Discrete Total Variation with Finite Elements and Applications to Imaging

TL;DR

This paper introduces a new discrete total variation (DTV) variant for finite element functions, enabling efficient image reconstruction algorithms on complex meshes with improved properties over traditional TV-seminorms.

Contribution

A novel DTV definition based on a nodal quadrature formula is proposed, allowing efficient implementation of classical image reconstruction algorithms in finite element spaces.

Findings

DTV has favorable properties compared to original TV-seminorm.

Algorithms for TV-$L^2$ and TV-$L^1$ can be efficiently implemented in finite element spaces.

The approach extends classical image reconstruction methods to complex geometries.

Abstract

The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart--Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-$L^2$ and TV-$L^1$, can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.