# L1-magic : Recovery of Sparse Signals via Convex programming ...

L1-magic : Recovery of Sparse Signals via Convex programming by Emmanuel Cands 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

Justin Romberg School of Electrical and Computer Engineering Georgia Tech http://users.ece.gatech.edu/~justin/ Justin_Romberg.html Min-L1 with equality constraints The program P1 min x 1 subject to Ax b also known as basis persuit, finds the vector with smallest l1 norm x 1 : xi i that explaines the observations b. 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 R e c o v e re d s ig n a l Original vector size 512 points but only 20 non zero elements, 120 measurements taken 1 N=512 K=20 M=120

0.5 0 -0.5 -1 0 100 200 300 400 500 600 0 100 200 300 400 500 600

o r ig in a l s ig n a l 1 0.5 0 -0.5 -1 R e c o v e re d s ig n a l Original vector size 512 points but only 20 non zero elements, 80 measurements taken 1.5 N=512 K=20 M=80 1 0.5 0 -0.5 -1 0 100 200 300

400 500 600 0 100 200 300 400 500 600 o r ig in a l s i g n a l 1 0.5 0 -0.5 -1

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