained to a fixed or moving surface or curve']",
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"text": "['The constraint techniques we use on individual particles work just as well for collections of particles']",
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"text": "['we could handle cloth/cloth intersections using the technique described']",
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"text": "['section 5']",
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"text": "['the cost is potentially large']",
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"text": "['we have chosen to deal with cloth/cloth contacts using penalty forces']",
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"text": "['The implicit solver easily tolerates these stiff forces']",
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"text": "['our goal is to determine a new position x(t 0 + h) and velocity x(t 0 + h) at time t 0 + h']",
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"text": "['the known position x(t 0 ) and velocity x(t 0 ) of the system at time t 0']",
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22482
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"text": "['This is accomplished simply by defining the system’s velocity v as v = x and then writing', '3 ', 'dt d x x = ', 'dt d v x = M −1 f(x, v v)']",
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"text": "['the step size h must be quite small to ensure stability when using this method']",
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"text": "['previously discussed']",
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"text": "['The difference in the two methods is that the forward method’s step is based solely on conditions at time t 0 while the backward method’s step is written in terms of conditions at the terminus of the step itself']",
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"text": "['The explicit forward Euler method applied to equation (3) approximates x and v as x v = h M v −1 0 f 0', 'where the force f 0 is defined by f 0 = f(x 0 , v 0 )']",
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"text": "['x and v are approximated by', '4 ', 'v x = h M −', '1 f(x 0 v + 0 + x, v v 0 + v)']",
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"text": "['The forward method requires only an evaluation of the function f']",
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"text": "['the backward method requires that we solve for values of x and v that satisfy equation (4)']",
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"text": "['rather than solve this equation exactly (which would require iteration) we apply a Taylor series expansion to f and make the firstorder approximation ∂f ∂f f(x 0 + x, v 0 + v) = f 0 + ∂x x + ∂v v']",
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"text": "['Equation (4) is a nonlinear equation']",
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"text": "['the backward Euler step consists of evaluating f 0 , ∂f/∂x and ∂f/∂v; forming the system in equation (6); solving the system for v; and then updating x and v']",
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"text": "['the derivative ∂f/∂x is evaluated for the state (x 0 , v 0 ) and similarly for ∂f/∂v']",
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24046,
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"text": "['v x = h M −1 (f 0 + v 0 ∂x ∂f + x v + ∂v ∂f v)']",
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23940,
24014
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"text": "['Substituting this approximation into equation (4) yields the linear system']",
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24112,
24231
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"text": "['Taking the bottom row of equation (5) and substituting x = h(v 0 + v) yields v = hM −1 f 0 + ∂x ∂f h(v 0 + v) + ∂v ∂f v']",
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24329,
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"text": "['I − hM −1 ∂v ∂f − h 2 M −1 ∂x ∂f v = hM −1 f 0 + h ∂x ∂f v 0']",
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24234,
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"text": "['Letting I denote the identity matrix, and regrouping, we obtain']",
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"text": "['which we then solve for v']",
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24434,
24478
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"text": "['Given v, we trivially compute x = h(v 0 + v)']",
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"text": "['The method is called “backward” Euler']",
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24912,
25027
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"text": "['starting from the output state (x 0 + x, v 0 + v) and using a forward Euler step to run the system backward in time']",
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25152,
25212
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"text": "['Forward Euler takes no notice of wildly changing derivatives']",
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25218,
25248
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"text": "['proceeds forward quite blindly']",
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25275,
25370
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"text": "['forces one to find an output state whose derivative at least points back to where you came from']",
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25250,
25264
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"text": "['Backward Euler']",
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25846,
25888
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"text": "['and then taking derivatives is impractical']",
"type": "own_claim",
"id": "T222"
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"offsets": [
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25732,
25787
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],
"text": "['Expressing the energy E as a single monolithic function']",
"type": "own_claim",
"id": "T223"
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{
"offsets": [
[
25924,
25994
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],
"text": "['A better approach is decompose E into a sum of sparse energy functions']",
"type": "own_claim",
"id": "T224"
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"offsets": [
[
26117,
26178
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"text": "['even decomposing E into sparse energy functions is not enough']",
"type": "own_claim",
"id": "T225"
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"offsets": [
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26180,
26230
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"text": "['Energy functions are an undesirable starting point']",
"type": "own_claim",
"id": "T226"
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"offsets": [
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26239,
26305
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"text": "['sensible damping functions cannot be derived from energy functions']",
"type": "data",
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26316,
26497
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"text": "['we define internal behavior by formulating a vector condition C(x) which we want to be zero, and then defining the associated energy as k C(x) T C(x) where k is a stiffness constant']",
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"text": "['An added bonus is that starting from this vector-based energy description tends to result in a simpler, more compact, and more easily coded formulation for ∂f/∂x than proceeding from an energy function in which the structure of C has been lost']",
"type": "own_claim",
"id": "T229"
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"offsets": [
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28069,
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"text": "['We can now easily describe the internal forces acting on the cloth, by just writing condition functions']",
"type": "own_claim",
"id": "T230"
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28174,
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"text": "['Forces and their derivatives are easily derived']",
"type": "own_claim",
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28228,
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"text": "['equations (7)']",
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28246,
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"text": "['(8)']",
"type": "data",
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"offsets": [
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28515,
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"text": "['it will be convenient to pretend momentarily that we have a single continuous function w(u, v) that maps from plane coordinates to world space']",
"type": "own_claim",
"id": "T234"
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"offsets": [
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28962,
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"text": "['Some previous continuum formulations have modeled stretch energy along an axis as essentially (w u T w u − 1) 2']",
"type": "background_claim",
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"offsets": [
[
29119,
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"text": "['15']",
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29124,
29126
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"text": "['16']",
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29129,
29131
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"text": "['17']",
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"text": "['4']",
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"text": "['We find this to be needlessly stiff']",
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29183,
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"text": "['near the rest state, the force gradient']",
"type": "own_claim",
"id": "T241"
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"text": "['is quite small, which partially negates the advantage implicit integration has in exploiting knowledge of the force gradient']",
"type": "own_claim",
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"text": "['A quadratic model for energy is, numerically, a better choice']",
"type": "own_claim",
"id": "T243"
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"offsets": [
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30978,
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"text": "['We have found the ability to control shrink/stretch anisotropically to be an indispensable modeling tool']",
"type": "own_claim",
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"offsets": [
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30887,
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"text": "['we might decrease b v near the end of a sleeve, inducing a tight cuff']",
"type": "own_claim",
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{
"offsets": [
[
30961,
30976
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"text": "['on a sweatshirt']",
"type": "data",
"id": "T246"
},
{
"offsets": [
[
31195,
31304
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],
"text": "['We can measure the extent to which cloth has sheared in a triangle by considering the inner product w u T w v']",
"type": "own_claim",
"id": "T247"
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"offsets": [
[
31326,
31346
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"text": "['this product is zero']",
"type": "own_claim",
"id": "T248"
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"offsets": [
[
31314,
31324
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],
"text": "['rest state']",
"type": "data",
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"offsets": [
[
31437,
31458
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"text": "['we need not normalize']",
"type": "own_claim",
"id": "T250"
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"offsets": [
[
31354,
31435
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],
"text": "['the stretch term prevents the magnitudes of w u and w v from changing overly much']",
"type": "data",
"id": "T251"
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"offsets": [
[
31494,
31564
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],
"text": "['the product w u T w v is a reasonable approximation to the shear angle']",
"type": "own_claim",
"id": "T252"
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{
"offsets": [
[
31463,
31492
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],
"text": "['the small angle approximation']",
"type": "data",
"id": "T253"
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"offsets": [
[
32577,
32637
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],
"text": "['the condition C should not be scaled by the triangles’ areas']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
32514,
32575
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],
"text": "['increases at the same rate that the triangle’s area decreases']",
"type": "own_claim",
"id": "T255"
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{
"offsets": [
[
32440,
32467
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],
"text": "['the square of the curvature']",
"type": "own_claim",
"id": "T256"
},
{
"offsets": [
[
32861,
32927
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],
"text": "['Rectangular meshes make it simple to treat bending anisotropically']",
"type": "own_claim",
"id": "T257"
},
{
"offsets": [
[
32929,
33022
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],
"text": "['The uv coordinates associated with particles make this possible for triangular meshes as well']",
"type": "own_claim",
"id": "T258"
},
{
"offsets": [
[
33278,
33356
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],
"text": "['The stiffness weighting for this edge should simply be k u ( u) 2 + k v ( v) 2']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
[
33244,
33275
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],
"text": "['u = u i − u j and v = v i − v j']",
"type": "data",
"id": "T260"
},
{
"offsets": [
[
33172,
33231
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],
"text": "['the edge between the triangles be between particles i and j']",
"type": "data",
"id": "T261"
},
{
"offsets": [
[
33030,
33122
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],
"text": "['material for which bending in the u and v directions are weighted by stiffnesses k u and k v']",
"type": "data",
"id": "T262"
},
{
"offsets": [
[
33724,
33775
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],
"text": "['These forces and their gradients are easily derived']",
"type": "own_claim",
"id": "T263"
},
{
"offsets": [
[
33828,
33894
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],
"text": "['The energies we have just described are functions of position only']",
"type": "own_claim",
"id": "T264"
},
{
"offsets": [
[
33938,
34041
]
],
"text": "['is critically dependent on well-chosen damping forces that are a function of both position and velocity']",
"type": "background_claim",
"id": "T265"
},
{
"offsets": [
[
33896,
33927
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],
"text": "['Robust dynamic cloth simulation']",
"type": "background_claim",
"id": "T266"
},
{
"offsets": [
[
34056,
34135
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],
"text": "['the strong stretch force must be accompanied by a suitably strong damping force']",
"type": "own_claim",
"id": "T267"
},
{
"offsets": [
[
34139,
34229
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],
"text": "['we are to prevent anomalous in-plane oscillations from arising between connected particles']",
"type": "data",
"id": "T268"
},
{
"offsets": [
[
34240,
34343
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],
"text": "['this strong damping force must confine itself solely to damping in-plane stretching/compressing motions']",
"type": "own_claim",
"id": "T269"
},
{
"offsets": [
[
34345,
34436
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],
"text": "['stretch damping should not arise due to motions that are not causing stretch or compression']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
34472,
34596
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],
"text": "['treatment of cloth used a simple viscous damping function which dissipated kinetic energy, independent of the type of motion']",
"type": "background_claim",
"id": "T271"
},
{
"offsets": [
[
34438,
34458
]
],
"text": "['Terzopoulos et al.’s']",
"type": "background_claim",
"id": "T272"
},
{
"offsets": [
[
34461,
34463
]
],
"text": "['16']",
"type": "data",
"id": "T273"
},
{
"offsets": [
[
34466,
34468
]
],
"text": "['17']",
"type": "data",
"id": "T274"
},
{
"offsets": [
[
34621,
34695
]
],
"text": "['improved upon this somewhat, borrowing a formulation due to Platt and Barr']",
"type": "background_claim",
"id": "T275"
},
{
"offsets": [
[
34598,
34613
]
],
"text": "['Carignan et al.']",
"type": "background_claim",
"id": "T276"
},
{
"offsets": [
[
34616,
34617
]
],
"text": "['4']",
"type": "data",
"id": "T277"
},
{
"offsets": [
[
34698,
34700
]
],
"text": "['11']",
"type": "data",
"id": "T278"
},
{
"offsets": [
[
34766,
34840
]
],
"text": "['does not match the quartic energy functions of their continuum formulation']",
"type": "background_claim",
"id": "T279"
},
{
"offsets": [
[
34713,
34735
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],
"text": "['their damping function']",
"type": "background_claim",
"id": "T280"
},
{
"offsets": [
[
34858,
34966
]
],
"text": "['we describe a general treatment for damping that is independent of the specific energy function being damped']",
"type": "own_claim",
"id": "T281"
},
{
"offsets": [
[
34845,
34857
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],
"text": "['this section']",
"type": "data",
"id": "T282"
},
{
"offsets": [
[
35102,
35135
]
],
"text": "['This is an easy trap to fall into']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
34968,
35082
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],
"text": "['It is tempting to formulate a damping function for an energy function E(x) by measuring the velocity of the energy']",
"type": "own_claim",
"id": "T284"
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{
"offsets": [
[
35141,
35169
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],
"text": "['it gives nonsensical results']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
35269,
35349
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],
"text": "['E ̇ is zero when E is at its minimum, regardless of the system’s velocity x = v']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
35236,
35254
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],
"text": "['E ̇ = (∂E/∂x) T x']",
"type": "data",
"id": "T287"
},
{
"offsets": [
[
35171,
35228
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],
"text": "['At an equilibrium point of E, the gradient ∂E/∂x vanishes']",
"type": "own_claim",
"id": "T288"
},
{
"offsets": [
[
35363,
35416
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],
"text": "['E ̇ is always too small near the system’s rest state']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
35427,
35485
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],
"text": "['basing the damping force on E ̇ is not what we want to do']",
"type": "own_claim",
"id": "T290"
},
{
"offsets": [
[
35503,
35644
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],
"text": "['the damping function should be defined not in terms of the energy E, but in terms of the condition C(x) we have been using to define energies']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
35646,
35750
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],
"text": "['The force f arising from the energy acts only in the direction ∂C(x)/∂x, and so should the damping force']",
"type": "own_claim",
"id": "T292"
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{
"offsets": [
[
35766,
35865
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],
"text": "['the damping force should depend on the component of the system’s velocity in the ∂C(x)/∂x direction']",
"type": "own_claim",
"id": "T293"
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"offsets": [
[
36082,
36104
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"text": "['d = −k d ∂C(x) ∂x C(x)']",
"type": "own_claim",
"id": "T294"
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{
"offsets": [
[
35986,
36049
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"text": "['the damping force d associated with a condition C have the form']",
"type": "own_claim",
"id": "T295"
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{
"offsets": [
[
35883,
35942
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],
"text": "['the damping strength should depend on (∂C(x)/∂x) T x = C(x)']",
"type": "own_claim",
"id": "T296"
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"offsets": [
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36132,
36184
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"text": "['This neatly parallels the fact that f = −k s ∂x C(x)']",
"type": "own_claim",
"id": "T297"
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{
"offsets": [
[
36418,
36486
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"text": "['we can now add accompanying damping forces by applying equation (11)']",
"type": "own_claim",
"id": "T298"
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{
"offsets": [
[
36324,
36416
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"text": "['the condition functions C we have defined in this section for stretch, bend and shear forces']",
"type": "data",
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"offsets": [
[
37028,
37106
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"text": "['we find it expedient simply to leave this term out, thereby restoring symmetry']",
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"id": "T300"
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{
"offsets": [
[
37015,
37026
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"text": "['section 5.2']",
"type": "data",
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"offsets": [
[
36681,
36749
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],
"text": "['∂x ∂d i j = −k d ∂C(x) ∂x i ∂ C(x) ∂x ̇ j T + ∂ ∂x C(x) i ∂x j C(x)']",
"type": "own_claim",
"id": "T302"
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36639,
36648
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"text": "['we obtain']",
"type": "own_claim",
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{
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36608,
36637
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],
"text": "['Differentiating equation (11)']",
"type": "data",
"id": "T304"
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{
"offsets": [
[
36867,
36906
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],
"text": "['we cannot expect ∂d/∂x to be symmetrica']",
"type": "own_claim",
"id": "T305"
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{
"offsets": [
[
36782,
36831
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],
"text": "['∂d/∂x is not a second derivative of some function']",
"type": "data",
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{
"offsets": [
[
37108,
37165
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],
"text": "['This simplification is clearly not physically justifiable']",
"type": "own_claim",
"id": "T307"
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"offsets": [
[
37171,
37226
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],
"text": "['we have not observed any ill effects from this omission']",
"type": "own_claim",
"id": "T308"
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"offsets": [
[
37556,
37608
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],
"text": "['the result is symmetrical without dropping any terms']",
"type": "own_claim",
"id": "T309"
},
{
"offsets": [
[
37354,
37378
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],
"text": "['C(x) ̇ = (∂C(x)/∂x) T v']",
"type": "data",
"id": "T310"
},
{
"offsets": [
[
37388,
37419
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],
"text": "['∂ C(x) ̇ ∂ ∂C(x) T ∂C(x) = v =']",
"type": "own_claim",
"id": "T311"
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{
"offsets": [
[
37422,
37433
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],
"text": "['∂v ∂v ∂x ∂x']",
"type": "own_claim",
"id": "T312"
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{
"offsets": [
[
37464,
37539
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],
"text": "['∂v ∂d i j = −k d ∂C(x) ∂x i ∂ C(x) ∂v ̇ j T = −k d ∂C(x) ∂x i ∂C(x) ∂x j T']",
"type": "own_claim",
"id": "T313"
},
{
"offsets": [
[
37991,
38078
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],
"text": "['The techniques we describe in this section could be used for multi-particle constraints']",
"type": "own_claim",
"id": "T314"
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{
"offsets": [
[
38089,
38144
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"text": "['constraints that share particle would need to be merged']",
"type": "own_claim",
"id": "T315"
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"offsets": [
[
38152,
38186
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],
"text": "['a set of four-particle constraints']",
"type": "own_claim",
"id": "T316"
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{
"offsets": [
[
38248,
38355
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],
"text": "['might merge to form a single constraint on arbitrarily many particles, which would be expensive to maintain']",
"type": "own_claim",
"id": "T317"
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{
"offsets": [
[
38196,
38246
]
],
"text": "['vertex/triangle or edge/edge contacts in the cloth']",
"type": "data",
"id": "T318"
},
{
"offsets": [
[
38374,
38424
]
],
"text": "['we handle cloth/cloth contacts with strong springs']",
"type": "own_claim",
"id": "T319"
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{
"offsets": [
[
38505,
38531
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],
"text": "['and “position alteration,”']",
"type": "own_claim",
"id": "T320"
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{
"offsets": [
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38557,
38566
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],
"text": "['section 6']",
"type": "data",
"id": "T321"
},
{
"offsets": [
[
38810,
38920
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],
"text": "['it is the particle’s acceleration, or equivalently, the change in the particle’s velocity, that is constrained']",
"type": "own_claim",
"id": "T322"
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{
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[
38766,
38808
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"text": "['the differential nature of our formulation']",
"type": "data",
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[
38983,
39032
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"text": "['we are explicitly setting the particle’s velocity']",
"type": "own_claim",
"id": "T324"
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{
"offsets": [
[
39100,
39192
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],
"text": "['we are constraining the particle’s velocity along either two or one mutually orthogonal axes']",
"type": "own_claim",
"id": "T325"
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{
"offsets": [
[
38925,
38976
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],
"text": "['the particle is constrained in all three dimensions']",
"type": "data",
"id": "T326"
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{
"offsets": [
[
39056,
39098
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],
"text": "['the constraint is in two or one dimensions']",
"type": "data",
"id": "T327"
},
{
"offsets": [
[
39384,
39532
]
],
"text": "['An obvious and quite exact method for constraining a particle is to reduce the number of coordinates describing the particle’s position and velocity']",
"type": "own_claim",
"id": "T328"
},
{
"offsets": [
[
39534,
39593
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],
"text": "['A completely constrained particle would have no coordinates']",
"type": "own_claim",
"id": "T329"
},
{
"offsets": [
[
39601,
39671
]
],
"text": "['a particle with one dimension of constraint would have two coordinates']",
"type": "own_claim",
"id": "T330"
},
{
"offsets": [
[
39673,
39689
]
],
"text": "['This is possible']",
"type": "own_claim",
"id": "T331"
},
{
"offsets": [
[
39694,
39729
]
],
"text": "['it complicates the system immensely']",
"type": "own_claim",
"id": "T332"
},
{
"offsets": [
[
39734,
39782
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],
"text": "['we change the number of coordinates per particle']",
"type": "data",
"id": "T333"
},
{
"offsets": [
[
39784,
39877
]
],
"text": "['we alter the size of the derivative matrices in equation (6), as well as the sparsity pattern']",
"type": "own_claim",
"id": "T334"
},
{
"offsets": [
[
40056,
40080
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],
"text": "['this is most unappealing']",
"type": "own_claim",
"id": "T335"
},
{
"offsets": [
[
39986,
40054
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],
"text": "['the transient nature of contact constraints between cloth and solids']",
"type": "data",
"id": "T336"
},
{
"offsets": [
[
40082,
40161
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],
"text": "['The computation of the derivative matrices’ entries is also greatly complicated']",
"type": "own_claim",
"id": "T337"
},
{
"offsets": [
[
40171,
40290
]
],
"text": "['we must now introduce extra Jacobian matrices that relate a particle’s reduced coordinates to its motion in world-space']",
"type": "own_claim",
"id": "T338"
},
{
"offsets": [
[
40301,
40431
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],
"text": "['correct constraint-release behavior between cloth and solid objects is difficult to achieve using a reduced coordinate formulation']",
"type": "own_claim",
"id": "T339"
},
{
"offsets": [
[
40458,
40508
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],
"text": "['we immediately rejected this method of constraints']",
"type": "own_claim",
"id": "T340"
},
{
"offsets": [
[
40556,
40627
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],
"text": "['We could constrain particles through the use of strong energy functions']",
"type": "own_claim",
"id": "T341"
},
{
"offsets": [
[
40845,
40872
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],
"text": "['do not carry as much weight']",
"type": "own_claim",
"id": "T342"
},
{
"offsets": [
[
40765,
40823
]
],
"text": "['the usual objections to enforcing constraints with springs']",
"type": "own_claim",
"id": "T343"
},
{
"offsets": [
[
40904,
40970
]
],
"text": "['found it to be a not unreasonable constraint enforcement mechanism']",
"type": "own_claim",
"id": "T344"
},
{
"offsets": [
[
40711,
40763
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],
"text": "['our entire formulation is geared to handle stiffness']",
"type": "data",
"id": "T345"
},
{
"offsets": [
[
40874,
40898
]
],
"text": "['We tried this for a time']",
"type": "data",
"id": "T346"
},
{
"offsets": [
[
40981,
41031
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],
"text": "['penalty methods do not enforce constraints exactly']",
"type": "background_claim",
"id": "T347"
},
{
"offsets": [
[
41037,
41088
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],
"text": "['they do add some additional stiffness to the system']",
"type": "background_claim",
"id": "T348"
},
{
"offsets": [
[
41181,
41263
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],
"text": "['we turned away from penalty methods except in the case of cloth/cloth interactions']",
"type": "own_claim",
"id": "T349"
},
{
"offsets": [
[
41096,
41150
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],
"text": "['the mechanism we describe enforces constraints exactly']",
"type": "data",
"id": "T350"
},
{
"offsets": [
[
41156,
41179
]
],
"text": "['adds no extra stiffness']",
"type": "data",
"id": "T351"
},
{
"offsets": [
[
41316,
41363
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],
"text": "['We could introduce additional constraint forces']",
"type": "own_claim",
"id": "T352"
},
{
"offsets": [
[
41394,
41436
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],
"text": "['into our system to satisfy the constraints']",
"type": "own_claim",
"id": "T353"
},
{
"offsets": [
[
41536,
41555
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],
"text": "['and extra equations']",
"type": "own_claim",
"id": "T354"
},
{
"offsets": [
[
41438,
41517
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],
"text": "['This involves augmenting the linear system of equation (6) with extra variables']",
"type": "own_claim",
"id": "T355"
},
{
"offsets": [
[
41600,
41663
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],
"text": "['this turns a positive definite system into an indefinite system']",
"type": "own_claim",
"id": "T356"
},
{
"offsets": [
[
41682,
41747
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],
"text": "['iterative methods such as CG will need to square the system first']",
"type": "own_claim",
"id": "T357"
},
{
"offsets": [
[
41757,
41782
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],
"text": "['doubling the running time']",
"type": "own_claim",
"id": "T358"
},
{
"offsets": [
[
41787,
41846
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],
"text": "['degrading the numerical conditionining of the linear system']",
"type": "own_claim",
"id": "T359"
},
{
"offsets": [
[
41862,
41969
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],
"text": "['an iterative method will generally not enforce the constraints exactly without a large number of iterations']",
"type": "background_claim",
"id": "T360"
},
{
"offsets": [
[
42065,
42128
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],
"text": "['the constraint method we describe steps past these difficulties']",
"type": "own_claim",
"id": "T361"
},
{
"offsets": [
[
42133,
42179
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],
"text": "['we turned away from using Lagrange multipliers']",
"type": "own_claim",
"id": "T362"
},
{
"offsets": [
[
41972,
42026
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],
"text": "['A direct method for solving the augmented system would']",
"type": "own_claim",
"id": "T363"
},
{
"offsets": [
[
42037,
42055
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],
"text": "['avoid this problem']",
"type": "own_claim",
"id": "T364"
},
{
"offsets": [
[
42242,
42320
]
],
"text": "['The idea behind our constraint enforcement mechanism is described quite simply']",
"type": "own_claim",
"id": "T365"
},
{
"offsets": [
[
42331,
42410
]
],
"text": "['the actual implementation is somewhat more complicated, to maximize performance']",
"type": "own_claim",
"id": "T366"
},
{
"offsets": [
[
42412,
42490
]
],
"text": "['A dynamic simulation usually requires knowledge of the inverse mass of objects']",
"type": "own_claim",
"id": "T367"
},
{
"offsets": [
[
42510,
42560
]
],
"text": "['the appearance of M −1 , and not M in equation (6)']",
"type": "data",
"id": "T368"
},
{
"offsets": [
[
42690,
42752
]
],
"text": "['it becomes trivial to enforce constraints by altering the mass']",
"type": "own_claim",
"id": "T369"
},
{
"offsets": [
[
42668,
42688
]
],
"text": "['inverse mass is used']",
"type": "data",
"id": "T370"
},
{
"offsets": [
[
42861,
42898
]
],
"text": "['we give the particle an infinite mass']",
"type": "own_claim",
"id": "T371"
},
{
"offsets": [
[
42900,
42941
]
],
"text": "['making it ignore all forces exerted on it']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
42835,
42859
]
],
"text": "['we take 1/m i to be zero']",
"type": "data",
"id": "T373"
},
{
"offsets": [
[
42943,
43071
]
],
"text": "['Complete control over a particle’s acceleration is thus taken care of by storing a value of zero for the particle’s inverse mass']",
"type": "own_claim",
"id": "T374"
},
{
"offsets": [
[
43222,
43246
]
],
"text": "['we need not always do so']",
"type": "own_claim",
"id": "T375"
},
{
"offsets": [
[
43170,
43220
]
],
"text": "['we normally think of a particle’s mass as a scalar']",
"type": "own_claim",
"id": "T376"
},
{
"offsets": [
[
43629,
43728
]
],
"text": "['a particle is prevented from accelerating along p by using an inverse mass matrix 1 (I − m i pp T )']",
"type": "own_claim",
"id": "T377"
},
{
"offsets": [
[
43762,
43778
]
],
"text": "['(I − pp T )p = 0']",
"type": "data",
"id": "T378"
},
{
"offsets": [
[
43604,
43626
]
],
"text": "['a unit vector p ∈ IR 3']",
"type": "data",
"id": "T379"
},
{
"offsets": [
[
43843,
43967
]
],
"text": "['we prevent a particle from accelerating in either the p or q direction by using the inverse mass matrix 1 (I − pp T − qq T )']",
"type": "own_claim",
"id": "T380"
},
{
"offsets": [
[
43797,
43841
]
],
"text": "['two mutually orthogonal unit vectors p and q']",
"type": "data",
"id": "T381"
},
{
"offsets": [
[
43973,
44097
]
],
"text": "['By allowing constrained particles to have these sorts of inverse masses, we can build constraints directly into equation (6)']",
"type": "own_claim",
"id": "T382"
},
{
"offsets": [
[
44829,
44922
]
],
"text": "['We are not limited to constraining particles to have zero accelerations in certain directions']",
"type": "own_claim",
"id": "T383"
},
{
"offsets": [
[
44932,
45014
]
],
"text": "['we control exactly what the change in velocity is along the constrained directions']",
"type": "own_claim",
"id": "T384"
},
{
"offsets": [
[
45149,
45217
]
],
"text": "['we can choose any value of z i for a completely constrained particle']",
"type": "own_claim",
"id": "T385"
},
{
"offsets": [
[
45225,
45255
]
],
"text": "['all directions are constrained']",
"type": "data",
"id": "T386"
},
{
"offsets": [
[
45307,
45339
]
],
"text": "['it has no constrained directions']",
"type": "data",
"id": "T387"
},
{
"offsets": [
[
45260,
45300
]
],
"text": "['unconstrained particle must have z i = 0']",
"type": "own_claim",
"id": "T388"
},
{
"offsets": [
[
45357,
45412
]
],
"text": "['we rewrite equation (6) to directly enforce constraints']",
"type": "own_claim",
"id": "T389"
},
{
"offsets": [
[
45020,
45133
]
],
"text": "['every particle i, let z i be the change in velocity we wish to enforce in the particle’s constrained direction(s)']",
"type": "data",
"id": "T390"
},
{
"offsets": [
[
45538,
45597
]
],
"text": "['we will obtain a v which is consistent with our constraints']",
"type": "own_claim",
"id": "T391"
},
{
"offsets": [
[
45458,
45513
]
],
"text": "['I − hW ∂v ∂f − h 2 W ∂x ∂f v = hW f 0 + h ∂x ∂f v 0 + z']",
"type": "data",
"id": "T392"
},
{
"offsets": [
[
45417,
45425
]
],
"text": "['we solve']",
"type": "data",
"id": "T393"
},
{
"offsets": [
[
45531,
45536
]
],
"text": "['for v']",
"type": "data",
"id": "T394"
},
{
"offsets": [
[
45660,
45771
]
],
"text": "['partially constrained particles will have a v i whose component in the constrained direction(s) is equal to z i']",
"type": "own_claim",
"id": "T395"
},
{
"offsets": [
[
45599,
45651
]
],
"text": "['Completely constrained particles will have v i = z i']",
"type": "own_claim",
"id": "T396"
},
{
"offsets": [
[
45904,
45935
]
],
"text": "['it worked exactly as advertised']",
"type": "own_claim",
"id": "T397"
},
{
"offsets": [
[
45832,
45888
]
],
"text": "['We initially implemented constraints using equation (14)']",
"type": "data",
"id": "T398"
},
{
"offsets": [
[
45966,
46055
]
],
"text": "['we solved equation (14) using a direct method (Gaussian elimination) without any problems']",
"type": "own_claim",
"id": "T399"
},
{
"offsets": [
[
45941,
45964
]
],
"text": "['very small test systems']",
"type": "data",
"id": "T400"
},
{
"offsets": [
[
46077,
46185
]
],
"text": "['we planned to use the iterative, sparsity-exploiting CG method, which immediately presents us with a problem']",
"type": "own_claim",
"id": "T401"
},
{
"offsets": [
[
46061,
46075
]
],
"text": "['larger systems']",
"type": "data",
"id": "T402"
},
{
"offsets": [
[
46187,
46233
]
],
"text": "['equation (14) is not a symmetric linear system']",
"type": "data",
"id": "T403"
},
{
"offsets": [
[
46253,
46316
]
],
"text": "['neither is equation (6) unless all particles have the same mass']",
"type": "data",
"id": "T404"
},
{
"offsets": [
[
46340,
46366
]
],
"text": "['require symmetric matrices']",
"type": "background_claim",
"id": "T405"
},
{
"offsets": [
[
46319,
46329
]
],
"text": "['CG methods']",
"type": "background_claim",
"id": "T406"
},
{
"offsets": [
[
46370,
46472
]
],
"text": "['We could apply a CG method to the unsymmetric matrix of equation (14) by use of the “normal equations”']",
"type": "own_claim",
"id": "T407"
},
{
"offsets": [
[
46478,
46641
]
],
"text": "['this involves multiplying the matrix of equation (14) with its transpose which doubles the cost of each iteration while squaring the condition number of the system']",
"type": "own_claim",
"id": "T408"
},
{
"offsets": [
[
46644,
46646
]
],
"text": "['14']",
"type": "data",
"id": "T409"
},
{
"offsets": [
[
46678,
46763
]
],
"text": "['We decided that using a CG method to solve the unsymmetric problem was not acceptable']",
"type": "own_claim",
"id": "T410"
},
{
"offsets": [
[
46796,
46849
]
],
"text": "['applying a CG method to equation (6) is not difficult']",
"type": "own_claim",
"id": "T411"
},
{
"offsets": [
[
46859,
46894
]
],
"text": "['we can transform this equation to 6']",
"type": "data",
"id": "T412"
},
{
"offsets": [
[
46775,
46794
]
],
"text": "['without constraints']",
"type": "data",
"id": "T413"
},
{
"offsets": [
[
46904,
46958
]
],
"text": "['they work best on positive definite symmetric matrices']",
"type": "own_claim",
"id": "T414"
},
{
"offsets": [
[
47302,
47358
]
],
"text": "['we cannot apply the same transformation to equation (14)']",
"type": "own_claim",
"id": "T415"
},
{
"offsets": [
[
47368,
47381
]
],
"text": "['W is singular']",
"type": "data",
"id": "T416"
},
{
"offsets": [
[
47442,
47476
]
],
"text": "['we cannot multiply through by W −1']",
"type": "own_claim",
"id": "T417"
},
{
"offsets": [
[
47479,
47674
]
],
"text": "['The solution to the problem of asymmetry is to modify the CG method so that it can operate on equation (15), while procedurally applying the constraints inherent in the matrix W at each iteration']",
"type": "own_claim",
"id": "T418"
},
{
"offsets": [
[
48444,
48522
]
],
"text": "['these two conditions imply that unconstrained particles have r i close to zero']",
"type": "own_claim",
"id": "T419"
},
{
"offsets": [
[
48530,
48577
]
],
"text": "['completely constrained particles have v i = z i']",
"type": "own_claim",
"id": "T420"
},
{
"offsets": [
[
48632,
48710
]
],
"text": "['our modified CG method should produce the same result as the regular CG method']",
"type": "own_claim",
"id": "T421"
},
{
"offsets": [
[
48602,
48630
]
],
"text": "['no particles are constrained']",
"type": "data",
"id": "T422"
},
{
"offsets": [
[
48032,
48103
]
],
"text": "['our modified CG method will try to find v that satisfies two conditions']",
"type": "own_claim",
"id": "T423"
},
{
"offsets": [
[
48128,
48220
]
],
"text": "['the component of r i in the particle’s unconstrained direction(s) will be made equal to zero']",
"type": "own_claim",
"id": "T424"
},
{
"offsets": [
[
48307,
48390
]
],
"text": "['the component of v i in the particle’s constrained direction(s) will be exactly z i']",
"type": "own_claim",
"id": "T425"
},
{
"offsets": [
[
48290,
48305
]
],
"text": "['each particle i']",
"type": "data",
"id": "T426"
},
{
"offsets": [
[
48111,
48126
]
],
"text": "['each particle i']",
"type": "data",
"id": "T427"
},
{
"offsets": [
[
64383,
64429
]
],
"text": "['The variance in the running times was under 5%']",
"type": "own_claim",
"id": "T428"
},
{
"offsets": [
[
64431,
64520
]
],
"text": "['We doubt that simulators based on explicit integration methods could make a similar claim']",
"type": "own_claim",
"id": "T429"
},
{
"offsets": [
[
64184,
64334
]
],
"text": "['Using the stiffness parameters in figure 1 as a reference, we ran the simulation with those bend stiffnesses multiplied by 0.1, 1.0, 10, 100 and 1,000']",
"type": "data",
"id": "T430"
},
{
"offsets": [
[
64871,
64999
]
],
"text": "['This is slightly better than O(n 1.5 ) performance, which is in line with the convergence rates of the conjugate gradient method']",
"type": "own_claim",
"id": "T431"
},
{
"offsets": [
[
65007,
65040
]
],
"text": "['for systems such as equation (18)']",
"type": "own_claim",
"id": "T432"
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{
"offsets": [
[
65002,
65004
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],
"text": "['14']",
"type": "data",
"id": "T433"
},
{
"offsets": [
[
64749,
64869
]
],
"text": "['The running times were, respectively, 0.23 seconds/frame, 0.46 seconds/frame, 2.23 seconds/frame, and 10.3 seconds/frame']",
"type": "data",
"id": "T434"
},
{
"offsets": [
[
64631,
64747
]
],
"text": "['We ran the simulation in figure 1 with cloth resolutions of 500, 899, 2,602 (shown in figure 1 ) and 7,359 particles']",
"type": "data",
"id": "T435"
},
{
"offsets": [
[
61259,
61388
]
],
"text": "['The methods introduced in all of the previous sections usually allow us to take sizeable steps forward, without loss of stability']",
"type": "own_claim",
"id": "T436"
},
{
"offsets": [
[
61399,
61475
]
],
"text": "['there are still times when the step size must be reduced to avoid divergence']",
"type": "own_claim",
"id": "T437"
},
{
"offsets": [
[
61599,
61689
]
],
"text": "['these methods tend to concentrate on the accuracy of the simulation, and not the stability']",
"type": "background_claim",
"id": "T438"
},
{
"offsets": [
[
61477,
61549
]
],
"text": "['There are a large number of methods for altering the size of a time step']",
"type": "background_claim",
"id": "T439"
},
{
"offsets": [
[
61691,
61729
]
],
"text": "['Our goal is animation, not engineering']",
"type": "data",
"id": "T440"
},
{
"offsets": [
[
61736,
61853
]
],
"text": "['visually pleasing results, meaning a numerically stable solution, rather than overall accuracy, is the deciding voice']",
"type": "own_claim",
"id": "T441"
},
{
"offsets": [
[
61855,
61925
]
],
"text": "['The trick is to recognize instability before you see it on your screen']",
"type": "own_claim",
"id": "T442"
},
{
"offsets": [
[
61949,
62064
]
],
"text": "['Stiffness, and thus any potential instability, arises almost completely from the strong stretch forces in the cloth']",
"type": "background_claim",
"id": "T443"
},
{
"offsets": [
[
62410,
62434
]
],
"text": "['Subtlety is not required']",
"type": "own_claim",
"id": "T444"
},
{
"offsets": [
[
62449,
62524
]
],
"text": "['an unstable step invariably results in stretch changes that are quite large']",
"type": "own_claim",
"id": "T445"
},
{
"offsets": [
[
62539,
62554
]
],
"text": "['easily detected']",
"type": "own_claim",
"id": "T446"
},
{
"offsets": [
[
62530,
62533
]
],
"text": "['are']",
"type": "own_claim",
"id": "T447"
},
{
"offsets": [
[
63226,
63325
]
],
"text": "['the simulator settles down to wasting only one out of every 40 steps in attempting too large a step']",
"type": "own_claim",
"id": "T448"
},
{
"offsets": [
[
63327,
63373
]
],
"text": "['This method, though simple, has served us well']",
"type": "own_claim",
"id": "T449"
},
{
"offsets": [
[
63191,
63224
]
],
"text": "['the user chooses too large a step']",
"type": "data",
"id": "T450"
},
{
"offsets": [
[
63118,
63180
]
],
"text": "['the simulator will try increasing the step size every 40 steps']",
"type": "data",
"id": "T451"
},
{
"offsets": [
[
49240,
49286
]
],
"text": "['We wholeheartedly refer the reader to Shewchuk']",
"type": "background_claim",
"id": "T452"
},
{
"offsets": [
[
49295,
49327
]
],
"text": "['for information on the CG method']",
"type": "background_claim",
"id": "T453"
},
{
"offsets": [
[
49290,
49292
]
],
"text": "['14']",
"type": "data",
"id": "T454"
},
{
"offsets": [
[
49396,
49503
]
],
