1. 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.

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

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

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