A higher order TV-type variational problem related to the denoising and inpainting of images

TL;DR

This paper surveys higher order TV-type variational problems in image denoising and inpainting, exploring their theoretical properties and potential advantages over classical TV models, including reduced staircasing effects.

Contribution

It extends previous TV-regularization results to higher derivatives, analyzing solutions in higher order BV spaces and their regularity, with implications for image processing.

Findings

Higher order TV models preserve edges better.

Solutions exhibit partial regularity in higher order BV spaces.

Higher order models reduce staircasing effects.

Abstract

We give a comprehensive survey on a class of higher order variational problems which are motivated by applications in mathematical imaging. The overall aim of this note is to investigate if and in which manner results from the first author's previous work on variants of the TV-regularization model (see e.g. [BF1], [BF2], [BF3] and [FT]) can be extended to functionals which involve higher derivatives. This seems to be not only of theoretical interest, but also relevant to applications since higher order TV-denoising appears to maintain the advantages of the classical model as introduced by Rudin, Osher and Fatemi in [ROF] while avoiding the unpleasant "staircasing" effect (see e.g. [BKP] or [LLT]). Our paper features results concerning generalized solutions in spaces of functions of higher order bounded variation, dual solutions as well as partial regularity of minimizers.