Nested iterative algorithms for convex constrained image recovery problems

TL;DR

This paper introduces nested iterative algorithms combining forward-backward and Douglas-Rachford methods for convex constrained image recovery, with proven convergence and applicability to wavelet-based denoising under various noise models.

Contribution

It develops novel nested algorithms for convex constrained image recovery, including extensions for non-Lipschitz gradients, with convergence proofs and practical effectiveness demonstrated.

Findings

Algorithms successfully applied to wavelet-based image restoration

Effective for Gaussian and Poisson noise models

Convergence of algorithms is theoretically established

Abstract

The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be non-smooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.