Coupled variational problems of linear growth related to the denoising and inpainting of images

TL;DR

This paper investigates coupled variational models with linear growth for image denoising and inpainting, establishing existence and regularity of solutions using relaxation theory and convex analysis.

Contribution

It provides new analytical results on coupled variational problems with linear growth, confirming conjectures and extending previous work in mathematical imaging.

Findings

Existence of solutions established

Regularity results proven

Framework applicable to image processing tasks

Abstract

In this note we present some results that were already conjectured in the work [9] by Bildhauer, Fuchs and Weickert, where they have investigated analytical aspects of coupled variational models with applications to mathematical imaging. Here we focus on variants of linear growth, which require a treatment in the framework of relaxation theory and convex analysis. Following basic ideas from [6] and [7], we establish existence and regularity of (dual-)solutions.