Sparse and Robust Signal Reconstruction

Abstract

Many problems in signal processing and statistical inference are based on finding a sparse solution to an undetermined linear system. The reference approach to this problem of finding sparse signal representations, on overcomplete dictionaries, leads to convex unconstrained optimization problems, with a quadratic term l2, for the adjustment to the observed signal, and a coefficient vector l1 norm. This work focuses on the development and experimental analysis of an algorithm for the solution of lq lp optimization problems, where p is in the range of zero to one and q is in the range of one to two, of which l2 l1 is an instance. The developed algorithm belongs to the majorization minimization class, where the solution of the problem is given by the minimization of a progression of majorizers of the original function. Each iteration corresponds to the solution of an l2 l1 problem, solved by the projected gradient algorithm. When tested on synthetic data and image reconstruction problems, the results show good performance of the implemented algorithm, both in compressed sensing and signal restoration scenarios.