How to implement compressed sensing reconstruction?

Question

I am new to the field of Compressive Sensing. I'm trying to implement an example in this link. This example have described and implemented a sample tone reconstruction carefully, but unfortunately, there is no use of l1-magic toolbox to reconstruct the signal using compressive sensing minimization.

I know that l1eq_pd function in l1-magic calculates x in Ax=b for compressed sensing but when I use this function it returns an error Error using linsolve: Matrix must be positive definite.

Have anyone solved this minimization before?

Does anyone know any substitute for this toolbox?

How can I format my code in stack exchange (below code) for Matlab using <!-- language: lang-or-tag-here --> ?

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Appendix 1:

My Matlab code:

%% signal initialization

n = 1/40000:1/40000:1/8;
f = sin(1394*pi*n) + sin(3266*pi*n);

%% Random Sampling

% b = Phi * f    (random sampling)
% c = Psi * f    where Psi = IDCT (sparsifying)

m = floor(rand(1,500)*length(f));
b = f(m);       % random samples of (f)
c = idct(f);    % sparsed (f)
plot(f,'b');hold
plot(m,b,'k.');title('Original Signal(f) and random sampless(b)');
axis([1,1000,-3,3])
legend('Original Signal','Randomly Sampled Signal')
figure, plot(c), axis([0,650,-10,10]);title('Sparse samples (c = IDCT(f))');

% f = Psi * c
% Psi = DCT
% f = DCT(c)
% c = IDCT(f)

%% solution 1 (x)

% Ax = b
% x = A\b

D = dct(eye(length(n),length(n)));
A = D(m,:);
% sound(f)
x = (A\b')';
b_hat = dct(x);
% sound(b_hat)

%% solution 2 (y)

% Ay = b
% y = pinv(A) * b

y = (pinv(A)*b')';
figure,plot(y),axis([0,650,-10,10])
figure,plot(dct(y)),axis([1,1000,-1.5,1.5])

%% solution 3 (s1)

s1 = l1eq_pd(y',A,A',b',5e-3,20); % L1-magic toolbox

This piece of code is a good example to understand compressed sensing and works correctly except the last line I am questioning.

Answer 1

I have had encountered this problem before. In using l1-magic toolbox, the first thing you need to do is to modify the file l1eq_pd.m.

To do so find all linsolve instructions and remove the third parameter, for example on line 141 change the following line from

[dv,hcond] = linsolve(H11p, w1p,opts);

to,

[dv,hcond] = linsolve(H11p, w1p);

This is because the structure opts specify the property of the matrix A in solving Ax=b linear system of equations and omitting the structure opts in the linsolve function will allow Matlab to choose the appropriate solver automatically [Reference]

If the errors persists, your own code should be scrutinized.

I suggest to have a look on this examples codes.