Module src.sara

Implementation of the SARA algorithm [Carrillo2012, Onose2017].

src.sara.sara(y, epsilon, A, At, pU, G, W, Psi, Psit, param, warsmtart_name, checkpoint_name, alph, flag_dimensionality_reduction, Lambda, varargin)

Implementation of the SARA imaging algorithm.

Implementation of the reweighting / PDFB algorithm underlying the SARA imaging approach [Onose2017]. At each iteration of the outer reweighting scheme, the PDFB solves a problem of the form

\[\min_{x \in \mathbb{R}_+^{N}} \mu \|\Psi^\dagger x \|_{\omega, 1} + \sum_{b} \iota_{ \{\| y_{b} - \Phi_{b}(\cdot) \|_2 \leq \varepsilon_{b}\} } (x).\]

Parameters

y (composite, cell) – Blocks of visibilities {L}{nblocks_l}.
epsilon (composite, cell) – Data fidelity constraints {L}{nblocks_l}.
A (function handle) – Measurement operator.
At (function handle) – Adjoint of the measurement operator.
pU (composite, cell) – Preconditioning matrices {L}{nblocks_l}.
G (composite, cell of sparse matrices) – Degridding matrices {L}{nblocks_l}.
W (composite, cell of int[:]) – Masks for selection of the blocks of visibilities.
Psi (anonymous function) – Function handle representing the serial implementation of the SARA dictionary.
Psit (anonymous function) – Function handle representing the serial implementation of the adjoint SARA dictionary.
param (struct) – Algorithm parameters (see full description below).
warmstart_name (string) – Name of a valid .mat file to initialize the solver (warm-start).
checkpoint_name (string) – Name defining the name for checkpoint files saved throughout the iterations.
alph (double) – Regularization parameter.
flag_dimensionality_reduction ([type]) – [description]
Lambda (composite, cell of complex[:, :]) – Dimensionality reduction matrix. (?)
varargin{1} (double[:, :]) – Initial value for the primal variable.
varargin{2} (double[:, :]) – Ground truth wideband image (only when debugging synthetic data experiments).

Returns

xsol – Reconstructed image.

Return type

double[:]

Important

The param structure should contain the following fields.

param.verbose (string): Print log or not
param.save_intermediate_results_mat (bool): Flag to save all estimates in a .mat file after each re-weighted
param.nu1 (double): Upper bound on the norm of the SARA operator \(\Psi^\dagger\).
param.nu2 (double): Upper bound on the norm of the measurement operator \(\Phi\)
param.gamma (double): Regularization parameter.
param.pdfb_min_iter (int): Minimum number of iterations (PDFB).
param.pdfb_max_iter (int): Maximum number of iterations (PDFB).
param.pdfb_rel_var (double): Relative variation tolerance (PDFB).
param.pdfb_fidelity_tolerance (double): Tolerance to check data constraints are satisfied (PDFB).
param.reweighting_max_iter (int): Maximum number of reweighting steps.
param.reweighting_min_iter (int): Minimum number of reweighting steps (to reach “noise” level).
param.reweighting_rel_var (double): Tolerance relative variation (reweighting).
param.reweighting_alpha (double): Starting reweighting parameter.
param.reweighting_sig (double): Noise level (in wavelet space)
param.elipse_proj_max_iter (int): Maximum number of iterations for the FB algo that implements the projection onto the ellipsoid.
param.elipse_proj_min_iter (int): Minimum number of iterations for the FB algo that implements the projection onto the ellipsoid.
parma.elipse_proj_eps (double): Stopping criterion for the projection.

The following fileds are added to param in the course of the algorithm to be able to restart it from a previous state

param.reweighting_alpha (double): Current state of the reweighting parameter.
param.init_reweight_step_count (int): Iteration index of the current reweighting step when the checkpoint file is written.
param.init_reweight_last_iter_step (int): Global iteration index (unrolled over all pdfb iterations performed in the reweighting scheme) from which to restart the algorithm.
param.init_t_start (int): Global iteration index (unrolled over all pdfb iterations performed in the reweighting scheme) from which to restart the algorithm (param.init_t_start = param.init_reweight_last_iter_step + 1).

The following fileds are involved in the adaptive estimation of the noise level

param.adjust_flag_noise (bool): Flag to activate the adaptive procedure to estimate the noise level.
param.adjust_noise_min_iter (int): Minimum number of iterations to enable the adjustement of the estimate of the noise level.
param.adjust_noise_rel_var (double): Tolerance relative variation (reweighting) to to enable the adjustement of the estimate of the noise level.
param.adjust_noise_start_iter (int): Number of iterations to force triggering noise adjustement.

Note

The checkpoint file saved throughout the iterations is composed of the following variables (to be verified).

param (struct): Current state of the parameter structure (updated with auxiliary parameters describing the state of the solver when the checkpoint file has been written).
xsol (double[:]): Current state of the image estimate.
res (double[:, :]): Current state of the residual image.
g (double[:]): Auxiliary variable involved in the update of the primal variable.
epsilon (cell): (Updated) Value of the l2-ball radii.
t_block (cell of int): Index of the last iteration where the weigths have been updated.
proj (cell of complex[:]): Auxiliary variable involved in the update of the data fidelity dual variables.
norm_res (cell of double): Norm of the residual image (per block).
v1 (double[:]): Dual variables associated with the sparsity prior.
v2 (cell of complex[:]): Dual variables associated with the data fidelity terms (each cell corresponding to a data block).
weights1 (double[:]): Weights associated with the sparsity prior.
end_iter (int): Last iteration (unrolled over all pdfb iterations).
t_l11 (double): Time to update the sparsity dual variable.
t_master (double): Time to perform update/computations on the master process.
t_data (double): Time to update the data fidelity dual variables.
rel_val (double): Relative variation of the solution across the iterations.