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

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.