# 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. May 2, 2015 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. May 3, 2015 at 7:01
• No, it doesn't make any sense. May 6, 2015 at 18:13