Fast Convex Optimizer for for images (compressed sensing)?

I have the following cost function

minimize ||x||1, 

subject to

|| A(x) - b||2 < epsilon

Where A is some operator (A(x) ~= A*x, unfortunately), x is my image, and b is my measurement.

At the moment, I am using cvx to solve. This works perfectly for little "images" (32 x 32), but for any real images (128x128 or 256 x 256) it is super slow!

Does anyone know of a better solver or a better way to solve this kind of problem? I feel like there has to be a better way with all of the compressed sensing done on images these days.

Thanks so much for your help! I really appreciate your help!

Christine

[Matt J]

I'm going to speculate that the slowness is not due to the quality of the solver, but may be due more to the quality of your implementation of the operator A(x). You should show us how that currently looks. In your other recent posts, you asked about A(x) operators that turned out to be linear, so it would be good to update us anyway on why A(x) ~= A*x

[SwishSwisher]

Thanks again for your help Matt!

A(x) is the following:

 A = (@x) Fi*(F*x*F.'*P)*Fi.'

The .*P is why I can't use A*x.

[SwishSwisher]

Actually after more research, I found that CVX is for convenience/accuracy, not for speed and not recommended for images. It is wonderful at 32 x 32!

If only A(x) == A*x. Then I'd have so many options.

[SwishSwisher]

Maybe I'm answering my own question. But SALSA might work.

http://cascais.lx.it.pt/~mafonso/salsa.html