Total variation denoising in $l^1$ anisotropy

TL;DR

This paper investigates solutions to anisotropic total variation denoising problems, showing that for PCR (piecewise constant on rectangles) data, solutions remain PCR, enabling finite algorithms and preserving continuity.

Contribution

It establishes that solutions to anisotropic TV denoising problems preserve PCR structure, allowing finite algorithms and continuity preservation for PCR data.

Findings

Solutions preserve PCR structure when data is PCR.

Finite algorithms can be used for PCR data.

Continuity is preserved in solutions.

Abstract

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.