A Total Variation Diminishing Interpolation Operator and Applications

TL;DR

This paper introduces a new interpolation operator for finite elements that preserves total variation, leading to improved error estimates in total variation denoising and flows.

Contribution

It constructs a total variation diminishing interpolation operator on Cartesian meshes, enhancing stability and approximation accuracy in finite element methods.

Findings

Operator does not increase total variation

Provides second order approximation in L^1

Improves error estimates in TV denoising and flows

Abstract

We construct, on continuous $Q_1$ finite elements over Cartesian meshes, an interpolation operator that does not increase the total variation. The operator is stable in $L^1$ and exhibits second order approximation properties. With the help of it we provide improved error estimates for discrete minimizers of the total variation denoising problem and for total variation flows.