def compute_kernel(self):
_2L = 2 * self.L
kernel = np.zeros((_2L, _2L, _2L), dtype=self.as_type)
filters_f = self.src.filters.evaluate_grid(self.L)
sq_filters_f = np.array(filters_f ** 2, dtype=self.as_type)
for i in range(0, self.n, self.batch_size):
pts_rot = rotated_grids(self.L, self.src.rots[:, :, i:i+self.batch_size])
weights = sq_filters_f[:, :, self.src.filters.indices[i:i+self.batch_size]]
weights *= self.src.amplitudes[i:i+self.batch_size] ** 2
if self.L % 2 == 0:
weights[0, :, :] = 0
weights[:, 0, :] = 0
pts_rot = m_reshape(pts_rot, (3, -1))
weights = m_flatten(weights)
kernel += 1 / (self.n * self.L ** 4) * anufft3(weights, pts_rot, (_2L, _2L, _2L), real=True)
# Ensure symmetric kernel
kernel[0, :, :] = 0
kernel[:, 0, :] = 0
kernel[:, :, 0] = 0
logger.info('Computing non-centered Fourier Transform')
kernel = mdim_ifftshift(kernel, range(0, 3))
kernel_f = fft2(kernel, axes=(0, 1, 2))
kernel_f = np.real(kernel_f)
return FourierKernel(kernel_f, centered=False)
def im_backproject(im, rot_matrices):
"""
Backproject images along rotation
:param im: An L-by-L-by-n array of images to backproject.
:param rot_matrices: An 3-by-3-by-n array of rotation matrices corresponding to viewing directions.
:return: An L-by-L-by-L volumes corresponding to the sum of the backprojected images.
"""
L, _, n = im.shape
ensure(L == im.shape[1], "im must be LxLxK")
ensure(n == rot_matrices.shape[2],
"No. of rotation matrices must match the number of images")
pts_rot = rotated_grids(L, rot_matrices)
pts_rot = m_reshape(pts_rot, (3, -1))
im_f = centered_fft2(im) / (L**2)
if L % 2 == 0:
im_f[0, :, :] = 0
im_f[:, 0, :] = 0
im_f = m_flatten(im_f)
plan = Plan(sz=(L, L, L), fourier_pts=pts_rot)
vol = np.real(plan.adjoint(im_f)) / L
return vol
def evaluate_t(self, v):
"""
Evaluate coefficient in dual basis
:param v: The coefficient array to be evaluated. The first dimensions must equal `self.sz`.
:return: The evaluation of the coefficient array `v` in the dual basis of `basis`.
This is an array of vectors whose first dimension equals `self.basis_count` and whose remaining dimensions
correspond to higher dimensions of `v`.
"""
x, sz_roll = unroll_dim(v, self.d + 1)
x = m_reshape(x,
new_shape=tuple([np.prod(self.sz)] +
list(x.shape[self.d:])))
r_idx = self.basis_coords['r_idx']
ang_idx = self.basis_coords['ang_idx']
mask = m_flatten(self.basis_coords['mask'])
ind = 0
ind_radial = 0
ind_ang = 0
v = np.zeros(shape=tuple([self.basis_count] + list(x.shape[1:])))
for ell in range(0, self.ell_max + 1):
k_max = self.k_max[ell]
idx_radial = ind_radial + np.arange(0, k_max)
nrms = self._norms[idx_radial]
radial = self._precomp['radial'][:, idx_radial]
radial = radial / nrms
sgns = (1, ) if ell == 0 else (1, -1)
for _ in sgns:
ang = self._precomp['ang'][:, ind_ang]
ang_radial = np.expand_dims(ang[ang_idx],
axis=1) * radial[r_idx]
idx = ind + np.arange(0, k_max)
v[idx] = ang_radial.T @ x[mask]
ind += len(idx)
ind_ang += 1
ind_radial += len(idx_radial)
v = roll_dim(v, sz_roll)
return v
def evaluate(self, v):
"""
Evaluate coefficient vector in basis
:param v: A coefficient vector (or an array of coefficient vectors) to be evaluated.
The first dimension must equal `self.basis_count`.
:return: The evaluation of the coefficient vector(s) `v` for this basis.
This is an array whose first dimensions equal `self.z` and the remaining dimensions correspond to
dimensions two and higher of `v`.
"""
v, sz_roll = unroll_dim(v, 2)
r_idx = self.basis_coords['r_idx']
ang_idx = self.basis_coords['ang_idx']
mask = m_flatten(self.basis_coords['mask'])
ind = 0
ind_radial = 0
ind_ang = 0
x = np.zeros(shape=tuple([np.prod(self.sz)] + list(v.shape[1:])))
for ell in range(0, self.ell_max + 1):
k_max = self.k_max[ell]
idx_radial = ind_radial + np.arange(0, k_max)
nrms = self._norms[idx_radial]
radial = self._precomp['radial'][:, idx_radial]
radial = radial / nrms
sgns = (1, ) if ell == 0 else (1, -1)
for _ in sgns:
ang = self._precomp['ang'][:, ind_ang]
ang_radial = np.expand_dims(ang[ang_idx],
axis=1) * radial[r_idx]
idx = ind + np.arange(0, k_max)
x[mask] += ang_radial @ v[idx]
ind += len(idx)
ind_ang += 1
ind_radial += len(idx_radial)
x = m_reshape(x, self.sz + x.shape[1:])
x = roll_dim(x, sz_roll)
return x
def precomp(self):
"""
Precomute the basis functions on a polar Fourier 3D grid.
