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Example_BPDN

PURPOSE ^

% Example_BPDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_BPDN
 Example to demonstrate use of BPDN solver.  A random Fourier sampling
 measurement operator is considered.  Daubechies 8 wavelets are used for
 the sparsifying operator.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_BPDN
0002 % Example to demonstrate use of BPDN solver.  A random Fourier sampling
0003 % measurement operator is considered.  Daubechies 8 wavelets are used for
0004 % the sparsifying operator.
0005 
0006 
0007 %% Clear workspace
0008 
0009 clc
0010 clear;
0011 
0012 
0013 %% Define paths
0014 
0015 addpath misc/
0016 addpath prox_operators/
0017 
0018 
0019 %% Define parameters
0020 
0021 % Coverages (half the plane for Fourier sampling)
0022 p = [0.50];
0023 
0024 % Noise level (on the measurements)
0025 input_snr = 30;
0026 
0027 
0028 %% Read image
0029 
0030 % Load image
0031 im = im2double(imread('cameraman.tif'));
0032 
0033 % Normalise
0034 im = im/max(max(im));
0035 
0036 % Enforce positivity
0037 im(im<0) = 0;
0038 
0039 
0040 %% Define sparsity operator
0041 
0042 dwtmode('per');
0043 
0044 % Decomposition level of the wavelet transform.
0045 nlevel = 4;
0046 
0047 % Daubechies 8 wavelet operator
0048 [C,S] = wavedec2(im,nlevel,'db8');
0049 
0050 Psit = @(x) wavedec2(x, nlevel, 'db8');
0051 Psi = @(x) waverec2(x, S, 'db8');
0052 
0053 
0054 %% Run simulations
0055 
0056 % Random Fourier sampling example
0057 %  Define mask
0058 %  Uniform sampling of the half Fourier plane
0059 mask = rand(size(im)) < p;
0060 mask(:,1:floor(size(im,2)/2))=0;
0061 mask = ifftshift(mask);
0062 mask(1,1)=0;
0063 mask(floor(size(im,1)/2):end,1)=0;
0064 
0065 ind = find(mask==1);
0066 Ma = sparse(1:numel(ind), ind, ...
0067   ones(numel(ind), 1), numel(ind), numel(im));
0068 
0069 % Composition (Masking o Fourier)
0070 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0071 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0072 
0073 % Apply measurement operator
0074 y = A(im);
0075 
0076 % Add Gaussian i.i.d. noise
0077 sigma_noise = 10^(-input_snr/20)*std(im(:));
0078 noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0079 y = y + noise;
0080 
0081 % Tolerance on noise
0082 epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise;
0083 epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise;
0084 tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball
0085 
0086 % Solve optimisation problem
0087 
0088 % Parameters for BPDN
0089 param.verbose = 1; % Print log or not
0090 param.gamma = 1e-1; % Convergence parameter
0091 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0092 param.max_iter = 200; % Max. number of iterations for the L1 problem
0093 param.nu_B2 = 1; % Bound on the norm of the operator A
0094 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0095 param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection
0096 param.pos_B2 = 1; %Positivity flag
0097 param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball
0098 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0099 %Optional parameters when Psit is not a tight frame:
0100 %param.nu_L1 = 1; % Bound on the norm of the operator Psit
0101 %param.max_iter_L1 = 200; %Max. number of iterations of the L1 prox
0102 %param.rel_obj_L1 = 1e-3; % Tolerance for the prox L1
0103 
0104 % Solve BPDN problem with positivity constraint
0105 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0106 
0107 % Compute SNR
0108 RSNR1 = 20*log10(norm(im,'fro') ...
0109   / norm(im-sol1,'fro'));
0110 
0111 % Example with only reality constraint
0112 param.pos_B2 = 0; %Positivity flag
0113 param.real_B2 = 1; %Reality flag
0114 
0115 % Solve BPDN problem
0116 sol2 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0117 
0118 % Compute SNR
0119 RSNR2 = 20*log10(norm(im,'fro') ...
0120   / norm(im-sol2,'fro'));
0121 
0122 
0123 %% Show results
0124 
0125 figure
0126 imagesc(sol1), axis off, axis image, colorbar
0127 title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB'])
0128 colormap gray
0129 
0130 figure
0131 imagesc(sol2), axis off, axis image, colorbar
0132 title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB'])
0133 colormap gray

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