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 compute_kernel(self): _2L = 2 * self.L kernel = np.zeros((_2L, _2L, _2L), dtype=self.dtype) sq_filters_f = self.src.eval_filter_grid(self.L, power=2) for i in range(0, self.n, self.batch_size): _range = np.arange(i, min(self.n, i + self.batch_size), dtype=int) pts_rot = rotated_grids(self.L, self.src.rots[_range, :, :]) weights = sq_filters_f[:, :, _range] weights *= self.src.amplitudes[_range]**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) * anufft(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 FB basis from those in standard 2D coordinate basis :param v: The coefficient array to be evaluated. The last 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 last dimension equals `self.count` and whose first dimensions correspond to first dimensions of `v`. """ if v.dtype != self.dtype: logger.warning( f"{self.__class__.__name__}::evaluate_t" f" Inconsistent dtypes v: {v.dtype} self: {self.dtype}" ) if isinstance(v, Image): v = v.asnumpy() v = v.T # RCOPT x, sz_roll = unroll_dim(v, self.ndim + 1) x = m_reshape( x, new_shape=tuple([np.prod(self.sz)] + list(x.shape[self.ndim :])) ) 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.count] + list(x.shape[1:])), dtype=v.dtype) for ell in range(0, self.ell_max + 1): k_max = self.k_max[ell] idx_radial = ind_radial + np.arange(0, k_max) # include the normalization factor of angular part ang_nrms = self.angular_norms[idx_radial] radial = self._precomp["radial"][:, idx_radial] radial = radial / ang_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.T # RCOPT
def evaluate(self, v): """ Evaluate coefficients in standard 2D coordinate basis from those in FB basis :param v: A coefficient vector (or an array of coefficient vectors) to be evaluated. The last dimension must equal `self.count`. :return: The evaluation of the coefficient vector(s) `v` for this basis. This is an array whose last dimensions equal `self.sz` and the remaining dimensions correspond to first dimensions of `v`. """ if v.dtype != self.dtype: logger.warning( f"{self.__class__.__name__}::evaluate" f" Inconsistent dtypes v: {v.dtype} self: {self.dtype}") # Transpose here once, instead of several times below #RCOPT v = v.reshape(-1, self.count).T 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:])), dtype=v.dtype) for ell in range(0, self.ell_max + 1): k_max = self.k_max[ell] idx_radial = ind_radial + np.arange(0, k_max, dtype=np.int) # include the normalization factor of angular part ang_nrms = self.angular_norms[idx_radial] radial = self._precomp["radial"][:, idx_radial] radial = radial / ang_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, dtype=np.int) x[mask] += ang_radial @ v[idx] ind += len(idx) ind_ang += 1 ind_radial += len(idx_radial) x = x.T.reshape(-1, *self.sz) # RCOPT return x
def __init__(self, sz, mask=None, dtype=np.float32): """ Initialize an object for Dirac basis :param sz: The shape of the vectors for which to define the basis. :param mask: A boolean _mask of size sz indicating which coordinates to include in the basis (default np.full(sz, True)). """ if mask is None: mask = np.full(sz, True) self._mask = m_flatten(mask) super().__init__(sz, dtype=dtype)
def evaluate_t(self, v): """ Evaluate coefficient in FB basis from those in standard 3D coordinate 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.count` and whose remaining dimensions correspond to higher dimensions of `v`. """ v = v.T x, sz_roll = unroll_dim(v, self.ndim + 1) x = m_reshape(x, new_shape=tuple([np.prod(self.sz)] + list(x.shape[self.ndim:]))) 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.count] + list(x.shape[1:])), dtype=self.dtype) 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 for _ in range(-ell, ell + 1): ang = self._precomp["ang"][:, ind_ang] ang_radial = np.expand_dims(ang[ang_idx], axis=1) * radial[r_idx] idx = ind + np.arange(0, len(idx_radial)) v[idx] = np.real(ang_radial.T @ x[mask]) ind += len(idx) ind_ang += 1 ind_radial += len(idx_radial) v = roll_dim(v, sz_roll) return v.T
def evaluate(self, v): """ Evaluate coefficients in standard 3D coordinate basis from those in FB basis :param v: A coefficient vector (or an array of coefficient vectors) to be evaluated. The first dimension must equal `self.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 = v.T 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:])), dtype=self.dtype) 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 for _ in range(-ell, ell + 1): ang = self._precomp["ang"][:, ind_ang] ang_radial = np.expand_dims(ang[ang_idx], axis=1) * radial[r_idx] idx = ind + np.arange(0, len(idx_radial)) 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.T
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 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 testDiracEvaluate(self): v = np.array([ [ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, -1.08106869e-17, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, -6.40456062e-03, -3.32961020e-03, -1.36887927e-02, -5.42770488e-03, 7.63680861e-03, 0.00000000e+00], [ 0.00000000e+00, 3.16377602e-03, -9.31273350e-03, 9.46128404e-03, 1.93239220e-02, 3.79891953e-02, 1.06841173e-02, -2.36467925e-03], [ 0.00000000e+00, 1.72736955e-03, -1.00710814e-02, 4.93520304e-02, 3.77702656e-02, 6.57365438e-02, 3.94739462e-03, -4.41228496e-03], [ 4.01551066e-18, -3.08071647e-03, -1.61670565e-02, 8.66886286e-02, 5.09898409e-02, 7.19313349e-02, 1.68313715e-02, 5.19180892e-03], [ 0.00000000e+00, 2.87262215e-03, -3.37732956e-02, 4.51706505e-02, 5.72215879e-02, 4.63553081e-02, 1.86552175e-03, 1.12608805e-02], [ 0.00000000e+00, 2.77905016e-03, -2.77499404e-02, -4.02645374e-02, -1.54969139e-02, -1.66229153e-02, -2.07389259e-02, 6.64060546e-03], [ 0.00000000e+00, 0.00000000e+00, 5.20080934e-03, -1.06788196e-02, -1.14761672e-02, -1.27443126e-02, -1.15563484e-02, 0.00000000e+00] ]) coeffs = m_flatten(v) result = self.basis.evaluate(coeffs) self.assertTrue(np.allclose(result, v))
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 }