# coherence calculation in sparse sensing

i have an image I of size 32*32. I perform the DCT of this image using the matlab function DCT2(I). I get a sparse representation of my image which is again a 32*32 image. I construct a circulant matrix A of size 200*1024, i.e i need only 200 samples out of the total 1024. Mathematically spaeaking, what i did is take my original signal x, sparsify using dct2 call it B, then compress is using A. So i get measurements y=A.B.x . Now how do i calculate the incoherence measure from A and B. Can somebody help.

• Your set-up is somewhat unconventional in terms of compressed sensing, because you would usually formulate your image x as sparse in the sense x = B.z (using your notation), where z is a sparse (DCT-domain) vector and B then needs to be the IDCT transform to get the image x. You would usually take the measurements of x: y = A.x = A.B.z Is there any particular reason for the somewhat "backwards" set-up you consider here? – Thomas Arildsen Dec 10 at 8:33

Update

I would like to rephrase the original problem with slightly different notation and claim that there is no need to measure the coherence of the product of the circulant matrix A and the DCT matrix B as asked by the OP.

In the compressive sensing setting, the coherence of a sensing matrix $$A$$ is useful when we are using it for constructing measurements $$y = A \alpha$$ where $$\alpha$$ is a sparse vector. Then, the coherence properties of $$A$$ can be used to give an estimate of number of measurements required in $$y$$ so that $$\alpha$$ can be recovered stably from $$y$$. $$\alpha$$ is actually a sparse representation of some signal $$x$$ in some orthonormal basis $$\Psi$$ which is represented by the equation $$x = \Psi \alpha$$. In this particular case, $$x$$ is the image 32x32, $$\Psi$$ is the IDCT2 transform, $$\alpha$$ is the sparse representation of $$x$$ obtained by DCT2 transformation which is the equation $$\alpha = \Psi^T x$$. Thus, the matrix $$B$$ from OP is $$\Psi^T$$. So the relationship between the measurements $$y$$ and the original signal $$x$$ is $$y = A \Psi^T x$$. The sparse recovery of $$x$$ from $$y$$ would happen in two steps.

1. Construct $$\alpha$$ from $$y$$ by using some algorithm $$\Delta$$. We represent this algorithmic recovery step as: $$\alpha = \Delta(A, y)$$. $$\Delta$$ could be basis pursuit or OMP or anything else.
2. Construct $$x$$ from $$\alpha$$ by using the equation $$x = \Psi \alpha$$.

In this overall recovery process, the matrix $$A B = A \Psi^T$$ plays no role. Hence, there is no need to measure the coherence of $$AB$$.

The old answer was incorrect. The coherence of a matrix $$A$$ doesn't change if it is pre-multiplied by an orthogonal matrix $$B$$ as $$BA$$. Coherence of a matrix can change in arbitrary ways if it is post-multiplied by an orthogonal matrix as $$AB$$. I am not aware of any particular results of relating the coherence of $$A$$ with $$AB$$.