def tikhonov(*, psfs, measurements, tikhonov_lam=0.129, tikhonov_order=1):
"""Perform Tikhonov regularization based image reconstruction for PSSI.
Solves x_hat = argmin_{x} { ||Ax-y||_2^2 + lam * ||Dx||_2^2 }. D is the
discrete derivative operator of order `tikhonov_order`.
Args:
psfs (PSFs): PSFs object containing psfs and other csbs state data
measured_noisy (ndarray): 4d array of noisy measurements
tikhonov_lam (float): regularization parameter of tikhonov
tikhonov_order (int): [0,1 or 2] order of the discrete derivative
operator used in the tikhonov regularization
Returns:
4d array of the reconstructed images
"""
num_sources = psfs.psfs.shape[1]
rows, cols = measurements.shape[1:]
# expand psf_dfts
exp_psf_dfts = np.repeat(np.fft.fft2(
size_equalizer(psfs.psfs, ref_size=[rows, cols])),
psfs.copies.astype(int),
axis=0)
psfs.psf_GAM = block_mul(block_herm(exp_psf_dfts), exp_psf_dfts)
# DFT of the kernel corresponding to (D^TD)
LAM = get_LAM(rows=rows, cols=cols, order=tikhonov_order)
return np.real(
np.fft.ifft2(
block_mul(
block_inv(psfs.psf_GAM + tikhonov_lam *
np.einsum('ij,kl', np.eye(num_sources), LAM)),
block_mul(
block_herm(exp_psf_dfts),
np.fft.fft2(np.fft.fftshift(measurements, axes=(1, 2)))))))
def test_block_mul(self):
from mas.block import block_mul
# matrix x matrix
result = block_mul(self.x, self.y)
self.assertEqual(result.shape, (3, 3, 5, 5))
# matrix x col vec
result = block_mul(self.x, self.y[:, 0, :, :])
self.assertEqual(result.shape, (3, 5, 5))
def iteration_end(psfs, lowest_psf_group_index):
"""
"""
psfs.initialized_data['GAM'] -= block_mul(
block_herm(psfs.initialized_data['psf_dfts']
[lowest_psf_group_index:lowest_psf_group_index + 1]),
psfs.initialized_data['psf_dfts']
[lowest_psf_group_index:lowest_psf_group_index + 1])
def cost(psfs, psf_group_index, **kwargs):
"""
"""
iteration_SIG_e_dft = (SIG_e_dft(psfs, kwargs['lam']) - block_mul(
block_herm(
psfs.initialized_data['psf_dfts'][psf_group_index:psf_group_index +
1]),
psfs.initialized_data['psf_dfts'][psf_group_index:psf_group_index +
1]))
if kwargs['no_dc']:
cov = block_inv(iteration_SIG_e_dft)
cov[:, :, 0, 0] = 0 # set the DC values to zero
return np.real(np.sum(np.trace(cov)))
return np.real(np.sum(np.trace(block_inv(iteration_SIG_e_dft))))
def primal1_update_gaussian(*, pre_primal1, SIG_inv, ATy, nu):
"""Function that updates the first primal variable of ADMM based on the
least squares data fidelity term which inherently assumes that the
measurements have additive Gaussian noise.
Args:
pre_primal1 (ndarray): intermediate ADMM variable which is typically
W(primal2 - dual) where W is the regularization transform
SIG_inv (ndarray): spectrum of (A^TA+nu*W^TW + ...)^{-1}
ATy (ndarray): FA^Ty term where F is DFT matrix
nu (float): augmented Lagrangian parameter of ADMM (step size)
kwargs (dict): keyword arguments of parameters related to the regularizer
Returns:
ndarray of updated primal1
"""
return np.real(
np.fft.ifft2(block_mul(SIG_inv, ATy + nu * np.fft.fft2(pre_primal1))))
def init(psfs, **kwargs):
"""
"""
_, _, rows, cols = psfs.psfs.shape
# psf_dfts = np.fft.fft2(psfs.psfs, axes=(2, 3))
psf_dfts = psfs.psf_dfts
initialized_data = {
"psf_dfts":
psf_dfts,
"GAM":
block_mul(
block_herm(
# scale rows of psf_dfts by copies
# split across sqrt to prevent rounding error
np.einsum('i,ijkl->ijkl', np.sqrt(psfs.copies), psf_dfts)),
np.einsum('i,ijkl->ijkl', np.sqrt(psfs.copies), psf_dfts),
),
"LAM":
get_LAM(rows=rows, cols=cols, order=kwargs['order'])
}
psfs.initialized_data = initialized_data
def strollr(*, sources, psfs, measurements, recon_init, iternum, periter, lr,
theta, s, lam, patch_shape, transform, learning, window_size,
group_size, group_size_s):
"""Function that implements PSSI deconvolution using both sparsifying transform
learning and emposing low rankness on the grouped patches of the images.
