Total variation of the normal vector field as shape prior

Abstract

An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the twodimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is non-differentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.Notice that we restrict the discussion to the isotropic case here, i.e., | · | 2 denotes the Euclidean norm. Moreover, Du is the derivative of u and {e 1 , e 2 } denotes the standard Euclidean basis. The seminorm (1.1) extends to less regular, so-called BV functions (bounded variation), whose distributional gradient exists only in the sense of measures. We refer the reader to Giusti, 1984;Attouch, Buttazzo, Michaille, 2006 for an extensive discussion of BV functions. The utility of (1.1) as a regularizer, or prior, lies in the fact that it favors piecewise constant solutions.In this paper, we introduce a novel regularizer based on the total variation, which can be used, for instance, in shape optimization applications as well as geometric inverse problems. In the latter class, the unknown, which one seeks to recover, is a shape Ω ⊂ R 3 , which might represent the location of a source or inclusion inside a given, larger domain, or the geometry of an inclusion or a scatterer. The boundary of Ω will be denoted by Γ.The novel functional, which we term the total variation of the normal field along a smooth surface Γ, is defined by |n| T V (Γ) := Γ |(D Γ n) ξ 1 | 2 g + |(D Γ n) ξ 2 | 2 g 1/2 ds (1.2)Date: August 22, 2019.

Discrete total variation of the normal vector field as shape prior with applications in geometric inverse problems

“…To this end, we adapt the well-known split Bregman method to our setting. This leads to a discrete realization of the approach presented in Bergmann, Herrmann, et al, 2019, where E are the edges of the unknown part Γ h of the boundary ∂Ω h . We will consider a concrete example in Section 4.1.…”

“…It is most commonly applied to functions with values in R or R n . In the companion paper Bergmann, Herrmann, et al, 2019, we introduced the total variation of the normal vector field n along smooth surfaces Γ ⊂ R 3 : |n| T V (Γ) := Γ |(D Γ n) ξ 1 | 2 g + |(D Γ n) ξ 2 | 2 g 1/2 ds.…”

“…Finally, | · | g denotes the norm induced by a Riemannian metric on S 2 . It was shown in Bergmann, Herrmann, et al, 2019 that (1.1) can be alternatively expressed as…”

“…Similarly as for the case of smooth surfaces discussed in Bergmann, Herrmann, et al, 2019, solving discrete shape optimization problems (1.3) is challenging due to the non-trivial dependency of n on the vertex positions of the discrete surface Γ h , as well as the non-smoothness of |n| DT V (Γ h ) . We therefore propose in Section 3 a version of the split Bregman method proposed in Goldstein, Osher, 2009, an algorithm from the alternating direction method of multipliers (ADMM) class in which the jumps in the normal vector are treated as a separate variable.…”

“…In our numerical example in Section 5, we will utilize a volume mesh consisting of tetrahedra, whose surface mesh consists of triangles; see Figure 2.1. Since the surface Γ h is non-smooth, the definition (1.1) of the total variation of the normal proposed in the companion paper Bergmann, Herrmann, et al, 2019 for smooth surfaces does not apply. Since the normal vector field n is piecewise constant here, its variation is concentrated in spontaneous changes across edges between facets, rather than gradual changes expressed by the derivative D Γ n. We therefore propose to replace (1.1) by…”