A Majorize-Minimize Subspace Approach for $\ell_2-\ell_0$ Image Regularization

In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes edge-preserving measures or frame-analysis potentials commonly used in image processing. As shown by our asymptotic results, the $\ell_2-\ell_0$ penalties we consider may be employed to provide approximate solutions to $\ell_0$-penalized optimization problems. One of the advantages of the proposed approach is that it allows us to derive an efficient majorize-minimize subspace algorithm. The convergence of the algorithm is investigated by using recent results in nonconvex optimization. The fast convergence properties of the proposed optimization method are illustrated through image processing examples. In particular, its effectiveness is demonstrated on several data recovery problems.