Presentation on theme: "Caltech October 2005 Compressive Sensing Tutorial PART 2"— Presentation transcript:
1 L1-magic : Recovery of Sparse Signals via Convex programming by Emmanuel Candès and Justin Romberg Caltech October 2005Compressive Sensing Tutorial PART 2Svetlana Avramov-ZamurovicJanuary 22, 2009.
2 Definitions X desired vector (N elements), K sparse Y measurements (M elements), K<M<NΨ orthonormal basis (NxN), X= ΨsΦ measurement matrix (MxN)L1 norm= sum(abs(all vector X elements))Linear programmingFind sparse solution that satisfies measurements, Y= ΦX and minimizes the L1 norm of X
3 MATLAB programs http://sparselab.stanford.edu/ Gabriel Peyré CNRS, CEREMADE, Université Paris Dauphine.Justin Romberg School of Electrical and Computer Engineering Georgia Tech
4 Min-L1 with equality constraints When x, A, b have real-valued entries, (P1) can be recast as an LP.% load random states for repeatable experimentsrand_state=1;randn_state=1;rand('state', rand_state);randn('state', randn_state);N = 512;% signal lengthT = 20;% number of spikes in the signalK = 120;% number of observations to makex = zeros(N,1);q = randperm(N);x(q(1:T)) = sign(randn(T,1));% random +/- 1 signal% %SAZ original signal to be recovereddisp('Creating measurment matrix...');A = randn(K,N);A = orth(A')';disp('Done.');y = A*x;% observations SAZ measurementsx0 = A'*y;% initial guess = min energyxp = l1eq_pd(x0, A, , y, 1e-3); % solve the LP