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.
Definitions
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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
MATLAB programs
http://sparselab.stanford.edu/
Gabriel Peyré
CNRS, CEREMADE, Université Paris Dauphine.
http://www.ceremade.dauphine.fr/~peyre/
Justin Romberg
School of Electrical and Computer Engineering
Georgia Tech
http://users.ece.gatech.edu/~justin/Justin_R
omberg.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
Original vector size 512 points but only 20 non zero elements, 120 measurements taken
Recovered signal
1
N=512
K=20
M=120
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original signal
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Original vector size 512 points but only 20 non zero elements, 80 measurements taken
1.5
N=512
K=20
M=80
Recovered signal
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original signal
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