Group sparse optimization for images recovery using capped folded concave functions

Abstract

This paper considers the image recovery problem by taking group sparsity into account as the prior knowledge. This problem is formulated as a group sparse optimization over the intersection of a polyhedron and a possibly degenerate ellipsoid. It is a convexly constrained optimization problem with a group cardinality objective function. We use a capped folded concave function to approximate the group cardinality function and show that the solution set of the continuous approximation problem and the set of group sparse solutions are the same. Moreover, we use a penalty method to replace the constraints in the approximation problem by adding a convex nonsmooth penalty function in the objective function. We show the existence of positive penalty parameters such that the solution sets of the unconstrained penalty problem and the group sparse problem are the same. We propose a smoothing penalty algorithm and show that any accumulation point of the sequence generated by the algorithm is a directional stationary point of the continuous approximation problem. Numerical experiments for recovery of group sparse image are presented to illustrate the efficiency of the smoothing penalty algorithm with adaptive capped folded concave functions.