Under-determined non-cartesian MR reconstruction with non-convex sparsity promoting analysis prior.

This work explores the problem of solving the MR reconstruction problem when the number of K-space samples acquired in a non-Cartesian grid is considerably less than the resolution (number of pixels) of the image. Mathematically this leads to the solution of an under-determined and ill-posed inverse problem. The inverse problem can only be solved when certain additional/prior assumption is made about the solution. In this case, the prior is the sparsity of the MR image in the wavelet domain. The non-convex lp-norm () of the wavelet coefficient is a suitable metric for sparsity. Such a prior can appear in two forms--in the synthesis prior formulation, the wavelet coefficients of the image is solved for while in the analysis prior formulation the actual image is solved for. Traditionally the synthesis prior formulation is more popular. However, in this work we will show that the analysis prior formulation on redundant wavelet transform provides better MR reconstruction results compared to the synthesis prior formulation.