Compressive Sensing matrix

Question

I am working with compressive sensing recovery with image and I test with various sensing matrices:

Case 1: Sensing matrix A of size MxN is i.i.d Gaussian matrix.

Case 2: Sensing matrix A is size of MxN is DCT matrix (it means that I pick up M rows from an NxN DCT matrix).

I see that the case 2 gives better reconstructed images than case 1 (I used total variation for recovery). I try to search this scenario, and really that no one uses an DCT base as a sensing matrix. Could you explain why? Thank you very much.

Answer 1

Let see my test: an image denote x, and its measurement y is sensed by:

y=Ax;

Now, we have two case: case 1: A is Gaussian matrix; case 2: A is DCT matrix.

Note that in some papers, The acquisition base $\Phi =A\Psi $; however, prof. Romberg states that $\Psi$ can be moved to the decoder for a simpler encoder. It means that image x is directly sensed by the sensing matrix A.

I don't know why the sensing matrix generated by DCT matrix is better than Gaussian matrix. :(

Answer 2

Because you are comparing apples and oranges. In the Gaussian case, you are expecting the image to be sparse in the Haar basis, whereas in the DCT case, you are expecting the gradient of the image to be sparse in the DCT basis.