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L1-magic : Recovery of Sparse Signals via Convex programming by Emmanuel Cand è s and Justin Romberg

L1-magic : Recovery of Sparse Signals via Convex programming by Emmanuel Cand è s and Justin Romberg. Caltech October 2005 Compressive Sensing Tutorial PART 2 Svetlana Avramov-Zamurovic January 22, 2009. Definitions. X desired vector (N elements), K sparse Y measurements (M elements), K<M<N

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L1-magic : Recovery of Sparse Signals via Convex programming by Emmanuel Cand è s and Justin Romberg

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  1. L1-magic : Recovery of Sparse Signals via Convex programmingby Emmanuel Candès and Justin Romberg Caltech October 2005 Compressive Sensing Tutorial PART 2 Svetlana Avramov-Zamurovic January 22, 2009.

  2. Definitions • X desired vector (N elements), K sparse • Y measurements (M elements), K<M<N • Ψ orthonormal basis (NxN), X= Ψs • Φ measurement matrix (MxN) • L1 norm= sum(abs(all vector X elements)) • Linear programming • Find sparse solution that satisfies measurements, Y= ΦX and minimizes the L1 norm of X

  3. MATLAB programs http://sparselab.stanford.edu/ Gabriel PeyréCNRS, CEREMADE, Université Paris Dauphine. http://www.ceremade.dauphine.fr/~peyre/ Justin RombergSchool of Electrical and Computer EngineeringGeorgia Tech http://users.ece.gatech.edu/~justin/Justin_Romberg.html

  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 experiments rand_state=1;randn_state=1;rand('state', rand_state);randn('state', randn_state); N = 512;% signal length T = 20;% number of spikes in the signal K = 120;% number of observations to make x = zeros(N,1);q = randperm(N);x(q(1:T)) = sign(randn(T,1)); % random +/- 1 signal% %SAZ original signal to be recovered disp('Creating measurment matrix...');A = randn(K,N);A = orth(A')';disp('Done.'); y = A*x;% observations SAZ measurements x0 = A'*y;% initial guess = min energy xp = l1eq_pd(x0, A, [], y, 1e-3); % solve the LP http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

  5. N=512 K=20 M=120 N=512 K=20 M=80

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