Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging

TL;DR

This paper introduces a discrete second-order total generalized variation (TGV) for piecewise constant functions on triangular meshes, extending TGV applications beyond pixel images to irregular data structures like finite element discretizations.

Contribution

It develops a novel discrete TGV framework for triangular meshes, enabling its use in non-standard grid data and finite element contexts, with analysis of its kernel structure.

Findings

Effective denoising and inpainting on non-standard grids

Application to 3D scanner data

Reduction of staircasing effect in TGV

Abstract

We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on non-standard grids, including data from a 3D scanner.