The algorithm details can be found on [arXiv:1808.01316]. For efficiency,
multiprocessing has been used in some parts of the algorithm.
Args:
sources (ndarray): 4d array of sources
psfs (PSFs): PSFs object containing psfs and related data
measurements (ndarray): 4d array of measurements
recon_init (ndarray): initialization for the reconstructed image(s)
iternum (int): number of iterations of ADMM
periter (int): iteration period of displaying the reconstructions
lr (float): augmented Lagrangian parameter of the low-rank term
theta (float): penalty parameter of the low-rank term
s (float): augmented Lagrangian parameter of the sparsity term
lam (float): penalty parameter of the sparsity term
patch_shape (tuple): tuple of the shape of the patches used
transform (ndarray): ndarray of the sparsifying transform used
learning (bool): boolean variable of whether the transform gets updated
window_size (tuple): tuple of the window size over which the group of
similar patches to the centered reference patch is searched
group_size (int): the number of patches in each group (of similar patches)
group_size_s (int): the number of patches in each group (for sparsity)
"""
num_sources = psfs.psfs.shape[1]
rows, cols = measurements.shape[1:]
psize = np.size(np.empty(patch_shape))
if type(theta) is np.float:
theta = np.ones(num_sources) * theta
if type(lam) is np.float:
lam = np.ones(num_sources) * lam
recon = recon_init
################# pre-compute some arrays for efficiency #################
patcher_spectrum = np.einsum('ij,kl->ijkl', np.eye(num_sources),
psize * np.ones((rows, cols)))
psfs.psf_dfts = np.repeat(np.fft.fft2(
size_equalizer(psfs.psfs, ref_size=[rows, cols])),
psfs.copies.astype(int),
axis=0)
psfs.psf_GAM = block_mul(block_herm(psfs.psf_dfts), psfs.psf_dfts)
psfdfts_h_meas = block_mul(block_herm(
psfs.psf_dfts), np.fft.fft2(np.fft.fftshift(
measurements,
axes=(1, 2)))) # this is reshaped FA^Ty term where F is DFT matrix
SIG_inv = block_inv(psfs.psf_GAM + (s + lr) * patcher_spectrum)
for iter in range(iternum):
# ----- Low-Rank Approximation -----
patches = patch_extractor(recon, patch_shape=patch_shape)
patch_means = np.mean(patches, axis=0)
patches_zeromean = (patches - patch_means).T
indices = np.zeros((patches.shape[1], group_size), dtype=np.int)
pool = multiprocessing.Pool()
lowrank_i = functools.partial(lowrank,
patches_zeromean=patches_zeromean,
window_size=window_size,
imsize=(rows, cols),
threshold=theta,
group_size=group_size)
D, indices = zip(*pool.map(lowrank_i, np.arange(patches.shape[1])))
D = np.array(D)
indices = np.array(indices, dtype=np.int)
pool.close()
pool.join()
D += np.einsum('ik,j->ijk', patch_means[indices],
np.ones(patches.shape[0]))
if s > 0:
# ----- Sparse Coding -----
patches_3d = np.zeros(
(patches.shape[0] * group_size_s, patches.shape[1]))
for i in range(group_size_s):
patches_3d[i * patches.shape[0]:(i + 1) *
patches.shape[0], :] = patches[:, indices[:, i]]
sparse_codes = transform @ patches_3d
#FIXME - implement variable threshold for each image
# sparse_indices = (sparse_codes > recon.lam).astype(np.int)
# if not sparse_indices.any():
# sparse_codes = np.zeros_like(sparse_codes)
#
# sparse_codes = sparse_codes * sparse_indices
for i in range(num_sources):
ind = np.arange(i * rows * cols, (i + 1) * rows * cols)
sparse_codes[:, ind] = hard_thresholding(sparse_codes[:, ind],
threshold=np.sqrt(
lam[i] / s))
if learning is True:
u, s, v_T = np.linalg.svd(sparse_codes @ patches_3d.T)
transform = u @ v_T
# ----- Image Update -----
Whb1 = np.zeros_like(patches)
Whb = transform.T @ sparse_codes
for i in range(group_size_s):
Whb1 = indsum(
Whb1,
Whb[i * patches.shape[0]:(i + 1) * patches.shape[0], :],
indices[:, i])
indvals = np.array(
list(Counter(indices[:, :group_size_s].flatten()).values()))
