Candes.m

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This Matlab demonstration of ''compressive sampling'' (<tt>a.k.a.</tt> ''compressed sensing'') by [http://www.acm.caltech.edu/~emmanuel Emmanuel Candes]<br> comes from his [http://www.ima.umn.edu/recordings/New_Directions_Short_Course/ND6.4-15.07/candes6-6-07.ram June 6 2007] video on the [[Optimization Videos]] page.
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<pre>
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%Emmanuel Candes, California Institute of Technology, June 6 2007, IMA Summerschool.
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%Transcribed by Jon Dattorro.
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%Fails using SDP solver SDPT3 on 7th consecutive run after Matlab R2007b startup. CVX version 1.2 (build 656).
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%Fails using SDP solver Sedumi on 4th consecutive run after Matlab R2007b startup. CVX version 1.2 (build 656).
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clear all, close all
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n = 512; % Size of signal
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m = 64; % Number of samples (undersample by a factor 8)
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k = 0:n-1; t = 0:n-1;
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F = exp(-i*2*pi*k'*t/n)/sqrt(n); % Fourier matrix
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freq = randsample(n,m);
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A = [real(F(freq,:));
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imag(F(freq,:))]; % Incomplete Fourier matrix
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S = 28;
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support = randsample(n,S);
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x0 = zeros(n,1); x0(support) = randn(S,1);
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b = A*x0;
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% Solve l1 using CVX
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cvx_quiet(true);
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%cvx_solver('sedumi');
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cvx_begin
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variable x(n);
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minimize(norm(x,1));
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A*x == b;
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cvx_end
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norm(x - x0)/norm(x0)
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figure, plot(1:n,x0,'b*',1:n,x,'ro'), legend('original','decoded')
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</pre>
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Code between <code>cvx_begin</code> and <code>cvx_end</code> requires [http://www.stanford.edu/~boyd/cvx CVX].
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<code>randsample()</code> is from Matlab Statistics Toolbox.
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Failure modes are reparable by [[Convex Iteration]]:
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<pre>
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%Emmanuel Candes, California Institute of Technology, June 6 2007, IMA Summerschool.
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%Convex Iteration implementation by Jon Dattorro.
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%Failure modes repaired.
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clear all, close all
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n = 512; % Size of signal
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m = 64; % Number of samples (undersample by a factor 8)
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k = 0:n-1; t = 0:n-1;
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F = exp(-i*2*pi*k'*t/n)/sqrt(n); % Fourier matrix
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freq = randsample(n,m);
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A = [real(F(freq,:));
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imag(F(freq,:))]; % Incomplete Fourier matrix
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S = 28;
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support = randsample(n,S);
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x0 = zeros(n,1); x0(support) = randn(S,1);
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b = A*x0;
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cvx_quiet(true);
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%cvx_solver('sedumi');
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%convex iteration
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y = ones(n,1);
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while 1
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% Solve l0 using CVX and Convex Iteration
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cvx_begin
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variable x(n);
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minimize(norm(y.*x,1));
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A*x == b;
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cvx_end
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% update search direction y
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[x_sorted, indices] = sort(abs(x), 'descend');
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y = ones(n,1);
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y(indices(1:S)) = 0;
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cardx = sum(abs(x) > 1e-6)
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if cardx <= S, break, end
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end
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norm(x - x0)/norm(x0)
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figure, plot(1:n,x0,'b*',1:n,x,'ro'), legend('original','decoded')
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</pre>
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Revision as of 23:29, 15 November 2008

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