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Experiment1

PURPOSE ^

% Experiment1

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiment1
 In this experiment we evaluate the performance of SARA for spread 
 spectrum acquisition. We use a 256x256 version of Lena as a test image. 
 Number of measurements is M = 0.2N and input SNR is set to 30 dB. These
 parameters can be changed by modifying the variables p (for the
 undersampling ratio) and input_snr (for the input SNR).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiment1
0002 % In this experiment we evaluate the performance of SARA for spread
0003 % spectrum acquisition. We use a 256x256 version of Lena as a test image.
0004 % Number of measurements is M = 0.2N and input SNR is set to 30 dB. These
0005 % parameters can be changed by modifying the variables p (for the
0006 % undersampling ratio) and input_snr (for the input SNR).
0007 
0008 
0009 %% Clear workspace
0010 
0011 clc
0012 clear;
0013 
0014 
0015 %% Define paths
0016 
0017 addpath misc/
0018 addpath prox_operators/
0019 addpath test_images/
0020 
0021 
0022 
0023 %% Read image
0024 
0025 imagename = 'lena_256.tiff';
0026 
0027 % Load image
0028 im = im2double(imread(imagename));
0029 
0030 % Normalise
0031 im = im/max(max(im));
0032 
0033 % Enforce positivity
0034 im(im<0) = 0;
0035 
0036 %% Parameters
0037 
0038 input_snr = 30; % Noise level (on the measurements)
0039 
0040 %Undersampling ratio M/N
0041 p=0.2;
0042 
0043 
0044 %% Sparsity operators
0045 
0046 %Wavelet decomposition depth
0047 nlevel=4;
0048 
0049 dwtmode('per');
0050 [C,S]=wavedec2(im,nlevel,'db8'); 
0051 ncoef=length(C);
0052 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0053 ncoef1=length(C1);
0054 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0055 ncoef2=length(C2);
0056 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0057 ncoef3=length(C3);
0058 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0059 ncoef4=length(C4);
0060 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0061 ncoef5=length(C5);
0062 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0063 ncoef6=length(C6);
0064 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0065 ncoef7=length(C7);
0066 
0067 %SARA
0068 
0069 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0070     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0071     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0072 
0073 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0074     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0075     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0076     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0080 
0081 %Db8 wavelet basis
0082 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0083 Psi2 = @(x) waverec2(x,S,'db8');
0084 
0085 %Curvelet
0086 %CurveLab needs to be installed to run Curvelet simulations
0087 realv = 1;
0088 Cv = fdct_usfft(im,realv);
0089 Mod = sopt_mltb_struct2size(Cv);
0090 
0091 Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); 
0092 Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv);
0093 
0094 %% Spread spectrum operator
0095 % Mask
0096 mask = rand(size(im)) < p; 
0097 ind = find(mask==1);
0098 % Masking matrix (sparse matrix in matlab)
0099 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0100     
0101 %Spread spectrum sequence
0102     
0103 ss=rand(size(im));
0104 C=(2*(ss<0.5)-1);
0105 
0106 A = @(x) Ma*reshape(fft2(C.*x)/sqrt(numel(ind)), numel(x), 1);
0107 At = @(x) C.*(ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0108     
0109 % Sampling
0110 y = A(im);
0111 % Add Gaussian i.i.d. noise
0112 sigma_noise = 10^(-input_snr/20)*std(im(:));
0113 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0114     
0115     
0116 % Tolerance on noise
0117 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0118 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0119     
0120     
0121 % Parameters for BPDN
0122 param.verbose = 1; % Print log or not
0123 param.gamma = 1e-1; % Converge parameter
0124 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0125 param.max_iter = 200; % Max. number of iterations for the L1 problem
0126 param.nu_B2 = 1; % Bound on the norm of the operator A
0127 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0128 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0129 param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active
0130 param.max_iter_B2=300;
0131 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0132 param.nu_L1 = 1;
0133 param.max_iter_L1 = 20;
0134 param.rel_obj_L1 = 1e-2;
0135     
0136     
0137 % Solve BPSA problem
0138     
0139 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0140     
0141 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0142     
0143 % SARA
0144 % It uses the solution to BPSA as a warm start
0145 maxiter=10;
0146 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8));
0147 tol=1e-3;
0148   
0149 sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1);
0150 
0151 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0152 
0153 % Solve BPBb8 problem
0154     
0155 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0156     
0157 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0158     
0159 % RWBPDb8
0160 % It uses the solution to BPDBb8 as a warm start
0161 maxiter=10;
0162 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0163 tol=1e-3;
0164   
0165 sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3);
0166       
0167 RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro'));
0168 
0169 % Parameters for Curvelet
0170 
0171 % Parameters for BPDN
0172 param3.verbose = 1; % Print log or not
0173 param3.gamma = 1e-1; % Converge parameter
0174 param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0175 param3.max_iter = 200; % Max. number of iterations for the L1 problem
0176 param3.nu_B2 = 1; % Bound on the norm of the operator A
0177 param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0178 param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0179 param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0180 param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0181 
0182     
0183 
0184 
0185 % Solve BP Curvelet problem
0186     
0187 sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3);
0188     
0189 RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro'));
0190     
0191 % RW-Curvelet
0192 % It uses the solution to BPDBb8 as a warm start
0193 maxiter=10;
0194 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0195 tol=1e-3;
0196   
0197 sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5);
0198      
0199 RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro'));
0200 
0201     
0202 % Parameters for TVDN
0203 param1.verbose = 1; % Print log or not
0204 param1.gamma = 1e-1; % Converge parameter
0205 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0206 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0207 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0208 param1.nu_B2 = 1; % Bound on the norm of the operator A
0209 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0210 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0211 param1.max_iter_B2 = 300;
0212 param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0213     
0214 % Solve TV problem
0215     
0216 sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0217     
0218 RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro'));
0219     
0220 % RWTV
0221 % It uses the solution to TV as a warm start
0222 maxiter=10;
0223 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0224 tol=1e-3;
0225   
0226 sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7);
0227     
0228 RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro'));
0229 
0230 
0231 %Show reconstructed images
0232 
0233 figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray;
0234 title(['BPSA, SNR=',num2str(RSNR1), 'dB'])
0235 figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray;
0236 title(['SARA, SNR=',num2str(RSNR2), 'dB'])
0237 
0238 figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray;
0239 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0240 figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray;
0241 title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB'])
0242 
0243 figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray;
0244 title(['Curvelet, SNR=',num2str(RSNR5), 'dB'])
0245 figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray;
0246 title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB'])
0247 
0248 figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray;
0249 title(['TV, SNR=',num2str(RSNR7), 'dB'])
0250 figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray;
0251 title(['RW-TV, SNR=',num2str(RSNR8), 'dB'])
0252 
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