# Matrix expansion into bases

I have an image (that is a matrix), let's say of dimensions NxN. I then want to expand this matrix into M basis matrices (for the moment I'm still unsure how many M of these basis matrices I should assume) of the same dimensions NxN, that is analog to basis expansion of a vector into basis vectors. But as for the base matrices, I want them to be either only the translation (fixed scaling) or only the scaling (fixed translation) of a given mother basis matrix. Is such an expansion justifiable, or probably even feasible in the first place? I'm interested in searching for a set of basis matrices whose corresponding expansion coefficients form a sparse vector.

• I'm a bit confused here -- you want to do an expansion into wavelets using either translation or dilation but not both? That defeats the point of using wavelets in the first place. – Batman May 2 '15 at 22:39
• That's exactly why I asked here. So it doesn't make much sense doing that? What I'm dealing with is actually, I have a 2D data (image) measured in Fourier space, so the measured data is in the form of a spectrum. I also have a priori knowledge about the original image in the real space of being sparse and I want to benefit from this property. That's why I actually prefer basis matrices which related only by translation with each other, because translation in Fourier space merely amounts to a linear modulation of the spectral phases. – Tommy 77 May 3 '15 at 7:01
• No, it doesn't make any sense. – Batman May 6 '15 at 18:13