An iterative algorithm for linear inverse problems with compound regularizers

In several imaging inverse problems, it may be of interest to encourage the solution to have characteristics which are most naturally expressed by the combination of more than one regularizer. The resulting optimization problems can not be dealt with by the current state-of-the-art algorithms, which are designed for single regularizers (such as total variation or sparseness-inducing penalties, but not both simultaneously). In this paper, we introduce an iterative algorithm to solve the optimization problem resulting from image (or signal) inverse problems with two (or more) regularizers. We illustrate the new algorithm in a problem of restoration of "group sparse" images, i.e., images displaying a special type of sparseness in which the active pixels tend to cluster together. Experimental results show the effectiveness of the proposed algorithm in solving the corresponding optimization problem.