"text": "['the effect of the matrix W in equation (14) is to filter out velocity changes in the constrained directions']",
"type": "own_claim",
"id": "T455"
},
{
"offsets": [
[
50761,
50798
]
],
"text": "['our stopping criterion should as well']",
"type": "own_claim",
"id": "T456"
},
{
"offsets": [
[
50700,
50759
]
],
"text": "['our constrained formulation ignores certain components of b']",
"type": "data",
"id": "T457"
},
{
"offsets": [
[
50803,
50829
]
],
"text": "['we add filtering to line 3']",
"type": "own_claim",
"id": "T458"
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{
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"text": "['should not include error due to the constraints']",
"type": "own_claim",
"id": "T459"
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{
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"text": "['we add filtering at lines 4 and 8']",
"type": "own_claim",
"id": "T460"
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{
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"text": "['products involving P −1 are trivially computed']",
"type": "own_claim",
"id": "T461"
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{
"offsets": [
[
51107,
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"text": "['We use a simple preconditioner P by making P be a diagonal matrix with P ii = 1/A ii']",
"type": "data",
"id": "T462"
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{
"offsets": [
[
51296,
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"text": "['we doubt there is a large speedup to be gained']",
"type": "own_claim",
"id": "T463"
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{
"offsets": [
[
51243,
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"text": "['More elaborate preconditioners could be used']",
"type": "own_claim",
"id": "T464"
},
{
"offsets": [
[
51503,
51605
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"text": "['obvious questions are “does it work?” followed by “how does it compare with the unmodified CG method?”']",
"type": "own_claim",
"id": "T465"
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{
"offsets": [
[
51489,
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"text": "['modified-pcg']",
"type": "data",
"id": "T466"
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{
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51669,
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"text": "['our method always converges, which answers the first question']",
"type": "own_claim",
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"text": "['in practice']",
"type": "data",
"id": "T468"
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{
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51606,
51654
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"text": "['Proofs about CG methods are difficult in general']",
"type": "background_claim",
"id": "T469"
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{
"offsets": [
[
51890,
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"text": "['we did not notice any substantial change in the number of iterations required by the method']",
"type": "own_claim",
"id": "T470"
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{
"offsets": [
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52013,
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"text": "['the two methods have similar convergence behavior']",
"type": "own_claim",
"id": "T471"
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{
"offsets": [
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51983,
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"text": "['Empirically']",
"type": "data",
"id": "T472"
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{
"offsets": [
[
51851,
51888
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"text": "['we began using procedure modified-pcg']",
"type": "data",
"id": "T473"
},
{
"offsets": [
[
51768,
51844
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],
"text": "['we used a penalty method and applied the standard CG method to equation (15)']",
"type": "data",
"id": "T474"
},
{
"offsets": [
[
52098,
52218
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],
"text": "['the running time is close to O(n 1.5 ), which is what unmodified CG would be expected to deliver on this sort of problem']",
"type": "own_claim",
"id": "T475"
},
{
"offsets": [
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52221,
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"text": "['14']",
"type": "data",
"id": "T476"
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{
"offsets": [
[
52064,
52083
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"text": "['Result in section 8']",
"type": "data",
"id": "T477"
},
{
"offsets": [
[
52362,
52470
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"text": "['we need to know what the actual force of constraint is, in order to determine when to terminate a constraint']",
"type": "own_claim",
"id": "T478"
},
{
"offsets": [
[
52308,
52327
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"text": "['contact constraints']",
"type": "own_claim",
"id": "T479"
},
{
"offsets": [
[
52486,
52584
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"text": "['we need to know the constraint force actually exerted in order to model frictional forces properly']",
"type": "own_claim",
"id": "T480"
},
{
"offsets": [
[
52599,
52680
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"text": "['it is easy to add one more step to modified-pcg to determine the constraint force']",
"type": "own_claim",
"id": "T481"
},
{
"offsets": [
[
52938,
53060
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],
"text": "['we can compute constraint forces at the end of modified-pcg by performing one last matrixvector product to compute A v − b']",
"type": "own_claim",
"id": "T482"
},
{
"offsets": [
[
52834,
52930
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],
"text": "['e i is exactly the extra constraint force that must have been supplied to enforce the constraint']",
"type": "own_claim",
"id": "T483"
},
{
"offsets": [
[
52682,
52823
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],
"text": "['When modified-pcg terminates, the residual error e = A v − b has the property that e i need not be close to zero if particle i is constrained']",
"type": "data",
"id": "T484"
},
{
"offsets": [
[
53191,
53280
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"text": "['The particles’ accelerations are inherently dependent on one another through the matrix A']",
"type": "own_claim",
"id": "T485"
},
{
"offsets": [
[
53284,
53297
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"text": "['equation (16)']",
"type": "data",
"id": "T486"
},
{
"offsets": [
[
53124,
53188
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"text": "['the extra matrix-vector product to compute e really is necessary']",
"type": "own_claim",
"id": "T487"
},
{
"offsets": [
[
53063,
53119
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],
"text": "['The vector r in modified-pcg is equal to filter(A v − b)']",
"type": "data",
"id": "T488"
},
{
"offsets": [
[
53315,
53397
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],
"text": "['the correct approach to determing constraint release is combinatoric, as in Baraff']",
"type": "own_claim",
"id": "T489"
},
{
"offsets": [
[
53400,
53401
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],
"text": "['2']",
"type": "data",
"id": "T490"
},
{
"offsets": [
[
53405,
53443
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],
"text": "['We reject this approach as impractical']",
"type": "own_claim",
"id": "T491"
},
{
"offsets": [
[
53450,
53468
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],
"text": "['the dimension of A']",
"type": "data",
"id": "T492"
},
{
"offsets": [
[
53479,
53617
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],
"text": "['we allow contacts to release when the constraint force between a particle and a solid switches from a repulsive force to an attractive one']",
"type": "own_claim",
"id": "T493"
},
{
"offsets": [
[
53632,
53660
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],
"text": "['this has proven to work well']",
"type": "own_claim",
"id": "T494"
},
{
"offsets": [
[
53619,
53630
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],
"text": "['In practice']",
"type": "data",
"id": "T495"
},
{
"offsets": [
[
53662,
53697
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],
"text": "['Friction presents a similar problem']",
"type": "own_claim",
"id": "T496"
},
{
"offsets": [
[
54159,
54216
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],
"text": "['Much has been written about collision detection for cloth']",
"type": "background_claim",
"id": "T497"
},
{
"offsets": [
[
54218,
54297
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],
"text": "['we have nothing substantial to add to the subject of collision detection per se']",
"type": "own_claim",
"id": "T498"
},
{
"offsets": [
[
55114,
55132
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],
"text": "['a frictional force']",
"type": "own_claim",
"id": "T499"
},
{
"offsets": [
[
55077,
55093
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],
"text": "['The force is not']",
"type": "own_claim",
"id": "T500"
},
{
"offsets": [
[
55230,
55269
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],
"text": "['it reasonably emulates dynamic friction']",
"type": "own_claim",
"id": "T501"
},
{
"offsets": [
[
55185,
55219
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],
"text": "['it is in actuality a damping force']",
"type": "own_claim",
"id": "T502"
},
{
"offsets": [
[
55141,
55180
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],
"text": "['it is proportional to the slip velocity']",
"type": "own_claim",
"id": "T503"
},
{
"offsets": [
[
55271,
55342
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],
"text": "['Applying static friction forces to cloth contacts is far more difficult']",
"type": "own_claim",
"id": "T504"
},
{
"offsets": [
[
55348,
55383
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],
"text": "['is a problem we have not solved yet']",
"type": "own_claim",
"id": "T505"
},
{
"offsets": [
[
55385,
55480
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],
"text": "['The forces, and their derivatives with respect to position and velocity, are of course included']",
"type": "own_claim",
"id": "T506"
},
{
"offsets": [
[
55484,
55497
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],
"text": "['equation (15)']",
"type": "data",
"id": "T507"
},
{
"offsets": [
[
56036,
56109
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],
"text": "['Both cloth/cloth and cloth/solid collisions give rise to the same problem']",
"type": "own_claim",
"id": "T508"
},
{
"offsets": [
[
56119,
56136
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],
"text": "['two contacts form']",
"type": "data",
"id": "T509"
},
{
"offsets": [
[
56434,
56545
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],
"text": "['collisions resulting in a reasonably substantial interpenetration depth can occur between one step and the next']",
"type": "own_claim",
"id": "T510"
},
{
"offsets": [
[
56392,
56432
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],
"text": "['our simulator proceeds in discrete steps']",
"type": "data",
"id": "T511"
},
{
"offsets": [
[
56556,
56591
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],
"text": "['this situation needs to be remedied']",
"type": "own_claim",
"id": "T512"
},
{
"offsets": [
[
56138,
56376
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],
"text": "['For both types of collisions, our detection algorithm reports an intersection, and then takes action to remedy the situation: either by enforcing a constraint (cloth/solid collisions) or by adding a penalty force (cloth/cloth) collisions.']",
"type": "data",
"id": "T513"
},
{
"offsets": [
[
56802,
56835
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],
"text": "['the situation is more complicated']",
"type": "own_claim",
"id": "T514"
},
{
"offsets": [
[
56769,
56791
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],
"text": "['cloth/solid collisions']",
"type": "data",
"id": "T515"
},
{
"offsets": [
[
56621,
56658
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],
"text": "['this would not appear to be a problem']",
"type": "own_claim",
"id": "T516"
},
{
"offsets": [
[
56660,
56763
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],
"text": "['the spring forces that are added work to counter the colliding velocities and then push the cloth apart']",
"type": "data",
"id": "T517"
},
{
"offsets": [
[
56597,
56619
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],
"text": "['cloth/cloth collisions']",
"type": "data",
"id": "T518"
},
{
"offsets": [
[
57114,
57134
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],
"text": "['This is unacceptable']",
"type": "own_claim",
"id": "T519"
},
{
"offsets": [
[
57018,
57112
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],
"text": "['the cloth particle will continue to remain embedded somewhere below the solid object’s surface']",
"type": "own_claim",
"id": "T520"
},
{
"offsets": [
[
56840,
57016
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],
"text": "['we simply enforce a constraint which causes the colliding cloth particle to have a velocity consistent with the solid object’s velocity, and continue to enforce that constraint']",
"type": "data",
"id": "T521"
},
{
"offsets": [
[
57136,
57182
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],
"text": "['One solution is to use Baumgarte stabilization']",
"type": "own_claim",
"id": "T522"
},
{
"offsets": [
[
57185,
57187
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],
"text": "['18']",
"type": "data",
"id": "T523"
},
{
"offsets": [
[
57192,
57348
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],
"text": "['which schedules the particle’s acceleration so that the position and velocity error of the particle with respect to the surface decay asymptotically to zero']",
"type": "background_claim",
"id": "T524"
},
{
"offsets": [
[
57391,
57407
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],
"text": "['found it lacking']",
"type": "own_claim",
"id": "T525"
},
{
"offsets": [
[
57350,
57385
]
],
"text": "['We experimented with this technique']",
"type": "data",
"id": "T526"
},
{
"offsets": [
[
57424,
57485
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],
"text": "['a fast rise to the surface was prone to noise and “jumpiness”']",
"type": "own_claim",
"id": "T527"
},
{
"offsets": [
[
57487,
57556
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"text": "['this could be eliminated, but at the cost of decreasing the step size']",
"type": "own_claim",
"id": "T528"
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{
"offsets": [
[
57558,
57610
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],
"text": "['A slower rise to the surface caused visual artifacts']",
"type": "own_claim",
"id": "T529"
},
{
"offsets": [
[
58125,
58187
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],
"text": "['simply changing particle positions produced disastrous results']",
"type": "own_claim",
"id": "T530"
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{
"offsets": [
[
58028,
58074
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],
"text": "['it should not interfere with a one-step solver']",
"type": "own_claim",
"id": "T531"
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{
"offsets": [
[
57893,
57996
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"text": "['this would be problematic when using a multi-step differential equation solver which expects continuity']",
"type": "own_claim",
"id": "T532"
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{
"offsets": [
[
58006,
58025
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"text": "['a RungeKutta method']",
"type": "data",
"id": "T533"
},
{
"offsets": [
[
58083,
58108
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],
"text": "['the backward Euler method']",
"type": "data",
"id": "T534"
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{
"offsets": [
[
57612,
57639
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"text": "['We tried a simpler solution']",
"type": "own_claim",
"id": "T535"
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{
"offsets": [
[
57641,
57885
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"text": "['when intersections occurred, rather than wait for a scheduled constraint or a penalty force to eliminate the intersection, we simply altered the positions of the cloth particles, effecting an instantaneous (and discontinuous) change in position']",
"type": "data",
"id": "T536"
},
{
"offsets": [
[
58385,
58460
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"text": "['This resulted in visibly “jumpy” behavior of the cloth in localized regions']",
"type": "own_claim",
"id": "T537"
},
{
"offsets": [
[
58256,
58383
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],
"text": "['altering particle positions arbitrarily introduced excessively large deformation energies in an altered particle’s neighborhood']",
"type": "data",
"id": "T538"
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{
"offsets": [
[
58189,
58250
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"text": "['The stretch energy term in a cloth system is extremely strong']",
"type": "background_claim",
"id": "T539"
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{
"offsets": [
[
58678,
58781
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"text": "['the ability to make arbitrary small changes in a particle’s position continued to attract our attention']",
"type": "own_claim",
"id": "T540"
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{
"offsets": [
[
58783,
58869
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"text": "['The entire process of implicit integration can be considered to be a filtering process']",
"type": "own_claim",
"id": "T541"
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{
"offsets": [
[
58873,
58874
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"text": "['7']",
"type": "data",
"id": "T542"
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{
"offsets": [
[
58901,
59022
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"text": "['a mechanism for filtering energy changes caused by displacing particles might make position alteration a viable technique']",
"type": "own_claim",
"id": "T543"
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{
"offsets": [
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59051,
59109
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"text": "['some sort of extra implicit step could be used as a filter']",
"type": "own_claim",
"id": "T544"
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{
"offsets": [
[
59115,
59196
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"text": "['forming and solving an additional linear system at each step seemed too expensive']",
"type": "own_claim",
"id": "T545"
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{
"offsets": [
[
59207,
59293
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"text": "['we can make use of the filtering effect of implicit integration without any extra work']",
"type": "own_claim",
"id": "T546"
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{
"offsets": [
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59915,
59962
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"text": "['we simply modify the top row of equation (4) to']",
"type": "own_claim",
"id": "T547"
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{
"offsets": [
[
59995,
60021
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"text": "['x i = h(v 0i + v i ) + y i']",
"type": "own_claim",
"id": "T548"
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{
"offsets": [
[
59880,
59913
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"text": "['This presents an obvious solution']",
"type": "own_claim",
"id": "T549"
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{
"offsets": [
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59721,
59803
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"text": "['the particle’s neighbors receive no advance notification of the change in position']",
"type": "own_claim",
"id": "T550"
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{
"offsets": [
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59805,
59878
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"text": "['they are confronted with the alteration at the beginning of the next step']",
"type": "data",
"id": "T551"
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{
"offsets": [
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59650,
59709
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"text": "['changing positions after a step has been taken doesn’t work']",
"type": "own_claim",
"id": "T552"
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{
"offsets": [
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60232,
60266
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"text": "['we must follow this change through']",
"type": "own_claim",
"id": "T553"
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{
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60194,
60230
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"text": "['modified the top row of equation (4)']",
"type": "data",
"id": "T554"
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"offsets": [
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60442,
60499
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"text": "['M − h ∂v ∂f − h 2 ∂x ∂f v = h f 0 + h ∂x ∂f v 0 + ∂x ∂f y']",
"type": "own_claim",
"id": "T555"
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{
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60274,
60287
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"text": "['equation (17)']",
"type": "data",
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60306,
60329
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"text": "['derivation of section 3']",
"type": "data",
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60338,
60372
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"text": "['symmetric transform from section 5']",
"type": "data",
"id": "T558"
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60519,
60679
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"text": "['This modification gives us complete control over both the position and velocity of a constrained particle in just one step, without any extra computational cost']",
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"id": "T559"
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{
"offsets": [
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60844,
60924
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"text": "['We can also add correction terms to particles involved in cloth/cloth collisions']",
"type": "own_claim",
"id": "T560"
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{
"offsets": [
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60926,
61051
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"text": "['Without a constraint on those particles’ velocities there is no guarantee that they will go exactly where we want in one step']",
"type": "own_claim",
"id": "T561"
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{
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61057,
61193
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"text": "['the ability to induce sizeable jumps in position without excessively stiff spring forces adds greatly to the stability of the simulation']",
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A17 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A17_M09_Motion_Graphs_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.2172/7134495
<Title>Motion Graphs</Title>
To appear in Proceedings of SIGGRAPH ’02
Lucas Kovar Michael Gleicher ∗ Frédéric Pighin † University of Wisconsin-Madison University of Wisconsin-Madison University of Southern California Institute for Creative Technologies
∗
∗ e-mail: {kovar,gleicher}@cs.wisc.edu † e-mail: [email protected]
<Abstract>In this paper we present a novel method for creating realistic, controllable motion. Given a corpus of motion capture data, we automatically construct a directed graph called a motion graph that encapsulates connections among the database. The motion graph consists both of pieces of original motion and automatically generated transitions. Motion can be generated simply by building walks on the graph. We present a general framework for extracting particular graph walks that meet a user’s specifications. We then show how this framework can be applied to the specific problem of generating different styles of locomotion along arbitrary paths.</Abstract>
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism Keywords: motion synthesis, motion capture, animation with constraints
<H1>1 Introduction</H1>
Realistic human motion is an important part of media like video games and movies. More lifelike characters make for more immersive environments and more believable special effects. At the same time, realistic animation of human motion is a challenging task, as people have proven to be adept at discerning the subtleties of human movement and identifying inaccuracies. One common solution to this problem is motion capture. However, while motion capture is a reliable way of acquiring realistic human motion, by itself it is a technique for reproducing motion. Motion capture data has proven to be difficult to modify, and editing techniques are reliable only for small changes to a motion. This limits the utility of motion capture if the data on hand isn’t sufficiently similar to what is desired, then often there is little that can be done other than acquire more data, a time-consuming and expensive process. This in particular is a problem for applications that require motion to be synthesized dynamically, such as interactive environments. Our goal is to retain the realism of motion capture while also giving a user the ability to control and direct a character. For example, we would like to be able to ask a character to walk around a room without worrying about having a piece of motion data that contains the correct number of steps and travels in the right directions. We also need to be able to direct characters who can perform multiple actions, rather than those who are only capable of walking around. This paper presents a method for synthesizing streams of motions based on a corpus of captured movement while preserving the quality of the original data. Given a set of motion capture data, we compile a structure called a motion graph that encodes how the captured clips may be re-assembled in different ways. The motion graph is a directed graph wherein edges contain either pieces of original motion data or automatically generated transitions. The nodes then serve as choice points where these small bits of motion join seamlessly. Because our methods automatically detect and create transitions between motions, users needn’t capture motions specifically designed to connect to one another. If desired, the user can tune the high-level structure of the motion graph to produce desired degrees of connectivity among different parts. Motion graphs transform the motion synthesis problem into one of selecting sequences of nodes, or graph walks. By drawing upon algorithms from graph theory and AI planning, we can extract graph walks that satisfy certain properties, thereby giving us control over the synthesized motions. To demonstrate the potential of our approach, we introduce a simple example. We were donated 78.5 seconds of motion capture, or about 2400 frames of animation, of a performer randomly walking around with both sharp and smooth turns. Since the motion was donated, we did not carefully plan out each movement, as the literature suggests is critical to successful application of motion capture data
[Washburn 2001]. From this data we constructed a motion graph and used an algorithm described later in this paper to extract motions that travelled along paths sketched on the ground. Characteristic movements of the original data like sharp turns were automatically used when appropriate, as seen in Figure 1 . It is possible to place additional constraints on the desired motion. For example, we noticed that part of the motion had the character sneaking around. By labelling these frames as special, we were able to specify that at certain points along the path the character must only use sneaking movements, and at other parts of the motion it must use normal walking motions, as is also shown in Figure 1 . The remainder of this paper is organized as follows. In Section 2 we describe related work. In Section 3 we describe how a motion graph is constructed from a database of motion capture. In Section 4 we set forth a general framework for extracting motion from the motion graph that meets user specifications. Section 5 discusses the specific problem of generating movements along a path and how it is handled in our framework. We conclude in Section 6 with a discussion of the scalability of our approach to large data sets and potential future work.
1
To appear in Proceedings of SIGGRAPH ’02
Figure 1: The top images show original motion capture data; two are walking motions and one is a sneaking motion. The black curves show the paths travelled by the character. The bottom images show new motion generated by a motion graph built out of these examples plus their mirror images. Images 1 and 2 show the result of having the motion graph fit walking motion to the smooth yellow paths. The black curve is the actual position of the center of mass on each frame. Image 3 shows motion formed by having the character switch from walking to sneaking halfway down the path.
<H1>2 Related Work</H1>
Much previous work with motion capture has revolved around editing individual clips of motion. Motion warping
[Witkin and Popović 1995]
can be used to smoothly add small changes to a motion. Retargeting [Gleicher 1998; Lee and Shin 1999]
maps the motion of a performer to a character of different proportions while retaining important constraints like footplants. Various signal processing operations
[Bruderlin and Williams 1995] can be applied to motion data. Our work is different from these efforts in that it involves creating continuous streams of motion, rather than modifying specific clips. One strategy for motion synthesis is to perform multi-target blends among a set of examples, yielding a continuous space of parameterized motion. Wiley and Hahn [1997]
used linear interpolation to create parameterizations of walking at various inclinations and reaching to various locations. Rose et al. [1998]
used radial basis functions to blend among clips representing the same motion performed in different styles. These works have a focus complementary to ours: while they are mainly concerned with generating parameterizations of individual clips, we are concerned with constructing controllable sequences of clips. Another popular approach to motion synthesis is to construct statistical models. Pullen and Bregler [2000]
used kernel-based probability distributions to synthesize new motion based on the statistical properties of example motion. Coherency was added to the model by explicitly accounting for correlations between parameters. Bowden [2000], Galata et al. [2001]
, and Brand and Hertzmann [2000]
all processed motion capture data by constructing abstract “states” which each represent entire sets of poses.
Transition probabilities between states were used to drive motion synthesis. Since these statistical models synthesize motion based on abstractions of data rather than actual data, they risk losing important detail. In our work we have tighter guarantees on the quality of generated motion. Moreover, these systems did not focus on the satisfaction of high-level constraints. We generate motion by piecing together example motions from a database. Numerous other researchers have pursued similar strategies. Perlin [1995] and Perlin and Goldberg [1996]
used a rulebased system and simple blends to attach procedurally generated motion into coherent streams. Faloutsos et al. [2001]
used support vector machines to create motion sequences as compositions of actions generated from a set of physically based controllers. Since our system involves motion capture data, rather than procedural or physically based motion, we require different approaches to identifying and generating transitions. Also, these systems were mainly concerned with appropriately generating individual transitions, whereas we address the problem of generating entire motions (with many transitions) that meet user-specified criteria. Lamouret and van de Panne [1996]
developed a system that used a database to extract motion meeting high-level constraints. However, their system was applied to a simple agent with five degrees of freedom, whereas we generate motion for a far more sophisticated character. Molina-Tanco and Hilton [2000]
used a state-based statistical model similar to those mentioned in the previous paragraph to rearrange segments of original motion data. These segments were attached using linear interpolation. The user could create motion by selecting keyframe poses, which were connected with a highprobability sequence of states. Our work considers more general and sophisticated sets of constraints. Work similar to ours has been done in the gaming industry to meet the requirements of online motion generation. Many companies use move trees
[Mizuguchi et al. 2001]
, which (like motion graphs) are graph structures representing connections in a database of motion. However, move trees are created manually — short motion clips are collected in carefully scripted capture sessions and blends are created by hand using interactive tools. Motion graphs are constructed automatically. Also, move trees are typically geared for rudimentary motion planning (“I want to turn left, so I should follow this transition”), as opposed to more complicated objectives. The generation of transitions is an important part of our approach. Early work in this area was done by
Perlin [1995]
, who presented a simple method for smoothly interpolating between two clips to create a blend. Lee [2000]
defined orientation filters that allowed these blending operations to be performed on rotational data in a more principled fashion. Rose et al. [1996]
presented a more complex method for creating transitions that preserved kinematic constraints and basic dynamic properties. Our main application of motion graphs is to control a character’s locomotion. This problem is important enough to have received a great deal of prior attention. Because a character’s path isn’t generally known in advance, synthesis is required.
Procedural and physically based synthesis methods have been developed for a few activities such as walking
[Multon et al. 1999; Sun and Metaxas 2001] and running
[Hodgins et al. 1995; Bruderlin and Calvert 1996]. While techniques such as these can generate flexible motion paths, the current range of movement styles is limited. Also, these methods do not produce the quality of motion attainable by hand animation or motion capture. While Gleicher [2001] presented a method for editing the path traversed in a clip of motion capture, it did not address the need for continuous streams of motion, nor could it choose which clip is correct to fit a path (e.g. that a turning motion is better when we have a curved path). Our basic approach — detecting transitions, constructing a graph, and using graph search techniques to find sequences satisfying user demands — has been applied previously to other problems. Schödl et al. [2000]
developed a similar method for synthesizing seamless streams of video from example footage and driving these streams according to high-level user input. Since writing this paper, we have learned of similar work done concurrently by a number of research groups. Arikan and Forsythe [2002]
constructed from a motion database a hierarchical graph similar to ours and used a randomized search algorithm to extract motion that meets user constraints. Lee et al. [2002]
also constructed a graph and generated motion via three user interfaces: a list of choices, a sketch-based interface similar to what we use for path fitting (Section 5), and a live video feed. Pullen and Bregler [2002]
keyframed a subset of a character’s degrees of freedom and matched small segments of this keyframed animation with the lower frequency bands of motion data. This resulted in sequences of short clips forming complete motions. Li et al [2002] generated a two-level statistical model of motion. At the lower level were linear dynamic systems representing characteristic movements called “textons”, and the higher level contained transition probabilities among textons. This model was used both to generate new motion based on user keyframes and to edit existing motion.
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<H1>3 Motion Graph Construction</H1>
In this section, we define the motion graph structure and the procedure for constructing it from a database of clips. A clip of motion is defined as a regular sampling of the character’s parameters, which consist of the position of the root joint and quaternions representing the orientations of each joint. We
Figure 2: Consider a motion graph built from two initial clips. (top) We can trivially
insert a node to divide an initial clip into two smaller clips. (bottom) We can also insert a transition joining either two different initial clips or different parts of the same initial clip.
also allow clips (or, more generally, sets of frames) to be annotated with other information, such as descriptive labels (“walking,” “karate”) and constraint information (left heel must be planted on these frames). A motion graph is a directed graph where all edges correspond to clips of motion. Nodes serve as choice points connecting these clips, i.e., each outgoing edge is potentially the successor to any incoming edge. A trivial motion graph can be created by placing all the initial clips from the database as arcs in the graph. This creates a disconnected graph with 2n nodes, one at the beginning and end of each clip. Similarly, an initial clip can be broken into two clips by inserting a node, since the later part of the motion is a valid successor to the earlier part (see Figure 2 ). A more interesting graph requires greater connectivity. For a node to have multiple outgoing edges, there must be multiple clips that can follow the clip(s) leading into the node. Since it is unlikely that two pieces of original data are sufficiently similar, we need to create clips expressly for this purpose. Transitions are clips designed such that they can seamlessly connect two segments of original data. By introducing nodes within the initial clips and inserting transition clips between otherwise disconnected nodes, we can create a wellconnected structure with a wide range of possible graph walks (see Figure 2 ). Unfortunately, creating transitions is a hard animation problem. Imagine, for example, creating a transition between a run and a backflip. In real life this would require several seconds for an athlete to perform, and the transition motion looks little like the motions it connects. Hence the problem of automatically creating such a transition is arguably as difficult as that of creating realistic motion in the first place. On the other hand, if two motions are “close” to each other then simple blending techniques can reliably generate a transition. In light of this, our strategy is to identify portions of the initial clips that are sufficiently similar that straightforward blending is almost certain to produce valid transitions. The remainder of this section is divided into three parts. First we describe our algorithm for detecting a set of candidate transition points. In the following two sections we discuss how we select among these candidate transitions and how blends are created at the chosen transition points. Finally, we explain how to prune the graph to eliminate problematic edges.
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<H2>3.1 Detecting Candidate Transitions</H2>
As in our system, motion capture data is typically represented as vectors of parameters specifying the root position and joint rotations of a skeleton on each frame. One might attempt to locate transition points by computing some vector norm to measure the difference between poses at each pair of frames. However, such a simple approach is ill-advised, as it fails to address a number of important issues:
1. Simple vector norms fail to account for the meanings of the parameters. Specifically, in the joint angle representation some parameters have a much greater overall effect on the character than others (e.g., hip orientation vs. wrist orientation). Moreover, there is no meaningful way to assign fixed weights to these parameters, as the effect of a joint rotation on the shape of the body depends on the current configuration of the body. 2. A motion is defined only up to a rigid 2D coordinate transformation. That is, the motion is fundamentally unchanged if we translate it along the floor plane or rotate it about the vertical axis. Hence comparing two motions requires identifying compatible coordinate systems. 3. Smooth blends require more information than can be obtained at individual frames. A seamless transition must account not only for differences in body posture, but also in joint velocities, accelerations, and possibly higher-order derivatives.
Our similarity metric incorporates each of these considerations. To motivate it, we note that the skeleton is only a means to an end. In a typical animation, a polygonal mesh is deformed according to the skeleton’s pose. This mesh is all that is seen, and hence it is a natural focus when considering how close two frames of animation are to each other. For this reason we measure the distance between two frames of animation in terms of a point cloud driven by the skeleton. Ideally this point cloud is a downsampling of the mesh defining the character. To calculate the distance D( i , ¡ j ) between two frames i and ¡ j , we consider the point clouds formed over two windows of frames of user-defined length k, one bordered at the beginning by i and the other bordered at the end by ¡ j . That is, each point cloud is the composition of smaller point clouds representing the pose at each frame in the window. The use of windows of frames effectively incorporates derivative information into the metric, and is similar to the approach in
[Schödl et al. 2000]. The size of the windows are the same as the length of the transitions, so D( i , ¡ j ) is affected by every pair of frames that form the transition. We use a value of k corresponding to a window of about a third of a second in length, as in
[Mizuguchi et al. 2001] The distance between i and ¡ j may be calculated by computing a weighted sum of squared distances between corresponding points p i and p i in the two point clouds. To address the problem of finding coordinate systems for these point clouds (item 2 in the above list), we calculate the minimal weighted sum of squared distances given that an arbitrary rigid 2D transformation may be applied to the second point cloud:
1
θ min ,x ,z ∑ i w i p i − T θ ,x 0 ,z 0 p i 2
where the linear transformation T θ ,x 0 ,z 0 rotates a point p about the y (vertical) axis by θ degrees and then translates it by (x 0 , z 0 ). The
Figure 3: An example error function for two motions. The entry at (i, j) contains the error for making a transition from the i th frame of the first motion to the j th frame of
the second. White values correspond to lower errors and black values to higher errors. The colored dots represent local minima.
index is over the number of points in each point cloud. The weights w i may be chosen both to assign more importance to certain joints (e.g., those with constraints) and to taper off towards the end of the window. This optimization has a closed-form solution:
2
θ = arctan ∑ ∑ i i w w i i (x (x i i z x i i − + x z i i z z i i ) ) − − ∑ ∑ i i 1 1 w w i i (xz (xx − + x zz z) )
3
1 x 0 = ∑ w (x − x cos( θ ) − z sin θ )
4
1 z 0 = ∑ w (z + x sin( θ ) − z cos θ )
where x = ∑ i w i x i and the other barred terms are defined similarly. We compute the distance as defined above for every pair of frames in the database, forming a sampled 2D error function. Figure 3 shows a typical result. To make our transition model more compact, we find all the local minima of this error function, thereby extracting the “sweet spots” at which transitions are locally the most opportune. This tactic was also used in
[Schödl et al. 2000]. These local minima are our candidate transition points.
<H2>3.2 Selecting Transition Points</H2>
A local minimum in the distance function does not necessarily imply a high-quality transition; it only implies a transition better than its neighbors. We are specifically interested in local minima with small error values. The simplest approach is to only accept local minima below an empirically determined threshold. This can be done without user intervention. However, often users will want to set the threshold themselves to pick an acceptable tradeoff between having good transitions (low threshold) and having high connectivity (high threshold). Different kinds of motions have different fidelity requirements. For example, walking motions have very exacting requirements on the transitions — people have seen others walk nearly every day since birth and consequently have a keen sense of what a walk should look like. On the other hand, most people are less familiar with ballet motions and would be less likely to detect inaccuracies in such motion. As a result, we allow a user to apply different thresholds to different pairs of motions; transitions among ballet motions may have a higher acceptance threshold than transitions among walking motions.
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<H2>3.3 Creating Transitions</H2>
If D( i , ¡ j ) meets the threshold requirements, we create a tran sition by blending frames i to i+k−1 with frames ¡ j−k+1 to ¡ j , inclusive. The first step is to apply the appropriate aligning 2D transformation to motion . Then on frame p of the transition ¡ (0 ≤ p < k) we linearly interpolate the root positions and perform spherical linear interpolation on joint rotations:
5
R p = α (p)R + [1 − α (p)]R i+p j−k+1+p
6
q ip = sler p(q i ¢ , q i £ , α (p)) i+p j−k+1+p
where R p is the root position on the p th transition frame and q ip is the rotation of the i th joint on the p th transition frame. To maintain continuity we choose the blend weights α (p) according to the conditions that α (p) = 1 for p ≤ −1, α (p) = 0 for p ≥ k, and that α (p) has C 1 continuity everywhere. This requires
7
α (p) = 2( p + 1 ) 3 − 3( p + 1 ) 2 + 1, −1 < p < k k k
Other transition schemes, such as
[Rose et al. 1996], may be used in place of this one. The use of linear blends means that constraints in the original motion may be violated. For example, one of the character’s feet may slide when it ought to be planted. This can be corrected by using constraint annotations in the original motions. We treat constraints as binary signals: on a given frame a particular constraint either exists or it does not. Blending these signals in analogy to equations 5 and 6 amounts to using the constraints from in the first half of the transition and the constraints from in the second half. In this ¡ manner each transition is automatically annotated with constraint information, and these constraints may later be enforced as a postprocessing step when motion is extracted form the graph. We will discuss constraint enforcement in more detail in the next section. Descriptive labels attached to the motions are carried along into transitions. Specifically, if a transition frame is a blend between a frame with a set of labels L 1 and another frame with a set of labels L 2 , then it has the union of these labels L 1 ∪ L 2 .
1 2 3 4 5 6 7 8
Figure 4: A simple motion graph. The largest strongly connected component is [1, 2, 3, 6, 7, 8]. Node 4 is a sink and 5 is a dead end.
