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**L1-magic : Recovery of Sparse Signals via Convex programming by Emmanuel Candès and Justin Romberg**

Caltech October 2005 Compressive Sensing Tutorial PART 2 Svetlana Avramov-Zamurovic January 22, 2009.

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**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

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**MATLAB programs http://sparselab.stanford.edu/**

Gabriel Peyré CNRS, CEREMADE, Université Paris Dauphine. Justin Romberg School of Electrical and Computer Engineering Georgia Tech

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**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

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

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