indkeys = np.argsort(
np.array(
list(Counter(indices[:, :group_size_s].flatten()).keys())))
Whb1 = Whb1 / indvals[indkeys]
Fc = np.fft.fft2(
patch_aggregator(Whb1,
patch_shape=patch_shape,
image_shape=(num_sources, rows, cols)))
else:
Fc = np.zeros_like(psfdfts_h_meas)
if lr > 0:
VhD = np.zeros_like(patches)
for i in range(group_size):
VhD = indsum(VhD, D[:, :, i].T, indices[:, i])
indvals = np.array(list(Counter(indices.flatten()).values()))
indkeys = np.argsort(
np.array(list(Counter(indices.flatten()).keys())))
VhD = VhD / indvals[indkeys]
Fd = np.fft.fft2(
patch_aggregator(VhD,
patch_shape=patch_shape,
image_shape=(num_sources, rows, cols)))
else:
Fd = np.zeros_like(psfdfts_h_meas)
recon = np.real(
np.fft.ifft2(block_mul(SIG_inv,
s * Fc + lr * Fd + psfdfts_h_meas)))
if (iter + 1) % periter == 0 or iter == iternum - 1:
deconv_plotter(sources=sources, recons=recon, iter=iter)
return recon
def sparsepatch(*,
sources,
psfs,
measurements,
recon_init,
iternum,
plot=True,
periter,
nu,
lam,
patch_shape,
transform,
learning):
"""Function that implements PSSI deconvolution with a patch based sparsifying
transform for sparse recovery.
P1 and P3 formulations described in [doi:10.1137/141002293] have been
implemented without the transform update step.
Args:
sources (ndarray): 4d array of sources
psfs (PSFs): PSFs object containing psfs and related data
measurements (ndarray): 4d array of measurements
recon_init (ndarray): initialization for the reconstructed image(s)
iternum (int): number of iterations of ADMM
plot (boolean): if set to True, display the reconstructions as the iterations go
periter (int): iteration period of displaying the reconstructions
nu (float): augmented Lagrangian parameter
lam (float): penalty parameter of the sparsity term
patch_shape (tuple): tuple of the shape of the patches used
transform (ndarray): ndarray of the sparsifying transform used
learning (bool): boolean variable of whether the transform gets updated
"""
num_sources = psfs.psfs.shape[1]
rows, cols = measurements.shape[1:]
psize = np.size(np.empty(patch_shape))
# mse_inner = np.zeros((num_sources,recon.maxiter))
if type(lam) is np.float or type(lam) is np.int:
lam = np.ones(num_sources) * lam
################## initialize the reconstruction ##################
recon = recon_init
################# pre-compute some arrays for efficiency #################
psfs.psf_dfts = np.repeat(np.fft.fft2(
size_equalizer(psfs.psfs, ref_size=[rows, cols])),
psfs.copies.astype(int),
axis=0)
psfs.psf_GAM = block_mul(block_herm(psfs.psf_dfts), psfs.psf_dfts)
psfdfts_h_meas = block_mul(block_herm(
psfs.psf_dfts), np.fft.fft2(np.fft.fftshift(
measurements,
axes=(1, 2)))) # this is reshaped FA^Ty term where F is DFT matrix
LAM = psize * np.ones((rows, cols))
spectrum = nu * np.einsum('ij,kl->ijkl', np.eye(num_sources), LAM)
SIG_inv = block_inv(psfs.psf_GAM + spectrum)
for iter in range(iternum):
# ----- Sparse Coding -----
patches = patch_extractor(recon, patch_shape=patch_shape)
sparse_codes = transform @ patches
for i in range(num_sources):
sparse_codes[:, i * rows * cols:(i + 1) * rows *
cols] = hard_thresholding(
sparse_codes[:, i * rows * cols:(i + 1) * rows *
cols],
threshold=np.sqrt(lam[i] / nu))
# ----- Image Update -----
Fc = np.fft.fft2(
patch_aggregator(transform.conj().T @ sparse_codes,
patch_shape=patch_shape,
image_shape=(num_sources, rows, cols)))
recon = np.real(
np.fft.ifft2(block_mul(SIG_inv, nu * Fc + psfdfts_h_meas)))
if learning is True:
u, s, vT = np.linalg.svd(sparse_codes @ patches.T)
transform = u @ vT
