A non-convex regularization approach for compressive sensing

Compressive sensing (CS) aims at reconstructing high dimensional data from a small number of samples or measurements. In this paper, we propose the minimization of a non-convex functional for the solution of the CS problem. The considered functional incorporates information on the self-similarity of the image by measuring the rank of some appropriately constructed matrices of fairly small dimensions. However, since the rank minimization is a NP hard problem, we consider, as a surrogate function for the rank, a non-convex, but smooth function. We provide a theoretical analysis of the proposed functional and develop an iterative algorithm to compute one of its stationary points. We prove the convergence of such algorithm and show, with some selected numerical experiments, that the proposed approach achieves good performances, even when compared with the state of the art.