Proximal methods for image restoration using a class of non-tight frame representations

The objective of this paper is to develop a convex optimization approach for solving image deconvolution problems involving frame representations. Until now, most of the proposed frame-based variational methods assumed either Lipschitz differentiability properties or tight representations. These assumptions are relaxed here, thus offering the possibility of considering a broader class of image restoration problems. The proposed algorithms allow us to solve both frame analysis and frame synthesis problems for various noise distributions. The proposed approach is proved to be effective for restoring data corrupted by Poisson noise by using (non-tight) discrete dual-tree wavelet representations.