A Proximal Decomposition Method for Solving Convex Variational Inverse Problems

TL;DR

This paper introduces a proximal decomposition algorithm capable of efficiently solving convex inverse problems involving multiple nonsmooth functions, with proven convergence and demonstrated effectiveness in signal and image processing tasks.

Contribution

It presents a novel, fully decomposable algorithm that handles an arbitrary number of nonsmooth convex functions, improving flexibility over existing methods.

Findings

Algorithm converges for multiple nonsmooth functions

Effective in signal processing applications

Outperforms existing methods in inverse problems

Abstract

A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of nonsmooth functions and establish its convergence. The algorithm fully decomposes the problem in that it involves each function individually via its own proximity operator. A significant improvement over the methods currently in use in the area of inverse problems is that it is not limited to two nonsmooth functions. Numerical applications to signal and image processing problems are demonstrated.