import numpy as np


def sigma_matrix2(sig_x, sig_y, theta):
    """Calculate the rotated sigma matrix (two dimensional matrix).
    Args:
        sig_x (float):
        sig_y (float):
        theta (float): Radian measurement.
    Returns:
        ndarray: Rotated sigma matrix.
    """
    d_matrix = np.array([[sig_x**2, 0], [0, sig_y**2]])
    u_matrix = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
    return np.dot(u_matrix, np.dot(d_matrix, u_matrix.T))


def mesh_grid(kernel_size):
    """Generate the mesh grid, centering at zero.
    Args:
        kernel_size (int):
    Returns:
        xy (ndarray): with the shape (kernel_size, kernel_size, 2)
        xx (ndarray): with the shape (kernel_size, kernel_size)
        yy (ndarray): with the shape (kernel_size, kernel_size)
    """
    ax = np.arange(-kernel_size // 2 + 1., kernel_size // 2 + 1.)
    xx, yy = np.meshgrid(ax, ax)
    xy = np.hstack((xx.reshape((kernel_size * kernel_size, 1)), yy.reshape(kernel_size * kernel_size,
                                                                           1))).reshape(kernel_size, kernel_size, 2)
    return xy, xx, yy


def pdf2(sigma_matrix, grid):
    """Calculate PDF of the bivariate Gaussian distribution.
    Args:
        sigma_matrix (ndarray): with the shape (2, 2)
        grid (ndarray): generated by :func:`mesh_grid`,
            with the shape (K, K, 2), K is the kernel size.
    Returns:
        kernel (ndarrray): un-normalized kernel.
    """
    inverse_sigma = np.linalg.inv(sigma_matrix)
    kernel = np.exp(-0.5 * np.sum(np.dot(grid, inverse_sigma) * grid, 2))
    return kernel

def bivariate_Gaussian(kernel_size, sig_x, sig_y, theta, grid=None, isotropic=True):
    """Generate a bivariate isotropic or anisotropic Gaussian kernel.
    In the isotropic mode, only `sig_x` is used. `sig_y` and `theta` is ignored.
    Args:
        kernel_size (int):
        sig_x (float):
        sig_y (float):
        theta (float): Radian measurement.
        grid (ndarray, optional): generated by :func:`mesh_grid`,
            with the shape (K, K, 2), K is the kernel size. Default: None
        isotropic (bool):
    Returns:
        kernel (ndarray): normalized kernel.
    """
    if grid is None:
        grid, _, _ = mesh_grid(kernel_size)
    if isotropic:
        sigma_matrix = np.array([[sig_x**2, 0], [0, sig_x**2]])
    else:
        sigma_matrix = sigma_matrix2(sig_x, sig_y, theta)
    kernel = pdf2(sigma_matrix, grid)
    kernel = kernel / np.sum(kernel)
    return kernel