Sparse and Robust Signal Reconstruction

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 focus the development and experimental analysis of an algorithm for the solution of lq-lp optimization problems, where p ∈ ]0, 1] ∧ q ∈ [1, 2], 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 shows a good performance of the implemented algorithm, both in compressed sensing and signal restoration scenarios.