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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
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