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

An analogue of the total-variation prior for the normal vector field along the boundary of smooth and non-smooth shapes in 3D is introduced. Its analysis in the smooth case is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. This novel functional is subsequently extended to piecewise flat, triangulated surfaces as they occur for instance in finite element computations. The ensuing discrete functional agrees with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of shape optimization problems in which the total variation of the normal appears as a regularizer. Both the continuous and discrete settings are detailed. Unlike most other priors such as surface area, the new functional allows for piecewise flat shapes in the discrete setting. As an application, a geometric inverse problem of inclusion detection type is considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.