Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors

TL;DR

This paper introduces a novel variational regularization method for inverse and imaging problems involving vector-valued functions, using derivative-free double integrals inspired by Sobolev semi-norms, with proven existence, stability, and convergence of solutions.

Contribution

It proposes a new class of regularization functionals based on double integrals for vector-valued functions, extending Sobolev semi-norm concepts without derivatives.

Findings

Existence of minimizers established.

Stability and convergence results proven.

Applicable to inverse and imaging problems with vector-valued functions.

Abstract

We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain, Br\'ezis and Mironescu, "Another Look at Sobolev Spaces". In: Optimal Control and Partial Differential Equations-Innovations and Applications, IOS press, Amsterdam, 2001. For the proposed regularization functionals we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.