<H2>3.4 Pruning The Graph</H2>
In its current state there are no guarantees that the graph can synthesize motion indefinitely, since there may be nodes (called dead ends) that are not part of any cycle (see Figure 4 ). Once such a node is entered there is a bound on how much additional motion can be generated. Other nodes (called sinks) may be part of one or more cycles but nonetheless only be able to reach a small fraction of the total number of nodes in the graph. While arbitrarily long motion may still be generated once a sink is entered, this motion is confined to a small part of the database. Finally, some nodes may have incoming edges such that no outgoing edges contain the same set of descriptive labels. This is dangerous since logical discontinuities may be forced into a motion. For example, a character currently in a “boxing” motion may have no choice but to transition to a “ballet” motion. To address these problems, we prune the graph such that, starting from any edge, it is possible to generate arbitrarily long streams of motion of the same type such that as much of the database as possible is used. This is done as follows. Every frame of original data is associated with a (possibly empty) set of labels. Say there are n unique sets. For each set, form the subgraph consisting of all edges whose frames have exactly this set of labels. Compute the strongly connected components (SCCs) of this subgraph, where an SCC is a maximal set of nodes such that there is a connecting graph walk for any ordered pair of nodes (u, v). The SCCs can be computed in O(V + E) time using an algorithm due to Tarjan. We eliminate from this subgraph (and hence the original motion graph) any edge that does not attach two nodes in the largest SCC. Once this process is completed for all n label sets, any nodes with no edges are discarded. A warning is given to the user if the largest SCC for a given set of labels contains below a threshold number of frames. Also, a warning is given if for any ordered pair of SCCs there is no way to transition from the first to the second. In either case, the user may wish to adjust the transition thresholds (Section 3.2) to give the graph greater connectivity.
<H1>4 Extracting Motion</H1>
By this stage we have finished constructing the motion graph. After describing exactly how a graph walk can be converted into displayable motion, we will consider the general problem of extracting motion that satisfies user constraints. Our algorithm involves solving an optimization problem, and so we conclude this section with some general recommendations on how to pose the optimization.
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<H2>4.1 Converting Graph Walks To Motion</H2>
Since every edge on the motion graph is a piece of motion, a graph walk corresponds to a motion generated by placing these pieces one after another. The only issue is to place each piece in the correct location and orientation. In other words, each frame must be transformed by an appropriate 2D rigid transformation. At the start of a graph walk this transformation is the identity. Whenever we exit a transition edge, the current transformation is multiplied by the transformation that aligned the pieces of motion connected by the transition (Section 3.1). As noted in Section 3.3, the use of linear blends to create transitions can cause artifacts, the most common of which is feet that slide when they ought to be planted. However, every graph walk is automatically annotated with constraint information (such as that the foot must be planted). These constraints are either specified directly in the original motions or generated as in Section 3.3, depending on whether the frame is original data or a transition.
These constraints may be satisfied using a variety of methods, such as
[Gleicher 1998] or
[Lee and Shin 1999]. In our work we used the method described in
[Kovar et al. 2002].
<H2>4.2 Searching For Motion</H2>
We are now in a position to consider the problem of finding motion that satisfies user-specified requirements. It is worth first noting that only very special graph walks are likely to be useful. For example, while a random graph walk will generate a continuous stream of motion, such an algorithm has little use other than an elaborate screen saver. As a more detailed example, consider computing an all-pairs shortest graph walk table for the graph. That is, given a suitable metric — say, time elapsed or distance travelled — we can use standard graph algorithms like Floyd-Warshall to find for each pair of nodes u and v the connecting graph walk that minimizes the metric. With this in hand we could, for example, generate the motion that connects one clip to another as quickly as possible. This is less useful than it might appear at first. First, there are no guarantees that the shortest graph walk is short in an absolute sense. In our larger test graphs (between a few and several thousand nodes) the average shortest path between any two nodes was on the order of two seconds. This is not because the graphs were poorly connected. Since the transitions were about one-third of a second apiece, this means there were on average only five or six transitions separating any two of the thousands of nodes. Second, there is no control over what happens during the graph walk — we can’t specify what direction the character travels in or where she ends up. More generally, the sorts of motions that a user is likely to be interested in probably don’t involve minimizing metrics as simple as total elapsed time. However, for complicated metrics there is typically no simple way of finding the globally optimal graph walk. Hence we focus instead on local search methods that try to find a satisfactory graph walk within a reasonable amount of time. We now present our framework for extracting graph walks that conform to a user’s specifications. We cast motion extraction as a search problem and use branch and bound to increase the efficiency of this search. The user supplies a scalar function g(w, e) that evaluates the additional error accrued by appending an edge e to the existing path w, which may be the empty path 0. / The total error f (w) of the path is defined as follows:
8
n f (w) = f ([e 1 , . . . , e n ]) = ∑ g([e 1 , . . . , e i−1 ], e i ) i=1
where w is comprised of the edges e 1 , . . . , e n . We require g(w, e) to be nonnegative, which means that we can never decrease the total error by adding more edges to a graph walk. In addition to f and g, the user must also supply a halting condition indicating when no additional edges should be added to a graph walk. A graph walk satisfying the halting condition is called complete. The start of the graph walk may either be specified by the user or chosen at random. Our goal is find a complete graph walk w that minimizes f . To give the user control over what sorts of motions should be considered in the search, we allow restrictions on what edges may be appended to a given walk w. For example, the user may decide that within a particular window of time a graph walk may only contain “sneaking” edges. A naıve solution is to use depth-first search to evaluate f for all complete graph walks and then select the best one. However, the number of possible graph walks grows exponentially with the average size of a complete graph walk. To address this we use a branch and bound strategy to cull branches of the search that are incapable of yielding a minimum. Since g(w, e) by assumption never decreases, f (w) is a lower bound on f (w + v) for any v, where w + v is the graph walk composed of v appended to w. Thus we can keep track of the current best complete graph walk w opt and immediately halt any branch of the search for which the graph walk’s error exceeds f (w opt ). Also, the user may define a threshold error ε such that if f (w) < ε , then w is considered to be “good enough” and the search is halted. Branch and bound is most successful when we can attain a tight lower bound early in the search process. For this reason it is worthwhile to have a heuristic for ordering the edges we explore out of a particular node. One simple heuristic is to order the children greedily — that is, given a set of unexplored children c 1 , . . . , c n , we search the one that minimizes g(w, c i ). While branch and bound reduces the number of graph walks we have to test against f , it does not change the fact that the search process is inherently exponential — it merely lowers the effective branching factor. For this reason we generate a graph walk incrementally. At each step we use branch and bound to find an optimal graph walk of n frames. We retain the first m frames of this graph walk and use the final retained node as a starting point for another search. This process continues until a complete graph walk is generated. In our implementation we used values of n from 80 to 120 frames (2 3 2 to 4 seconds) and m from 25 to 30 frames (about one second). Sometimes it is useful to have a degree of randomness in the search process, such as when one is animating a crowd. There are a couple of easy ways to add randomness to the search process without sacrificing a good result. The first is to select a start for the search at random. The second is retain the r best graph walks at the end of each iteration of the search and randomly pick among the ones whose error is within some tolerance of the best solution.
<H2>4.3 Deciding What To Ask For</H2>
Since the motion extracted from the graph is determined by the function g, it is worth considering what sorts of functions are likely to produce desirable results. To understand the issues involved, we consider a simple example. Imagine we want to lay down two clips on the floor and create a motion that starts at the first clip and ends at the second. Both clips must end up in the specified position and orientation. We can formally state this problem as follows: given a starting node N in the graph and a target edge e, find a graph walk this section. The halting condition was to play a specific clip of two kicking motions. The error of a complete graph walk (which necessarily ended with the kicking clip) was determined by how far away this kicking clip was from being in a particular position and orientation. The character spends approximately seven seconds making minute adjustments to its orientation in an attempt to better align itself with the final clip. The highlighted line shows the the path of the target clip in its desired position and orientation. that ends with e such that the transformation T applied to e is as close as possible to a given transformation T . What one will receive is a motion like in Figure 5 , where the initial clip is a walking motion and the final clip is a kick. The character turns around in place several times in an attempt to better line up with the target clip. While it’s conceivable that given a larger database we would have found a better motion, the problem here is with the function we passed into the search algorithm. First, it gives no guidance as to what should be done in the middle of the motion; all that matters is that the final clip be in the right position and orientation. This means the character is allowed to do whatever is possible in order to make the final fit, even if the motion is nothing that a real person would do. Second, the goal is probably more specific than necessary. If it doesn’t matter what kick the character does, then it should be allowed to choose a kick that doesn’t require such effort to aim. More generally, there are two lessons we can draw from this example. First, g should give some sort of guidance throughout the entire motion, as arbitrary motion is almost never desirable. Second, g should be no more restrictive than necessary, in order to give the search algorithm more goals to seek. Note the tradeoff here — guiding the search toward a particular result must be balanced against unduly preventing it from considering all available options.
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Figure 5: The above motion was generated using the search algorithm discussed in
<H1>5 Path Synthesis</H1>
We have cast motion extraction as an optimization problem, and we have given some reasons why the formulation of this optimization can be difficult. To demonstrate that it is nonetheless possible to come up with optimization criteria that allow us to solve a real problem, we apply the preceding framework to path synthesis. This problem is simple to state: given a path P specified by the user, generate motion such that the character travels along P. In this section we present our algorithm for path synthesis, present results, and discuss applications of the technique.
<H2>5.1 Implementing Path Synthesis</H2>
Given the framework in the previous section, our only tasks are to define an error function g(w, e) and appropriate halting criteria. The basic idea is to estimate the actual path P travelled by the character during a graph walk and measure how different it is from P. The graph walk is complete when P is sufficiently long. A simple way to determine P is to project the root onto the floor at each frame, forming a piecewise linear curve 1 . Let P(s) be the point on P whose arc-length distance from the start of P is s. The i th frame of the graph walk, w i , is at some arc length s(w i ) from the start of P . We define the corresponding point on P as the point at the same arc length, P(s(w i )). For the j th frame of e, we calculate the squared distance between P (s(e j )) and P(s(e j )). g(w, e) is the sum of these errors:
9
n g(w, e) = ∑ P (s(e i )) − P(s(e i )) 2 i=1
Note that s(e i ) depends on the total arc length of w, which is why this equation is a function of w as well as e. The halting condition for path synthesis is when the current total length of P meets or exceeds that of P. Any frames on the graph walk at an arc length longer than the total length of P are mapped to the last point on P. The error function g(w, e) was chosen for a number of reasons. First, it is efficient to compute, which is important in making the search algorithm practical. Second, the character is given incentive to make definite progress along the path. If we were to have required the character to merely be near the path, then it would have no reason not to alternate between travelling forwards and backwards. Finally, this metric allows the character to travel at whatever speed is appropriate for what needs to be done. For example, a sharp turn will not cover distance at the same rate as walking straight forward. Since both actions are equally important for accurate path synthesis, it is important that one not be given undue preference over the other. One potential problem with this metric is that a character who stands still will never have an incentive to move forward, as it can accrue zero error by remaining in place. While we have not encountered this particular problem in practice, it can be countered by requiring at least a small amount of forward progress γ on each frame. More exactly, we can replace in Equation 9 the function s(e i ) with t(e i ) = max(t(e i−1 ) + s(e i ) − s(e i−1 ),t(e i−1 ) + γ ). Typically the user will want all generated motion to be of a single type, such as walking. This corresponds to confining the search to the subgraph containing the appropriate set of descriptive labels. More interestingly, one can require different types of motion on different parts of the path. For example, one might want the character to walk along the first half of the path and sneak down the rest. The necessary modifications to accomplish this are simple. We will consider the case of two different motion types; the generalization to higher numbers is trivial. We divide the original path into two smaller adjoining paths, P 1 and P 2 , based on where the transition from type T 1 to type T 2 is to occur. If the character is currently fitting P 2 , then the algorithm is identical to the single-type case. If the character is fitting P 1 , then we check to see if we are a threshold distance from the end of P 1 . If not, we continue to only consider edges of type T 1 . Otherwise we allow the search to try both edges of type T 1 and T 2 ; in the latter case we switch to fitting P 2 . Note that we only allow this switch to occur once on any given graph walk, which prevents the resulting motion from randomly switching between the two actions.
<H2>5.2 Results</H2>
While the examples shown in Figure 1 suggest that our technique is viable, it perhaps isn’t surprising that we were able to find accurate fits to the given paths. As shown in the upper portion of the 1 In our implementation we defined the path as a spline approximating this piecewise linear path, although this has little impact on the results. figure, the input motion had a fair amount of variation, including straight-ahead marches, sharp turns, and smooth changes of curvature. However, our algorithm is still useful when the input database is not as rich. Refer to Figure 6 . We started with a single 12.8second clip of an actor sneaking along the indicated path. To stretch this data further, we created a mirror-image motion and then built a motion graph out of the two. From these we were able to construct the new motions shown at the bottom of the figure, both of which are themselves approximately 13 seconds in length. Figure 7 shows fits to a more complicated path. The first example uses walking motions and the second uses martial arts motions; the latter demonstrates that our approach works even on motions that are not obviously locomotion. For the walking motion, the total computation time was nearly the same as the length of the generated animation (58.1 seconds of calculation for 54.9 seconds animation). The martial arts motion is 87.7 seconds long and required just 15.0 seconds of computation. In general, in our test cases the duration of a generated motion was either greater than or approximately equal to the amount of time needed to produce it. Both motion graphs had approximately 3000 frames (100 seconds) of animation. Finally, Figure 8 shows paths containing constraints on the allowable motion type. In the first section of each path the character is required to walk, in the second it must sneak, and in the third it is to perform martial arts moves. Not only does the character follow the path well, but transitions between action types occur quite close to their specified locations. This example used a database of approximately 6000 frames (200 seconds). All examples were computed on a 1.3GHz Athlon. For our largest graph (about 6000 frames), approximately twenty-five minutes were needed to compute the locations of all candidate transitions points. Approximately five minutes of user time were required to select transition thresholds, and it took less than a minute to calculate blends at these transitions and prune the resulting graph.
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To appear in Proceedings of SIGGRAPH ’02
<H2>5.3 Applications Of Path Synthesis</H2>
Directable locomotion is a general enough need that the preceding algorithm has many applications. Interactive Control. We can use path synthesis techniques to give a user interactive control over a character. For example, when the user hits the left arrow key the character might start travelling east. To accomplish this, we can use the path fitting algorithm to find the sequence of edges starting from our current location on the graph that best allow the character to travel east. The first edge on the resulting graph walk is the next clip that will be played. This process may then be repeated. To make this practical, we can precompute for every node in the graph a sequence of graph walks that fit straight-line paths in a sampling of directions (0 degrees, 30 degrees, . . . ). The first edges on these paths are then stored for later use; they are the best edges to follow given the direction the character is supposed to travel in. High-Level Keyframing. If we want a character to perform certain actions in a specific sequence and in specific locations, we can draw a path with subsections requiring the appropriate action types. This allows us to generate complex animations without the tedium of manual keyframing. For this reason we term this process “highlevel” keyframing — the user generates an animation based on what should be happening and where. Motion Dumping. If an AI algorithm is used to determine that a character must travel along a certain path or start performing certain actions, the motion graph may be used to “dump” motion on top of the algorithm’s result. Hence motion graphs may be used as a back-end for animating non-player characters in video games and interactive environments — the paths and action types can be specified by a high-level process and the motion graph would fill in the details. Crowds. While our discussion so far has focused on a single character, there’s no reason why it couldn’t be applied to several characters in parallel. Motion graphs may be used as a practical tool for crowd generation. For example, a standard collision-avoidance algorithm could be used to generate a path for each individual, and the motion graph could then generate motion that conforms to this path. Moreover, we can use the techniques described at the end of Section 4.2 to add randomness to the generated motion.
<H1>6 Discussion</H1>
In this paper we have presented a framework for generating realistic, controllable motion through a database of motion capture. Our approach involves automatically constructing a graph that encapsulates connections among different pieces of motion in the database and then searching this graph for motions that satisfy user constraints. We have applied our framework to the problem of path synthesis. As we had limited access to data, our largest examples used a database of several thousand frames of motion. While we believe this was sufficient to show the potential of our method, a character with a truly diverse set of actions might require hundreds or thousands of times more data. Hence the scalability of our framework bears discussion. The principle computational bottleneck in graph construction is locating candidate transitions (Section 3.1). This requires comparing every pair of the F frames in the database and therefore involves O(F 2 ) operations. However, this calculation is trivial to parallelize, and distances between old frames needn’t be recomputed if additions are made to the database. It is the exception rather than the rule that two pieces of motion are sufficiently similar that a transition is possible, and hence motion graphs tend to be sparse. In our experience the necessary amount of storage is approximately proportional to the size of the database. The number of edges leaving a node in general grows with the size of the graph, meaning the branching factor in our search algorithm may grow as well. However, we expect that future motion graphs will be larger mainly because the character will be able to perform more actions. That is, for example, having increasing amounts of walking motion isn’t particularly useful once one can direct a character along nearly any path. Hence the branching factor in a particular subgraph will remain stationary once that subgraph is sufficiently large. We anticipate that typical graph searches will be restricted to one or two subgraphs, and so we expect that the search will remain practical even for larger graphs. We conclude with a brief discussion of future work. One limitation of our approach is that the transition thresholds must be specified by hand, since (as discussed in Section 3.2) different kinds of motions have different fidelity requirements. Setting thresholds in databases involving many different kinds of motions may be overly laborious, and so we are investigating methods for automating this process. A second area of future work is to incorporate parameterizable motions
[Wiley and Hahn 1997; Rose et al. 1998] into our system, rather than having every node correspond to a static piece of motion. This would add flexibility to the search process and potentially allow generated motion to better satisfy user constraints. Finally, we are interested in applying motion graphs to problems other than path synthesis.
8
To appear in Proceedings of SIGGRAPH ’02
Figure 6: The leftmost image shows the original motion and its reflection and the following images show motion generated by our path synthesis algorithm. The thick yellow lines were the paths to be fit and the black line is an approximation of the actual path of the character. Note how we are able to accurately fit nontrivial paths despite the limited variation in the path of the original motion.
Figure 7: The left image shows a walking motion generated to fit to a path that spells “Hello” in cursive. The right image shows a karate motion fit to the same path. The total calculation time for the walking motion was 58.1 seconds and the animation itself is 54.9 seconds. The 87.7-second karate motion was computed in just 15.0 seconds. All computation was done on a 1.3gHz Athlon.
Figure 8: These images are both fits to paths wherein the character is required to walk, then sneak, and finally perform martial arts moves. The desired transition points are indicated by where the curve changes color. Note that the character both fits the path accurately and switches to the appropriate motion type close to the desired location.
9
To appear in Proceedings of SIGGRAPH ’02
<H1>Acknowledgements</H1>
We would like to acknowledge Andrew Gardner, Alex Mohr, and John Schreiner for assisting in video production, proofreading, and other technical matters. We also thank the University of Southern California’s School of Film and Television for their support and the reviewers for their many useful suggestions. Our work was made possible through generous motion data donations from Spectrum Studios (particularly Demian Gordon), House of Moves, and The Ohio State University. This work was supported in part by NSF grants CCR-9984506 and IIS-0097456, the U.S. Army 2 , the Wisconsin Alumni Research Fund’s University Industrial Relations program, equipment donations from IBM, NVidia, and Intel, and software donations from Discreet, Alias/Wavefront, and Pixar.
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10
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],
"text": "['used support vector machines to create motion sequences as compositions of actions generated from a set of physically based controllers']",
"type": "background_claim",
"id": "T75"
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{
"offsets": [
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8891,
8910
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],
"text": "['Perlin and Goldberg']",
"type": "data",
"id": "T76"
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"offsets": [
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9284,
9357
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],
"text": "['we require different approaches to identifying and generating transitions']",
"type": "own_claim",
"id": "T77"
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"offsets": [
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8881,
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],
"text": "['1995']",
"type": "data",
"id": "T78"
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{
"offsets": [
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8912,
8916
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],
"text": "['1996']",
"type": "data",
"id": "T79"
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{
"offsets": [
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9192,
9282
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],
"text": "['our system involves motion capture data, rather than procedural or physically based motion']",
"type": "data",
"id": "T80"
},
{
"offsets": [
[
9024,
9040
]
],
"text": "['Faloutsos et al.']",
"type": "data",
"id": "T81"
},
{
"offsets": [
[
9042,
9046
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],
"text": "['2001']",
"type": "data",
"id": "T82"
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{
"offsets": [
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9365,
9453
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],
"text": "['these systems were mainly concerned with appropriately generating individual transitions']",
"type": "background_claim",
"id": "T83"
},
{
"offsets": [
[
9463,
9514
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],
"text": "['we address the problem of generating entire motions']",
"type": "own_claim",
"id": "T84"
},
{
"offsets": [
[
9539,
9572
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],
"text": "['that meet user-specified criteria']",
"type": "own_claim",
"id": "T85"
},
{
"offsets": [
[
9608,
9696
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],
"text": "['developed a system that used a database to extract motion meeting high-level constraints']",
"type": "background_claim",
"id": "T86"
},
{
"offsets": [
[
9574,
9599
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],
"text": "['Lamouret and van de Panne']",
"type": "data",
"id": "T87"
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{
"offsets": [
[
9601,
9605
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],
"text": "['1996']",
"type": "data",
"id": "T88"
},
{
"offsets": [
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9707,
9778
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],
"text": "['their system was applied to a simple agent with five degrees of freedom']",
"type": "background_claim",
"id": "T89"
},
{
"offsets": [
[
9788,
9845
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],
"text": "['we generate motion for a far more sophisticated character']",
"type": "own_claim",
"id": "T90"
},
{
"offsets": [
[
9879,
10014
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],
"text": "['used a state-based statistical model similar to those mentioned in the previous paragraph to rearrange segments of original motion data']",
"type": "background_claim",
"id": "T91"
},
{
"offsets": [
[
10016,
10071
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],
"text": "['These segments were attached using linear interpolation']",
"type": "background_claim",
"id": "T92"
},
{
"offsets": [
[
9847,
9870
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],
"text": "['Molina-Tanco and Hilton']",
"type": "data",
"id": "T93"
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{
"offsets": [
[
9872,
9876
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],
"text": "['2000']",
"type": "data",
"id": "T94"
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{
"offsets": [
[
10073,
10193
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],
"text": "['The user could create motion by selecting keyframe poses, which were connected with a highprobability sequence of states']",
"type": "background_claim",
"id": "T95"
},
{
"offsets": [
[
10195,
10264
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],
"text": "['Our work considers more general and sophisticated sets of constraints']",
"type": "own_claim",
"id": "T96"
},
{
"offsets": [
[
10266,
10376
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],
"text": "['Work similar to ours has been done in the gaming industry to meet the requirements of online motion generation']",
"type": "own_claim",
"id": "T97"
},
{
"offsets": [
[
10378,
10407
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],
"text": "['Many companies use move trees']",
"type": "background_claim",
"id": "T98"
},
{
"offsets": [
[
10410,
10431
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],
"text": "['Mizuguchi et al. 2001']",
"type": "data",
"id": "T99"
},
{
"offsets": [
[
10435,
10440
]
],
"text": "['which']",
"type": "background_claim",
"id": "T100"
},
{
"offsets": [
[
10462,
10531
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],
"text": "['are graph structures representing connections in a database of motion']",
"type": "background_claim",
"id": "T101"
},
{
"offsets": [
[
10542,
10573
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],
"text": "['move trees are created manually']",
"type": "background_claim",
"id": "T102"
},
{
"offsets": [
[
10652,
10702
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],
"text": "['blends are created by hand using interactive tools']",
"type": "data",
"id": "T103"
},
{
"offsets": [
[
10576,
10647
]
],
"text": "['short motion clips are collected in carefully scripted capture sessions']",
"type": "data",
"id": "T104"
},
{
"offsets": [
[
10704,
10747
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],
"text": "['Motion graphs are constructed automatically']",
"type": "own_claim",
"id": "T105"
},
{
"offsets": [
[
10755,
10818
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],
"text": "['move trees are typically geared for rudimentary motion planning']",
"type": "background_claim",
"id": "T106"
},
{
"offsets": [
[
10923,
10989
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],
"text": "['The generation of transitions is an important part of our approach']",
"type": "own_claim",
"id": "T107"
},
{
"offsets": [
[
11048,
11136
]
],
"text": "['presented a simple method for smoothly interpolating between two clips to create a blend']",
"type": "background_claim",
"id": "T108"
},
{
"offsets": [
[
11028,
11034
]
],
"text": "['Perlin']",
"type": "data",
"id": "T109"
},
{
"offsets": [
[
11036,
11040
]
],
"text": "['1995']",
"type": "data",
"id": "T110"
},
{
"offsets": [
[
11143,
11147
]
],
"text": "['2000']",
"type": "data",
"id": "T111"
},
{
"offsets": [
[
11158,
11280
]
],
"text": "['orientation filters that allowed these blending operations to be performed on rotational data in a more principled fashion']",
"type": "background_claim",
"id": "T112"
},
{
"offsets": [
[
11312,
11424
]
],
"text": "['a more complex method for creating transitions that preserved kinematic constraints and basic dynamic properties']",
"type": "background_claim",
"id": "T113"
},
{
"offsets": [
[
11138,
11141
]
],
"text": "['Lee']",
"type": "data",
"id": "T114"
},
{
"offsets": [
[
11282,
11293
]
],
"text": "['Rose et al.']",
"type": "data",
"id": "T115"
},
{
"offsets": [
[
11426,
11502
]
],
"text": "['Our main application of motion graphs is to control a character’s locomotion']",
"type": "own_claim",
"id": "T116"
},
{
"offsets": [
[
11295,
11299
]
],
"text": "['1996']",
"type": "data",
"id": "T117"
},
{
"offsets": [
[
11504,
11585
]
],
"text": "['This problem is important enough to have received a great deal of prior attention']",
"type": "background_claim",
"id": "T118"
},
{
"offsets": [
[
11648,
11669
]
],
"text": "['synthesis is required']",
"type": "background_claim",
"id": "T119"
},
{
"offsets": [
[
11672,
11762
]
],
"text": "['Procedural and physically based synthesis methods have been developed for a few activities']",
"type": "background_claim",
"id": "T120"
},
{
"offsets": [
[
11595,
11646
]
],
"text": "['a character’s path isn’t generally known in advance']",
"type": "data",
"id": "T121"
},
{
"offsets": [
[
11771,
11778
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],
"text": "['walking']",
"type": "data",
"id": "T122"
},
{
"offsets": [
[
11827,
11834
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],
"text": "['running']",
"type": "data",
"id": "T123"
},
{
"offsets": [
[
11781,
11799
]
],
"text": "['Multon et al. 1999']",
"type": "data",
"id": "T124"
},
{
"offsets": [
[
11801,
11821
]
],
"text": "['Sun and Metaxas 2001']",
"type": "data",
"id": "T125"
},
{
"offsets": [
[
11837,
11856
]
],
"text": "['Hodgins et al. 1995']",
"type": "data",
"id": "T126"
},
{
"offsets": [
[
11858,
11884
]
],
"text": "['Bruderlin and Calvert 1996']",
"type": "data",
"id": "T127"
},
{
"offsets": [
[
11893,
11952
]
],
"text": "['techniques such as these can generate flexible motion paths']",
"type": "background_claim",
"id": "T128"
},
{
"offsets": [
[
11954,
12001
]
],
"text": "['the current range of movement styles is limited']",
"type": "background_claim",
"id": "T129"
},
{
"offsets": [
[
12009,
12106
]
],
"text": "['these methods do not produce the quality of motion attainable by hand animation or motion capture']",
"type": "background_claim",
"id": "T130"
},
{
"offsets": [
[
12140,
12207
]
],
"text": "['a method for editing the path traversed in a clip of motion capture']",
"type": "background_claim",
"id": "T131"
},
{
"offsets": [
[
12124,
12128
]
],
"text": "['2001']",
"type": "data",
"id": "T132"
},
{
"offsets": [
[
12114,
12122
]
],
"text": "['Gleicher']",
"type": "data",
"id": "T133"
},
{
"offsets": [
[
12209,
12269
]
],
"text": "['it did not address the need for continuous streams of motion']",
"type": "background_claim",
"id": "T134"
},
{
"offsets": [
[
12271,
12326
]
],
"text": "['nor could it choose which clip is correct to fit a path']",
"type": "background_claim",
"id": "T135"
},
{
"offsets": [
[
12394,
12412
]
],
"text": "['Our basic approach']",
"type": "own_claim",
"id": "T136"
},
{
"offsets": [
[
12538,
12583
]
],
"text": "['has been applied previously to other problems']",
"type": "own_claim",
"id": "T137"
},
{
"offsets": [
[
12415,
12436
]
],
"text": "['detecting transitions']",
"type": "data",
"id": "T138"
},
{
"offsets": [
[
12438,
12458
]
],
"text": "['constructing a graph']",
"type": "data",
"id": "T139"
},
{
"offsets": [
[
12464,
12535
]
],
"text": "['using graph search techniques to find sequences satisfying user demands']",
"type": "data",
"id": "T140"
},
{
"offsets": [
[
12617,
12758
]
],
"text": "['a similar method for synthesizing seamless streams of video from example footage and driving these streams according to high-level user input']",
"type": "background_claim",
"id": "T141"
},
{
"offsets": [
[
12600,
12604
]
],
"text": "['2000']",
"type": "data",
"id": "T142"
},
{
"offsets": [
[
12585,
12598
]
],
"text": "['Schödl et al.']",
"type": "data",
"id": "T143"
},
{
"offsets": [
[
12760,
12866
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],
"text": "['Since writing this paper, we have learned of similar work done concurrently by a number of research groups']",
"type": "own_claim",
"id": "T144"
},
{
"offsets": [
[
12972,
13052
]
],
"text": "['used a randomized search algorithm to extract motion that meets user constraints']",
"type": "background_claim",
"id": "T145"
},
{
"offsets": [
[
13073,
13144
]
],
"text": "['also constructed a graph and generated motion via three user interfaces']",
"type": "background_claim",
"id": "T146"
},
{
"offsets": [
[
13450,
13516
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],
"text": "['This resulted in sequences of short clips forming complete motions']",
"type": "background_claim",
"id": "T147"
},
{
"offsets": [
[
13293,
13448
]
],
"text": "['keyframed a subset of a character’s degrees of freedom and matched small segments of this keyframed animation with the lower frequency bands of motion data']",
"type": "background_claim",
"id": "T148"
},
{
"offsets": [
[
13534,
13583
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],
"text": "['generated a two-level statistical model of motion']",
"type": "background_claim",
"id": "T149"
},
{
"offsets": [
[
13759,
13858
]
],
"text": "['This model was used both to generate new motion based on user keyframes and to edit existing motion']",
"type": "background_claim",
"id": "T150"
},
{
"offsets": [
[
12868,
12887
]
],
"text": "['Arikan and Forsythe']",
"type": "data",
"id": "T151"
},
{
"offsets": [
[
12889,
12893
]
],
"text": "['2002']",
"type": "data",
"id": "T152"
},
{
"offsets": [
[
12896,
12967
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],
"text": "['constructed from a motion database a hierarchical graph similar to ours']",
"type": "background_claim",
"id": "T153"
},
{
"offsets": [
[
13054,
13064
]
],
"text": "['Lee et al.']",
"type": "data",
"id": "T154"
},
{
"offsets": [
[
13066,
13070
]
],
"text": "['2002']",
"type": "data",
"id": "T155"
},
{
"offsets": [
[
13146,
13163
]
],
"text": "['a list of choices']",
"type": "data",
"id": "T156"
},
{
"offsets": [
[
13165,
13229
]
],
"text": "['a sketch-based interface similar to what we use for path fitting']",
"type": "data",
"id": "T157"
},
{
"offsets": [
[
13266,
13284
]
],
"text": "['Pullen and Bregler']",
"type": "data",
"id": "T158"
},
{
"offsets": [
[
13231,
13240
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],
"text": "['Section 5']",
"type": "data",
"id": "T159"
},
{
"offsets": [
[
13247,
13264
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],
"text": "['a live video feed']",
"type": "data",
"id": "T160"
},
{
"offsets": [
[
13286,
13290
]
],
"text": "['2002']",
"type": "data",
"id": "T161"
},
{
"offsets": [
[
13518,
13526
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],
"text": "['Li et al']",
"type": "data",
"id": "T162"
},
{
"offsets": [
[
13528,
13532
]
],
"text": "['2002']",
"type": "data",
"id": "T163"
},
{
"offsets": [
[
13585,
13686
]
],
"text": "['At the lower level were linear dynamic systems representing characteristic movements called “textons”']",
"type": "data",
"id": "T164"
},
{
"offsets": [
[
13692,
13757
]
],
"text": "['the higher level contained transition probabilities among textons']",
"type": "data",
"id": "T165"
},
{
"offsets": [
[
14836,
14916
]
],
"text": "['A motion graph is a directed graph where all edges correspond to clips of motion']",
"type": "own_claim",
"id": "T166"
},
{
"offsets": [
[
15420,
15474
]
],
"text": "['A more interesting graph requires greater connectivity']",
"type": "own_claim",
"id": "T167"
},
{
"offsets": [
[
15680,
15730
]
],
"text": "['we need to create clips expressly for this purpose']",
"type": "own_claim",
"id": "T168"
},
{
"offsets": [
[
15606,
15678
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],
"text": "['it is unlikely that two pieces of original data are sufficiently similar']",
"type": "data",
"id": "T169"
},
{
"offsets": [
[
15480,
15518
]
],
"text": "['a node to have multiple outgoing edges']",
"type": "data",
"id": "T170"
},
{
"offsets": [
[
15520,
15598
]
],
"text": "['there must be multiple clips that can follow the clip(s) leading into the node']",
"type": "own_claim",
"id": "T171"
},
{
"offsets": [
[
15947,
16028
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],
"text": "['we can create a wellconnected structure with a wide range of possible graph walks']",
"type": "own_claim",
"id": "T172"
},
{
"offsets": [
[
15882,
15945
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],
"text": "['inserting transition clips between otherwise disconnected nodes']",
"type": "data",
"id": "T173"
},
{
"offsets": [
[
15835,
15877
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],
"text": "['introducing nodes within the initial clips']",
"type": "data",
"id": "T174"
},
{
"offsets": [
[
16034,
16042
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],
"text": "['Figure 2']",
"type": "data",
"id": "T175"
},
{
"offsets": [
[
16061,
16109
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],
"text": "['creating transitions is a hard animation problem']",
"type": "own_claim",
"id": "T176"
},
{
"offsets": [
[
16335,
16471
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],
"text": "['the problem of automatically creating such a transition is arguably as difficult as that of creating realistic motion in the first place']",
"type": "own_claim",
"id": "T177"
},
{
"offsets": [
[
16492,
16599
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],
"text": "['if two motions are “close” to each other then simple blending techniques can reliably generate a transition']",
"type": "own_claim",
"id": "T178"
},
{
"offsets": [
[
16619,
16783
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],
"text": "['our strategy is to identify portions of the initial clips that are sufficiently similar that straightforward blending is almost certain to produce valid transitions']",
"type": "own_claim",
"id": "T179"
},
{
"offsets": [
[
17304,
17450
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],
"text": "['motion capture data is typically represented as vectors of parameters specifying the root position and joint rotations of a skeleton on each frame']",
"type": "background_claim",
"id": "T180"
},
{
"offsets": [
[
17452,
17590
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],
"text": "['One might attempt to locate transition points by computing some vector norm to measure the difference between poses at each pair of frames']",
"type": "background_claim",
"id": "T181"
},
{
"offsets": [
[
17601,
17638
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],
"text": "['such a simple approach is ill-advised']",
"type": "own_claim",
"id": "T182"
},
{
"offsets": [
[
17643,
17691
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],
"text": "['it fails to address a number of important issues']",
"type": "own_claim",
"id": "T183"
},
{
"offsets": [
[
17704,
17774
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],
"text": "['Simple vector norms fail to account for the meanings of the parameters']",
"type": "own_claim",
"id": "T184"
},
{
"offsets": [
[
17961,
18031
]
],
"text": "['there is no meaningful way to assign fixed weights to these parameters']",
"type": "own_claim",
"id": "T185"
},
{
"offsets": [
[
18036,
18140
]
],
"text": "['the effect of a joint rotation on the shape of the body depends on the current configuration of the body']",
"type": "data",
"id": "T186"
},
{
"offsets": [
[
17790,
17903
]
],
"text": "['in the joint angle representation some parameters have a much greater overall effect on the character than others']",
"type": "data",
"id": "T187"
},
{
"offsets": [
[
18346,
18418
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],
"text": "['comparing two motions requires identifying compatible coordinate systems']",
"type": "own_claim",
"id": "T188"
},
{
"offsets": [
[
18223,
18338
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],
"text": "['the motion is fundamentally unchanged if we translate it along the floor plane or rotate it about the vertical axis']",
"type": "own_claim",
"id": "T189"
},
{
"offsets": [
[
18145,
18212
]
],
"text": "['A motion is defined only up to a rigid 2D coordinate transformation']",
"type": "own_claim",
"id": "T190"
},
{
"offsets": [
[
18423,
18503
]
],
"text": "['Smooth blends require more information than can be obtained at individual frames']",
"type": "own_claim",
"id": "T191"
},
{
"offsets": [
[
18505,
18664
]
],
"text": "['A seamless transition must account not only for differences in body posture, but also in joint velocities, accelerations, and possibly higher-order derivatives']",
"type": "own_claim",
"id": "T192"
},
{
"offsets": [
[
18739,
18806
]
],
"text": "['To motivate it, we note that the skeleton is only a means to an end']",
"type": "own_claim",
"id": "T193"
},
{
"offsets": [
[
18808,
18893
]
],
"text": "['In a typical animation, a polygonal mesh is deformed according to the skeleton’s pose']",
"type": "background_claim",
"id": "T194"
},
{
"offsets": [
[
19028,
19148
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],
"text": "['For this reason we measure the distance between two frames of animation in terms of a point cloud driven by the skeleton']",
"type": "own_claim",
"id": "T195"
},
{
"offsets": [
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18895,
18924
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],
"text": "['This mesh is all that is seen']",
"type": "background_claim",
"id": "T196"
},
{
"offsets": [
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18936,
19026
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],
"text": "['it is a natural focus when considering how close two frames of animation are to each other']",
"type": "background_claim",
"id": "T197"
},
{
"offsets": [
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19592,
19684
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],
"text": "['The use of windows of frames effectively incorporates derivative information into the metric']",
"type": "own_claim",
"id": "T198"
},
{
"offsets": [
[
19150,
19227
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],
"text": "['Ideally this point cloud is a downsampling of the mesh defining the character']",
"type": "own_claim",
"id": "T199"
},
{
"offsets": [
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19690,
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],
"text": "['is similar to the approach']",
"type": "background_claim",
"id": "T200"
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{
"offsets": [
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19722,
19740
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],
"text": "['Schödl et al. 2000']",
"type": "data",
"id": "T201"
},
{
"offsets": [
[
21841,
21935
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],
"text": "['To make our transition model more compact, we find all the local minima of this error function']",
"type": "own_claim",
"id": "T202"
},
{
"offsets": [
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21945,
22025
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],
"text": "['extracting the “sweet spots” at which transitions are locally the most opportune']",
"type": "own_claim",
"id": "T203"
},
{
"offsets": [
[
22058,
22076
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],
"text": "['Schödl et al. 2000']",
"type": "data",
"id": "T204"
},
{
"offsets": [
[
22206,
22299
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],
"text": "['A local minimum in the distance function does not necessarily imply a high-quality transition']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
22301,
22355
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],
"text": "['it only implies a transition better than its neighbors']",
"type": "own_claim",
"id": "T206"
},
{
"offsets": [
[
22429,
22523
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],
"text": "['The simplest approach is to only accept local minima below an empirically determined threshold']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
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22525,
22567
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],
"text": "['This can be done without user intervention']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
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22578,
22694
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],
"text": "['often users will want to set the threshold themselves to pick an acceptable tradeoff between having good transitions']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
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22711,
22739
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],
"text": "['and having high connectivity']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
22696,
22709
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],
"text": "['low threshold']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
22741,
22755
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],
"text": "['high threshold']",
"type": "data",
"id": "T212"
},
{
"offsets": [
[
22758,
22821
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],
"text": "['Different kinds of motions have different fidelity requirements']",
"type": "own_claim",
"id": "T213"
},
{
"offsets": [
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22836,
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],
"text": "['walking motions have very exacting requirements on the transitions']",
"type": "data",
"id": "T214"
},
{
"offsets": [
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22905,
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],
"text": "['people have seen others walk nearly every day since birth and consequently have a keen sense of what a walk should look like']",
"type": "data",
"id": "T215"
},
{
"offsets": [
[
23050,
23162
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],
"text": "['most people are less familiar with ballet motions and would be less likely to detect inaccuracies in such motion']",
"type": "data",
"id": "T216"
},
{
"offsets": [
[
23177,
23252
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],
"text": "['we allow a user to apply different thresholds to different pairs of motions']",
"type": "own_claim",
"id": "T217"
},
{
"offsets": [
[
24490,
24514
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],
"text": "['Other transition schemes']",
"type": "background_claim",
