Non-smooth variational regularization for processing manifold-valued data

TL;DR

This paper explores variational regularization techniques for processing data on nonlinear manifolds, including TV, TGV, Mumford-Shah models, and wavelet methods, with algorithms for denoising and inverse problems.

Contribution

It extends variational regularization methods to manifold-valued data, developing discrete energies and algorithms for denoising and inverse problems, including wavelet regularization.

Findings

Developed discrete energies for manifold denoising

Proposed algorithms for variational minimization on manifolds

Extended methods to inverse measurement problems

Abstract

Many methods for processing scalar and vector valued images, volumes and other data in the context of inverse problems are based on variational formulations. Such formulations require appropriate regularization functionals that model expected properties of the object to reconstruct. Prominent examples of regularization functionals in a vector-space context are the total variation (TV) and the Mumford-Shah functional, as well as higher-order schemes such as total generalized variation models. Driven by applications where the signals or data live in nonlinear manifolds, there has been quite some interest in developing analogous methods for nonlinear, manifold-valued data recently. In this chapter, we consider various variational regularization methods for manifold-valued data. In particular, we consider TV minimization as well as higher order models such as total generalized variation (TGV). Also, we discuss (discrete) Mumford-Shah models and related methods for piecewise constant data. We develop discrete energies for denoising and report on algorithmic approaches to minimize them. Further, we also deal with the extension of such methods to incorporate indirect measurement terms, thus addressing the inverse problem setup. Finally, we discuss wavelet sparse regularization for manifold-valued data.