Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems

TL;DR

This paper introduces a novel total variation prior for the normal vector field on 3D shapes, enabling piecewise flat shape regularization and improving shape reconstruction in inverse problems.

Contribution

It proposes a new total variation functional for normal vectors on triangulated surfaces, compatible with piecewise flat shapes, and develops a split Bregman method for shape optimization.

Findings

Effective mesh denoising demonstrated

Accurate shape identification in inverse problems

Functional aligns with discrete mean curvature

Abstract

An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.