"id": "T218"
},
{
"offsets": [
[
24545,
24577
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],
"text": "['may be used in place of this one']",
"type": "background_claim",
"id": "T219"
},
{
"offsets": [
[
24526,
24542
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],
"text": "['Rose et al. 1996']",
"type": "data",
"id": "T220"
},
{
"offsets": [
[
24579,
24665
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],
"text": "['The use of linear blends means that constraints in the original motion may be violated']",
"type": "own_claim",
"id": "T221"
},
{
"offsets": [
[
24680,
24745
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],
"text": "['one of the character’s feet may slide when it ought to be planted']",
"type": "data",
"id": "T222"
},
{
"offsets": [
[
24747,
24824
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],
"text": "['This can be corrected by using constraint annotations in the original motions']",
"type": "own_claim",
"id": "T223"
},
{
"offsets": [
[
24826,
24864
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],
"text": "['We treat constraints as binary signals']",
"type": "own_claim",
"id": "T224"
},
{
"offsets": [
[
24866,
24935
]
],
"text": "['on a given frame a particular constraint either exists or it does not']",
"type": "data",
"id": "T225"
},
{
"offsets": [
[
25113,
25200
]
],
"text": "['In this ¡ manner each transition is automatically annotated with constraint information']",
"type": "own_claim",
"id": "T226"
},
{
"offsets": [
[
25206,
25310
]
],
"text": "['these constraints may later be enforced as a postprocessing step when motion is extracted form the graph']",
"type": "own_claim",
"id": "T227"
},
{
"offsets": [
[
25897,
25991
]
],
"text": "['In its current state there are no guarantees that the graph can synthesize motion indefinitely']",
"type": "own_claim",
"id": "T228"
},
{
"offsets": [
[
25999,
26017
]
],
"text": "['there may be nodes']",
"type": "data",
"id": "T229"
},
{
"offsets": [
[
26037,
26067
]
],
"text": "['that are not part of any cycle']",
"type": "data",
"id": "T230"
},
{
"offsets": [
[
26073,
26081
]
],
"text": "['Figure 4']",
"type": "data",
"id": "T231"
},
{
"offsets": [
[
26085,
26176
]
],
"text": "['Once such a node is entered there is a bound on how much additional motion can be generated']",
"type": "own_claim",
"id": "T232"
},
{
"offsets": [
[
26205,
26335
]
],
"text": "['may be part of one or more cycles but nonetheless only be able to reach a small fraction of the total number of nodes in the graph']",
"type": "own_claim",
"id": "T233"
},
{
"offsets": [
[
26178,
26189
]
],
"text": "['Other nodes']",
"type": "own_claim",
"id": "T234"
},
{
"offsets": [
[
26343,
26412
]
],
"text": "['arbitrarily long motion may still be generated once a sink is entered']",
"type": "own_claim",
"id": "T235"
},
{
"offsets": [
[
26414,
26469
]
],
"text": "['this motion is confined to a small part of the database']",
"type": "own_claim",
"id": "T236"
},
{
"offsets": [
[
26471,
26585
]
],
"text": "['Finally, some nodes may have incoming edges such that no outgoing edges contain the same set of descriptive labels']",
"type": "own_claim",
"id": "T237"
},
{
"offsets": [
[
26587,
26604
]
],
"text": "['This is dangerous']",
"type": "own_claim",
"id": "T238"
},
{
"offsets": [
[
26611,
26662
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],
"text": "['logical discontinuities may be forced into a motion']",
"type": "own_claim",
"id": "T239"
},
{
"offsets": [
[
26677,
26777
]
],
"text": "['a character currently in a “boxing” motion may have no choice but to transition to a “ballet” motion']",
"type": "data",
"id": "T240"
},
{
"offsets": [
[
26779,
26992
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],
"text": "['To address these problems, we prune the graph such that, starting from any edge, it is possible to generate arbitrarily long streams of motion of the same type such that as much of the database as possible is used']",
"type": "own_claim",
"id": "T241"
},
{
"offsets": [
[
28138,
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],
"text": "['By this stage we have finished constructing the motion graph']",
"type": "own_claim",
"id": "T242"
},
{
"offsets": [
[
28375,
28429
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],
"text": "['Our algorithm involves solving an optimization problem']",
"type": "own_claim",
"id": "T243"
},
{
"offsets": [
[
28724,
28812
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],
"text": "['a graph walk corresponds to a motion generated by placing these pieces one after another']",
"type": "own_claim",
"id": "T244"
},
{
"offsets": [
[
28671,
28722
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],
"text": "['every edge on the motion graph is a piece of motion']",
"type": "data",
"id": "T245"
},
{
"offsets": [
[
28814,
28891
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],
"text": "['The only issue is to place each piece in the correct location and orientation']",
"type": "own_claim",
"id": "T246"
},
{
"offsets": [
[
28909,
28981
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],
"text": "['each frame must be transformed by an appropriate 2D rigid transformation']",
"type": "own_claim",
"id": "T247"
},
{
"offsets": [
[
29250,
29391
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],
"text": "['the use of linear blends to create transitions can cause artifacts, the most common of which is feet that slide when they ought to be planted']",
"type": "own_claim",
"id": "T248"
},
{
"offsets": [
[
29237,
29248
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],
"text": "['Section 3.3']",
"type": "data",
"id": "T249"
},
{
"offsets": [
[
29402,
29473
]
],
"text": "['every graph walk is automatically annotated with constraint information']",
"type": "own_claim",
"id": "T250"
},
{
"offsets": [
[
29488,
29512
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],
"text": "['the foot must be planted']",
"type": "data",
"id": "T251"
},
{
"offsets": [
[
29515,
29599
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],
"text": "['These constraints are either specified directly in the original motions or generated']",
"type": "own_claim",
"id": "T252"
},
{
"offsets": [
[
29606,
29617
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],
"text": "['Section 3.3']",
"type": "data",
"id": "T253"
},
{
"offsets": [
[
29619,
29682
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],
"text": "['depending on whether the frame is original data or a transition']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
[
29685,
29746
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],
"text": "['These constraints may be satisfied using a variety of methods']",
"type": "background_claim",
"id": "T255"
},
{
"offsets": [
[
29758,
29771
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],
"text": "['Gleicher 1998']",
"type": "data",
"id": "T256"
},
{
"offsets": [
[
29778,
29795
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],
"text": "['Lee and Shin 1999']",
"type": "data",
"id": "T257"
},
{
"offsets": [
[
29798,
29838
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],
"text": "['In our work we used the method described']",
"type": "own_claim",
"id": "T258"
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{
"offsets": [
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29844,
29861
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],
"text": "['Kovar et al. 2002']",
"type": "data",
"id": "T259"
},
{
"offsets": [
[
29928,
30037
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],
"text": "['We are now in a position to consider the problem of finding motion that satisfies user-specified requirements']",
"type": "own_claim",
"id": "T260"
},
{
"offsets": [
[
30069,
30122
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],
"text": "['only very special graph walks are likely to be useful']",
"type": "own_claim",
"id": "T261"
},
{
"offsets": [
[
30143,
30206
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],
"text": "['a random graph walk will generate a continuous stream of motion']",
"type": "own_claim",
"id": "T262"
},
{
"offsets": [
[
30208,
30277
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],
"text": "['such an algorithm has little use other than an elaborate screen saver']",
"type": "own_claim",
"id": "T263"
},
{
"offsets": [
[
30725,
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],
"text": "['This is less useful than it might appear at first']",
"type": "own_claim",
"id": "T264"
},
{
"offsets": [
[
30783,
30865
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],
"text": "['there are no guarantees that the shortest graph walk is short in an absolute sense']",
"type": "own_claim",
"id": "T265"
},
{
"offsets": [
[
30867,
30892
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],
"text": "['In our larger test graphs']",
"type": "data",
"id": "T266"
},
{
"offsets": [
[
30936,
31015
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],
"text": "['the average shortest path between any two nodes was on the order of two seconds']",
"type": "data",
"id": "T267"
},
{
"offsets": [
[
31145,
31240
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],
"text": "['there were on average only five or six transitions separating any two of the thousands of nodes']",
"type": "own_claim",
"id": "T268"
},
{
"offsets": [
[
31077,
31132
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],
"text": "['the transitions were about one-third of a second apiece']",
"type": "data",
"id": "T269"
},
{
"offsets": [
[
31017,
31069
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],
"text": "['This is not because the graphs were poorly connected']",
"type": "own_claim",
"id": "T270"
},
{
"offsets": [
[
31250,
31309
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],
"text": "['there is no control over what happens during the graph walk']",
"type": "own_claim",
"id": "T271"
},
{
"offsets": [
[
31312,
31389
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],
"text": "['we can’t specify what direction the character travels in or where she ends up']",
"type": "own_claim",
"id": "T272"
},
{
"offsets": [
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31407,
31543
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],
"text": "['the sorts of motions that a user is likely to be interested in probably don’t involve minimizing metrics as simple as total elapsed time']",
"type": "own_claim",
"id": "T273"
},
{
"offsets": [
[
31554,
31653
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],
"text": "['for complicated metrics there is typically no simple way of finding the globally optimal graph walk']",
"type": "own_claim",
"id": "T274"
},
{
"offsets": [
[
31661,
31779
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],
"text": "['we focus instead on local search methods that try to find a satisfactory graph walk within a reasonable amount of time']",
"type": "own_claim",
"id": "T275"
},
{
"offsets": [
[
33970,
34072
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],
"text": "['Branch and bound is most successful when we can attain a tight lower bound early in the search process']",
"type": "own_claim",
"id": "T276"
},
{
"offsets": [
[
34090,
34185
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],
"text": "['it is worthwhile to have a heuristic for ordering the edges we explore out of a particular node']",
"type": "own_claim",
"id": "T277"
},
{
"offsets": [
[
33283,
33384
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],
"text": "['the number of possible graph walks grows exponentially with the average size of a complete graph walk']",
"type": "own_claim",
"id": "T278"
},
{
"offsets": [
[
33155,
33272
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],
"text": "['A naıve solution is to use depth-first search to evaluate f for all complete graph walks and then select the best one']",
"type": "own_claim",
"id": "T279"
},
{
"offsets": [
[
32815,
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],
"text": "['Our goal is find a complete graph walk w that minimizes f']",
"type": "own_claim",
"id": "T280"
},
{
"offsets": [
[
34359,
34435
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],
"text": "['branch and bound reduces the number of graph walks we have to test against f']",
"type": "own_claim",
"id": "T281"
},
{
"offsets": [
[
34438,
34515
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],
"text": "['it does not change the fact that the search process is inherently exponential']",
"type": "own_claim",
"id": "T282"
},
{
"offsets": [
[
34518,
34565
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],
"text": "['it merely lowers the effective branching factor']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
35020,
35095
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],
"text": "['Sometimes it is useful to have a degree of randomness in the search process']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
35110,
35134
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],
"text": "['one is animating a crowd']",
"type": "data",
"id": "T285"
},
{
"offsets": [
[
35136,
35241
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],
"text": "['There are a couple of easy ways to add randomness to the search process without sacrificing a good result']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
35243,
35298
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],
"text": "['The first is to select a start for the search at random']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
35300,
35477
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],
"text": "['The second is retain the r best graph walks at the end of each iteration of the search and randomly pick among the ones whose error is within some tolerance of the best solution']",
"type": "own_claim",
"id": "T288"
},
{
"offsets": [
[
35622,
35709
]
],
"text": "['it is worth considering what sorts of functions are likely to produce desirable results']",
"type": "own_claim",
"id": "T289"
},
{
"offsets": [
[
35553,
35620
]
],
"text": "['the motion extracted from the graph is determined by the function g']",
"type": "data",
"id": "T290"
},
{
"offsets": [
[
36970,
37051
]
],
"text": "['it’s conceivable that given a larger database we would have found a better motion']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
37053,
37126
]
],
"text": "['the problem here is with the function we passed into the search algorithm']",
"type": "own_claim",
"id": "T292"
},
{
"offsets": [
[
36178,
36212
]
],
"text": "['The error of a complete graph walk']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
36261,
36365
]
],
"text": "['was determined by how far away this kicking clip was from being in a particular position and orientation']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
35967,
36102
]
],
"text": "['We can formally state this problem as follows: given a starting node N in the graph and a target edge e, find a graph walk this section']",
"type": "own_claim",
"id": "T295"
},
{
"offsets": [
[
36367,
36517
]
],
"text": "['The character spends approximately seven seconds making minute adjustments to its orientation in an attempt to better align itself with the final clip']",
"type": "data",
"id": "T296"
},
{
"offsets": [
[
36860,
36962
]
],
"text": "['The character turns around in place several times in an attempt to better line up with the target clip']",
"type": "data",
"id": "T297"
},
{
"offsets": [
[
37135,
37209
]
],
"text": "['it gives no guidance as to what should be done in the middle of the motion']",
"type": "own_claim",
"id": "T298"
},
{
"offsets": [
[
37211,
37291
]
],
"text": "['all that matters is that the final clip be in the right position and orientation']",
"type": "own_claim",
"id": "T299"
},
{
"offsets": [
[
37304,
37386
]
],
"text": "['the character is allowed to do whatever is possible in order to make the final fit']",
"type": "own_claim",
"id": "T300"
},
{
"offsets": [
[
37396,
37445
]
],
"text": "['the motion is nothing that a real person would do']",
"type": "data",
"id": "T301"
},
{
"offsets": [
[
37455,
37504
]
],
"text": "['the goal is probably more specific than necessary']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
37509,
37555
]
],
"text": "['it doesn’t matter what kick the character does']",
"type": "data",
"id": "T303"
},
{
"offsets": [
[
37562,
37639
]
],
"text": "['it should be allowed to choose a kick that doesn’t require such effort to aim']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
37657,
37708
]
],
"text": "['there are two lessons we can draw from this example']",
"type": "own_claim",
"id": "T305"
},
{
"offsets": [
[
37717,
37781
]
],
"text": "['g should give some sort of guidance throughout the entire motion']",
"type": "own_claim",
"id": "T306"
},
{
"offsets": [
[
37969,
38099
]
],
"text": "['guiding the search toward a particular result must be balanced against unduly preventing it from considering all available options']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
37786,
37828
]
],
"text": "['arbitrary motion is almost never desirable']",
"type": "data",
"id": "T308"
},
{
"offsets": [
[
37838,
37884
]
],
"text": "['g should be no more restrictive than necessary']",
"type": "own_claim",
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"text": "['To demonstrate that it is nonetheless possible to come up with optimization criteria that allow us to solve a real problem, we apply the preceding framework to path synthesis']",
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"text": "['This problem is simple to state: given a path P specified by the user, generate motion such that the character travels along P']",
"type": "own_claim",
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"text": "['The basic idea is to estimate the actual path P travelled by the character during a graph walk and measure how different it is from P']",
"type": "own_claim",
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"text": "['A simple way to determine P is to project the root onto the floor at each frame, forming a piecewise linear curve 1']",
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"text": "['The error function g(w, e) was chosen for a number of reasons']",
"type": "own_claim",
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"text": "['is important in making the search algorithm practical']",
"type": "own_claim",
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"text": "['the character is given incentive to make definite progress along the path']",
"type": "own_claim",
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"text": "['it would have no reason not to alternate between travelling forwards and backwards']",
"type": "own_claim",
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"text": "['we were to have required the character to merely be near the path']",
"type": "data",
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"text": "['this metric allows the character to travel at whatever speed is appropriate for what needs to be done']",
"type": "own_claim",
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"text": "['a sharp turn will not cover distance at the same rate as walking straight forward']",
"type": "own_claim",
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"text": "['One potential problem with this metric is that a character who stands still will never have an incentive to move forward']",
"type": "own_claim",
"id": "T324"
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"text": "['both actions are equally important for accurate path synthesis']",
"type": "data",
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"text": "['it is important that one not be given undue preference over the other']",
"type": "own_claim",
"id": "T326"
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{
"offsets": [
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"text": "['it can accrue zero error by remaining in place']",
"type": "data",
"id": "T327"
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"text": "['we have not encountered this particular problem in practice']",
"type": "own_claim",
"id": "T328"
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"text": "['it can be countered by requiring at least a small amount of forward progress γ on each frame']",
"type": "own_claim",
"id": "T329"
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{
"offsets": [
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41343,
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"text": "['we can replace in Equation 9 the function s(e i ) with t(e i ) = max(t(e i−1 ) + s(e i ) − s(e i−1 ),t(e i−1 ) + γ )']",
"type": "own_claim",
"id": "T330"
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{
"offsets": [
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41461,
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"text": "['Typically the user will want all generated motion to be of a single type']",
"type": "own_claim",
"id": "T331"
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41543,
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"text": "['walking']",
"type": "data",
"id": "T332"
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{
"offsets": [
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],
"text": "['More interestingly, one can require different types of motion on different parts of the path']",
"type": "own_claim",
"id": "T333"
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{
"offsets": [
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41865,
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"text": "['The necessary modifications to accomplish this are simple']",
"type": "own_claim",
"id": "T334"
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{
"offsets": [
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42178,
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],
"text": "['the character is currently fitting P 2']",
"type": "data",
"id": "T335"
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{
"offsets": [
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42224,
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],
"text": "['the algorithm is identical to the single-type case']",
"type": "own_claim",
"id": "T336"
},
{
"offsets": [
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41552,
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],
"text": "['This corresponds to confining the search to the subgraph containing the appropriate set of descriptive labels']",
"type": "own_claim",
"id": "T337"
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{
"offsets": [
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],
"text": "['we only allow this switch to occur once on any given graph walk']",
"type": "own_claim",
"id": "T338"
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{
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42638,
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"text": "['prevents the resulting motion from randomly switching between the two actions']",
"type": "own_claim",
"id": "T339"
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"text": "['our technique is viable']",
"type": "own_claim",
"id": "T340"
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{
"offsets": [
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42796,
42804
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],
"text": "['Figure 1']",
"type": "data",
"id": "T341"
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{
"offsets": [
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42876,
42929
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],
"text": "['we were able to find accurate fits to the given paths']",
"type": "own_claim",
"id": "T342"
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{
"offsets": [
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43122,
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"text": "['the input motion had a fair amount of variation']",
"type": "own_claim",
"id": "T343"
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{
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42947,
42967
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],
"text": "['upper portion of the']",
"type": "data",
"id": "T344"
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{
"offsets": [
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43114,
43120
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],
"text": "['figure']",
"type": "data",
"id": "T345"
},
{
"offsets": [
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43181,
43203
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],
"text": "['straight-ahead marches']",
"type": "data",
"id": "T346"
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{
"offsets": [
[
43205,
43216
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],
"text": "['sharp turns']",
"type": "data",
"id": "T347"
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{
"offsets": [
[
43222,
43249
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],
"text": "['smooth changes of curvature']",
"type": "data",
"id": "T348"
},
{
"offsets": [
[
43260,
43328
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],
"text": "['our algorithm is still useful when the input database is not as rich']",
"type": "own_claim",
"id": "T349"
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{
"offsets": [
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43339,
43347
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],
"text": "['Figure 6']",
"type": "data",
"id": "T350"
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{
"offsets": [
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43858,
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"text": "['our approach works even on motions that are not obviously locomotion']",
"type": "own_claim",
"id": "T351"
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{
"offsets": [
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43791,
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"text": "['the second uses martial arts motions']",
"type": "data",
"id": "T352"
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"offsets": [
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"text": "['in our test cases the duration of a generated motion was either greater than or approximately equal to the amount of time needed to produce it']",
"type": "own_claim",
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"text": "['the character follow the path well']",
"type": "own_claim",
"id": "T356"
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44712,
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"text": "['transitions between action types occur quite close to their specified locations']",
"type": "own_claim",
"id": "T357"
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{
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44956,
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"text": "['approximately twenty-five minutes were needed to compute the locations of all candidate transitions points']",
"type": "own_claim",
"id": "T354"
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{
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44432,
44440
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],
"text": "['Figure 8']",
"type": "data",
"id": "T355"
},
{
"offsets": [
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44917,
44934
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],
"text": "['our largest graph']",
"type": "data",
"id": "T358"
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{
"offsets": [
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45064,
45149
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],
"text": "['Approximately five minutes of user time were required to select transition thresholds']",
"type": "own_claim",
"id": "T359"
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45155,
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],
"text": "['it took less than a minute to calculate blends at these transitions and prune the resulting graph']",
"type": "own_claim",
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{
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45387,
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"text": "['Directable locomotion is a general enough need that the preceding algorithm has many applications']",
"type": "background_claim",
"id": "T361"
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{
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45507,
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"text": "['We can use path synthesis techniques to give a user interactive control over a character']",
"type": "own_claim",
"id": "T362"
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{
"offsets": [
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45610,
45689
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"text": "['when the user hits the left arrow key the character might start travelling east']",
"type": "data",
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46454,
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"text": "['we can draw a path with subsections requiring the appropriate action types']",
"type": "own_claim",
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{
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46357,
46452
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"text": "['we want a character to perform certain actions in a specific sequence and in specific locations']",
"type": "data",
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{
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46530,
46615
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"text": "['This allows us to generate complex animations without the tedium of manual keyframing']",
"type": "own_claim",
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{
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46899,
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"text": "['the motion graph may be used to “dump” motion on top of the algorithm’s result']",
"type": "own_claim",
"id": "T367"
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{
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46775,
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"text": "['an AI algorithm is used to determine that a character must travel along a certain path or start performing certain actions']",
"type": "data",
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{
"offsets": [
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46985,
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"text": "['motion graphs may be used as a back-end for animating non-player characters in video games and interactive environments']",
"type": "own_claim",
"id": "T369"
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{
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47107,
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"text": "['the paths and action types can be specified by a high-level process and the motion graph would fill in the details']",
"type": "data",
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{
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47294,
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],
"text": "['there’s no reason why it couldn’t be applied to several characters in parallel']",
"type": "own_claim",
"id": "T371"
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{
"offsets": [
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47374,
47440
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],
"text": "['Motion graphs may be used as a practical tool for crowd generation']",
"type": "own_claim",
"id": "T372"
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{
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47636,
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"text": "['we can use the techniques described']",
"type": "own_claim",
"id": "T373"
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{
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47698,
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"text": "['to add randomness to the generated motion']",
"type": "own_claim",
"id": "T374"
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{
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47679,
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"text": "['end of Section 4.2']",
"type": "data",
"id": "T375"
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{
"offsets": [
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47455,
47548
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],
"text": "['a standard collision-avoidance algorithm could be used to generate a path for each individual']",
"type": "own_claim",
"id": "T376"
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{
"offsets": [
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47554,
47624
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"text": "['the motion graph could then generate motion that conforms to this path']",
"type": "own_claim",
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47807,
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"text": "['we have presented a framework for generating realistic, controllable motion through a database of motion capture']",
"type": "own_claim",
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47921,
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"text": "['Our approach involves automatically constructing a graph that encapsulates connections among different pieces of motion in the database']",
"type": "own_claim",
"id": "T379"
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{
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48061,
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"text": "['then searching this graph for motions that satisfy user constraints']",
"type": "own_claim",
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{
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48228,
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"text": "['our largest examples used a database of several thousand frames of motion']",
"type": "own_claim",
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{
"offsets": [
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48197,
48226
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],
"text": "['we had limited access to data']",
"type": "data",
"id": "T382"
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{
"offsets": [
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48309,
48375
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],
"text": "['we believe this was sufficient to show the potential of our method']",
"type": "own_claim",
"id": "T383"
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{
"offsets": [
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48377,
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"text": "['a character with a truly diverse set of actions might require hundreds or thousands of times more data']",
"type": "own_claim",
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{
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48487,
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"text": "['the scalability of our framework bears discussion']",
"type": "own_claim",
"id": "T385"
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{
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48538,
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"text": "['The principle computational bottleneck in graph construction is locating candidate transitions']",
"type": "own_claim",
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{
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48634,
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"text": "['Section 3.1']",
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48648,
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"text": "['This requires comparing every pair of the F frames in the database and therefore involves O']",
"type": "own_claim",
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{
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48746,
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"text": "['operations']",
"type": "own_claim",
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48767,
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"text": "['this calculation is trivial to parallelize']",
"type": "own_claim",
"id": "T390"
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{
"offsets": [
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48815,
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"text": "['distances between old frames needn’t be recomputed']",
"type": "own_claim",
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48869,
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"text": "['additions are made to the database']",
"type": "data",
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49038,
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"text": "['motion graphs tend to be sparse']",
"type": "own_claim",
"id": "T393"
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{
"offsets": [
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48905,
49026
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"text": "['It is the exception rather than the rule that two pieces of motion are sufficiently similar that a transition is possible']",
"type": "own_claim",
"id": "T394"
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{
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49089,
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"text": "['the necessary amount of storage is approximately proportional to the size of the database']",
"type": "own_claim",
"id": "T395"
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{
"offsets": [
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49180,
49258
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"text": "['The number of edges leaving a node in general grows with the size of the graph']",
"type": "own_claim",
"id": "T396"
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{
"offsets": [
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49268,
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"text": "['the branching factor in our search algorithm may grow as well']",
"type": "own_claim",
"id": "T397"
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{
"offsets": [
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49340,
49456
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],
"text": "['we expect that future motion graphs will be larger mainly because the character will be able to perform more actions']",
"type": "own_claim",
"id": "T398"
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{
"offsets": [
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49480,
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"text": "['having increasing amounts of walking motion isn’t particularly useful once one can direct a character along nearly any path']",
"type": "data",
"id": "T399"
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{
"offsets": [
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49611,
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"text": "['the branching factor in a particular subgraph will remain stationary']",
"type": "own_claim",
"id": "T400"
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"text": "['that subgraph is sufficiently large']",
"type": "data",
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"offsets": [
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"text": "['we expect that the search will remain practical even for larger graphs']",
"type": "own_claim",
"id": "T402"
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"text": "['We anticipate that typical graph searches will be restricted to one or two subgraphs']",
"type": "own_claim",
"id": "T403"
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{
"offsets": [
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49939,
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"text": "['One limitation of our approach is that the transition thresholds must be specified by hand']",
"type": "own_claim",
"id": "T404"
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{
"offsets": [
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50067,
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"text": "['different kinds of motions have different fidelity requirements']",
"type": "data",
"id": "T405"
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{
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50054,
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"text": "['Section 3.2']",
"type": "data",
"id": "T406"
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50132,
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"text": "['Setting thresholds in databases involving many different kinds of motions may be overly laborious']",
"type": "own_claim",
"id": "T407"
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"text": "['we are investigating methods for automating this process']",
"type": "own_claim",
"id": "T408"
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{
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"text": "['into our system']",
"type": "own_claim",
"id": "T409"
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"text": "['A second area of future work is to incorporate parameterizable motions']",
"type": "own_claim",
"id": "T410"
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{
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"text": "['having every node correspond to a static piece of motion']",
"type": "own_claim",
"id": "T411"
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{
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"text": "['Wiley and Hahn 1997']",
"type": "data",
"id": "T412"
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{
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],
"text": "['Rose et al. 1998']",
"type": "data",
"id": "T413"
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{
"offsets": [
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50495,
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"text": "['This would add flexibility to the search process and potentially allow generated motion to better satisfy user constraints']",
"type": "own_claim",
"id": "T414"
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{
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50628,
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],
"text": "['we are interested in applying motion graphs to problems other than path synthesis']",
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A35 | <?xml version="1.0" encoding="UTF-8" standalone="no"?>
<Document xmlns:gate="http://www.gate.ac.uk" name="A35_C05_Comparing_Efficiency_of_Integration_Methods_for_Cloth_Simulation_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1109/cgi.2001.934683
<Title>Comparing Efficiency of Integration Methods for Cloth Simulation</Title>
Pascal VOLINO
Nadia MAGNENAT-THALMANN MIRALab
C.U.I.
University of Geneva CH
Switzerland
Web: http://miralabwww.unige.ch Email: [pascal|thalmann]@cui.unige.ch Phone: -41 (22) 705 77 63 -41 (22) 705 77 80
<Abstract>Any cloth simulation system needs efficient numerical methods for integrating the equations that describe the mechanical behavior of the discrete representation of the cloth. Choosing the adequate method should be done with full knowledge of the advantages and weaknesses of the main techniques. This paper presents a quantitative comparison of the efficiency of the most common integration techniques used for cloth simulation, and raises the key considerations for optimal implementations depending on the practical kind of simulation problematic.</Abstract>
Keywords: Cloth simulation, numerical integration, implicit integration, Euler, Midpoint, Runge-Kutta.
<H1>1. Introduction</H1>
The correct choice of the simulation method and its implementation is a very important issue in the design of an efficient cloth simulation system.
Among the available methods, there are finite elements methods
[ EIS 96 ], continuum mechanics
[ TER 87 ] or particle systems
[ BRE 94 ].
We will focus on the latter, which has shown to bring the best compromise between accuracy and speed for highly deformable objects such as cloth
[ VOL 95 ] [ VOL 97 ]. A particle system represents the mechanical system as a set of punctual masses. The cloth surface shape is represented by the geometry between neighboring particles. The mechanical behavior is represented as interaction forces between the particles, which depend on the relative position and speed of the particles, measuring deformation and deformation speed. Various models exist for this representation, which rank from the simple spring-mass representation (spring forces between particle couples depending on the distance between the particles) to accurate surface or volume models (involving complex interactions between several neighboring particles). The laws ruling these interactions also rank from linear to highly nonlinear involving discontinuities and hysteretic curves. The evolution of the system is computed numerically from these equations that form a large and sparse ordinary differential equation system, which, through adequate modeling, is also first-order. This numerical system has to be integrated numerically, for finally obtaining the evolution of the mechanical system along time, usually as a sequence of successive positions of the object along regular time intervals.
Various numerical methods related
Fax:
to integration of numerical ordinary first-order differential systems are available for this purpose
[ PRE 92 ].
The aim of this study is not to describe the implementation of these methods, which has already been carried out extensively in
[ EBE 96 ] [ VOL 97 ] [ BAR 98 ] [ VOL 00 ], and with some adaptations in
[ DES 99 ] [ EBE 00 ] [ KAN 00 ]. It rather intends to evaluate quantitatively the performance of the main integration methods in terms of speed and accuracy. Using a “typical” cloth object made of a common fabric material, we compare the computation speed and accuracy of each integration methods depending several simulation contexts, giving the reader an overview of the performance he can expect from each method.
<H1>2. Integration Methods</H1>
The choice of the adequate integration method has to be carried out using various considerations related to the kind of problem to be simulated. Among those considerations, there are: * The size of the problem, mostly related to the number of particles used to describe the mechanical system. * The desired accuracy, which reflects the allowable numerical tolerance between the computed solution and the theoretical evolution expected from the mechanical model. * The simulation context, which can either be an extensive computation of the motion along time requiring accurate evaluation of all the dynamical factors, or a simple relaxation process where the simulation has to converge to the static rest state as quickly as possible. * The stiffness of the problem, mainly related to the rigidity of the particle interactions and the size of the chosen time step, which translated into the “difficulty” the numerical method has to compute the evolution numerically, and which practically causes inaccuracy and instability problems. * The time an iteration takes to compute, and the number of mechanical derivations (computation of particle forces from their position and speed) the methods requires to compute one. The literature is abundant about various integration methods which aim to solve linear systems of first-order ordinary differential equations
[ PRE 92 ]. One can easily turn the second-order systems relating dynamical mechanical systems into first-order systems by constructing a state vector defined by the concatenation of position and speed states of the system, such as to fit the requirements of any of these algorithms. Among all the available methods, we can identify several main classes: * Explicit methods, which compute the state of the next time step out of a direct extrapolation of the previous states using derivative evaluations. * Implicit methods, which deduce the state of the next time step from an equation system expressing some “reversibility” from the extrapolated solution. * Low-order methods, which use a reduced number of evaluations for computing simple low-order extrapolations, leading to quickly computed, but inaccurate iterations. * High-order methods, which use several evaluations to compute high-order solutions that get much more accurate as the time step is reduced.
<H2>2.1. Scope of the Study</H2>
We shall restrict our consideration to three different methods which explore the range of these classes, and which seem to fit the best the requirements set for cloth simulation problems, in terms of implementation simplicity and efficiency for particle systems using large numbers of particles that interact sparsely and with a constant topology. The methods that we consider for this application are: * The explicit Midpoint method, which is a simple low-order explicit method. It requires two mechanical derivations per iteration and returns a second-order accurate solution relative to the time step. It also requires two storages of the state vector. We preferred this method to the still simpler first-order Euler method, because of the obvious gains of accuracy and stability which, despite the additional mechanical evaluation, makes it largely more efficient. We implemented this method for garment simulation in
[ VOL 95 ]. * The explicit Runge-Kutta method, implemented in its fifth-order version with error evaluation
[ PRE 92 ]. It requires five mechanical derivations per iteration, as well as five storages of the state vector. This method is supposed to provide high accuracy, which increases significantly as the time step is reduced. This method was experimented in [ EBE 96 ] and [ VOL 97 ]. * The Backward Euler method, which is the implicit implementation of its simple forward counterpart. It requires one mechanical evaluation and the resolution of a sparse linear system per iteration, as well as one storage of the system state additionally to those required for the system resolution algorithm. This method is supposed to provide approximate results that are not subject to numerical instability as the time step is increased. We implemented this method combined with a Conjugate Gradient algorithm using linear system matrix products computed on the fly, as described in
[ VOL 00 ], and thus able to take into account the anisotropy and nonlinearities of the mechanical model as the actual Hessian matric is used for each current state of the mechanical system. No initial matrix setup is required, suppressing also the need of separating linear and nonlinear components as discussed in [ EBE 00 ]. We have also carried out some preliminary tests with the Rosenbrook method, which is an implicit implementation of a fourth-order Runge-Kutta method. It is supposed to combine the stability of implicit methods with the accuracy of high-order methods. We implemented this method using the algorithm described in
[ PRE 92 ]
, but preliminary experiments have shown very deceptive results, and the gain of accuracy did not compensate the large calculations required for each iteration, whereas increased instability problems did not allow time steps much larger than those used for good accuracy with backward Euler. We did not consider in our tests the methods aimed toward simplifications which might highly approximate and degrade the dynamic behavior of deformable models, such as implicit integration with precomputed inverse matrices [ DES 99 ]
which involves high simplification and linrarization of the Hessian matrix and which also becomes very unpractical for large matrix sizes (the inverse of a sparse matrix is not necessarily sparse). We simulated such algorithm using accurate resolution on an accordingly approximated constant matrix, and we found that these approximations produced more simulation errors (on dynamic behavior of wrinkles and motion damping particularly) than producing a quick and rough linear system solution using a reduced number of Conjugate Gradient iterations with an accurate matrix. Even more drastic simplifications [ KAN 00 ] reduce the matrices to their diagonal component.
