Description of Example_weighted_BPDN
Description: % Exampled_weighted_L1

Home > matlab > Example_weighted_BPDN.m

# Example_weighted_BPDN

## PURPOSE

% Exampled_weighted_L1

## SYNOPSIS

This is a script file.

## DESCRIPTION

    % Exampled_weighted_L1
     Example to demonstrate use of BPDN solver when incorporating weights
     (performs one re-weighting of previous solution).

## CROSS-REFERENCE INFORMATION

This function calls:

- sopt_mltb_solve_BPDN sopt_mltb_solve_BPDN - Solve BPDN problem

This function is called by:

## SOURCE CODE

    0001 %% Exampled_weighted_L1
    0002 % Example to demonstrate use of BPDN solver when incorporating weights
    0003 % (performs one re-weighting of previous solution).
    0004
    0005
    0006 %% Clear workspace
    0007 clc;
    0008 clear;
    0009
    0010 %% Define paths
    0011 addpath misc/
    0012 addpath prox_operators/
    0013
    0014 %% Parameters
    0015 N = 64;
    0016 input_snr = 30; % Noise level (on the measurements)
    0017 randn('seed', 1); rand('seed', 1);
    0018
    0019 %% Create an image with few spikes
    0020 im = zeros(N); ind = randperm(N^2); im(ind(1:100)) = 1;
    0021 %
    0022 figure(1);
    0023 subplot(221), imagesc(im); axis image; axis off;
    0024 colormap gray; title('Original image'); drawnow;
    0025
    0026 %% Create a mask
    0027 % Mask
    0028 mask = rand(size(im)) < 0.095; ind = find(mask==1);
    0029 % Masking matrix (sparse matrix in matlab)
    0030 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
    0031 % Masking operator
    0032 A = @(x) Ma*x(:);
    0033 At = @(x) reshape(Ma'*x(:), size(im));
    0034
    0035 %% Reconstruct from a few Fourier measurements
    0036
    0037 % Composition (Masking o Fourier)
    0038 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
    0039 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
    0040
    0041 % Sparsity operator
    0042 Psit = @(x) x; Psi = Psit;
    0043
    0044 % Select 33% of Fourier coefficients
    0045 y = A(im);
    0046
    0047 % Add Gaussian i.i.d. noise
    0048 sigma_noise = 10^(-input_snr/20)*std(im(:));
    0049 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
    0050
    0051 % Display the downsampled image
    0052 figure(1);
    0053 subplot(222); imagesc(real(At(y))); axis image; axis off;
    0054 colormap gray; title('Measured image'); drawnow;
    0055
    0056 % Tolerance on noise
    0057 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
    0058
    0059 % Parameters for BPDN
    0060 param.verbose = 1; % Print log or not
    0061 param.gamma = 1e-1; % Converge parameter
    0062 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
    0063 param.max_iter = 300; % Max. number of iterations for the TVDN problem
    0064 param.nu_B2 = 1; % Bound on the norm of the operator A
    0065 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
    0066 param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
    0067 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
    0068 param.pos_l1 = 1; %
    0069
    0070 % Solve BPDN problem (without weights)
    0071 sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
    0072
    0073 % Show reconstructed image
    0074 figure(1);
    0075 subplot(223); imagesc(real(sol)); axis image; axis off;
    0076 colormap gray; title(['First estimate - ', ...
    0077     num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow;
    0078
    0079 % Refine the estimate
    0080 param.weights = 1./(abs(sol)+1e-5);
    0081 sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
    0082
    0083 % Show reconstructed image
    0084 figure(1);
    0085 subplot(224); imagesc(real(sol)); axis image; axis off;
    0086 colormap gray; title(['Second estimate - ', ...
    0087     num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow;