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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 programmingby Emmanuel Candès and Justin Romberg

Caltech October 2005

Compressive Sensing Tutorial PART 2

Svetlana Avramov-Zamurovic

January 22, 2009.

• 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

Gabriel PeyréCNRS, CEREMADE, Université Paris Dauphine.

Justin RombergSchool of Electrical and Computer EngineeringGeorgia Tech

http://users.ece.gatech.edu/~justin/Justin_Romberg.html

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

K=20

M=120

N=512

K=20

M=80