<H2>2.2. Implementation</H2>
All these methods were implemented in a single framework, which allows the simulation of cloth objects, using two different discrete mechanical representations: * A complete surface elasticity model which allows the simulation of anisotropic elasticity (weft and warp Young modulus, shearing, Poisson coefficient) and associated viscosity. Bending is also implemented, but not taken into account in this study. The base element of this simulation is a triangle of the mesh describing the surface, and the elasticity laws are computed as interactions between the three vertices of a triangle reflecting all the mechanical behavior curves which, for this study, are restricted to be linear. * A simplified spring-mass model which represents an approximated equivalent elasticity model using linear viscoelastic springs connecting the particle couples describing each edge of the surface mesh. This model is one of the simplest that a cloth simulation application would use. The implementation also supports collision detection and response, which were disabled for these tests. An object-oriented framework written in C++ integrate all these technologies into a single application allowing simulation of cloth objects of any shape with specified parameters. The application is run on a SGI Octane having a 200 MHz R100000 processor, and enough memory for working without swapping. Performance timings are done on the mechanical computation only, and do not take into account display and data structure management.
<H1>3. Performance</H1>
Performance is a key issue in choosing the adequate integration method, as cloth simulation usually involves very large mechanical systems described by a huge number of variables, and the numerical resolution of the system is therefore critical to the total computation time. Performance actually depends on several factors: * The computation time taken for one iteration of the algorithm. This depends on the complexity of the method, and also related to the number of times the forces of the system have to de derived from the system state using the laws of mechanics. * The time step for one iteration, which represents the time discretization required to reach a given accuracy or numerical stability for a given method. * The desired accuracy of the resolution, which may be coarse if only static equilibrium is to be computed, or high if the whole dynamic evolution of the system is desired. Accuracy increases along with time step reduction as better as the method is high-order. * The numerical stability of the method, which also limits the time step for a given method and a given mechanical system. These factors describe our investigation field in the following sections.
1 0 100 1000 10000 100000 1 Midpoint Runge Kutta Inv Euler (1) 0.1 Inv Euler (2) Inv Euler (4) Inv Euler (8) 0.01 0.001 0.0001 1 0 100 1000 10000 100000 1 0 Midpoint Runge Kutta 1 Inv Euler (1) Inv Euler (2) Inv Euler (4) Inv Euler (8) 0.1 0.01 0.001 Fig.1. Computation time per iteration for the various integration methods. on SGI Octane R10000 200MHz for simplified model (top) and accurate model.(botton) Time (vertical, seconds) per Polygons (horizontal, number).
<H2>3.1. Computation Time</H2>
The total computation time is the time required for computing one iteration times the number of iterations. Our first investigation is to evaluate the iteration computation time for each of these methods. For these measurements, we have simulated a square of fabric with a given discretization both with the accurate and simplified models, using the Midpoint, the RungeKutta and the Backward Euler methods, with 1, 2, 4, 8 iterations in the Conjugate Gradient algorithm for the latter, and measured computation time (Fig.1). From these tests, we can see that with our implementation, the computation times per mesh polygon are roughly the following:
Method Accurate Simplified Midpoint 32 μs 8 μs Runge-Kutta 95 μs 25 μs Back. Euler (No CG iter.) 31 μs 5 μs Back. Euler (Per CG iter.) 16 μs 2.5 μs
The most important fact to note is that the application of the Backward Euler method with a reduced number of Conjugate Gradient iterations compares very well with the traditional explicit methods. With one iteration only, it is barely worse than the very simple explicit Midpoint method. Our implementation, described in [ VOL 00 ] does not explicitly construct the matrix of the system to be resolved by the Conjugate Gradient, but computes “on the fly” the product of this matrix with vectors when needed by the Conjugate Gradient algorithm. This gives a very efficient implementation when using a low number of Conjugate Gradient iterations (no heavy preprocessing for building the matrix), which is often sufficient for most applications. These tests will help us to choose the method that gives the best compromise between accuracy and computation speed, as discussed in the next section.
<H2>3.2. Dynamic Accuracy</H2>
For measuring accuracy and numerical stability of the algorithms, we need to set up a “standard” material on which the experiments are carried out, as well as the rules allowing to extrapolate the results to any material of different size and parameters. In the scope of our study, we restrict the experimentation to linear metric elasticity of an isotropic cloth material, described by a Young modulus E and a surface density d. For the simulation, the surface square is discretized into elements which roughly have the length l, and the computation is carried out with time steps of size t. Defining the Condition Coefficient Thanks to the linearity of the equations describing linear elasticity, we reduce the number of parameters describing a problem using proportionality laws, and we compute a “condition coefficient” K which illustrates the acceleration of a mesh element with normalization to the problem mass, simulation time step and element size, as follows: K = E d 1 l 2 t 2 (1) This non-dimensional coefficient actually measures the “numerical difficulty” of computing an iteration of time step t with the given problem. We checked experimentally with our implementation that any scaling of a simulation along distance, time and mass which leaves K unchanged does not change anything to the simulation result. A typical cloth simulation problem could involve a cotton fabric cloth surface, which typically have a density d = 0.1 kg.m -2 and a Young modulus E = 2 0 N . m 1 . Given a discretization into elements averaging one centimeter and a simulation time step of ten milliseconds, the condition coefficient of the problem computed with (1) is K = 2 0 0 . It is possible to define similar coefficients related to bending and viscosity modulus. The corresponding K coefficients are respectively multiplied by additional l -2 and t factors. In simulations that consider simultaneously all these forms of mechanical behaviors, the dominant K coefficient rules the “numerical difficulty” of the problem. The Free Fall Test Our first test intends to measure the accuracy of the various methods in the context of accurate dynamic simulation. In such kind of simulation, the interest is to reproduce exactly the motion of a cloth object along time, the accuracy of its evolution being the key of the realism of an animation involving simulated cloth. In this experiment, we let a horizontal fabric square of 1.0 m side length, initially still, fall by its own weight, under a gravitation field of 1 0 m . s 2 acceleration (Fig.2). The discretization was set to roughly 400 elements ( l = 0 . 0 5 m ), and its Young modulus was altered in order to obtain various K values, with a simulation time step of t = 0.01 s. When using implicit methods, we perform a preconditioning of the system state variables of the linear system to be resolved using the inverse square root of the mass of the corresponding particle. This allows the iterations of the Conjugate Gradient algorithm to distribute the resolution numerical errors as evenly as possible between the particles, so that to obtain for instance a fall speed that does not depend on the mass of the particle. We measure the time it takes for this fabric piece to fall a height of 1 m . Without any additional external forces considered (no aerodynamic interactions), we expect this to happen in a constant time of 0.45 s. Fig.3. Fall time (vertical, seconds) with various K values (right) and different integration methods (left). Theoretical time: 0.45 s. Null values indicate numerical instability. Several interesting facts arise from this experiment. As a matter of numerical stability, the Midpoint method supports K values up to almost 3 whereas the RungeKutta method supports K values up to almost 100. This indicates that with Runge-Kutta, it is possible to use simulation time steps which are almost six times larger than with Midpoint. Given the fact that a Runge-Kutta iteration takes only three times more computation than a Midpoint iteration (Fig.1), the Runge-Kutta method seems to be computationally two times more efficient than the Midpoint method. As a matter of simulation accuracy, both Midpoint and Runge-Kutta seem to preserve accuracy correctly within their range of numerical stability. While the implicit Euler method seems stable for any K value, its accuracy is however very degraded by high K values and reduced numbers of Conjugate Gradient iterations. More precisely, we see that accuracy is well preserved with one Conjugate Gradient iteration up to a K value of 4, and increasing the iteration number n times also increases the K value n 2 times for the same accuracy. From this, we can see that the Inverse Euler method needs at least four Conjugate Gradient iterations to reach the accuracy of the Runge-Kutta method. We also see that similar requirement of accuracy bring the two methods in parity in terms of computation time (Fig.1). However, it should be noted that the experiment was carried out using a uniformly discretized mesh, and uniform mechanical parameters. Real-world simulations do not have this regularity, and numerical instability with explicit methods occur in the stiffest regions of the mesh, which, even if they are marginal in the whole mechanical system, may totally “explode” and destroy the simulation and therefore will rule the size of the largest time step possible. With implicit methods, the resulting inaccuracies may be unnoticed when taking a time step adapted to the average stiffness. Anyhow, this experiment shows clearly that when accurate reproduction of dynamic motion is required, it is not possible to increase the time step of implicit methods as much as desired, as this cause very noticeable inaccuracy as weak forces will be “neglected” relatively to stiff forces. While this is not an issue for draping problems where only the final state is desired, this aspect has to be taken into account when accurate reproduction of the whole evolution is wanted. While implicit Euler is a good choice for robust simulation where accuracy is not really an issue, the explicit Runge-Kutta offers good possibilities of ensuring high accuracy because of its high-order solution, and also because is provides good possibilities integration error evaluation for efficient time step control which, by the way, is too context-sensitive for being pre-evaluated using only the knowledge of K. Discretization and Computation Time The condition coefficient value is a good indicator of the time step and accuracy that can be expected for a given problem with a given iteration time step. Considering a simulation involving elements n times smaller, maintaining accuracy and stability (preserving K constant in formula (1)) would require a time step n times smaller, and therefore n times as many iterations for simulating the mechanical system along a constant duration. Given the fact that there are also n 2 times more elements to handle, the total computation time is finally multiplied by a drastic n 3 (even n 4 if curvature stiffness rule the simulation accuracy). While this factor is what cause explicit methods to become so inefficient with refined discretizations as this scaling has to be strictly observed for preventing instability, implicit methods are a bit more tolerant if only “visual” accuracy matters, accuracy which is not related to the size of the elements.
Fig.2. The free fall test: A horizontal fabric square, initially at rest, is dropped from 1 m height in a gravity field of 10 m.s -2 .
10 9 8 7 6 5 4 3 2 1 0 64 512 4096 (1) Euler Euler (2) (4) Euler Euler (8) (16) Euler Kutta Midpoint 1 8 Inv Inv Inv Inv Runge Inv
<H2>3.3. Draping Speed</H2>
Draping is another context of simulation, where only the final static equilibrium state of the mechanical system is to be computed. Here, the interest is to converge to the equilibrium state as quickly as possible, with minimum computation charge. As the full evolution of the cloth along time is not an interest, accuracy can be traded away for computation speed. From the dynamic study described above, implicit methods should be quite strong on this point, as they do not suffer from numerical instability, and allow large time steps to be used at the expense of dynamic accuracy which can here be neglected. The Draping Test For this test, we let an initially horizontal fabric square, attached along one of its edges, fall by its own weight (Fig.4). The fabric is a cotton sample of 1 m side length, with a Young modulus E = 2 0 N . m 1 and a density d = 0 . 1 k g . m 2 , discretized into 10000 polygons l = 0.01 m, and put in a gravitation field of 1 0 m s 2 . Without any damping, we expect that in its first oscillation, the fabric reach a roughly vertical position after slightly more than half a second. Our purpose is here to find the computation time necessary to obtain the fabric in its vertical position. For this, we count the number of computation iterations necessary for obtaining the fabric in its vertical position in its first oscillation, not being interested by the realism of this motion (Fig.5).
Fig.4. The draping test: An horizontal 1 m square of fabric fixed along a side falling in a gravitation field of 10 m s-2 . 700 600 500 400 300 200 100 0 Inv (1) Euler (2) Euler Inv Euler (4) Inv Inv Euler (8) (16) Euler Kutta Runge Midpoint 10 0.1 0.001 Inv
Fig.5. Number of iterations (vertical) required to get the fabric at vertical position, with various time steps (right, seconds), and integration methods (left). Null values indicate numerical instability. Our first finding is that the explicit methods seem quite not adapted for draping. Runge-Kutta requires more than 500 iterations for performing the simulation without instability, with the maximum allowed time step t = 0 . 0 0 1 s , which in fact corresponds to the maximum value of K coefficient experimented in the previous section. The backward Euler method is robust enough to handle the problem without instability for any time step. However, we see that larger time steps do not proportionally translate into fewer steps for performing the draping. As the time step becomes larger, and as the corresponding K coefficient exceeds the theoretical limit observed in the previous section, we quickly observe a “saturation” of the number of iterations to a constant which seems to be inversely proportional to the number of Conjugate Gradient iterations that were performed. From this it is clear that when K exceeds the dynamic accuracy limit of a given implicit integration method, the time step does not really reflect a time interval anymore. In such case, the implicit method will only evaluate an approximation of the rest state of the mechanical system by linear extrapolation from the Hessian matrix, whose accuracy depends on the number of Conjugate Gradient iterations that were used to resolve the corresponding linear system. Hence, there is no real way to “cheat” on the time step for speeding up draping, even if dynamic accuracy is not a concern: The total number of Conjugate Gradient iterations for performing all the simulation iterations of a draping problem cannot go below an incompressible number, related to a kind of “total computational difficulty” for solving a draping problem, which in the case of our experiment seems around 100. Still, this experiment shows the drastic advantage of using implicit methods for draping problems: With our implementation using the accurate elasticity model and the computation times measured in section 3.1, the draping could be computed in 30 seconds with Backward Euler with any large time step, compared to 150 seconds with Runge-Kutta when using an “optimal” time step.
<H2>3.4. Dealing with Nonlinear Models</H2>
Most mechanical simulations work with numerical equations that are not linear. There are two main reasons for such nonlinearity: * The equations describing the mechanical behavior laws are not linear. For instance, the strain-stress relation describing elasticity may actually be complex curves, which furthermore may take into account timedependent and hysteretic behaviors. * During the simulation, the orientation of the mechanical elements change, and this modifies the expressions of the mechanical laws in the world coordinates. While rarely causing numeric “explosions” as with explicit methods, nonlinearity may disrupt the stability of simulations integrated with implicit models with large disturbing vibrations, particularly when using large time steps that cause iterations to converge to the equilibrium state of the mechanical objects rather than simulating accurately their mechanical behavior. This can for instance be observed when simulating stretched flat surfaces without curvature forces. The reason for that is that the hypothetical equilibrium state is derived from the knowledge of the Hessian matrix, which relates the firstorder evolution of the forces as the deformations change. Nonlinearity causes this matrix to change between the successive iterations, and this evaluation to be inaccurate, despite high system resolution accuracy that can be reached with numerous Conjugate Gradient iterations. The solution for this is to approximate the Hessian matrix for taking into account the changes that may be observed from the change of the system state between successive iterations. While an underestimation of de derivatives may lead to an equilibrium state valuation too far from the current state, and by this cause instability, an overestimation of the derivatives will place this evaluation nearer to the current state, therefore stabilizing the simulation, at the expense of extra numerical damping and slow convergence. This is particularly true for drastic linearisations as for example used in [ DES 99 ]. Knowledge of the expected state changes between successive time steps are required to perform this approximation correctly. With nonlinear mechanical behavior, one solution is to take the steepest parts of the curves as derivatives, whereas for the element orientation problem, isotropic derivatives considering force evolution equally in any directions may be considered. However, the more drastic these approximations are, the less accurate the simulation will be for dynamic simulations, and the slower the simulation will converge for draping problems. A nice solution described in [ EBE 00 ], which makes sense when efficiency relies on the use of a constant Hessian matrix, is to perform the implicit resolution on a linear constant approximation, and to simulate the nonlinear and variable component, unlikely to cause stiffness problems, using an explicit method.
<H2>3.5. Real Case Simulation</H2>
In order to test the efficiency of our model in the context of garment animation, the algorithms have been integrated in a 3D design framework allowing the management of complex garment objects in interaction with animated virtual characters. This integration has been carried out in the form of a 3DStudio Max plugin (Fig.6), running on a 500 MHz PentiumIII PC. We have simulated a 2000 Polygon garment made of the cotton material described in Section 3. The mesh elements are roughly five centimeters in size, and therefore the resulting condition coefficient K is roughly 8 with a simulation time step of 10 milliseconds. The cloth simulation process has two stages: * The garment assembly stage, where the patterns are pulled together and seamed around the body. This is a draping problem involving to obtain a rest position of the garment as quickly as possible. * The garment animation stage, where the motion of the garment is computed as the body is animated. The dynamical motion of the cloth is important here. The garment assembly and seaming operations could be performed almost four times faster with the Backward Euler (2 minutes) than with Runge-Kutta (8 minutes), knowing that collision detection and response account for more than the half of the computation time, and actually limits the time step size when contact starts between the cloth and the body. For the dynamical animation, comparable accuracy could be obtained between Runge-Kutta and Backward Euler using eight iterations of the Conjugate Gradient, which gave similar computation times. The backward Euler method however allowed the increase of the time step up to 0.1 seconds, where inaccuracies began to show up: Mostly, excessive “heaviness” of the fabric that failed to follow the motion of the body properly, garments folds that would not disappear quickly, as well as additional artifacts caused by inaccurate collision response.
Fig.6. Garment simulation in the 3DStudio Max environment.
<H1>4. Conclusion</H1>
Recent literature has emphasized on the relevance of implicit methods for cloth simulation. The implicit Euler method seems effectively a good candidate for most situations involving cloth simulation, because of the robustness resulting from not being prone to numerical instability. This is particularly true when simulating very heterogeneous mechanical systems (elements of various sizes and various mechanical properties) where, using explicit models, the most critical elements would rule the time step size for all the simulation. Contrary the perception of the implicit model iteration being slow because of the linear system resolution it involves, the inverse Euler iteration often proves to be faster than the explicit Runge-Kutta method of higher order, if an adequate approximate linear system resolution is implemented. A limited number of Conjugate Gradient iterations seems suitable for this. Furthermore, while increasing the time step seems not limited by instability with implicit methods, it should be kept in mind that this is still done at the expense of accuracy of the whole simulation. The number of iterations should also be set sensitively to the stiffness of the mechanical problem, for limiting the potential inaccuracies that become particularly visible when an accurate simulation of a dynamical system is wanted. There is an obvious advantage of using implicit methods, and particularly the inverse Euler method, for draping problems where quick convergence to a rest position is required quickly. Our test have shown that the inverse Euler method allow to perform a draping problem almost ten times as fast as with the Runge-Kutta method. While not exactly reproducing real mechanical behavior, the simulation with large time steps provides a quite efficient convergence to equilibrium, and the numerical errors quite often act as extra damping, removing the need of adding them explicitly to the model. For dynamic problems where accurate evolution of the mechanical system along time is needed, the advantage of implicit methods is less obvious. Their stability gives a false sense of efficiency, allowing obtaining quickly a result by “cheating” on the time step size. However, playing back the generated animation, artifacts quickly show up: Excessive damping, wrinkles and folds that fail to disappear, and even objects failing to fall correctly by their own weight. These artifacts are still augmented by the approximations made to the Hessian matrix, possibly in the purpose of reducing instability, while excessive reduction of the Conjugate Gradient iterations produce additional inaccuracy and slow convergence. It seems that there is still some benefit in using the Backward Euler method than any other explicit method for dynamic simulations thanks to the reduced time it takes to compute one iteration, which also only requires one derivation of the particle forces from the state of the system. Our tests have shown a roughly doubled speed for the accuracy corresponding to the limit of stability of the Runge-Kutta method. We got substantial improvements through the implementation of the implicit Midpoint method
[ VOL 00 ]
, which however had the drawback of increasing the numerical instability problem, forcing additional use of isotropic force gradients, at the expense of accuracy. The explicit methods have still their interest, and should be reserved for simulations requiring high accuracy and particularly those where involving low mechanical damping and where mechanical energy conservation is important. Instability concerns will force parameters and time step size to ensure good accuracy for the simulation of all particles of the discrete mechanical representation, and therefore for the entire mechanical object. This may however require prohibitive computation times for very stiff and discretized models. The 5th-order Runge-Kutta method das proven to be a good solution [ EBE 96 ] [ VOL 97 ]
, because of its high accuracy, and because it furthermore provides integration error evaluation, which is a very good hint to the very sensitive problem of optimal time step size determination. The simpler Midpoint method may have some interest only in very particular cases involving very loose materials with rough discretization, or when numerous fast iterations with small time steps are required for other reasons (high motion sampling, collision detection, very discontinuous models). All these considerations should be carefully taken into account when designing a mechanical simulation engine, as they are the keys to efficient simulation, and therefore complex models that, for garment simulation, express fully visual experience of real fashion models. We intend to pursue our investigations for dealing with damping in a more accurate way. This still remains an important issue to dynamic realism of cloth simulation models, which has to take into account viscosity, the dissipative effect of hysteretic behavior, as well as collision damping and friction. The integration methods have to be tuned to take precisely these effects into account.
<H1>Bibliography</H1>
[BAR 98] : D. Baraff, A. W i t k i n , "Large Steps i n Cloth Simulation", Computer Graphics (SIGGRAPH’98 proceedings), Addison-Wesley, 32, pp 106-117, 1998.
[BRE 94] : D.E. Breen, D.H. House, M.J. W o z n y , "Predicting the Drape of Woven Cloth Using Interacting Particles", Computer Graphics (SIGGRAPH’94 proceedings), Addison-Wesley, pp 365-372, July 1994.
[DES 99] : M. Desbrun, P.Schröder, A. Barr, "Interactive Animation of Structured Deformable Objects", Proceedings of Graphics Interface, 1999.
[EBE 96] : B. Eberhardt, A. Weber, W. Strasser, "A Fast, Flexible, Particle-System Model for Cloth Draping", Computer Graphics in Textiles and Apparel (IEEE Computer Graphics and Applications), pp 52-59, Sept. 1996.
[EBE 00] : B. Eberhardt, O. Etzmuss, M. Hauth, "Implicit-Explicit Schemes for Fast Animation with Particles Systems", Proceedings of the Eurographics workshop on Computer Animation and Simulation, pp 137-151, 2000.
[EIS 96] : J.W. Eischen, S. Deng, T.G. C l a p p , "Finite-Element Modeling and Control o f Flexible Fabric Parts", Computer Graphics i n Textiles and Apparel (IEEE Computer Graphics and Applications), pp 71-80, Sept. 1996.
[KAN 00] : Y.M. Kang, J.H. Choi, H.G. C h o , D.H. Lee, C.J. Park, "Real-Time Animation Technique for Flexible and Thin Objects", WSCG proceedings, pp 322-329, 2000.
[PRE 92] : W.H. Press, W.T. Vetterling, S . A . Teukolsky, B.P. Flannery, "Numerical Recipes in C", Second edition, Cambridge University Press, 1992.
[TER 87] : D. Terzopoulos, J.C. Platt, H. Barr, "Elastically Deformable Models", Computer Graphics (SIGGRAPH’97 proceedings), Addison-Wesley, 21, pp 205-214, 1987.
[VOL 95] : P. Volino, M. Courchesne, N . Magnenat-Thalmann, "Versatile and Efficient Techniques for Simulating Cloth and Other Deformable Objects", Computer Graphics (SIGGRAPH’95 proceedings), Addison-Wesley, pp 137-144, 1995.
[VOL 97] : P. Volino, N. Magnenat-Thalmann, "Developing Simulation Techniques for a n Interactive Clothing System", Virtual Systems and Multimedia (VSMM’97 proceedings), Geneva, Switzerland, pp 109-118, 1997.
[VOL 00] : P. Volino, N. Magnenat-Thalmann, "Implementing fast Cloth Simulation with Collision Response", Computer Graphics International 2000, pp 257-266, 2000.
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"text": "['increased instability problems did not allow time steps much larger than those used for good accuracy with backward Euler']",
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"text": "['We did not consider in our tests the methods aimed toward simplifications which might highly approximate and degrade the dynamic behavior of deformable models']",
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"text": "['We simulated such algorithm using accurate resolution on an accordingly approximated constant matrix']",
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"text": "['these approximations produced more simulation errors']",
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"text": "['Performance is a key issue in choosing the adequate integration method']",
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"text": "['This depends on the complexity of the method, and also related to the number of times the forces of the system have to de derived from the system state using the laws of mechanics']",
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"text": "['Accuracy increases along with time step reduction as better as the method is high-order']",
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"text": "['The numerical stability of the method, which also limits the time step for a given method and a given mechanical system']",
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"text": "['The most important fact to note is that the application of the Backward Euler method with a reduced number of Conjugate Gradient iterations compares very well with the traditional explicit methods']",
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"text": "['Method Accurate Simplified Midpoint 32 μs 8 μs Runge-Kutta 95 μs 25 μs Back. Euler (No CG iter.) 31 μs 5 μs Back. Euler (Per CG iter.) 16 μs 2.5 μs']",
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"text": "['With one iteration only, it is barely worse than the very simple explicit Midpoint method']",
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"text": "['does not explicitly construct the matrix of the system to be resolved by the Conjugate Gradient, but computes “on the fly” the product of this matrix with vectors when needed by the Conjugate Gradient algorithm']",
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"text": "['Our implementation']",
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"text": "['VOL 00']",
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"text": "['This gives a very efficient implementation when using a low number of Conjugate Gradient iterations']",
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"text": "['which is often sufficient for most applications']",
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"id": "T108"
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"text": "['These tests will help us to choose the method that gives the best compromise between accuracy and computation speed']",
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"text": "['For measuring accuracy and numerical stability of the algorithms, we need to set up a “standard” material on which the experiments are carried out, as well as the rules allowing to extrapolate the results to any material of different size and parameters']",
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"text": "['Thanks to the linearity of the equations describing linear elasticity, we reduce the number of parameters describing a problem using proportionality laws, and we compute a “condition coefficient” K which illustrates the acceleration of a mesh element with normalization to the problem mass, simulation time step and element size, as follows: K = E d 1 l 2 t 2']",
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"id": "T111"
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"text": "['This non-dimensional coefficient actually measures the “numerical difficulty” of computing an iteration of time step t with the given problem']",
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"text": "['A typical cloth simulation problem could involve a cotton fabric cloth surface, which typically have a density d = 0.1 kg.m -2 and a Young modulus E = 2 0 N . m']",
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"text": "['the condition coefficient of the problem computed with (1) is K = 2 0 0']",
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"text": "['a discretization into elements averaging one centimeter and a simulation time step of ten milliseconds']",
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"text": "['It is possible to define similar coefficients related to bending and viscosity modulus']",
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"text": "['In simulations that consider simultaneously all these forms of mechanical behaviors, the dominant K coefficient rules the “numerical difficulty” of the problem']",
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"text": "['In such kind of simulation, the interest is to reproduce exactly the motion of a cloth object along time, the accuracy of its evolution being the key of the realism of an animation involving simulated cloth']",
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"text": "['we perform a preconditioning of the system state variables of the linear system to be resolved using the inverse square root of the mass of the corresponding particle']",
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"text": "['we expect this to happen in a constant time of 0.45 s']",
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"text": "['Fig.1']",
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"text": "['the Runge-Kutta method seems to be computationally two times more efficient than the Midpoint method']",
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"text": "['Real-world simulations do not have this regularity, and numerical instability with explicit methods occur in the stiffest regions of the mesh, which, even if they are marginal in the whole mechanical system, may totally “explode” and destroy the simulation and therefore will rule the size of the largest time step possible']",
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"text": "['is too context-sensitive for being pre-evaluated using only the knowledge of K']",
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"text": "['The condition coefficient value is a good indicator of the time step and accuracy that can be expected for a given problem with a given iteration time step']",
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"text": "['would require a time step n times smaller, and therefore n times as many iterations for simulating the mechanical system along a constant duration']",
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"text": "['only “visual” accuracy matters']",
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"text": "['the full evolution of the cloth along time is not an interest']",
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"text": "['they do not suffer from numerical instability, and allow large time steps to be used at the expense of dynamic accuracy']",
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"text": "['Runge-Kutta requires more than 500 iterations for performing the simulation without instability, with the maximum allowed time step t = 0 . 0 0 1 s , which in fact corresponds to the maximum value of K coefficient experimented in the previous section']",
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"text": "['The backward Euler method is robust enough to handle the problem without instability for any time step']",
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"text": "['larger time steps do not proportionally translate into fewer steps for performing the draping']",
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"text": "['we quickly observe a “saturation” of the number of iterations to a constant which seems to be inversely proportional to the number of Conjugate Gradient iterations that were performed']",
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"text": "['the time step becomes larger']",
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"text": "['the corresponding K coefficient exceeds the theoretical limit observed in the previous section']",
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"text": "['The total number of Conjugate Gradient iterations for performing all the simulation iterations of a draping problem cannot go below an incompressible number, related to a kind of “total computational difficulty” for solving a draping problem']",
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"text": "['With our implementation using the accurate elasticity model and the computation times measured in section 3.1, the draping could be computed in 30 seconds with Backward Euler with any large time step, compared to 150 seconds with Runge-Kutta when using an “optimal” time step']",
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"text": "['the strain-stress relation describing elasticity may actually be complex curves, which furthermore may take into account timedependent and hysteretic behaviors']",
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"text": "['During the simulation, the orientation of the mechanical elements change, and this modifies the expressions of the mechanical laws in the world coordinates']",
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"text": "['While rarely causing numeric “explosions” as with explicit methods, nonlinearity may disrupt the stability of simulations integrated with implicit models with large disturbing vibrations']",
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"text": "['particularly when using large time steps that cause iterations to converge to the equilibrium state of the mechanical objects rather than simulating accurately their mechanical behavior']",
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"text": "['This can']",
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"text": "['Nonlinearity causes this matrix to change between the successive iterations, and this evaluation to be inaccurate']",
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"text": "['despite high system resolution accuracy that can be reached with numerous Conjugate Gradient iterations']",
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"text": "['The solution for this is to approximate the Hessian matrix for taking into account the changes that may be observed from the change of the system state between successive iterations']",
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28851,
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"text": "['an underestimation of de derivatives may lead to an equilibrium state valuation too far from the current state, and by this cause instability']",
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29189,
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29267,
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"text": "['DES 99']",
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"text": "['Knowledge of the expected state changes between successive time steps are required to perform this approximation correctly']",
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29437,
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"text": "['one solution is to take the steepest parts of the curves as derivatives']",
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29406,
29435
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"text": "['the element orientation problem']",
"type": "data",
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29555,
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"text": "['isotropic derivatives considering force evolution equally in any directions may be considered']",
"type": "background_claim",
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"text": "['the more drastic these approximations are, the less accurate the simulation will be for dynamic simulations, and the slower the simulation will converge for draping problems']",
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"text": "['which makes sense when efficiency relies on the use of a constant Hessian matrix, is to perform the implicit resolution on a linear constant approximation, and to simulate the nonlinear and variable component, unlikely to cause stiffness problems, using an explicit method']",
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29865,
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"text": "['EBE 00']",
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"text": "['A nice solution described in']",
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"text": "['the resulting condition coefficient K is roughly 8 with a simulation time step of 10 milliseconds']",
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"text": "['2 minutes']",
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"text": "['The number of iterations should also be set sensitively to the stiffness of the mechanical problem, for limiting the potential inaccuracies that become particularly visible when an accurate simulation of a dynamical system is wanted']",
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"text": "['Our test have shown that the inverse Euler method allow to perform a draping problem almost ten times as fast as with the Runge-Kutta method']",
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"text": "['While not exactly reproducing real mechanical behavior, the simulation with large time steps provides a quite efficient convergence to equilibrium']",
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"text": "['they are the keys to efficient simulation, and therefore complex models that, for garment simulation, express fully visual experience of real fashion models']",
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<Title>Fluid Simulation using Laplacian Eigenfunctions</Title>
TYLER DE WITT
CHRISTIAN LESSIG
EUGENE FIUME University of Toronto
Graphics]:
<Abstract>We present an algorithm for the simulation of incompressible fluid phenomena that is computationally efficient and leads to visually convincing simulations with far fewer degrees of freedom than existing approaches. Rather than using an Eulerian grid or Lagrangian elements, we represent vorticity and velocity using a basis of global functions defined over the entire simulation domain. We show that choosing Laplacian eigenfunctions for this basis provides benefits, including correspondence with spatial scales of vorticity and precise energy control at each scale. We perform Galerkin projection of the Navier-Stokes equations to derive a time evolution equation in the space of basis coefficients. Our method admits closed form solutions on simple domains but can also be implemented efficiently on arbitrary meshes.</Abstract>
Categories and Subject Descriptors: I.3.7 [Computer ThreeDimensional Graphics and Realism—Animation General Terms: Animation, Fluid Simulation Additional Key Words and Phrases: Laplacian eigenfunctions, physically based animation, fluid, smoke, geometric mechanics
ACM
Reference Format:
de Witt, T., Lessig, C., and Fiume, E. 2011. Fluid Simulation using Laplacian Eigenfunctions. ACM Trans. Graph.
<H1>1. INTRODUCTION</H1>
Fluid motion is naturally captivating. Over the years, it has piqued the imagination and curiosity of artists, mathematicians and scientists. The fascination with fluid motion is exemplified by its long history in the computer graphics literature. Early work focused on obtaining motion that is visually interesting and convincing. More recent physically based techniques rely primarily on numerical approximation of the Navier-Stokes equations. Computer simulation of a model necessitates a finite representation of its spatial quantities. In the past, many approaches for choosing a finite representation have been proposed including the use of Eulerian grids, La-
Tyler de Witt acknowledges a NSERC-Canada grant. [email protected] , [email protected] , [email protected] Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or [email protected] . c YYYY ACM 0730-0301/YYYY/11-ARTXXX $10.00 DOI 10.1145/XXXXXXX.YYYYYYY http://doi.acm.org/10.1145/XXXXXXX.YYYYYYY
Fig. 1.
Examples of Laplacian eigenfunction velocity basis fields on various domains. They are stationary solutions to the Navier-Stokes equations. For simple domains Laplacian eigenfunctions have closed form expressions; for general meshes they are defined through eigendecomposition of a discrete operator.
grangian fluid mass particles, vorticity primitives and model reduction. Grid-based techniques are the most common approach. However, they suffer from high computational complexity, due to the general requirement at each simulation step to solve a system of equations whose size is proportional to the number of grid elements in the domain. Lagrangian techniques, such as mass particles, removes the dependence on the simulation domain. That said, the computation of pressure and other fluid quantities are expensive and approximations lead to noticeable violations of incompressibility. Vorticity primitives, including particles and filaments, are very effective at simulating smoke in inviscid media but have difficulties modelling diffusion and handling boundary conditions. Model reduction is a data-driven approach that exploits a precomputed set of example simulations to obtain a low dimensional representation for fluid motion. While this technique is very efficient at run-time, it suffers from significant costs for precomputation and storage, and is dependent on the performance of an existing simulator or other mechanism to obtain ground-truth data. We propose an algorithm for the interactive simulation of fluid motion that avoids many of the shortcomings of existing techniques. We employ a representation of fluid velocity and vorticity in a finite dimensional basis of Laplacian eigenfunctions. The resulting velocity basis fields are divergence free and respect boundary conditions, so that these constraints are enforced automatically without the need for additional computation. Our algorithm can be formulated as Galerkin projection of the vorticity form of the Navier-Stokes equations onto Laplacian eigenfunctions defined over the simulation domain. The resulting finite dimensional form of the equations describes the time evolution of the basis coefficients. We precompute the non-linear advection terms between pairs of basis functions and store the result as structure coefficients in a set of matrices. Viscosity and external forces are incorporated using linear terms, and the basis function coefficients are hence updated using a simple matrix-vector equation. Laplacian eigenfunctions form an orthogonal basis, allowing one to easily compute the energy of the fluid. Additionally, Laplacian eigenfunctions of increasing eigenvalue magnitude have a natural visual correspondence with decreasing scales of vorticity. Coupled with orthogonality, the correspondence allows precise control of a fluid’s turbulent spectrum through adjustment of basis coefficients. With Laplacian eigenfunctions the viscosity can be simulated accurately through a simple exponential decay of basis coefficients, but also allows arbitrary user-controlled or automatic adjustment of the spectrum to achieve a desired effect. For some practically important simulation domains such as a 2-D plane and 3-D rectangular cavity, Laplacian eigenfunctions have closed form expressions, allowing fully analytic simulation. In these cases, no mesh is required to store the fluid’s velocity. Instead, a velocity can be precisely evaluated at any spatial coordinate without the need for interpolation. Furthermore, closed form expressions allow symbolic evaluation of the precomputed advection operator, making this process fast and exact. However, our method is not limited to these domains, and we present a formulation on structured and irregular meshes using discrete exterior calculus, in which velocity and vorticity basis fields are eigenvectors of a discrete Laplacian operator. Additionally, our method supports the interaction of immersed moving obstacles and buoyancy through projection of forces to the velocity basis fields. Our method allows considerable flexibility in choosing the basis dimensionality. Even simulations with few degrees of freedom provide visually convincing results, avoiding the artifacts common to very low-dimensional representations in Eulerian or Lagrangian simulations. In this respect, our method provides a principled means of dimensionality reduction of the Navier-Stokes fluid equations. However, our method is not data-driven as seen in current model reduction techniques and hence avoids the need for an existing fluid simulator or pre-existing data. We believe our algorithm and choice of basis provides an exciting avenue and will be an important complement to the methods in the literature.
