A forward-backward view of some primal-dual optimization methods in image recovery

TL;DR

This paper unifies various primal-dual optimization algorithms for image recovery under a general forward-backward framework, enabling extensions with variable metrics and demonstrating their effectiveness in image restoration tasks.

Contribution

It shows that many primal-dual algorithms can be derived from a single forward-backward approach and introduces variable metric extensions for improved performance.

Findings

Unified framework for primal-dual algorithms

Extensions with variable metrics demonstrated

Application to image restoration tasks

Abstract

A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primal-dual proximal approaches have been developed which provide efficient solutions to large-scale optimization problems. The objective of this paper is to show that a number of existing algorithms can be derived from a general form of the forward-backward algorithm applied in a suitable product space. Our approach also allows us to develop useful extensions of existing algorithms by introducing a variable metric. An illustration to image restoration is provided.