Gaussian quadrature points and weights are also generated
in radical and phi dimensions.
"""
n_r = int(self.ell_max + 1)
n_theta = int(2 * self.sz[0])
n_phi = int(self.ell_max + 1)
r, wt_r = lgwt(n_r, 0.0, self.c)
z, wt_z = lgwt(n_phi, -1, 1)
r = m_reshape(r, (n_r, 1))
wt_r = m_reshape(wt_r, (n_r, 1))
z = m_reshape(z, (n_phi, 1))
wt_z = m_reshape(wt_z, (n_phi, 1))
phi = np.arccos(z)
wt_phi = wt_z
theta = 2 * pi * np.arange(n_theta).T / (2 * n_theta)
theta = m_reshape(theta, (n_theta, 1))
# evaluate basis function in the radial dimension
radial_wtd = np.zeros(shape=(n_r, np.max(self.k_max),
self.ell_max + 1))
for ell in range(0, self.ell_max + 1):
k_max_ell = self.k_max[ell]
rmat = r * self.r0[0:k_max_ell, ell].T / self.c
radial_ell = np.zeros_like(rmat)
for ik in range(0, k_max_ell):
radial_ell[:, ik] = sph_bessel(ell, rmat[:, ik])
nrm = np.abs(sph_bessel(ell + 1, self.r0[0:k_max_ell, ell].T) / 4)
radial_ell = radial_ell / nrm
radial_ell_wtd = r**2 * wt_r * radial_ell
radial_wtd[:, 0:k_max_ell, ell] = radial_ell_wtd
# evaluate basis function in the phi dimension
ang_phi_wtd_even = []
ang_phi_wtd_odd = []
for m in range(0, self.ell_max + 1):
n_even_ell = int(
np.floor((self.ell_max - m) / 2) + 1 -
np.mod(self.ell_max, 2) * np.mod(m, 2))
n_odd_ell = int(self.ell_max - m + 1 - n_even_ell)
phi_wtd_m_even = np.zeros((n_phi, n_even_ell), dtype=phi.dtype)
phi_wtd_m_odd = np.zeros((n_phi, n_odd_ell), dtype=phi.dtype)
ind_even = 0
ind_odd = 0
for ell in range(m, self.ell_max + 1):
phi_m_ell = norm_assoc_legendre(ell, m, z)
nrm_inv = np.sqrt(0.5 / pi)
phi_m_ell = nrm_inv * phi_m_ell
phi_wtd_m_ell = wt_phi * phi_m_ell
if np.mod(ell, 2) == 0:
phi_wtd_m_even[:, ind_even] = phi_wtd_m_ell[:, 0]
ind_even = ind_even + 1
else:
phi_wtd_m_odd[:, ind_odd] = phi_wtd_m_ell[:, 0]
ind_odd = ind_odd + 1
ang_phi_wtd_even.append(phi_wtd_m_even)
ang_phi_wtd_odd.append(phi_wtd_m_odd)
# evaluate basis function in the theta dimension
ang_theta = np.zeros((n_theta, 2 * self.ell_max + 1),
dtype=theta.dtype)
ang_theta[:, 0:self.ell_max] = np.sqrt(2) * np.sin(
theta @ m_reshape(np.arange(self.ell_max, 0, -1),
(1, self.ell_max)))
ang_theta[:, self.ell_max] = np.ones(n_theta, dtype=theta.dtype)
ang_theta[:,
self.ell_max + 1:2 * self.ell_max + 1] = np.sqrt(2) * np.cos(
theta @ m_reshape(np.arange(1, self.ell_max + 1),
(1, self.ell_max)))
ang_theta_wtd = (2 * pi / n_theta) * ang_theta
theta_grid, phi_grid, r_grid = np.meshgrid(theta,
phi,
r,
sparse=False,
indexing='ij')
fourier_x = m_flatten(r_grid * np.cos(theta_grid) * np.sin(phi_grid))
fourier_y = m_flatten(r_grid * np.sin(theta_grid) * np.sin(phi_grid))
fourier_z = m_flatten(r_grid * np.cos(phi_grid))
fourier_pts = 2 * pi * np.vstack(
(fourier_x[np.newaxis, ...], fourier_y[np.newaxis, ...],
fourier_z[np.newaxis, ...]))
return {
'radial_wtd': radial_wtd,
'ang_phi_wtd_even': ang_phi_wtd_even,
'ang_phi_wtd_odd': ang_phi_wtd_odd,
'ang_theta_wtd': ang_theta_wtd,
'fourier_pts': fourier_pts
}