def kernel_matrix(s_theta, s_phi, q_theta, q_phi,
kernel=inv_funk_radon_even_kernel, N=18):
"""Construct the kernel matrix, A.
The kernel projects sampling points to evaluation points. Therefore,
s_theta and s_phi are constrained by where you sampled, whereas q_theta and
q_phi can be arbitrary (that's where the kernel is being evaluated).
To phrase this another way, (s_theta, s_phi) define the location of the
kernels [and, hence, the weights], whereas (q_theta, q_phi) is where we
want to evaluate them.
Parameters
----------
s_theta, s_phi : (P,) ndarray
Sampling points (inclination and azimuthal angles).
q_theta, q_phi : (Q,) ndarray
Evaluation points (inclination and azimuthal angles).
N : int
Maximum degree of spherical harmonic subspace.
"""
P = len(s_theta)
Q = len(q_theta)
s_theta = s_theta[:, None]
s_phi = s_phi[:, None]
cos_theta = sphere.cos_inc_angle(s_theta, s_phi, q_theta, q_phi)
return kernel(cos_theta, N)
def kernel_plot(kernel, grid_density=150, N=14):
theta_grid = np.linspace(0, np.pi, grid_density)
phi_grid = np.linspace(0, 2 * np.pi, grid_density)
kernel_vals = kernel(sphere.cos_inc_angle(0, 0, theta_grid[:, None], phi_grid), N)
plot.surf_grid_3D(kernel_vals, theta_grid, phi_grid, scale_radius=True)
def kernel_reconstruct(kernels_theta, kernels_phi, weights,
grid_theta, grid_phi, kernel=std_kernel, N=8):
"""Kernel reconstruction of the PDF, on a grid on the sphere.
Parameters
----------
kernels_theta, kernels_phi : (N,) ndarray
Positions of the kernels.
weights : (N,) ndarray
Weights of the kernels.
grid_theta : (M,) ndarray
Inclination angles of grid.
grid_phi : (N,) ndarray
Azimuth angles of grid.
kernel : callable
Kernel function used in the reconstruction.
N : int
Maximum order of kernel polynomials.
Returns
-------
pdf : (M, N) ndarray
Reconstruction of the PDF on the specified grid.
"""
PDF_recon = np.zeros(np.broadcast(grid_theta, grid_phi).shape)
for (k_theta, k_phi, w) in zip(kernels_theta, kernels_phi, weights):
cos_theta = sphere.cos_inc_angle(grid_theta, grid_phi,
k_theta, k_phi)
PDF_recon += w * kernel(cos_theta, N)
return PDF_recon