# mse_inner[:,iter] = np.mean(
# (sources - recon.reconstructed)**2,
# axis=(1, 2, 3)
# )
#
# dfid = recon.nu*(1/np.size(measurements))*np.sum(abs(
# block_mul(
# psfs.selected_psf_dfts, np.fft.fft2(recon.reconstructed)
# ) - np.fft.fft2(measurements)
# )**2)
#
# sp_error = np.sum((recon.transform@patch_extractor(
# recon.reconstructed,
# patch_shape=recon.patch_shape
# )-sparse_codes)**2)
if plot == True and ((iter + 1) % periter == 0 or iter == iternum - 1):
deconv_plotter(sources=sources, recons=recon, iter=iter)
return recon
plotter4d(sources, title='Orig', figsize=(5.6, 8), cmap='gist_heat')
[k, num_sources, aa, bb] = psfs.selected_psfs.shape[:2] + sources.shape[2:]
LAM = get_LAM(rows=aa, cols=bb, order=tikhonov_order)
ssim = np.zeros((len(tikhonov_lam), num_instances, num_sources))
psnr = np.zeros((len(tikhonov_lam), num_instances, num_sources))
for i in range(len(tikhonov_lam)):
SIG_inv = block_inv(psfs.selected_GAM + tikhonov_lam[i] *
np.einsum('ij,kl', np.eye(num_sources), LAM))
for j in range(num_instances):
# DFT of the kernel corresponding to (D^TD)
recon = np.real(
np.fft.ifft2(
block_mul(
SIG_inv,
block_mul(psfs.selected_psf_dfts_h,
np.fft.fft2(measured_noisy_instances[j])))))
###### COMPUTE PERFORMANCE METRICS ######
mse = np.mean((sources - recon)**2, axis=(1, 2, 3))
psnr[i, j] = 20 * np.log10(
np.max(sources, axis=(1, 2, 3)) / np.sqrt(mse))
for p in range(num_sources):
ssim[i, j, p] = compare_ssim(sources[p, 0],
recon[p, 0],
data_range=np.max(recon[p, 0]) -
np.min(recon[p, 0]))
plotter4d(recon,
cmap='gist_heat',
figsize=(5.6, 8),
title='Recon. SSIM={}\n Recon. PSNR={}'.format(
def admm(*,
sources=None,
psfs,
measurements,
regularizer,
recon_init,
iternum,
plot=True,
periter=5,
nu,
lam,
**kwargs):
"""Function that implements PSSI deconvolution with the ADMM algorithm using
the specified regularization method.
Args:
sources (ndarray): 3d array of sources
psfs (PSFs): PSFs object containing psfs and related data
measurements (ndarray): 3d array of measurements
regularizer (function): function that specifies the regularization type
recon_init (ndarray): initialization for the reconstructed image(s)
iternum (int): number of iterations of ADMM
plot (boolean): if set to True, display the reconstructions as the iterations go
periter (int): iteration period of displaying the reconstructions
nu (float): augmented Lagrangian parameter (step size) of ADMM
lam (list): regularization parameter of dimension num_sources
kwargs (dict): keyword arguments to be passed to the regularizers
Returns:
ndarray of reconstructed images
"""
num_sources = psfs.psfs.shape[1]
rows, cols = measurements.shape[1:]
if type(lam) is np.float or type(lam) is np.float64 or type(lam) is np.int:
lam = np.ones(num_sources) * lam
################## initialize the primal/dual variables ##################
primal1 = recon_init
primal2 = None
dual = None
################# pre-compute some arrays for efficiency #################
psfs.psf_dfts = np.repeat(np.fft.fft2(
size_equalizer(psfs.psfs, ref_size=[rows, cols])),
psfs.copies.astype(int),
axis=0)
psfs.psf_GAM = block_mul(block_herm(psfs.psf_dfts), psfs.psf_dfts)
psfdfts_h_meas = block_mul(block_herm(
psfs.psf_dfts), np.fft.fft2(np.fft.fftshift(
measurements,
axes=(1, 2)))) # this is reshaped FA^Ty term where F is DFT matrix
SIG_inv = get_SIG_inv(regularizer=regularizer,
measurements=measurements,
psfs=psfs,
nu=nu,
**kwargs)
for iter in range(iternum):
######################### PRIMAL 1,2 UPDATES #########################
primal1, primal2, pre_primal2, dual = regularizer(
psfs=psfs,
measurements=measurements,
psfdfts_h_meas=psfdfts_h_meas,
SIG_inv=SIG_inv,
primal1=primal1,
primal2=primal2,
dual=dual,
nu=nu,
lam=lam,
**kwargs)
########################### DUAL UPDATE ###########################
dual += (pre_primal2 - primal2)
if plot == True and ((iter + 1) % periter == 0 or iter == iternum - 1):
deconv_plotter(sources=sources, recons=primal1, iter=iter)
return primal1