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Fig. 2. A buoyant smoke simulation using a basis of 128 Laplacian eigenfunctions.
<H1>2. RELATED WORK</H1>
Incompressible fluid dynamics is a vast subject. We survey some relevant work from geometric mechanics, computational fluid dynamics (CFD) and the computer graphics literature.
<H2>2.1 Geometric Mechanics</H2>
Euler’s equations describing the dynamics of a rotating rigid body date from the 18th century. In 1901,
Poincaré [1901]
showed that by considering various group manifolds as the configuration space, Euler’s equation could apply generally to a class of physical systems. For example, in the case of a rotating rigid body the group is the rotation group SO(3). Arnold [1966]
showed that an ideal incompressible fluid is described similarly as geodesic motion on SDiff, the Lie group of volume preserving diffeomorphisms. The notion of structure coefficients to describe the interaction of Lie algebra basis elements of these groups is directly related to the precomputed coefficient matrices used in our method. Many of these concepts are summarized by
Marsden and Ratiu [1999]. Representing vorticity using Laplacian eigenfunctions dates back at least to
Yudovich [1963]
, who used this method to prove existence and uniqueness theorems for the two dimensional NavierStokes equations. More recently,
Agrachev et al. [2005]
used vorticity Laplacian eigenfunctions to prove theorems in the mathematical control literature. This paper was our inspiration for investigating a Laplacian eigenfunction representation of vorticity as a practical means of fluid simulation in computer graphics applications.
<H2>2.2 Computational Fluid Dynamics</H2>
In the 1950’s, Silberman presented a fluid simulation algorithm for the earth’s atmosphere in a basis of spherical harmonics, which are Laplacian eigenfunctions on the surface of a sphere
[Silberman 1954]. This basis was applied to the vorticity stream function fluid equations in two dimensions, and the advection operator was evaluated symbolically. This method has come to be known in the CFD literature as the interaction coefficient method. Outside of atmospheric sciences, it is not widely used due to poor scaling for large basis dimensionality. Such performance considerations were the motivation for the development of spectral methods as pioneered by Orszag
[Orszag 1969]. Spectral methods are characterized by the use of a fast transform allowing efficient calculation of advection in the spatial domain, thereby avoiding convolution sums in the spectral domain. They are often used to study homogeneous turbulence
[Orszag and Patterson 1972; Rogallo et al. 1981]. Fourier series or Chebyshev polynomials are commonly employed, as spectral methods are limited to bases admitting a fast transform. Our method is most analogous to the interaction coefficient method of
[Silberman 1954]
, although we consider arbitrary domains. On arbitrary domains, Laplacian eigenfunctions do not in general admit a fast transform and hence do not share the inherent theoretical performance of a spectral method. However, Laplacian eigenfunctions have many other benefits as we describe in Section 3. Furthermore, theoretical performance scaling is less critical for the applications we consider and we show that visually detailed simulations are attainable at low cost. Divergence free finite element methods (DFFEM) employ bases of discrete divergence free velocity fields to solve fluid equations in a space that satisfies mass continuity a priori
[Gustafson and Hartman 1983]. Our method is similar in this respect. However, in contrast to DFFEM, for some simple domains Laplacian eigenfunctions do not require a discrete mesh. Also, to our knowledge no basis employed in DFFEM exhibits all of the advantageous proper- ties of Laplacian eigenfunctions, including orthogonality, stationarity with respect to Navier-Stokes equations, global support, and correspondence with spatial scales of vorticity.
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<H2>2.3 Computer Graphics</H2>
Fluid simulation methods in the computer graphics literature belong to roughly four categories: grid-based, mass particles, vortex elements and model reduction. Grid-based techniques for simulating the 3-D Navier-Stokes equations were introduced by
Foster and Metaxas [1996]
but were unstable due to the use of explicit integrators. Stam developed an unconditionally stable integration scheme using semi-Lagrangian advection and an implicit integrator
[Stam 1999]. However, the result produces artificial viscosity which dampens vortices prematurely, and requires an iterative linear solver to solve for a pressure field to enforce incompressibility.
Works aimed at mitigating or minimizing artificial diffusion include vorticity confinement
[Fedkiw et al. 2001] and high order advection schemes
[Selle et al. 2008].
To improve the performance of the iterative pressure solver, use of adaptive grids
[Losasso et al. 2004] and hierarchical coarse grids for projection
[Lentine et al. 2010]
have been proposed. Stam [2002]
used the 2-D Fourier transform of a velocity field to perform fast pressure projection, but this method is limited to simple domains and boundary conditions, and still dissipates energy. Bridson presented a simple means to generate procedural divergence free flows through the curl of a vector potential stream function
[Bridson et al. 2007]
but this work did not address physical dynamics. Elcott presented a method that preserves circulation on simplicial meshes, but does not preserve energy
[Elcott et al. 2007]. Mullen et al. developed a fluid integrator capable of perfect energy preservation or desired viscosity independent of grid-resolution
[Mullen et al. 2009]
, but this method is complex and requires a solution to a non-linear system at each timestep. Hybrid particle-grid methods such as FLIP
[Zhu and Bridson 2005]
are effective in eliminating numerical diffusion, but still require a grid to enforce incompressibility. Common to all these stable grid-based techniques previously mentioned is the need to solve a system of equations at each time integration step, the size of which is proportional to the number of grid elements. In contrast, the performance of our method is independent of the domain or grid resolution. In fact, for typical domains such as a 2-D rectangle or 3-D rectangular cavity, the global basis functions we employ have closed form expressions, removing the need for a velocity grid representation entirely. Our method allows controllable viscosity, and supports general domains through a formulation on discrete meshes. Particle methods track a fluid’s mass through Lagrangian elements.
Smoothed particle hydrodynamics (SPH) was introduced to graphics by
Desbrun and Gascuel [1996] and used subsequently to simulate water
[Müller et al. 2003; Adams et al. 2007]. Enforcing incompressibility in SPH methods is computationally expensive, making them impractical for a large number of particles. Our method satisfies incompressibility automatically as it operates directly in a space of divergence free fields. Vortex methods use Lagrangian elements such as particles or filaments to track vorticity, and advect these elements through the fluid’s velocity
[Gamito et al. 1995; Park and Kim 2005; Angelidis et al. 2006; Weißmann and Pinkall 2010]. A formulation using vorticity guarantees incompressibility, but the reconstruction of the velocity field is computationally expensive, typically involving the Biot-Savart formula. We also use a vorticity formulation, hence requiring no explicit enforcement of the incompressibility constraint. However, we use a superposition of global basis functions allowing the representation of arbitrary vorticity fields, whereas Lagrangian elements are limited to vorticity concentrated at points or on curves. Additionally, choosing Laplacian eigenfunctions as a basis allows the velocity field to be recovered trivially, removing the need for complicated and expensive reconstruction. Gupta and Narasimhan represented fluid velocity in a basis of Legendre polynomials allowing analytic evaluation of differential operators
[Gupta and Narasimhan 2007]. However, only boxboundary conditions were considered and the velocity basis fields are not strictly divergence free hence requiring a pressure projection step to enforce incompressibility. Model reduction has been applied to fluid simulation by
Treuille et al. [2006]. This technique chooses a reduced velocity basis defined on a mesh through observation of an existing fluid simulator. The resulting run-time performance is fast, but the precomputation time and memory requirements are large. Furthermore, it is unclear how well this technique generalizes to arbitrary flows, as behavior is limited to the examples present in training. Our method can be used directly as a means of dimensionality reduction through choice of the basis dimension N , but it differs from current approaches in many respects. We choose an appropriate velocity basis a priori instead of relying on observation of an existing fluid simulator. Up to a desired scale of vorticity, Laplacian eigenfunctions form a complete basis for divergence free fields. Adding basis functions increases the coverage in a well defined way. In contrast, a data driven basis can only approximate flows that are in some sense “close” to those observed in training, and there is no guarantee that additional training data will substantially increase the span of the resulting PCA basis. Our basis has a natural correspondence with spatial scales of vorticity that is lacking in
[Treuille et al. 2006]. Finally, Laplacian eigenfunctions have closed form expressions for some simple domains, in which case the precomputation time and memory requirements are vastly reduced in comparison.
Symbol Description N Basis dimension. {Φ k } Set of velocity basis fields. {φ k } Set of vorticity basis fields. u Fluid velocity vector field. ω Fluid vorticity vector field. ω 1 , ω 2 , . . . , ω N Basis coefficients. w = [ω 1 ω 2 . . . ω N ] T Column vector of basis coefficients. ∆ Vector Laplacian operator. λ k Scalar eigenvalue of k th basis field. Adv(·, ·) Advection operator. {C k } Structure coefficient matrices. C k [i, j] (i, j) th entry of the C k matrix. a x , a y , a z Canonical basis for R 3 .
Fig. 3. Nomenclature.
<H1>3. LAPLACIAN EIGENFUNCTIONS AS BASIS FIELDS</H1>
We express the velocity field of a fluid u on a domain D as
N u = ω i Φ i . i where Φ k are eigenfunctions of the vector Laplacian ∆ = grad (div ) − curl 2 . When acting on divergence free fields, the vector Laplacian reduces to ∆ = −curl 2 . We require the set of basis
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Fig. 4.
Laplacian eigenfunctions have a correspondence with spatial scales of vorticity, allowing basis coefficients to be interpreted as a discrete spectrum of vorticity. The right of the spectrum corresponds to larger magnitudes of eigenvalues and smaller scale vortices.
fields Φ k to be divergence free and satisfy a free slip condition at the boundary. Hence our basis fields are completely characterized by ∆Φ k = λ k Φ k
1
div (Φ k ) = 0 Φ k · n = 0 at ∂D
where λ k are eigenvalues and n is a vector normal to the boundary. The eigenfunctions of the Laplacian operator ∆ are domain dependent. For many simple domains, functions satisfying Eq. 1 have closed form expressions, which are available for example in the physics literature where they describe the magnetic fields of electromagnetic resonators
[Cheng 1999]. For instance, on a π × π square domain, Laplacian eigenfunctions satisfying Eq. 1 have the closed form expressions 1 Φ k = k 1 2 + k 2 2 (k 2 sin(k 1 x) cos(k 2 y)a x
2
−k
2
1 cos(k 1 x) sin(k 2 y)a y ) ,
where k = (k 1 , k 2 ) ∈ Z 2 is a tuple of integers known as the vector wave number. The vector fields Φ k are Laplacian eigenfunctions with eigenvalues λ k = −(k 1 2 + k 2 2 ). Examples are plotted in Figure 4 . We will continue to use the square domain as a concrete, illustrative example throughout the text, although closed form expressions also exist for many other domains including a 3-D rectangular prism
[de Witt 2010], a disc, the surface of a sphere, or a planar region with a wrap around boundary condition. For our simulation method, we also require the vorticity field ω = curl (u) and a vorticity basis {φ k } with φ k = curl Φ k . For example, the vorticity basis fields associated with Eq. 2 are just the curl of the velocity basis functions and given by
3
φ k = sin(k 1 x) sin(k 2 y)a z .
One can verify that the φ k are also Laplacian eigenfunctions of the domain. However, as u and ω are orthogonal, the vorticity basis functions have only a normal component at the boundary, and hence satisfy ∆φ k = λ k φ k
4
φ k × n = 0 at ∂D.
<H2>3.1 Basis Field Properties</H2>
We summarize some additional interesting and useful properties of our basis. One can verify that the example expressions of Eqs. 2 and 3 satisfy all the properties listed below. Velocity-Vorticity Duality. In general, reconstructing a velocity field from a vorticity field is computationally expensive, typically involving the use of the Biot-Savart Law
[Angelidis et al. 2006; Weißmann and Pinkall 2010]. The key benefit of a representation in
Laplacian eigenfunctions is that the inverse operator curl −1 applied to vorticity basis functions yields a simple expression: Φ k = curl −1 φ k = curl −1 1 ∆φ k λ k = 1 curl −1 −curl 2 φ k λ k
5
1 = − curl φ k . λ k
A further important observation is that due to linearity of the curl operator, the expansion of the vorticity ω in the φ i basis shares the same coefficients as the expansion of the velocity u in the Φ k basis
N N N ω = curl u = curl ω i Φ i = ω i curl Φ i = ω i φ i . i i i
This is notable since a single coefficient vector w = [ω 1 ω 2 . . . ω N ] uniquely identifies both the fluid’s velocity u and its vorticity ω. Either field can be easily reconstructed from the basis coefficients ω i . Orthogonality. Laplacian eigenfunctions on a domain form an orthogonal set. The total energy of a signal expressed in an orthogonal basis is the sum of the squares of its coefficients by Parseval’s identity. The fluid’s kinetic energy can thus be calculated as Spatial scales of vorticity. As shown in Figure 4 , larger eigenvalues of the Laplacian correspond to fields with smaller vortices. Basis coefficients can be interpreted as a discrete spatial spectrum of the fluid with higher “frequencies” corresponding to smaller scales of vorticity. This notion has been previously applied by Stam and Fiume using a Fourier basis to generate procedural stochastic turbulence
[Stam and Fiume 1993]. A decomposition into a spectrum of vorticity is important for at least two reasons. First, because computations require our basis to be finite, this ordered structure provides a principle by which to select the finite set. In choosing to truncate the spectrum at some finite N , the error we incur is well defined: we lose the ability to simulate vortices smaller than a given scale. Second, combined with orthogonality, our basis delivers a means of controlling the energy at different scales of vorticity by adjusting the magnitude of the basis coefficients. We use this property in Section 4 to accurately model viscous energy decay. It could also be used to initialize or arbitrarily change a fluid’s turbulent spectrum. Closed form expressions. For some simple domains, the basis fields have closed form expressions. This allows the velocity to be evaluated at any spatial coordinate without the need for a voxelized grid or interpolation. A grid may still be used for visualization, for example to track density or subsample the velocity from the closed form expressions to accelerate particle advection. However, this grid is independent of the simulation, and its resolution may be changed without changing the performance or behavior of the underling simulation. Although the benefits of closed form expressions are limited to simple geometries, a 2-D rectangle and 3D rectangular cavity both represent typical simulation domains. In Section 8 we compute basis fields numerically for general meshed domains through a discrete vector Laplacian operator.
N u 2 = ω i 2 . D i
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Fig. 5. A fluid’s motion is a curve through the Euclidean space of basis coefficients, shown for N = 3 as an illustrative example. From the NavierStokes equations, we derive an expression for the tangent vector w ̇ as a function of the current state vector.
<H1>4. DYNAMICS</H1>
A fluid’s velocity field will change continuously over time according to physical laws. In our basis representation, this can be described by the continuous change of the coefficient vector w. In this section we derive an expression for the time derivative w ̇ in terms of the basis coefficients only. The vorticity formulation of the Navier-Stokes equation is
6
ω ̇ = Adv(u, ω) + ν∆ω + curl (f )
where ω = curl u and f are external forces. For notational convenience, we choose Adv(·, ·) to represent the advection term, which is defined as Adv(u, ω) := curl (ω × u). We perform Galerkin projection of the Navier-Stokes equations onto a Laplacian eigenfunction basis by substituting the expansions ω = i ω i φ i , u = j ω j Φ j and ω ̇ = k ω ̇ k φ k into Eq. 6 and rearranging terms through linearity of operators N N N ω ̇ k φ k = ω i ω j Adv(Φ i , φ j ) + ν ∆ω i φ i + curl (f ). k i j i We discuss each right hand term separately. Advection. The Adv(Φ i , φ j ) terms represent the nonlinear advection of basis fields. As will be detailed in Section 6, we precompute these terms and the vorticity basis coefficients of the result are stored in a set of matrices C k . After equating coefficients, the contribution of the self advection term can be written as N N
7
ω ̇ k = ω i ω j C k [i, j], i j
summarized in matrix form as ω ̇ k = w T C k w, where C k are precomputed matrices and w is the column vector of basis coefficients [ω 1 ω 2 . . . ω N ] T . Viscosity. Because φ k are Laplacian eigenfunctions, the viscous term becomes ν i ∆ω k φ k = ν k λ k ω k φ k . The effect of viscosity on each basis coefficient is hence described by the linear first order differential equation ω ̇ k = νλ k ω k which conveniently has the closed form solution ω ̇ k (t) = ω k (0)e νλ k t . This says that the magnitude of each basis coefficient decays with a time constant equal to the eigenvalue, which is physically correct, as small vortices dissipate faster than large vortices. External forces. External forces can be incorporated by projecting curl (f ) on the vorticity basis, to obtain coefficients f i =
Fig. 6.
A basis of Laplacian eigenfunctions is orthogonal, hence surfaces of constant energy are spheres in coefficient space. Left: Unconstrained explicit timesteps are subsequently projected to the manifold of constant energy by normalizing the position vector. Right: Although more expensive, it is also possible to derive an N -dimensional rotation matrix that constrains integration exactly to the state manifold.
N curl (f ), φ i satisfying curl (f ) = i f i φ i . Due to Eq. 5, f i can be equivalently obtained by directly projecting f to the velocity basis basis f i = f , Φ i , which often involves less computation. The contribution to ω ̇ k is then ω ̇ k = f k . Time Evolution Equation. In total, the time derivative of each basis coefficient is
8
ω ̇ k = w T C k w + νλ k ω k + f k .
<H1>5. TIME INTEGRATION</H1>
Any standard numerical integration scheme can be applied to integrate Eq. 8 forward in time. However, for computer graphics applications speed and energy stability are important requirements. We first describe our preferred integration scheme that meets these two requirements, and then discuss other available techniques. Our basis is orthogonal allowing kinetic energy to be calculated as a sum of squared coefficients. Additionally, orthogonality implies that surfaces of constant energy in the Euclidean space of coefficients are spheres. An inviscid fluid preserves kinetic energy, and should trace out a path on such a sphere. We choose a fast explicit integrator (such as forward Euler or Runge-Kutta method) to first perform an unconstrained timestep, followed by renormalization to enforce the energy constraint as depicted in Figure 6 . Renormalizing to preserve the kinetic energy is a technique available in any fluid simulation method and is not particular to our approach. However, when employing grid based velocity fields it is often undesirable as it can lead to visual artifacts. We have not observed such artifacts, possibly because our basis fields are globally supported and energy is never dissipated locally through a pressure projection step as for example in
[Stam 1999]. The effect of viscosity and projected forces will change the kinetic energy, so these terms are integrated following the energy renormalization. Physical viscosity is achieved by decaying each coefficient exponentially as described in Section 4. Our integration scheme is summarized in the pseudo-code of Alg. 1. Run time complexity. Computation is dominated by the evaluation of matrix vector products, making the run time complexity O(z), where z is the total number of non-zero entries in all the {C k } combined. In general, {C k } are dense and z is O(N 3 ), leading to a computational complexity similar to that of
[Treuille et al. 2006]. However, for some domains where closed form expressions are available including a 2-D rectangle and 3-D cavity, the regular-
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e 1 = N i w[i] 2 // Store kinetic energy of velocity field for k = 1 to N do w[k] ̇ = w T C k w // Matrix vector product w += w∆t ̇ // Explicit Euler integration e 2 = N i w[i] 2 // Calculate energy after time step w *= e 1 /e 2 // Renormalize energy for k = 1 to N do w[k] *= exp(λ k ∆t) // Dissipate energy for viscosity w[k] += f [k] // External forces Algorithm 1: Pseudo-code for our fluid simulator. Time integration is explicit and does not require the solution to a linear system.
ity of the boundary leads to very sparse {C k } matrices making the theoretical complexity O(N 2 ) and effectively ≈ O(N ) for practical ranges of N . Additional Integration Schemes. Eq. 8 is a symbolic expression for the first time derivative of vorticity. Differentiating this expression produces closed form expressions for time derivatives of arbitrary order. These can be useful for alternate integration schemes to improve accuracy or allow time reversibility. However, considering that stability has already been enforced it may not be a concern for graphics applications. Greater accuracy could also be easily achieved through high order explicit schemes using a small timestep. A final integration scheme that is theoretically interesting involves the calculation of an N dimensional rotation matrix R, which, when applied to the coefficient vector w, constrains its motion exactly to the constant energy N -sphere. This approximates the true geodesic motion of the Euler fluid equations near the current state. The position vector ω and the tangent vector ω ̇ span an N −1 dimensional rotation plane that uniquely identifies an N ×N skew symmetric matrix ξ. This matrix g is an element of so(N ), the Lie algebra of the N -dimensional rotation group SO(N ). Multiplying by ∆t and exponentiating the matrix yields the N × N rotation matrix R = exp(∆tξ). This method is more expensive than explicit integration with renormalization, and we have found that in comparison it offers very little gain in accuracy for small timesteps. However, it is of interest because it preserves the geometric viewpoint of a fluid as a high dimensional rotation group, and provides a more rigorous way of enforcing energy preservation compared to the renormalization correction step.
<H1>6. PRECOMPUTATION OF ADVECTION OPERATOR</H1>
The operator Adv(u, ω) := curl (ω × u) represents the advection of a fluid’s vorticity by its velocity field. It has many equivalent expressions, including the the Lie derivative L u ω, or the JacobiLie bracket of vector fields −[u, ω], Adv(Φ i , φ j ) := L Φ i φ j = −[Φ i , φ j ] = curl (φ j × Φ i ). In our context, all the preceding expressions are equivalent, and any can be used to evaluate the advection of pairs of basis fields. For domains admitting closed form expressions for Laplacian eigenfunctions, the evaluation can be performed symbolically and is hence
for i, j = 1 to N do p = Adv(φ i , Φ j ) // Project the result onto finite basis for k = 1 to N do C k [i, j] = Proj(p, φ k ) Algorithm 2: Pseudocode for precomputing entries of C k matrices.
exact. For discrete domains, it can be approximated numerically on a mesh as described in Section 8. For every pair of basis functions we evaluate the advection operator and express the result in the finite φ k basis. The basis coefficients of this projection are the structure coefficients that form the {C k } matrices and satisfy Adv(Φ i , φ j ) = C k [i, j]φ k . k The Laplacian eigenfunction basis is closed under the Jacobi-Lie bracket. Hence, we expect the result to factor perfectly into a linear combination of vorticity basis functions. For simulation, our basis must necessarily be finite dimensional. Despite closure, the advection operator may produce coefficients beyond the chosen finite bandlimit N which cannot be stored. This is unavoidable, as the nonlinear advection operator necessitates products of functions. Considering for example the Fourier basis, the multiplication of two N bandlimited functions is in general bandlimited by 2N . Physically this represents the cascading of energy to ever higher scales of turbulence. Projecting the result of the advection operator to our finite dimensional basis amounts to truncating the coefficients beyond the bandlimit N . However, this truncation is physically motivated, since in a real fluid the vortices will eventually reach a small enough scale and dissipate quickly through viscosity. A pseudo-code listing of the precomputation procedure is shown in Algorithm 2. Properties of C k matrices. Because the Jacobi-Lie bracket and vector cross product are anti-symmetric operators, the structure coefficient matrices have the property 1 1 C k [i, j] = − C k [j, i]. λ i λ j The antisymmetry reflects an important property of the our basis functions. The self advection Adv(Φ k , φ k ) of a vorticity basis field φ k by its velocity Φ k is identically zero, and hence u ̇ = 0, meaning that each velocity basis field is a stationary flow. This is analogous to the stable rotation of a rigid body about a principal axis
[Arnold 1966]. To illustrate the preceding discussion, the evaluation of the structure coefficients in closed form for a 2-D rectangle is provided in Appendix A as an example.
<H1>7. EXTERNAL FORCES</H1>
As discussed in Section 4, external forces can be incorporated by projecting f to the velocity basis basis f i = f , Φ i . The inner product for vector valued f and Φ i is defined by the summation of dot product of vectors at every point x within the domain f , Φ i = f (x) · Φ i (x). x We make use of external forces to allow immersed moving obstacles and to incorporate a simple buoyancy model.
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<H2>7.1 Moving Obstacles</H2>
The eigenfunctions of the Laplacian operator are defined by their domain and boundary conditions, making the velocity and vorticity basis fields domain dependent. Static irregular boundaries and obstacles are supported in our method through precomputation on general meshes as will be discussed in Section 8. However, moving obstacles change the shape and boundary conditions of the domain dynamically, and hence require special consideration. Our goal is to satisfy the internal boundary conditions of immersed objects at all times. This requirement can be simply stated: in addition to remaining divergence free, the fluid velocity at an object’s boundary should be equal to the normal component of the boundary’s velocity. This satisfies the free-slip condition when the object is at rest, and equates normal components of the fluid and boundary velocity when the object is in motion. Our solution is as follows. At each time-step we project the difference from the desired normal component onto the velocity basis Φ k and subtract the result from the current state vector w. The result is a divergence free field that best satisfies the desired boundary conditions. Note that this method is not perfect, as the projected forces only approximate the desired forces to the extent that the basis fields can resolve them. In other words, to handle obstacles with small spatial features, one must increase N to use basis functions of a sufficiently high spatial frequency. However, for coarse objects, we have found this method to provide reasonable accuracy, and it is efficient enough to perform interactively without requiring precomputation. Treuille et al. [2006]
also correct the normal velocity component through projection to a divergence free field, and our technique is similar in this respect. However, in their case an additional set of fields they name the boundary basis are employed that are chosen based on the object’s geometry to best correct for normal velocity components. The boundary basis allows the free-slip constraint to be more accurately enforced in the vicinity of the boundary, but adds substantially to memory and precomputation expense. It also does nothing to improve the quality of object-fluid interaction since the underlying simulation basis, to which the boundary basis must be numerically projected, remains unchanged. In contrast, our basis fields exhibit a spectrum of spatial scales (akin to a Fourier Series) allowing some guarantee of resolving obstacle features with similar length scales. Although our method does not perfectly resolve the boundary, it avoids the use of multiple bases for simulation and boundaries as well as the associated expensive precomputation and memory requirements.
Fig. 7.
Irregular obstacles interact with the fluid simulation by projecting contact forces to the velocity basis to best satisfy boundary conditions.
Fig. 8. Velocity basis fields for a tetrahedral mesh obtained through eigendecomposition of a discrete Laplacian operator.
<H2>7.2 Buoyancy</H2>
In some of our examples we incorporate a simple buoyancy model. Smoke density or particle density are subsampled onto a grid. Buoyancy forces at each grid centre are calculated through the Boussinesq approximation. These forces are projected to the velocity basis through pointwise multiplication.
<H1>8. FORMULATION ON MESHED DOMAINS</H1>
Simple geometries admit basis fields with closed form expressions. However, our method also supports discrete domains defined on a mesh. For this, we require a set of basis fields defined on the mesh that are eigenfunctions of a discrete Laplacian operator, as well as a means to precompute their advection numerically. Discrete exterior calculus (DEC) provides a principled means of describing operators and quantities on simplicial meshes
[Desbrun et al. 2005]. It has been applied to fluid simulation in previous work, and we use a discrete formulation on tetrahedral meshes analogous to
[Mullen et al. 2009; Elcott et al. 2007]. Regular voxel meshes are also supported as a special case of this discretization. Discrete Basis Fields. Through DEC we define the discrete Laplacian operator ∆ = −curl 2 = −d ∗ d∗ which has a representation as a sparse, symmetric matrix. We compute the eigendecomposition of this matrix to produce the discrete velocity and vorticity basis fields. The velocity basis fields satisfy a free slip boundary condition and are divergence free, due to constraints imposed implicitly through the Laplacian operator matrix. For example, to enforce a free-slip velocity boundary condition, we omit (set to zero) the rows of the discrete Laplacian ∆ that calculate velocity flux on boundary faces. Defined as above, ∆ admits only divergence free solutions in its eigendecomposition. Hence the fields produced by the eigensolver satisfy the conditions of Eq. 1 in a discrete setting. Examples of basis fields for a tetrahedral mesh are shown in Figure 9 . Discrete Advection Operator. We also employ DEC to approximate the advection operator Adv(·, ·) using appropriate discretizations. This evaluation is similar to that employed in
[Mullen et al. 2009]. Other than the discrete representation and computations described above, the rest of our fluid simulation method remains the same. The operation and performance of the time integration scheme described in Section 5 does not change, since it operates only with the basis coefficients. However, additional expenses in the case of meshed domains include the storage of discrete basis fields, and the reconstruction of the velocity field through summation. As we show in Section 9, these costs are reasonable for typical operating parameters, but can become large for simulations employing very fine meshes and large basis dimensionality.
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Memory Domain Domain Basis Mesh Precomp. C k Type Dim. Ele. (s) (Mb) N T O(N 2 ) O(N 2 ) 3-D Cube 81 3.6 0.05 3-D Cube 172 15 0.3 Closed form 3-D Cube 325 55 1.1 3-D Cube 540 149 3.3 3-D Cube 81 3.6 0.05 3-D Cube 172 15 0.3 Closed form 3-D Cube 325 55 1.1 3-D Cube 540 149 3.3 N T O(T N 2 ) O(N 3 ) Head 64 ≈ 24 3 436 0.9 Tets Head 128 ≈ 24 3 1689 7.2 Armadillo 16 32 3 18 0.01 Armadillo 32 32 3 47 0.1 Voxels Armadillo 64 32 3 185 0.9 Armadillo 128 32 3 1109 7.8 Bunny 32 32 3 79 0.11 Bunny 64 32 3 269 0.95 Voxels Bunny 128 32 3 1305 7.8 Bunny 256 32 3 6911 62.4
Fig. 10. Precomputation time, storage requirements and runtime performance performed on a single CPU core. N is the basis dimensionality and T is the number of mesh elements. † Closed form velocity bases cached on a 32 3 grid. †† Per 1000 closed form evaluations. ††† Buoyancy forces calculated on a 16 3 subsampled grid.
<H1>9. RESULTS</H1>
The storage requirements, precomputation time, and runtime performance of our simulation method for both closed form and meshed domains are presented in Figure 10 . All experiments were performed on a single CPU core. Time integration was performed using an explicit fourth order Runge Kutta method. Closed form domains are limited in their boundaries, but have notable advantages in terms of runtimes, precomputation and memory requirements. For examples including external forces (such as buoyancy or moving obstacles), the cost of projecting the forces on to the basis is noted. This cost is proportional to the mesh resolution and the number of basis fields. In the case of the bunny, a subsampled 16 3 density grid is used for the buoyancy force calculations. Velocity Reconstruction. For discrete meshes, velocity field reconstruction requires summation of stored basis fields. This is proportional to the mesh resolution and the dimension of the basis. On closed form domains, there are two alternatives for velocity reconstruction. The basis fields may be pre-evaluated on a mesh and stored, just as in the discrete case. Alternatively, they may be computed on demand. Closed form evaluation is proportional to the number of basis functions and the number of advected quantities. Each alternative has its strengths. Caching the basis fields uses memory, but saves computation when many quantities are being advected through the field (density or millions of particles). If only a few particles need to be advected (leaves in wind, for example), then evaluating closed form expressions is accurate and fast and does not have additional memory requirements. A column in Table 10 lists the cost of 1000 closed form evaluations.
Runtime Basis Advection Vel. Reconst Vel. Reconst External Fields † (ms) (cached)† (closed form) Forces ††† (Mb) (ms) (ms) †† (ms)
O(T N ) ≈ O(N ) O(T N ) O(T N ) O(T N
)
)
66 18 26 141 37 56 267 70 105 443 125 172 18 30 37 67 n/a 70 126 125 216 O(T N ) O(N 3 ) O(T N ) O(T N 26 20 10 52 42 20 13 3.5 5 22 26 7 10 43 52 15 20 88 105 39 41 172 26 8 10 22 52 17 19 43 105 40 39 90 210 130 87 174
The supplemental videos demonstrate simulations performed on a variety of domains with varying basis dimensionality. A comparison to the stable fluids algorithm is included as a rough qualitative validation. We demonstrate flow on some simple tetrahedral meshes; however we chose a structured voxelized grid for the bunny example only to facilitate implementation. A robust tetrahedral mesh implementation would have similar performance characteristics and alleviate the boundary “stair case” artifacts. The effects of basis dimensionality are illustrated through the bunny example. Modes with small eigenvalue capture the low frequency motion of the fluid. Notably, the bunny’s ears do not begin to be resolved until after the 64th mode. In addition to using the bottom of the spectrum to capture the large scale motion, one may choose additional modes from much higher parts of the spectrum to incorporate smaller scales. This demonstrates the benefit of a basis that exhibits a spectrum of scales. Note that these high frequency modes interact and decay physically, in contrast to other post-processing turbulence models.
<H1>10. CURRENT LIMITATIONS</H1>
Our method is most applicable to gaseous phenomena and situations when the domain is entirely filled by fluid. Currently it is not readily adaptable to typical liquid simulations that require a constantly changing fluid domain with a free surface. We have shown that interesting dynamics can be captured in a reasonably sized basis dimension and simulated interactively. However, various issues prevent it from scaling well to very large basis dimension or grid resolutions. For irregular domains, the runtime is in general O(N 3 ). Large mesh resolutions also require large
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Fig. 9.
Top-left: An interactive painting application benefits from an accurate viscous fluid simulation that is independent of the density grid. Topright: Fluid simulation within a tetrahedral model of a head. Bottom-left: A turbulent 3-D flow in a cubic domain. Bottom-right: Initializing an inviscid simulation with basis functions that are spatially symmetric produces a perpetual symmetric flow.
precomputation times and storage for the basis fields. For discrete meshes, the cost of reconstructing the velocity field and projecting external forces grows linearly with the basis dimension and mesh resolution. Many of these issues are not present for domains with closed form expressions. However, in this case the shape of the boundary is limited. Also, when advecting many particles or projecting many forces the velocity basis fields must still be cached as the cost of closed form evaluations become prohibitive.
<H1>11. FUTURE WORK</H1>
Artistic Control. We believe our method has potential to be exploited for the expressive control of fluid phenomena. We have shown how to continuously change the basis coefficients to simulate the physical motion of a fluid. However, any smooth curve through coefficient space, physical or not, may be perceived as “fluid-like” as it represents a continuously changing volume preserving flow that respects all boundary conditions. In addition to constructing completely arbitrary flows, perturbing existing physical paths offers a means to deviate from physics while quantifying this deviation. Due to orthogonality of the basis and its correspondence with vorticity of varying scales, we have a unique mechanism for spectral energy control. This could be used to implement timevarying filters to amplify or attenuate parts of the spectrum, such as achieving crescendos of turbulence or gradual calming. Again, we have a means of quantifying the deviation from non-physical energy behavior, as we have shown how to decay the spectrum according to physical viscosity. Space-time control for fluids has been attempted previously in
[Treuille et al. 2003; McNamara et al. 2004; Fattal and Lischinski 2004]. Many of these methods can be expensive because the optimization scales sharply with the size of the grid, making them impractical for interesting domains. A low dimensional basis offers a good setting to implement control policies that would be intractable in higher dimensions as demonstrated for example by Barbi c et al.
[Barbi c et al. 2009]. Our method’s availability of closed form expressions for time derivatives could also prove useful in optimization algorithms. Our method is fast enough to be interactive, and is very memory efficient and well formulated on rectangular domains due to the available closed form expressions. This makes it particularly attractive for use in image based settings such as painting applications that simulate fluid phenomena, as we briefly demonstrate in the video. Additional potential uses in this vein include texture synthesis and non-photorealistic rendering. Improvements to our Method. Boundaries of moving obstacles are handled only approximately and could benefit from alternate methods. We have presented a fast and stable integration scheme; however, additional time integrators could be explored, particularly symmetric integrators to allow time reversibility as was achieved in
[Mullen et al. 2009]. Time reversibility could prove useful in fluid control applications, as was demonstrated for rigid bodies by
Twigg and James [2008]. We have evaluated the advection operator symbolically for closed form expressions on rectangular 2-D and 3-D domains. The same could be done for additional geometries, such as a 2-D disk or a spherical surface. Also, different boundary conditions (for example, a wrap-around boundary condition) remain to be considered, which could prove useful for tilings of fluid simulation domains
[Wicke et al. 2009]. Other Applications. Divergence free fields have many potential uses besides simulating natural phenomena. Fluid motion describes the optimal transport in an incompressible medium, and can be used to quantify volume preserving deformations. This has uses in image analysis and shape deformation. We plan to consider how the unique properties of our method could be exploited in these fields. In particular, the elegant formulation on rectangular domains could make it useful for medical image registration. Additionally, the availability of closed form expressions and flexibility in choosing the basis dimension make it an accurate and tractable model for optimization methods. This could be useful for the inverse modelling of real fluid flows for the purpose of parameter estimation, for example to estimate viscosity from sampled velocity measurements.
<H1>12. CONCLUSION</H1>
We have presented a fluid simulation method that uses eigenfunctions of the vector Laplacian as bases. We have described many of its unique properties and its use as a practical means of fluid simulation for computer graphics. The orthogonality of the basis functions and their correspondence to a spectrum of vorticity scales enables energy control at varying turbulent scales. We have used this property to enforce stability of integrators and simulate physical viscosity. Flexibility in choosing basis dimensionality and the ability to integrate directly in a space of basis coefficients permits computational efficiency, enabling interactive performance. The existence of closed form solutions for simple domains allows symbolic evaluation of the advection operator and the ability to sparsely evaluate velocities on demand. We have demonstrated some of the useful properties of our method, but many exciting avenues remain to be explored. We plan to investigate its use for the the expressive control of fluid motion, such as spectral energy control and space time optimization. We also believe there is potential for our method to be exploited in other research areas such as medical imaging and inverse flow modelling.
