Convergence rates for regularization functionals with polyconvex integrands

TL;DR

This paper establishes linear convergence rates for variational regularization methods using polyconvex integrands, extending analysis beyond convex functionals, with application to image registration.

Contribution

It introduces convergence rate analysis for nonconvex polyconvex regularization functionals, broadening theoretical understanding in variational regularization.

Findings

Linear convergence rates derived for polyconvex integrands

Application demonstrated in image registration problem

Generalized Bregman distance used for analysis

Abstract

Convergence rates results for variational regularization methods typically assume the regularization functional to be convex. While this assumption is natural for scalar-valued functions, it can be unnecessarily strong for vector-valued ones. In this paper we focus on regularization functionals with polyconvex integrands. Even though such functionals are nonconvex in general, it is possible to derive linear convergence rates with respect to a generalized Bregman distance, an idea introduced by Grasmair in 2010. As a case example we consider the image registration problem.