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arXiv · 1908.07916

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

Ronny Bergmann·Marc Herrmann·Roland Herzog·Stephan Schmidt·José Vidal Núñez
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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.

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Ronny Bergmann, Marc Herrmann, Roland Herzog, Stephan Schmidt, José Vidal Núñez. 2019-08-20. Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems. https://doi.org/10.1088/1361-6420/ab6d5c

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