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<H1>APPENDIX A. EVALUATING ADVECTION IN CLOSED FORM FOR 2-D DOMAIN</H1>
We evaluate the advection operator for pairs of basis functions defined on a 2-D rectangular domain, defined in Eqs. 3 and 2. We evaluate Adv(Φ i , φ j ) = curl (φ j × Φ i ) recalling that i, j are vector wave numbers i = (i 1 , i 2 ), j = (j 1 , j 2 ) and the eigenvalues λ i = −(i 2 1 + i 2 2 ). This simplifies to 1 Adv(Φ i , φ j ) = i 1 j 2 cos(i 1 x) cos(j 2 y) sin(j 1 x) sin(i 2 y) λ i 1 − i 2 j 1 cos(j 1 x) cos(i 2 y) sin(i 1 x) sin(j 2 y) a z . λ i The trigonometric identity cos(α) sin(β) = 1 sin(α + β) − 2 1 sin(α − β) allows factoring to a suitable expression which is 2 indeed spanned by {φ k }: 1 Adv(Φ i , φ j ) = 4λ i ((i 1 j 2 − i 2 j 1 )φ i 1 +j 1 ,i 2 +j 2 − (i 1 j 2 + i 2 j 1 )φ i 1 +j 1 ,i 2 −j 2 + (i 1 j 2 + i 2 j 1 )φ i 1 −j 1 ,i 2 +j 2 −(i 1 j 2 − i 2 j 1 )φ i 1 −j 1 ,i 2 −j 2 ) . The resulting coefficients are stored in the {C k } matrices 1 C i 1 +j 1 ,i 2 +j 2 [i, j] = − 4(i 1 2 + i 2 2 ) (i 1 j 2 − i 2 j 1 ) 1 C i 1 +j 1 ,i 2 −j 2 [i, j] = 4(i 1 2 + i 2 2 ) (i 1 j 2 + i 2 j 1 ) 1 C i 1 −j 1 ,i 2 +j 2 [i, j] = − 4(i 1 2 + i 2 2 ) (i 1 j 2 + i 2 j 1 ) 1 C i 1 −j 1 ,i 2 −j 2 [i, j] = 4(i 1 2 − i 2 2 ) (i 1 j 2 − i 2 j 1 ). This result demonstrates closure of the advection operator. The indices i, j are meant figuratively, as they represent tuples of integers. A suitable re-mapping from (i 1 , i 2 ) and (j 1 , j 2 ) to positive integers is necessary in an implementation. When outside of the storable finite range, they are discarded as described previously. Note the sums of indices i 1 + j 1 and i 2 + j 2 , which reflect the doubling in bandlimit due to multiplication of sinusoidal functions.
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P ARK , S. I. AND K IM , M. J. 2005. Vortex fluid for gaseous phenomena. In Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation. SCA ’05. ACM, New York, NY, USA, 261–270.
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Received July 2011; accepted July 2011
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
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ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
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"text": "['La-']",
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"text": "['showed that an ideal incompressible fluid is described similarly as geodesic motion on SDiff']",
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"text": "['The notion of structure coefficients to describe the interaction of Lie algebra basis elements of these groups is directly related to the precomputed coefficient matrices used in our method']",
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"text": "['it is not widely used']",
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"text": "['Such performance considerations were the motivation for the development of spectral methods as pioneered by Orszag']",
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"text": "['They are often used to study homogeneous turbulence']",
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"text": "['Rogallo et al. 1981']",
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"text": "['Orszag and Patterson 1972']",
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"text": "['Fourier series or Chebyshev polynomials are commonly employed']",
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"text": "['spectral methods are limited to bases admitting a fast transform']",
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"text": "['Our method is most analogous to the interaction coefficient method of']",
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"text": "['Silberman 1954']",
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"text": "['admit a fast transform']",
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"text": "['Laplacian eigenfunctions do not']",
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"text": "['On arbitrary domains']",
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"text": "['Laplacian eigenfunctions have many other benefits']",
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"text": "['Section 3']",
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"text": "['theoretical performance scaling is less critical']",
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"text": "['for the applications we consider']",
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"text": "['visually detailed simulations are attainable at low cost']",
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"text": "['Divergence free finite element methods']",
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"text": "['employ bases of discrete divergence free velocity fields to solve fluid equations']",
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"text": "['in a space that satisfies mass continuity a priori']",
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"text": "['Gustafson and Hartman 1983']",
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"text": "['Our method is similar in this respect']",
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"text": "['Bridson presented a simple means to generate procedural divergence free flows through the curl of a vector potential stream function']",
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"text": "['Laplacian eigenfunctions satisfying Eq.']",
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"text": "['Figure 4']",
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21413,
21419
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21502
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"text": "['a planar region with a wrap around boundary condition']",
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21755
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21787,
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21857,
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"text": "['u and ω are orthogonal']",
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22036,
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22090,
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"text": "['φ k × n = 0 at ∂D']",
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"text": "['the example expressions of Eqs. 2 and 3 satisfy all the properties listed below']",
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22401,
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22539,
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"text": "['Angelidis et al. 2006']",
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"text": "['Weißmann and Pinkall 2010']",
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"text": "['The key benefit of a representation in']",
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22639,
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"text": "['Laplacian eigenfunctions is that the inverse operator curl −1 applied to vorticity basis functions yields a simple expression']",
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"text": "['Φ k = curl −1 φ k = curl −1 1 ∆φ k λ k = 1 curl −1 −curl 2 φ k λ k']",
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22870,
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"text": "['the expansion of the vorticity ω in the φ i basis shares the same coefficients as the expansion of the velocity u in the Φ k basis']",
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"text": "['N N N ω = curl u = curl ω i Φ i = ω i curl Φ i = ω i φ i']",
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23229,
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"text": "['the error']",
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"text": "['subsample the velocity from the closed form expressions to accelerate particle advection']",
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"text": "['3D rectangular cavity']",
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"text": "['a 2-D rectangle']",
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"text": "['Eq.']",
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"text": "['5']",
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"text": "['Stam 1999']",
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"text": "['Section 8']",
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35401,
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35925,
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"text": "['the vortices will eventually reach a small enough scale and dissipate quickly through viscosity']",
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36383,
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"text": "['The self advection Adv(Φ k , φ k ) of a vorticity basis field φ k by its velocity Φ k is identically zero']",
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36500,
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36523,
36569
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36571,
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"text": "['This is analogous to the stable rotation of a rigid body about a principal axis']",
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36653,
36664
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"text": "['Arnold 1966']",
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36707,
36786
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"text": "['the evaluation of the structure coefficients in closed form for a 2-D rectangle']",
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36802,
36812
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"text": "['Appendix A']",
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37903,
37991
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38074,
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38538,
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<Document xmlns:gate="http://www.gate.ac.uk" name="A08_S09_Pose_Space_Deformation_A_Unified_Approach_to_Shape_Interpolation_and_Skeleton-Driven_Deformation_CITATION_PURPOSE_M_v1.xml">
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http://dx.doi.org/10.1145/344779.344862
<Title>Pose Space Deformation: A Unified Approach to Shape Interpolation and Skeleton-Driven Deformation</Title>
J. P. Lewis
∗
Matt Cordner
Nickson Fong
∗ [email protected]
Centropolis
<Abstract>Pose space deformation generalizes and improves upon both shape interpolation and common skeleton-driven deformation techniques. This deformation approach proceeds from the observation that several types of deformation can be uniformly represented as mappings from a pose space, defined by either an underlying skeleton or a more abstract system of parameters, to displacements in the object local coordinate frames. Once this uniform representation is identified, previously disparate deformation types can be accomplished within a single unified approach. The advantages of this algorithm include improved expressive power and direct manipulation of the desired shapes yet the performance associated with traditional shape interpolation is achievable. Appropriate applications include animation of facial and body deformation for entertainment, telepresence, computer gaming, and other applications where direct sculpting of deformations is desired or where real-time synthesis of a deforming model is required.</Abstract>
CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid and object modeling I.3.6 [Computer Graphics]: Methodology and Techniques—Interaction techniques I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation Keywords: Animation, Deformation, Facial Animation, Morphing, Applications.
<H1>1 Introduction</H1>
Free form deformation has been approached from several distinct perspectives. As an abstract and general problem, good methods have been obtained both using the well known technique that bears this name
[ 32 , 12 , 17 ]
and other kinematic surface deformation techniques, and with physical models that simulate the time evolution of a membrane or solid. The animation of human and creature skin deformation is arguably the most common and important application of free form deformation in computer graphics. While such creature animation can be considered a special case of general free form deformation, its importance and difficulty have lead researchers to propose a number of domain-specific algorithms that will be reviewed in Section 2. The problem of realistic facial animation is being actively and successfully addressed by image-based and hybrid techniques. These techniques are not yet suitable for all applications, however: while a purely image-based approach can achieve very realistic images, this advantage may be lost if one needs to introduce geometry and surface reflectance in order to re-light characters to match preexisting or dynamically computed environments. Film and entertainment applications require fanciful creatures that fall outside the scope of image-based approaches. Some of the most impressive examples of geometry-based (as opposed to image-based) human and creature animation have been obtained in the entertainment industry. These efforts traditionally use shape interpolation for facial animation and a standard but variously-named algorithm that we will term skeleton subspace deformation (SSD) for basic body deformation
[ 25 , 9 ]. While shape interpolation is well-liked by production animators, it is not suitable for skeleton-driven deformation. On the other hand SSD produces characteristic defects and is notoriously difficult to control. These issues, which will be detailed in the next section, lead us to look for a more general approach to surface deformation. We consider the following to be desirable characteristics of a skeletonbased surface deformation algorithm:
• The algorithm should handle the general problem of skeletoninfluenced deformation rather than treating each area of anatomy as a special case. New creature topologies should be accommodated without programming or considerable setup efforts. • It should be possible to specify arbitrary desired deformations at arbitrary points in the parameter space, with smooth interpolation of the deformation between these points. • The system should allow direct manipulation of the desired deformations
[ 33 ]. • The locality of deformation should be controllable, both spatially and in the skeleton’s configuration space (pose space). • In addition, we target a conventional animator-controlled work process rather than an approach based on automatic simulation. As such we require that animators be able to visualize the interaction of a reasonably high-resolution model with an environment in real time (with ‘high resolution’ defined in accord with current expectations). Real time synthesis is also required for applications such as avatars and computer games.
Our solution, termed pose space deformation, provides a uniform and expressive approach to both facial skin deformation and skeleton-driven deformation. It addresses the previously mentioned drawbacks of shape interpolation and SSD while retaining the simplicity and performance associated with these techniques. The next section reviews various approaches to free form deformation and describes shape interpolation and skeleton subspace deformation algorithms. The pose space deformation algorithm requires well behaved and efficient scattered data interpolation in high dimensional spaces; Section 3 considers this issue. The pose-space deformation algorithm itself is described in Section 4; examples and applications are shown in the last section.
<H1>2 Background</H1>
Recent research has delivered significant improvements in many areas of character animation, including surface representation, model capture, performance capture, and hybrid (partially image-based) rendering approaches. In this literature review we focus specifically on milestones in the surface deformation models and necessarily omit other important contributions.
<H2>2.1 Surface Deformation Models</H2>
Continuous deformation of a character skin was first addressed in Parke’s pioneering facial animation work
[ 26 ]. In this work, control vertices were deformed by custom algorithmic implementation of carefully selected high-level parameters (‘raise-upper-lip’, etc.). Komatsu [ 13 ]
and Magnenat-Thalmann et. al. [ 23 ]
demonstrated human body deformation driven by an underlying skeleton.
The region and shape of deformation is algorithmically defined in each of these approaches. Magnenat-Thalmann et. al. developed algorithms for each of the various joints in the hand. The discussion in Komatsu focuses on the elbow and shows how the skin crease on the acute side can be obtained by a suitable algorithmic manipulation of the surface control vertices. The algorithms in this early work do not suffer the ‘collapsing elbow’ characteristic of the SSD algorithm (below). On the other hand, the algorithms are specific to particular types of joints and are perhaps too simple to portray the complexity and individual variability of real anatomy. The short film Tony de Peltrie
[ 3 ]
popularized the use of shape interpolation for facial animation. Forsey [ 11 ]
describes a characteroriented deformation scheme in which the bending of a smooth surface can be controlled by anchoring levels of a multi-resolution spline surface to the underlying skeleton. These efforts are distinguished from the previous purely algorithmic approaches in giving the modeler control of and responsibility for the deformation. The specification and animation of surface deformation remains an active area of investigation
[ 17 , 10 ]. The Wires technique [ 22 ] is one interesting recent contribution; this approach is notable in providing a direct manipulation interface in a form immediately familiar to sculptors (armatures).
<H2>2.2 Multi-Layered and Physically Inspired Models</H2>
Chadwick, Haumann, and Parent [ 7 ]
introduced a multi-layered and physically inspired approach to skin deformation. In their model a free-form deformation abstractly represents underlying body tissues and mediates skin movement. Chadwick et. al. demonstrated expressive three-dimensional cartoon characters but deformation of a realistic character was not shown. Other researchers have investigated modeling the underlying body tissues in greater depth
[ 27 , 24 , 8 , 35 ]. Most recently, several groups have undertaken ambitious efforts to produce anatomically inspired multi-layered models of animals and humans with considerable verisimilitude. Nedel and Thalmann [ 19 ]
simulate the surface deformation of muscles using spring mesh dynamics; a modeled skin cross section is reshaped by a ray-casting procedure that finds the maximum displacement of the underlying tissue. Several papers by Wilhelms and coworkers have shown anatomically representative human and animal models. In Wilhelms and Van Gelder [ 36 ]
several classes of muscles are algorithmically modeled with attention to volume conservation; skin is a spring mesh anchored to underlying tissue or bone in appropriate areas. Scheepers et. al. [ 31 ]
produced convincing representations of muscles as well as preliminary but promising skin deformation.
<H2>2.3 Common Practice</H2>
In recent years character animation has moved beyond being a research topic and sophisticated deforming characters routinely appear in films and on television. Various techniques are employed,
p’ p p" q’ q q"
Figure 1: The skeleton subspace deformation algorithm. The deformed position of a
point p lies on the line p p defined by the images of that point rigidly transformed by the neighboring skeletal coordinate frames, resulting in the characteristic ‘collapsing elbow’ problem (solid line).
including manually animated FFDs and custom procedural approaches in the spirit of
[ 26 , 23 , 13 ]. Arguably the most common practice in character animation (as reflected in commercial software, animation books and courses, and some custom software) is founded on the twin techniques of shape interpolation and SSD
[ 18 , 9 ].
<H3>2.3.1 Shape Interpolation</H3>
Shape interpolation (also called shape blending and multi-target morphing) is probably the most widely used approach to skin deformation for facial animation
[ 3 , 18 , 9 ]. Surface control vertices are simply an animated linear combination (not necessarily convex, i.e., individual weights can be greater than one or less than zero) of the corresponding vertices on a number of key shapes S k : k=0 w k S k . A variation of this technique uses a single base shape S 0 and a number of delta shapes, S 0 + k=1 w k (S k − S 0 ). By writing the delta shape form as (1 − 1 w k )S 0 + 1 w k S k it is clear that the space of achievable shapes is identical in both variations. 1 An attractive feature of shape interpolation is that the desired expressions can be directly specified by sculpting. The limitations of shape interpolation. Given the popularity and effectiveness of this simple approach, it would be desirable to employ it on regions of the body other than the face. The blending of rigid shapes is inconsistent with regions of the body that are bending under the action of an underlying skeleton, however. Of course the key shapes could be deformed to the moving articulated figure using some other algorithm, but this defeats the purpose of proposing shape interpolation as the means of obtaining the deformation in question. Shape interpolation also has some drawbacks for its intended role of facial animation. For one, the interpolation is not always smooth. Consider interpolating from a smile (shape A) to a neutral pose (B) and then to a frown (C). An individual vertex travels in a straight line between A and B and again in a line between B and C. Selecting smoothly changing weights with dw/dt = 0 at the key shapes merely causes the deformation to “ease in” and stop at each key pose before continuing on – the time derivative of control point motion is smooth, but the motion path itself is only piecewise linear (parametric versus geometric continuity).
In practice animators object to the linear nature of the interpolation
[ 34 ] and have sometimes compensated by sculpting new key shapes as often as every three to five frames
[ 38 ]. These comments will be revisited in the discussion of the pose space approach later in the paper. 1 Provided that the weights sum to one. This is enforced in the delta shape formulation. It is not enforced in the (non-delta) shape interpolation formulation as written, but weights that do not sum to one are a separate effect – they cause the face to change overall scale.
Figure 2: The ‘collapsing elbow’ in action, c.f. Figure 1 .
<H3>2.3.2 Skeleton-Subspace Deformation</H3>
This simple algorithm has been repeatedly conceived and appears in commercial software packages under several rather uninformative names such as skinning, enveloping, etc. The algorithm is unpublished but is subsumed by more general published schemes such as
[ 23 ]. The position of a control vertex p on the deforming surface of an articulated object lies in the subspace defined by the rigid transformations of that point by some number of relevant skeletal coordinate frames ( Figure 1 ). This may be notated
p = w k L k (p) p (in more detail) p = w k L δ k L k 0 −1 L p 0 p
where L p 0 is the transform from the surface containing p to the world coordinate system, L 0 k is the transform from the stationary skeletal frame k to the world system (L 0 k −1 L 0 p together represent p in the coordinate system of skeletal frame k), and L δ k expresses the moving skeletal frame k in the world system. The deformation is controlled by the user through the weights w k . SSD is fairly versatile. For example, secondary animation effects such as muscle bulging and swelling of the chest can be achieved by variably weighting the surface to an abstract “bone” whose translation or scale is manually animated. The limitations of SSD. The first major shortcoming of SSD results directly from the fact that the deformation is restricted to the indicated subspace. In common situations such as shoulders and elbows the desired deformation does not lie in this subspace, hence no amount of adjusting the algorithm weights will produce good results. This fact leads to considerable frustration by users of the algorithm – the character of the deformation changes as the weights are changed, sometimes sustaining the incorrect assumption that some combination of weights will produce good results. In fact, the SSD algorithm can be easily identified in animations by its characteristic ‘collapsing joint’ defect (Figures 1, 2). This problem is extreme in the case of simulating the twist of a human forearm (the pose taken in turning a door handle, Figure 3). In this case the subspace basis consists of surface points rigidly transformed by the forearm frame (no axis rotation) and the wrist frame (axis rotation). With a rotation of 180 degrees this line crosses the axis of the arm, i.e., the forearm collapses entirely as the SSD weights transition at some point from the forearm to wrist frames. A second difficulty with SSD is that, unlike shape interpolation, it does not permit direct manipulation; artists instead directly or indirectly edit the meshes of weights w k (for each control vertex on a surface there is one weight per skeletal frame that affects the vertex). SSD algorithms consequently have the reputation for being tedious and difficult to control. Artists with a poor understanding of the underlying algorithm have difficulty distinguishing between results that can be further improved by adjusting weights and results that cannot be improved since the desired result lies outside the achievable subspace, resulting in the impression of unpredictability (“sometimes adjusting the weights helps, sometimes it doesn’t”). In some cases the SSD defects can be manually corrected using FFDs and other techniques, and one could consider a scheme whereby these fixes are procedurally invoked as the skeleton articulates. But although FFDs work well (and have a direct manipulation algorithm
[ 12 ]
) the layered FFDs do not reduce the difficulty in adjusting the underlying SSD. The algorithm introduced in the subsequent sections removes the need for such layered fix-it approaches and permits direct specification of the desired deformations.
Figure 3: The forearm in the ‘twist’ pose, as in turning a door handle, computed by SSD. As the twist approaches 180 ◦ the arm collapses.
<H3>2.3.3 Unified Approaches</H3>
Several published algorithms and commercial packages combine aspects of skeleton-driven deformation and shape interpolation in ways that anticipate our approach. In the pioneering work of Burtnyk and Wein, two dimensional characters were animated using a polygonal rubber sheet that afforded both skeletal and local deformation control
[ 6 ]. Van Overveld described a two-dimensional animation system in which animation is controlled by a skeleton and character deformation is driven from this skeleton through a scattered interpolation
[ 20 ]. This work is similar in spirit to ours but differs in that it used the image plane as a global interpolation domain rather than introducing a pose space. Litwinowicz and Williams’s system [ 16 ]
is also a precedent and introduced sophisticated scattered interpolation (again in the image domain). Several papers consider animation (and indeed image synthesis in general) as a special case of neural net learning and interpolation/extrapolation
[ 14 , 15 , 21 ]. While this viewpoint is valid, in practice it is perhaps excessively general, for example, a skeleton is merely learned rather than being an intrinsic part of the model. While employed at Industrial Light and Magic the first author of the present paper developed a system that attempted to blend shape interpolation and SSD algorithms; a small portion of it remains in use in their well known Caricature animation system. Drawbacks of this work included both a complicated dependence on the details of SSD and its overall conception as a “correction” to SSD. Some commercial packages allow blending between two sculpted deformations as a function of a single-joint rotation, thereby combining shape interpolation and skeleton-driven deformation in a limited but useful setting.
Figure 4: Shepard’s interpolant operating on a set of colinear points. The derivative is zero at the data points, and the curve extrapolates to the average of the data values.
<H2>2.4 Kinematic or Physical Simulation?</H2>
The depth of simulation is a prevalent issue in computer graphics, albeit one that is not always consciously considered. Early approaches to animation were purely kinematic; an emphasis on physically based modeling appeared in the literature later. Recent sophisticated approaches allow a hybrid of animator-controlled and physically governed animation as needed. In rendering we perhaps see the opposite trend – much of the literature a decade ago focused on ever deeper simulations of reality, whereas ‘shallower’ imagebased approaches are attracting attention at present. Similarly, in character deformation both deep and shallow approaches have their place. Deep models promise universally accurate simulation, and the importance of representing humans justifies the needed effort. The authors of these approaches acknowledge that producing anatomically plausible models is a daunting task, however. Pose space deformation is a shallow, purely kinematic approach to deformation (i.e. without reference to underlying forces, mass, volume), and it has consequent disadvantages. In particular, accuracy is reliant on the modeler/animator rather than being guaranteed by the simulation. On the other hand, our algorithm has clear advantages with respect to simplicity and generality, direct manipulation, real-time synthesis, and other criteria listed in the introduction.
<H1>3 Deformation as Scattered Interpolation</H1>
In abstract, we wish to express the deformation of a surface as a function of either the pose of an underlying skeleton, or equivalently as a function of some other set of parameters such as the {smile, raise-eyebrow,...} controls desirable in facial animation. We also wish to directly sculpt the desired deformation at various points in the parameter space, rather than working in a more abstract space such as the coefficients on various coordinate frames as required by the SSD algorithm. A scattered data interpolation method is required because deformations will be sculpted at arbitrary (rather than regularly spaced) poses. Since this interpolation is central to our application (the results of the interpolation will be directly visible in the animating deformation), we will consider the available scattered interpolation approaches before settling on a candidate.
<H2>3.1 Shepard’s Method</H2>
Shepard’s method [ 1 , 2 ]
is a frequently employed scattered data interpolation scheme in computer graphics. In this method the interpolated value is a weighted sum of the surrounding data points
Figure 5: Radial basis functions φ(x) = exp(−x 2 /2σ 2 ), σ = 10 interpolating the same set of colinear points as in Figure 4 . A different y scale is used to fit the curve. The curve extrapolates to zero.
normalized by the sum of the weights, d(x) ˆ = w w k (x)d k (x) k
with weights set to an inverse power of the distance: w k (x) = x− x k −p . (This is singular at the data points x k and should computed as (||x − x k + ) −p ). With p > 1 the interpolation surface is once differentiable. Unfortunately this simple scheme has some potentially undesirable properties. Far from the data the weights will be approximately the same, d(∞) ˆ = w ∞ d k /w ∞ 1 = d k /N , i.e. the interpolated surface converges to the average of the data values. A serious drawback for some applications is that the derivative of the surface is zero at the data points ( Figure 4 ).
<H2>3.2 Radial Basis Functions</H2>
Radial basis functions
[ 28 , 29 ]
have become a popular choice for scattered interpolation. The interpolant is a linear combination of nonlinear functions of distance from the data points:
1
N d(x) ˆ = w k φ( x − x k ) k
If N values of d are available then the weights can be easily solved by a linear system; this can be derived either by least squares fit or by subspace projection. Taking the latter approach, we reconsider the available data points as a single point d in an N dimensional space, and consider φ k () = φ( x j − x k ) as the kth basis vector. The best approximation to d in the space spanned by φ k () occurs (in direct analogy with the three-dimensional case) when the weights are such that the error d − Φw (with φ k () comprising the columns of Φ) is orthogonal to each of the φ k (): Φ T (Φw − d) = 0
so (the so-called “normal equation”) Φ T Φw = Φ T d can be solved for the familiar w = (Φ T Φ) −1 Φ T d
A least squares approach leads to the identical result. Any nonlinear function φ() will interpolate the data, including odd choices such as φ(x) = x (which is nonlinear since x = x − x k is the argument), provided that the columns of Φ are independent. On the other hand a smooth φ() will result in a smooth interpolant (a weighted sum of continuous functions is continuous). In fact radial basis functions have a universal convergence property similar to Fourier series, though the convergence definition is different. The preceding description maps a k-dimensional input space (arbitrary k) to a one dimensional range, i.e., it is the k-dimensional version of a height field. Surfaces can of course be interpolated by allowing different combinations of the same basis functions in different dimensions, i.e., vector valued w k . The distance can be generalized to Mahalanobis distance (effectively rotating and stretching the basis function)
[ 4 ].
<H2>3.3 Energy Functionals and Non-Convex Methods</H2>
Various visual reconstruction schemes can be adapted for scattered data interpolation. In these schemes the interpolated or approximated surface is found as the minimum of a functional such as where the first term penalizes deviation of the surface d ˆ from the available data d and the second regularizing term votes for surface smoothness e.g. by integrating the squared second derivative of the surface. With small λ many of these schemes can serve as scattered data interpolants; reference [ 5 ]
is a good introduction to these approaches. In some of the most powerful formulations of scattered interpolation the regularizer is considered to hold everywhere except at an unknown set of edges – this is the piecewise-smooth prior desirable in image reconstruction. Since the unknown edges may exist (or not exist) at any location in the domain, all combinations of possible edge locations must be considered and the interpolation cost is prima facie exponential in the surface resolution.
| d(x) ˆ − d(x)| 2 dx + λP ( d) ˆ
<H1>4 Pose Space Deformation</H1>
The crux of our approach is the identification of an appropriate space for defining deformations. As discussed above, the interpolation domain is (a subset of) the pose space of an articulated character, or equivalently the space defined by some set of parameters such as facial controls. In concept the range of the interpolation function could simply be the desired movement of the surface control vertices. To make the job easier for the interpolation we instead interpolate the desired deviation of a surface vertex (expressed in the local frame) from its initially computed position (the rigidly transformed position in the case of an articulated model). Several reasons for this choice will be mentioned shortly. Thus the deforming surface is defined by p + δ with p moved rigidly by the skeleton or other underlying system, and where configuration is the configuration of the set of joints or parameters controlled by the animator. Our scheme can be bootstrapped on top of an existing software system: the model is posed as desired and the desired surface at that pose is sculpted. Our algorithm computes the difference between the initial and resculpted model at that pose. This ‘deformation’ is associated with the joints or other parameters that have moved from their default positions to create the particular pose. One or more deformations will then be interpolated in this subspace using a scattered data approach. We now have enough criteria to select a particular interpolation scheme. Although it would be desirable to allow deformations to change both continuously and discontinuously with respect to the pose space, creature deformations that are discontinuous with respect to pose seem unlikely. As such the expensive energy functional and non-convex schemes are not necessary. In addition we want δ to approach zero away from the data, and the width of this falloff should be selectable. Together these comments support φ k (x) = exp( −( x−x 2σ 2 k ) 2 ) as one possible choice of radial basis ( Figure 5 ). Gaussian radial basis functions are reputed to be well behaved and our experience supports this judgement. Gaussian radial basis functions with adjustable placement and σ are discussed in the neural net literature and optimizing over these parameters is possible. This issue does not arise in our application, however, since the animator decides where in the parameter space to sculpt a pose (effectively deciding the basis function placement). The falloff σ is also specified explicitly by the animator, as described below.
δ = f interp (configuration)
<H2>4.1 Algorithm Summary</H2>
The steps in a pose space deformation (PSD) algorithm will now be described consecutively. Definitions. A pose is defined as the configuration of any pose controls (joints or abstract manipulators) that have changed from their default values. An abstract manipulator is a UI control or arbitrary piece of geometry whose movement will control the interpolation of some deformation, such as a muscle bulge or a desired facial attribute such as “happiness.” A self-relative configuration of the controls is actually considered, for example, an elbow involves two skeletal frames but only one joint angle. The pose space is the space spanned by the variations of these controls. If n = 2 pose controls are active and each has three degrees of freedom then a 3(n − 1) pose space is defined, and the particular position of the controls defines a point in that space. Sculpt. The artist first positions some set of pose controls and then sculpts a deformation for that pose. The artist also assigns a falloff (Gaussian σ), either as a symmetric radius across all controls or to each control individually (axis stretched falloff). Define δ(pose). Any control vertices that have moved from their rest position are found. This is done in the local coordinate frame, i.e., rigid body articulated motion results in zero δ. The δ values for the deformed vertices are computed (again in the local coordinate system) and they are saved in a database together with their corresponding location in a pose space. (At the boundary of several surface patches there may be shared vertices that need to be coincident to maintain surface continuity. Unlike some SSD implementations interpolation in pose space by definition cannot separate such vertices). Solve. When several such deformations have been saved (or when the artist is ready to try animating) it is necessary to solve the interpolation problem. For each control vertex that was moved during sculpting there are now one or more δ values at points in the pose space. Note that the dimension of the pose space can vary across vertices, for example, a particular vertex might be modified in three sculpted deformations but a neighboring vertex might have been modified in only two deformations. The interpolation is done independently for each control vertex (but see additional details below); in our experience using patch surfaces this has not been problematic. Singular Φ T Φ is interpreted as a user error; in practice this has turned out to be the result of saving new deformations without moving any pose controls rather than a result of actual numerical problems. Synthesis. The model is now moved to an arbitrary pose. The location in pose space is determined from the concatenated relative degrees of freedom of the pose controls (simply interpreted as independent dimensions). For each deforming control vertex a δ is interpolated from the delta values at the stored poses using Eq. (1). Evaluate and Repeat. At this point the model interpolates through the previously defined deformation(s). The most recently defined deformation may extend too far (or not far enough) in pose space, however. There is a rich literature of schemes for optimizing radial basis parameters including σ
[ 4 ]. On the other hand, animators consider detailed control of the animation to be part of their craft and are quite happy to have interpolation parameters exposed to them. We have found that this potentially abstract parameter is comprehensible so long as it is possible to explore the effect of different values. At a minimum axis-aligned scaling of the falloff should be available; we have not experimented with Mahalanobis rotation of the basis. Based on the evaluation the artist may decide to sculpt additional poses as needed to achieve the desired motion. A detail that was omitted previously will now be mentioned: when a deformed vertex is found the associated pose space is determined as described above. If there are previous deformations of this vertex in the same pose space then the new deformation is simply another point to interpolate. The new deformation’s pose space may, however, be different from the previous spaces associated with the vertex! In such a case a new pose space is started, and the δ is computed as a delta from the previous layered PSD synthesis rather than from the base model. This ensures that the previous deformations are interpolated while allowing the artist complete freedom in determining the extent of the deformation and the associated pose controls. While there is an issue of commutativity, in our experience artists consider this iterative layered refinement to be a natural process. In the preceeding discussion we have not described the representation of rotations and other transformations. This is a well known issue; well behaved transformations are fundamental and are hopefully addressed early in the development of any character animation system.
<H2>4.2 Cost</H2>
With n poses three matrices of size n must be inverted for each surface control vertex. Typically n will be between 1 and 10, say, so this cost is small. Also it is incurred at a convenient time – during setup (as a pose is saved) rather than during synthesis. For synthesis, the basis function φ(x) can be implemented by interpolated table lookup and the sqrt required in the Euclidean distance can be composed with φ(x) in the table. The cost of Eq. (1) is then not much greater than the cost of traditional shape interpolation, i.e., real time synthesis is possible with significant models on current machines.
<H1>5 Applications and Discussion 5.1 PSD for Skeleton-Driven Deformation</H1>
An articulated model such as a human will typically have a number of different deformation subspaces, each with one or several deformations; the deformations in different subspaces may overlap spatially e.g. to simulate the influence of different muscles. The deformations needed for an elbow, for example, will be interpolated in the one-dimensional subspace defined by the elbow joint angle. Deformations in a shoulder area will need to consider two or more degrees of freedom. The neck/chest/leg blend area of many quadrupeds is a more complex case – the motion of the skin surface in these regions may depend on the relative configuration of several leg bones as well as the rib cage and possibly the neck region of the spine. PSD handles all these cases simply and uniformly. Figures 8 and 9 are a simple comparison of PSD and SSD algorithms in action on human elbow and shoulder regions.
<H2>5.2 PSD for Facial Animation</H2>
The application of PSD to facial animation is best described by comparison with shape interpolation (SI). • In both approaches a set of key shapes (or delta shapes) are sculpted. The same set of shapes can be used in both approaches. • Whereas shape interpolation is (despite the name) a superposition of a set of shapes, PSD interpolates among these shapes. • The animator’s task in PSD is to choose the interpolation path (and adjust interpolation parameters such as falloff if desired). With SI the animator’s task is to choose the interpolation path but also solve the problem of representing this path by using a set of (non-orthogonal!) basis shapes. In practice this has been considered the major difficulty in applying SI when high quality animation demands large numbers of basis shapes
[ 38 ]. • In shape interpolation the key shapes and the animation parameter space are one and the same – the keys define the axes of the animation parameter space. In PSD the key shapes are positioned as desired in a space of desired dimensionality. One can assign each shape in PSD to a separate dimension, exactly as with SI. On the other hand, PSD allows one to sculpt intermediate expressions (half-smile) and situate them half-way along the relevant (full-smile) axis. Similarly a sculpted pose that represents the simultaneous activation of several parameters (e.g. happy but surprised, or smiling with a wink) can simply be saved at the appropriate location in the pose space. Psychological research has shown that human facial expressions are largely described by two “emotional” axes
[ 30 ] ( Figure 6 ); this two-dimensional space would be a convenient high-level pose space for controlling facial animation. • The PSD interpolation is smooth if so desired. To illustrate these comments consider Figure 7 , which abstractly represents both SI and PSD with an identical set of expressions (neutral, half-smile, full-smile, frown). In the SI side of the diagram expressions are arranged as independent (but not orthogonal) dimensions as required by SI. In the PSD diagram the expressions are situated in an expression space having a happy-unhappy axis; a second axis (arousal) and an expression (delighted) on that axis are added to show a multidimensional space. As illustrated, a PSD path from neutral to half-smile to full-smile is monotonic, as might be expected; the motion of a surface point over this interpolation is also smooth. To interpolate these emotions using SI requires a zig-zag pattern of weights: the half-smile weight goes from zero to one, and then back to zero as the full-smile weight goes from zero to one.The motion of a surface point would also be piecewise linear using SI in this scenario.
<H2>5.3 PSD for Secondary Animation</H2>
Switches and dials. Additional “dimensions” of deformation can be added at any time by adding a new parameter and associating additional poses with the movement of this parameter. For example, a limb can be modeled in a particular pose both in an unloaded state and with muscles sculpted to express carrying a heavy load. The ‘heavy’ pose can be associated with the ‘on’ state of an abstract parameter (e.g. an isolated bone moved into the vertical position); light and heavy loads can then be controlled by flipping this switch. Similarly one can imagine setting up a dial that causes the character to morph; this would of course require a significant set of additional deformation poses.
<H1>6 Conclusions</H1>
Pose space deformation is not the last word in surface deformation for character animation; high quality anatomically based models are certainly preferable. Nevertheless both anatomically based and purely kinematic models have their place. In the current computer animation culture animators generally practice their craft by direct and exhaustive specification of the desired motion combined with quick evaluation using real-time playback. Deeper simulation approaches intrinsically take away some of this control, and animators often argue (rightly or not) that automated processes are inferior or will not produce a human feel. The performance of current anatomically based models prohibits animation preview and other real-time applications such as telepresence and gaming (one published result is several orders of magnitude slower than real time), and the effort needed to produce an anatomically accurate model is not always justified, nor even appropriate if the model is of a fanciful creature whose surface appearance may be inconsistent with any plausible internal anatomy in any case. PSD unifies and improves upon two techniques that have been common graphics practice for more than a decade. This relatively simple algorithm uniformly handles a variety of deformation situations ranging from a simple elbow to secondary animation. The setup cost of the algorithm is insignificant, and the synthesis cost is only slightly more than that of shape interpolation, so real-time synthesis is possible at effective resolutions on current hardware. We expect that this algorithm will be a useful complement to current techniques.
alarmed excited afraid astonished angry delighted annoyed frustrated happy pleased content serene, miserable calm depressed relaxed bored tired sleepy
Figure 6: Schematic diagram of emotion space obtained by multi-dimensional scaling from pairwise similarity ratings, simplified from [ 30 ].
N=neutral S=smile F H F=frown H=half−smile arousal delighted F N H S N S happiness Shape Interpolation Pose Space Deformation
Figure 7: Abstract comparison of shape interpolation versus pose space deformation using the same set of facial expressions.
<H1>Acknowledgements</H1>
The authors wish to thank Sean Jenkins, David Miya, Amir Nasrabadi, Steven Puri, Kosta Saric, Steffen Wild, Lance Williams, Raymond Yeung, and the anonymous reviewers.
<H1>References</H1>
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Figure 8a. Comparison of PSD and SSD on an animating shoulder – PSD using only two sculpted poses.
Figure 8b. SSD on an animating shoulder. The shoulder area is especially problematic for SSD due to the large range of rotational movement.
Figure 10. Smooth interpolation of four expressions (frown, neutral, smirk, smile) arranged along a single axis in a pose space, c.f. the discussion of Figure 7 .
facial animation, personal
Figure 9. Comparison of PSD (at left) and SSD on the extreme pose of an elbow.
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"text": "['The first major shortcoming of SSD results directly from the fact that the deformation is restricted to the indicated subspace']",
"type": "background_claim",
"id": "T147"
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14932,
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],
"text": "['In common situations such as shoulders and elbows the desired deformation does not lie in this subspace']",
"type": "background_claim",
"id": "T148"
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{
"offsets": [
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15043,
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],
"text": "['no amount of adjusting the algorithm weights will produce good results']",
"type": "background_claim",
"id": "T149"
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15115,
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],
"text": "['This fact leads to considerable frustration by users of the algorithm']",
"type": "background_claim",
"id": "T150"
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{
"offsets": [
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15187,
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],
"text": "['the character of the deformation changes as the weights are changed, sometimes sustaining the incorrect assumption that some combination of weights will produce good results']",
"type": "background_claim",
"id": "T151"
},
{
"offsets": [
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15371,
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],
"text": "['the SSD algorithm can be easily identified in animations by its characteristic ‘collapsing joint’ defect']",
"type": "background_claim",
"id": "T152"
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{
"offsets": [
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],
"text": "['Figures 1, 2']",
"type": "data",
"id": "T153"
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"offsets": [
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15492,
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],
"text": "['This problem is extreme in the case of simulating the twist of a human forearm']",
"type": "background_claim",
"id": "T154"
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"offsets": [
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15613,
15621
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],
"text": "['Figure 3']",
"type": "data",
"id": "T155"
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"offsets": [
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15965,
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],
"text": "['A second difficulty with SSD is that']",
"type": "background_claim",
"id": "T156"
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16031,
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],
"text": "['it does not permit direct manipulation']",
"type": "background_claim",
"id": "T157"
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{
"offsets": [
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16071,
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],
"text": "['artists instead directly or indirectly edit the meshes of weights w k']",
"type": "background_claim",
"id": "T158"
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"offsets": [
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],
"text": "['SSD algorithms consequently have the reputation for being tedious and difficult to control']",
"type": "background_claim",
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16336,
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],
"text": "['Artists with a poor understanding of the underlying algorithm have difficulty distinguishing between results that can be further improved by adjusting weights and results that cannot be improved since the desired result lies outside the achievable subspace, resulting in the impression of unpredictability']",
"type": "background_claim",
"id": "T160"
},
{
"offsets": [
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16707,
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],
"text": "['In some cases the SSD defects can be manually corrected using FFDs and other techniques']",
"type": "background_claim",
"id": "T161"
},
{
"offsets": [
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16800,
16900
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],
"text": "['one could consider a scheme whereby these fixes are procedurally invoked as the skeleton articulates']",
"type": "background_claim",
"id": "T162"
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{
"offsets": [
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16915,
16929
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],
"text": "['FFDs work well']",
"type": "background_claim",
"id": "T163"
},
{
"offsets": [
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16982,
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],
"text": "['the layered FFDs do not reduce the difficulty in adjusting the underlying SSD']",
"type": "background_claim",
"id": "T164"
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{
"offsets": [
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17061,
17225
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],
"text": "['The algorithm introduced in the subsequent sections removes the need for such layered fix-it approaches and permits direct specification of the desired deformations']",
"type": "own_claim",
"id": "T165"
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{
"offsets": [
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17484,
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],
"text": "['Several published algorithms and commercial packages combine aspects of skeleton-driven deformation and shape interpolation in ways that anticipate our approach']",
"type": "background_claim",
"id": "T166"
},
{
"offsets": [
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17646,
17819
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],
"text": "['In the pioneering work of Burtnyk and Wein, two dimensional characters were animated using a polygonal rubber sheet that afforded both skeletal and local deformation control']",
"type": "background_claim",
"id": "T167"
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{
"offsets": [
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17823,
17824
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],
"text": "['6']",
"type": "data",
"id": "T168"
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"offsets": [
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17828,
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],
"text": "['Van Overveld described a two-dimensional animation system in which animation is controlled by a skeleton and character deformation is driven from this skeleton through a scattered interpolation']",
"type": "background_claim",
"id": "T169"
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{
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18025,
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],
"text": "['20']",
"type": "data",
"id": "T170"
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"offsets": [
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],
"text": "['This work is similar in spirit to ours']",
"type": "own_claim",
"id": "T171"
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{
"offsets": [
[
18074,
18183
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],
"text": "['differs in that it used the image plane as a global interpolation domain rather than introducing a pose space']",
"type": "own_claim",
"id": "T172"
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{
"offsets": [
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18185,
18218
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],
"text": "['Litwinowicz and Williams’s system']",
"type": "background_claim",
"id": "T173"
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"offsets": [
[
18221,
18223
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],
"text": "['16']",
"type": "data",
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"offsets": [
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18227,
18299
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],
"text": "['is also a precedent and introduced sophisticated scattered interpolation']",
"type": "background_claim",
"id": "T175"
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{
"offsets": [
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18329,
18362
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],
"text": "['Several papers consider animation']",
"type": "background_claim",
"id": "T176"
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"offsets": [
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18403,
18475
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],
"text": "['as a special case of neural net learning and interpolation/extrapolation']",
"type": "background_claim",
"id": "T177"
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18479,
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"text": "['14']",
"type": "data",
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18484,
18486
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],
"text": "['15']",
"type": "data",
"id": "T179"
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18489,
18491
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],
"text": "['21']",
"type": "data",
"id": "T180"
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18524
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],
"text": "['this viewpoint is valid']",
"type": "background_claim",
"id": "T181"
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{
"offsets": [
[
18526,
18571
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],
"text": "['in practice it is perhaps excessively general']",
"type": "background_claim",
"id": "T182"
},
{
"offsets": [
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18586,
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],
"text": "['a skeleton is merely learned rather than being an intrinsic part of the model']",
"type": "background_claim",
"id": "T183"
},
{
"offsets": [
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18665,
18829
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],
"text": "['While employed at Industrial Light and Magic the first author of the present paper developed a system that attempted to blend shape interpolation and SSD algorithms']",
"type": "background_claim",
"id": "T184"
},
{
"offsets": [
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18831,
18915
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],
"text": "['a small portion of it remains in use in their well known Caricature animation system']",
"type": "background_claim",
"id": "T185"
},
{
"offsets": [
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18917,
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],
"text": "['Drawbacks of this work included both a complicated dependence on the details of SSD and its overall conception as a “correction” to SSD']",
"type": "background_claim",
"id": "T186"
},
{
"offsets": [
[
19054,
19168
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],
"text": "['Some commercial packages allow blending between two sculpted deformations as a function of a single-joint rotation']",
"type": "background_claim",
"id": "T187"
},
{
"offsets": [
[
19178,
19271
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],
"text": "['combining shape interpolation and skeleton-driven deformation in a limited but useful setting']",
"type": "background_claim",
"id": "T188"
},
{
"offsets": [
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19569,
19634
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],
"text": "['The depth of simulation is a prevalent issue in computer graphics']",
"type": "background_claim",
"id": "T189"
},
{
"offsets": [
[
19643,
19688
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],
"text": "['one that is not always consciously considered']",
"type": "background_claim",
"id": "T190"
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{
"offsets": [
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19690,
19741
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],
"text": "['Early approaches to animation were purely kinematic']",
"type": "background_claim",
"id": "T191"
},
{
"offsets": [
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19743,
19816
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],
"text": "['an emphasis on physically based modeling appeared in the literature later']",
"type": "background_claim",
"id": "T192"
},
{
"offsets": [
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19818,
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],
"text": "['Recent sophisticated approaches allow a hybrid of animator-controlled and physically governed animation as needed']",
"type": "background_claim",
"id": "T193"
},
{
"offsets": [
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19933,
19979
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],
"text": "['In rendering we perhaps see the opposite trend']",
"type": "background_claim",
"id": "T194"
},
{
"offsets": [
[
19982,
20063
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],
"text": "['much of the literature a decade ago focused on ever deeper simulations of reality']",
"type": "background_claim",
"id": "T195"
},
{
"offsets": [
[
20073,
20142
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],
"text": "['‘shallower’ imagebased approaches are attracting attention at present']",
"type": "background_claim",
"id": "T196"
},
{
"offsets": [
[
20155,
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],
"text": "['in character deformation both deep and shallow approaches have their place']",
"type": "background_claim",
"id": "T197"
},
{
"offsets": [
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20231,
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],
"text": "['Deep models promise universally accurate simulation']",
"type": "background_claim",
"id": "T198"
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{
"offsets": [
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20288,
20353
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],
"text": "['the importance of representing humans justifies the needed effort']",
"type": "background_claim",
"id": "T199"
},
{
"offsets": [
[
20355,
20462
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],
"text": "['The authors of these approaches acknowledge that producing anatomically plausible models is a daunting task']",
"type": "background_claim",
"id": "T200"
},
{
"offsets": [
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20473,
20550
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],
"text": "['Pose space deformation is a shallow, purely kinematic approach to deformation']",
"type": "background_claim",
"id": "T201"
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{
"offsets": [
[
20612,
20647
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],
"text": "['and it has consequent disadvantages']",
"type": "background_claim",
"id": "T202"
},
{
"offsets": [
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20664,
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],
"text": "['accuracy is reliant on the modeler/animator rather than being guaranteed by the simulation']",
"type": "background_claim",
"id": "T203"
},
{
"offsets": [
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20775,
20940
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],
"text": "['our algorithm has clear advantages with respect to simplicity and generality, direct manipulation, real-time synthesis, and other criteria listed in the introduction']",
"type": "own_claim",
"id": "T204"
},
{
"offsets": [
[
21515,
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],
"text": "['A scattered data interpolation method is required']",
"type": "own_claim",
"id": "T205"
},
{
"offsets": [
[
21573,
21615
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],
"text": "['deformations will be sculpted at arbitrary']",
"type": "own_claim",
"id": "T206"
},
{
"offsets": [
[
21647,
21652
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],
"text": "['poses']",
"type": "own_claim",
"id": "T207"
},
{
"offsets": [
[
21660,
21708
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],
"text": "['this interpolation is central to our application']",
"type": "own_claim",
"id": "T208"
},
{
"offsets": [
[
21710,
21796
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],
"text": "['the results of the interpolation will be directly visible in the animating deformation']",
"type": "own_claim",
"id": "T209"
},
{
"offsets": [
[
21799,
21895
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],
"text": "['we will consider the available scattered interpolation approaches before settling on a candidate']",
"type": "own_claim",
"id": "T210"
},
{
"offsets": [
[
21976,
21977
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],
"text": "['1']",
"type": "data",
"id": "T211"
},
{
"offsets": [
[
21980,
21981
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],
"text": "['2']",
"type": "data",
"id": "T212"
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{
"offsets": [
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21957,
21973
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],
"text": "['Shepard’s method']",
"type": "background_claim",
"id": "T213"
},
{
"offsets": [
[
21985,
22066
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],
"text": "['is a frequently employed scattered data interpolation scheme in computer graphics']",
"type": "background_claim",
"id": "T214"
},
{
"offsets": [
[
22710,
22772
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],
"text": "['this simple scheme has some potentially undesirable properties']",
"type": "background_claim",
"id": "T215"
},
{
"offsets": [
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22946,
23051
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],
"text": "['A serious drawback for some applications is that the derivative of the surface is zero at the data points']",
"type": "background_claim",
"id": "T216"
},
{
"offsets": [
[
23054,
23062
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],
"text": "['Figure 4']",
"type": "data",
"id": "T217"
},
{
"offsets": [
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22774,
22868
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],
"text": "['Far from the data the weights will be approximately the same, d(∞) ˆ = w ∞ d k /w ∞ 1 = d k /N']",
"type": "background_claim",
"id": "T218"
},
{
"offsets": [
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22876,
22944
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],
"text": "['the interpolated surface converges to the average of the data values']",
"type": "background_claim",
"id": "T219"
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{
"offsets": [
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23132,
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],
"text": "['Radial basis functions']",
"type": "background_claim",
"id": "T220"
},
{
"offsets": [
[
23158,
23160
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],
"text": "['28']",
"type": "data",
"id": "T221"
},
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"offsets": [
[
23163,
23165
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],
"text": "['29']",
"type": "data",
"id": "T222"
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23169,
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"text": "['have become a popular choice for scattered interpolation']",
"type": "background_claim",
"id": "T223"
},
{
"offsets": [
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23405,
23432
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],
"text": "['N values of d are available']",
"type": "data",
"id": "T224"
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"offsets": [
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23438,
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],
"text": "['the weights can be easily solved by a linear system']",
"type": "background_claim",
"id": "T225"
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23491,
23564
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],
"text": "['this can be derived either by least squares fit or by subspace projection']",
"type": "background_claim",
"id": "T226"
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{
"offsets": [
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24125,
24179
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],
"text": "['A least squares approach leads to the identical result']",
"type": "background_claim",
"id": "T227"
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{
"offsets": [
[
24181,
24233
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],
"text": "['Any nonlinear function φ() will interpolate the data']",
"type": "background_claim",
"id": "T228"
},
{
"offsets": [
[
24396,
24444
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],
"text": "['a smooth φ() will result in a smooth interpolant']",
"type": "background_claim",
"id": "T229"
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"offsets": [
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24509,
24595
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"text": "['radial basis functions have a universal convergence property similar to Fourier series']",
"type": "background_claim",
"id": "T230"
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{
"offsets": [
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24604,
24643
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"text": "['the convergence definition is different']",
"type": "background_claim",
"id": "T231"
},
{
"offsets": [
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24050,
24116
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],
"text": "['Φ T Φw = Φ T d can be solved for the familiar w = (Φ T Φ) −1 Φ T d']",
"type": "background_claim",
"id": "T232"
},
{
"offsets": [
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23743,
23808
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],
"text": "['The best approximation to d in the space spanned by φ k () occurs']",
"type": "background_claim",
"id": "T233"
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{
"offsets": [
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23861,
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],
"text": "['when the weights are such that the error d − Φw']",
"type": "background_claim",
"id": "T234"
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{
"offsets": [
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23951,
24004
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],
"text": "['is orthogonal to each of the φ k (): Φ T (Φw − d) = 0']",
"type": "background_claim",
"id": "T235"
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{
"offsets": [
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24344,
24376
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],
"text": "['the columns of Φ are independent']",
"type": "data",
"id": "T236"
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"offsets": [
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24803,
24815
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],
"text": "['Surfaces can']",
"type": "background_claim",
"id": "T237"
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24826,
24928
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"text": "['be interpolated by allowing different combinations of the same basis functions in different dimensions']",
"type": "background_claim",
"id": "T238"
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{
"offsets": [
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24956,
25011
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],
"text": "['The distance can be generalized to Mahalanobis distance']",
"type": "background_claim",
"id": "T239"
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{
"offsets": [
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25072,
25073
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"text": "['4']",
"type": "data",
"id": "T240"
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"offsets": [
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25162,
25247
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"text": "['Various visual reconstruction schemes can be adapted for scattered data interpolation']",
"type": "background_claim",
"id": "T241"
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{
"offsets": [
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25569,
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],
"text": "['With small λ many of these schemes can serve as scattered data interpolants']",
"type": "background_claim",
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25658,
25659
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],
"text": "['5']",
"type": "data",
"id": "T243"
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"offsets": [
[
25663,
25705
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],
"text": "['is a good introduction to these approaches']",
"type": "background_claim",
"id": "T244"
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{
"offsets": [
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25707,
25858
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],
"text": "['In some of the most powerful formulations of scattered interpolation the regularizer is considered to hold everywhere except at an unknown set of edges']",
"type": "background_claim",
"id": "T245"
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{
"offsets": [
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25861,
25929
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"text": "['this is the piecewise-smooth prior desirable in image reconstruction']",
"type": "background_claim",
"id": "T246"
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{
"offsets": [
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25937,
25964
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],
"text": "['the unknown edges may exist']",
"type": "background_claim",
"id": "T247"
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{
"offsets": [
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25980,
26009
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],
"text": "['at any location in the domain']",
"type": "background_claim",
"id": "T248"
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{
"offsets": [
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26011,
26073
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],
"text": "['all combinations of possible edge locations must be considered']",
"type": "background_claim",
"id": "T249"
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"offsets": [
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26078,
26153
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],
"text": "['the interpolation cost is prima facie exponential in the surface resolution']",
"type": "background_claim",
"id": "T250"
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{
"offsets": [
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26261,
26357
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],
"text": "['The crux of our approach is the identification of an appropriate space for defining deformations']",
"type": "own_claim",
"id": "T251"
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{
"offsets": [
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26379,
26524
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],
"text": "['the interpolation domain is (a subset of) the pose space of an articulated character, or equivalently the space defined by some set of parameters']",
"type": "own_claim",
"id": "T252"
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{
"offsets": [
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26550,
26669
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],
"text": "['In concept the range of the interpolation function could simply be the desired movement of the surface control vertices']",
"type": "own_claim",
"id": "T253"
},
{
"offsets": [
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26985,
27026
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],
"text": "['the deforming surface is defined by p + δ']",
"type": "own_claim",
"id": "T254"
},
{
"offsets": [
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26716,
26780
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],
"text": "['we instead interpolate the desired deviation of a surface vertex']",
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"id": "T255"
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{
"offsets": [
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26812,
26848
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],
"text": "['from its initially computed position']",
"type": "own_claim",
"id": "T256"
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{
"offsets": [
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27200,
27268
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],
"text": "['Our scheme can be bootstrapped on top of an existing software system']",
"type": "own_claim",
"id": "T257"
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{
"offsets": [
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27270,
27348
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],
"text": "['the model is posed as desired and the desired surface at that pose is sculpted']",
"type": "own_claim",
"id": "T258"
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{
"offsets": [
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27771,
27893
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],
"text": "['it would be desirable to allow deformations to change both continuously and discontinuously with respect to the pose space']",
"type": "own_claim",
"id": "T259"
},
{
"offsets": [
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27895,
27974
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],
"text": "['creature deformations that are discontinuous with respect to pose seem unlikely']",
"type": "own_claim",
"id": "T260"
},
{
"offsets": [
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27984,
28056
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],
"text": "['the expensive energy functional and non-convex schemes are not necessary']",
"type": "own_claim",
"id": "T261"
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28169,
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],
"text": "['Together these comments support φ k (x) = exp( −( x−x 2σ 2 k ) 2 ) as one possible choice of radial basis']",
"type": "own_claim",
"id": "T262"
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"text": "['the width of this falloff should be selectable']",
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],
"text": "['Figure 5']",
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"text": "['Gaussian radial basis functions are reputed to be well behaved']",
"type": "background_claim",
"id": "T263"
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],
"text": "['Gaussian radial basis functions with adjustable placement and σ are discussed in the neural net literature']",
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"text": "['optimizing over these parameters is possible']",
"type": "background_claim",
"id": "T267"
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],
"text": "['This issue does not arise in our application']",
"type": "own_claim",
"id": "T268"
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],
"text": "['the animator decides where in the parameter space to sculpt a pose']",
"type": "own_claim",
"id": "T269"
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],
"text": "['a 3(n − 1) pose space is defined']",
"type": "own_claim",
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],
"text": "['n = 2 pose controls are active']",
"type": "data",
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],
"text": "['each has three degrees of freedom']",
"type": "data",
"id": "T272"
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],
"text": "['the particular position of the controls defines a point in that space']",
"type": "own_claim",
"id": "T273"
},
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],
"text": "['Any control vertices that have moved from their rest position are found']",
"type": "own_claim",
"id": "T274"
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"offsets": [
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],
"text": "['it is necessary to solve the interpolation problem']",
"type": "own_claim",
"id": "T275"
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[
30657,
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],
"text": "['several such deformations have been saved']",
"type": "data",
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},
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[
30798,
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],
"text": "['For each control vertex that was moved during sculpting there are now one or more δ values at points in the pose space']",
"type": "own_claim",
"id": "T277"
},
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],
"text": "['the dimension of the pose space can vary across vertices']",
"type": "own_claim",
"id": "T278"
},
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],
"text": "['a particular vertex might be modified in three sculpted deformations but a neighboring vertex might have been modified in only two deformations']",
"type": "data",
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[
31144,
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],
"text": "['The interpolation is done independently for each control vertex']",
"type": "own_claim",
"id": "T280"
},
{
"offsets": [
[
31283,
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],
"text": "['this has not been problematic']",
"type": "own_claim",
"id": "T281"
},
{
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[
31262,
31282
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],
"text": "['using patch surfaces']",
"type": "data",
"id": "T282"
},
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"offsets": [
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31361,
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],
"text": "['in practice this has turned out to be the result of saving new deformations without moving any pose controls rather than a result of actual numerical problems']",
"type": "own_claim",
"id": "T283"
},
{
"offsets": [
[
31314,
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],
"text": "['Singular Φ T Φ is interpreted as a user error']",
"type": "own_claim",
"id": "T284"
},
{
"offsets": [
[
31737,
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],
"text": "['For each deforming control vertex a δ is interpolated from the delta values at the stored poses using Eq. (1)']",
"type": "own_claim",
"id": "T285"
},
{
"offsets": [
[
31869,
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],
"text": "['At this point the model interpolates through the previously defined deformation(s)']",
"type": "own_claim",
"id": "T286"
},
{
"offsets": [
[
31953,
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],
"text": "['The most recently defined deformation may extend too far (or not far enough) in pose space']",
"type": "own_claim",
"id": "T287"
},
{
"offsets": [
[
32054,
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],
"text": "['There is a rich literature of schemes for optimizing radial basis parameters including σ']",
"type": "background_claim",
"id": "T288"
},
{
"offsets": [
[
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],
"text": "['4']",
"type": "data",
"id": "T289"
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"offsets": [
[
32170,
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],
"text": "['animators consider detailed control of the animation to be part of their craft and are quite happy to have interpolation parameters exposed to them']",
"type": "background_claim",
"id": "T290"
},
{
"offsets": [
[
32338,
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],
"text": "['this potentially abstract parameter is comprehensible']",
"type": "own_claim",
"id": "T291"
},
{
"offsets": [
[
32403,
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],
"text": "['it is possible to explore the effect of different values']",
"type": "data",
"id": "T292"
},
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"offsets": [
[
32461,
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],
"text": "['At a minimum axis-aligned scaling of the falloff should be available']",
"type": "own_claim",
"id": "T293"
},
{
"offsets": [
[
32596,
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],
"text": "['Based on the evaluation the artist may decide to sculpt additional poses as needed to achieve the desired motion']",
"type": "own_claim",
"id": "T294"
},
{
"offsets": [
[
32802,
32841
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],
"text": "['the associated pose space is determined']",
"type": "own_claim",
"id": "T295"
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"offsets": [
[
32775,
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],
"text": "['a deformed vertex is found']",
"type": "data",
"id": "T296"
},
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"offsets": [
[
32865,
32934
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],
"text": "['there are previous deformations of this vertex in the same pose space']",
"type": "data",
"id": "T297"
},
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"offsets": [
[
32940,
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]
],
"text": "['the new deformation is simply another point to interpolate']",
"type": "own_claim",
"id": "T298"
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"offsets": [
[
33000,
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],
"text": "['The new deformation’s pose space may']",
"type": "own_claim",
"id": "T299"
},
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"offsets": [
[
33047,
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],
"text": "['be different from the previous spaces associated with the vertex']",
"type": "own_claim",
"id": "T300"
},
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"offsets": [
[
33128,
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],
"text": "['a new pose space is started']",
"type": "own_claim",
"id": "T301"
},
{
"offsets": [
[
33161,
33261
]
],
"text": "['the δ is computed as a delta from the previous layered PSD synthesis rather than from the base model']",
"type": "own_claim",
"id": "T302"
},
{
"offsets": [
[
33263,
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]
],
"text": "['This ensures that the previous deformations are interpolated while allowing the artist complete freedom in determining the extent of the deformation and the associated pose controls']",
"type": "own_claim",
"id": "T303"
},
{
"offsets": [
[
33452,
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],
"text": "['there is an issue of commutativity']",
"type": "own_claim",
"id": "T304"
},
{
"offsets": [
[
33506,
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]
],
"text": "['artists consider this iterative layered refinement to be a natural process']",
"type": "own_claim",
"id": "T305"
},
{
"offsets": [
[
33692,
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],
"text": "['This is a well known issue']",
"type": "background_claim",
"id": "T306"
},
{
"offsets": [
[
33720,
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],
"text": "['well behaved transformations are fundamental and are hopefully addressed early in the development of any character animation system']",
"type": "own_claim",
"id": "T307"
},
{
"offsets": [
[
33901,
33987
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],
"text": "['With n poses three matrices of size n must be inverted for each surface control vertex']",
"type": "own_claim",
"id": "T308"
},
{
"offsets": [
[
33989,
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],
"text": "['Typically n will be between 1 and 10']",
"type": "own_claim",
"id": "T309"
},
{
"offsets": [
[
34035,
34053
]
],
"text": "['this cost is small']",
"type": "own_claim",
"id": "T310"
},
{
"offsets": [
[
34060,
34095
]
],
"text": "['it is incurred at a convenient time']",
"type": "own_claim",
"id": "T311"
},
{
"offsets": [
[
34098,
34110
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],
"text": "['during setup']",
"type": "data",
"id": "T312"
},
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"offsets": [
[
34337,
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],
"text": "['The cost of Eq. (1) is then not much greater than the cost of traditional shape interpolation']",
"type": "own_claim",
"id": "T313"
},
{
"offsets": [
[
34162,
34248
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],
"text": "['For synthesis, the basis function φ(x) can be implemented by interpolated table lookup']",
"type": "own_claim",
"id": "T314"
},
{
"offsets": [
[
34253,
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],
"text": "['the sqrt required in the Euclidean distance can be composed with φ(x) in the table']",
"type": "own_claim",
"id": "T315"
},
{
"offsets": [
[
34438,
34513
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],
"text": "['real time synthesis is possible with significant models on current machines']",
"type": "own_claim",
"id": "T316"
},
{
"offsets": [
[
34624,
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]
],
"text": "['An articulated model such as a human will typically have a number of different deformation subspaces, each with one or several deformations']",
"type": "own_claim",
"id": "T317"
},
{
"offsets": [
[
34765,
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],
"text": "['the deformations in different subspaces may overlap spatially']",
"type": "own_claim",
"id": "T318"
},
{
"offsets": [
[
35104,
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]
],
"text": "['The neck/chest/leg blend area of many quadrupeds is a more complex case']",
"type": "own_claim",
"id": "T322"
},
{
"offsets": [
[
35178,
35353
]
],
"text": "['the motion of the skin surface in these regions may depend on the relative configuration of several leg bones as well as the rib cage and possibly the neck region of the spine']",
"type": "own_claim",
"id": "T323"
},
{
"offsets": [
[
35355,
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],
"text": "['PSD handles all these cases simply and uniformly']",
"type": "own_claim",
"id": "T319"
},
{
"offsets": [
[
35586,
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],
"text": "['The application of PSD to facial animation is best described by comparison with shape interpolation']",
"type": "own_claim",
"id": "T320"
},
{
"offsets": [
[
35694,
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],
"text": "['In both approaches a set of key shapes (or delta shapes) are sculpted']",
"type": "own_claim",
"id": "T321"
},
{
"offsets": [
[
35765,
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],
"text": "['The same set of shapes can be used in both approaches']",
"type": "own_claim",
"id": "T324"
},
{
"offsets": [
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35830,
35852
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],
"text": "['shape interpolation is']",
"type": "background_claim",
"id": "T325"
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"offsets": [
[
35872,
35906
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],
"text": "['a superposition of a set of shapes']",
"type": "background_claim",
"id": "T326"
},
{
"offsets": [
[
35908,
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],
"text": "['PSD interpolates among these shapes']",
"type": "own_claim",
"id": "T327"
},
{
"offsets": [
[
35947,
36009
]
],
"text": "['The animator’s task in PSD is to choose the interpolation path']",
"type": "own_claim",
"id": "T328"
},
{
"offsets": [
[
36076,
36241
]
],
"text": "['With SI the animator’s task is to choose the interpolation path but also solve the problem of representing this path by using a set of (non-orthogonal!) basis shapes']",
"type": "background_claim",
"id": "T329"
},
{
"offsets": [
[
36243,
36315
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],
"text": "['In practice this has been considered the major difficulty in applying SI']",
"type": "background_claim",
"id": "T330"
},
{
"offsets": [
[
36321,
36381
]
],
"text": "['high quality animation demands large numbers of basis shapes']",
"type": "data",
"id": "T331"
},
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"offsets": [
[
36385,
36387
]
],
"text": "['38']",
"type": "data",
"id": "T332"
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"offsets": [
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36393,
36485
]
],
"text": "['In shape interpolation the key shapes and the animation parameter space are one and the same']",
"type": "background_claim",
"id": "T333"
},
{
"offsets": [
[
36488,
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]
],
"text": "['the keys define the axes of the animation parameter space']",
"type": "background_claim",
"id": "T334"
},
{
"offsets": [
[
36547,
36631
]
],
"text": "['In PSD the key shapes are positioned as desired in a space of desired dimensionality']",
"type": "own_claim",
"id": "T335"
},
{
"offsets": [
[
36633,
36709
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],
"text": "['One can assign each shape in PSD to a separate dimension, exactly as with SI']",
"type": "own_claim",
"id": "T336"
},
{
"offsets": [
[
36730,
36779
]
],
"text": "['PSD allows one to sculpt intermediate expressions']",
"type": "own_claim",
"id": "T337"
},
{
"offsets": [
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36793,
36837
]
],
"text": "['and situate them half-way along the relevant']",
"type": "own_claim",
"id": "T338"
},
{
"offsets": [
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36851,
36855
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],
"text": "['axis']",
"type": "own_claim",
"id": "T339"
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"offsets": [
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36867,
36948
]
],
"text": "['a sculpted pose that represents the simultaneous activation of several parameters']",
"type": "own_claim",
"id": "T340"
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"offsets": [
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37000,
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]
],
"text": "['can simply be saved at the appropriate location in the pose space']",
"type": "own_claim",
"id": "T341"
},
{
"offsets": [
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36955,
36974
]
],
"text": "['happy but surprised']",
"type": "data",
"id": "T342"
},
{
"offsets": [
[
36979,
36998
]
],
"text": "['smiling with a wink']",
"type": "data",
"id": "T343"
},
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"offsets": [
[
37067,
37175
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],
"text": "['Psychological research has shown that human facial expressions are largely described by two “emotional” axes']",
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"id": "T344"
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{
"offsets": [
[
37179,
37181
]
],
"text": "['30']",
"type": "data",
"id": "T345"
},
{
"offsets": [
[
37186,
37194
]
],
"text": "['Figure 6']",
"type": "data",
"id": "T346"
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{
"offsets": [
[
37198,
37301
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],
"text": "['this two-dimensional space would be a convenient high-level pose space for controlling facial animation']",
"type": "own_claim",
"id": "T347"
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{
"offsets": [
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37305,
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],
"text": "['The PSD interpolation is smooth']",
"type": "own_claim",
"id": "T348"
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{
"offsets": [
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37340,
37350
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],
"text": "['so desired']",
"type": "data",
"id": "T349"
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{
"offsets": [
[
37872,
37936
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],
"text": "['a PSD path from neutral to half-smile to full-smile is monotonic']",
"type": "own_claim",
"id": "T350"
},
{
"offsets": [
[
37390,
37398
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],
"text": "['Figure 7']",
"type": "data",
"id": "T351"
},
{
"offsets": [
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37960,
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]
],
"text": "['the motion of a surface point over this interpolation is also smooth']",
"type": "own_claim",
"id": "T352"
},
{
"offsets": [
[
38030,
38106
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],
"text": "['To interpolate these emotions using SI requires a zig-zag pattern of weights']",
"type": "background_claim",
"id": "T353"
},
{
"offsets": [
[
38108,
38221
]
],
"text": "['the half-smile weight goes from zero to one, and then back to zero as the full-smile weight goes from zero to one']",
"type": "background_claim",
"id": "T354"
},
{
"offsets": [
[
38222,
38308
]
],
"text": "['The motion of a surface point would also be piecewise linear using SI in this scenario']",
"type": "background_claim",
"id": "T355"
},
{
"offsets": [
[
38401,
38559
]
],
"text": "['Additional “dimensions” of deformation can be added at any time by adding a new parameter and associating additional poses with the movement of this parameter']",
"type": "own_claim",
"id": "T356"
},
{
"offsets": [
[
38574,
38701
]
],
"text": "['a limb can be modeled in a particular pose both in an unloaded state and with muscles sculpted to express carrying a heavy load']",
"type": "own_claim",
"id": "T357"
},
{
"offsets": [
[
38991,
39069
]
],
"text": "['this would of course require a significant set of additional deformation poses']",
"type": "own_claim",
"id": "T358"
},
{
"offsets": [
[
38921,
38989
]
],
"text": "['one can imagine setting up a dial that causes the character to morph']",
"type": "own_claim",
"id": "T359"
},
{
"offsets": [
[
38703,
38782
]
],
"text": "['The ‘heavy’ pose can be associated with the ‘on’ state of an abstract parameter']",
"type": "own_claim",
"id": "T360"
},
{
"offsets": [
[
38841,
38909
]
],
"text": "['light and heavy loads can then be controlled by flipping this switch']",
"type": "own_claim",
"id": "T361"
},
{
"offsets": [
[
39124,
39214
]
],
"text": "['Pose space deformation is not the last word in surface deformation for character animation']",
"type": "own_claim",
"id": "T362"
},
{
"offsets": [
[
39216,
39279
]
],
"text": "['high quality anatomically based models are certainly preferable']",
"type": "own_claim",
"id": "T363"
},
{
"offsets": [
[
39294,
39362
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],
"text": "['both anatomically based and purely kinematic models have their place']",
"type": "background_claim",
"id": "T364"
},
{
"offsets": [
[
39364,
39563
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],
"text": "['In the current computer animation culture animators generally practice their craft by direct and exhaustive specification of the desired motion combined with quick evaluation using real-time playback']",
"type": "background_claim",
"id": "T365"
},
{
"offsets": [
[
39565,
39638
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],
"text": "['Deeper simulation approaches intrinsically take away some of this control']",
"type": "background_claim",
"id": "T366"
},
{
"offsets": [
[
39644,
39665
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],
"text": "['animators often argue']",
"type": "background_claim",
"id": "T367"
},
{
"offsets": [
[
39683,
39753
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],
"text": "['that automated processes are inferior or will not produce a human feel']",
"type": "background_claim",
"id": "T368"
},
{
"offsets": [
[
39755,
39868
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],
"text": "['The performance of current anatomically based models prohibits animation preview and other real-time applications']",
"type": "background_claim",
"id": "T369"
},
{
"offsets": [
[
39982,
40087
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],
"text": "['the effort needed to produce an anatomically accurate model is not always justified, nor even appropriate']",
"type": "background_claim",
"id": "T370"
},
{
"offsets": [
[
40091,
40219
]
],
"text": "['the model is of a fanciful creature whose surface appearance may be inconsistent with any plausible internal anatomy in any case']",
"type": "data",
"id": "T371"
},
{
"offsets": [
[
40221,
40328
]
],
"text": "['PSD unifies and improves upon two techniques that have been common graphics practice for more than a decade']",
"type": "own_claim",
"id": "T372"
},
{
"offsets": [
[
40330,
40416
]
],
"text": "['This relatively simple algorithm uniformly handles a variety of deformation situations']",
"type": "own_claim",
"id": "T373"
},
{
"offsets": [
[
40469,
40517
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],
"text": "['The setup cost of the algorithm is insignificant']",
"type": "own_claim",
"id": "T374"
},
{
"offsets": [
[
40523,
40596
]
],
"text": "['the synthesis cost is only slightly more than that of shape interpolation']",
"type": "own_claim",
"id": "T375"
},
{
"offsets": [
[
40601,
40677
]
],
"text": "['real-time synthesis is possible at effective resolutions on current hardware']",
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"role": "Arg1"
},
"tail": {
"ref_id": "T363",
"role": "Arg2"
},
"type": "contradicts"
},
{
"id": "R198",
"head": {
"ref_id": "T368",
"role": "Arg1"
},
"tail": {
"ref_id": "T367",
"role": "Arg2"
},
"type": "parts_of_same"
},
{
"id": "R199",
"head": {
"ref_id": "T371",
"role": "Arg1"
},
"tail": {
"ref_id": "T370",
"role": "Arg2"
},
"type": "supports"
},
{
"id": "R200",
"head": {
"ref_id": "T375",
"role": "Arg1"
},
"tail": {
"ref_id": "T376",
"role": "Arg2"
},
"type": "supports"
}
] |