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

Bergmann, Ronny; Herrmann, Marc; Herzog, Roland; Schmidt, Stephan; Vidal-Núñez, José

doi:10.1088/1361-6420/ab6d5c

Cited by 5 publications

(10 citation statements)

References 41 publications

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“…Returning to curvature-based functionals, both our total absolute curvature |κ 1 | + |κ 2 | and the functional (κ 2 1 + κ 2 2 ) 1/2 have an interpretation as total variation of the normal vector field. The latter functional is studied in detail in a recent preprint by Bergmann et al [2019]. They also discuss the energies E, E ′ and touch upon several of our topics.…”

Section: Previous Workmentioning

confidence: 99%

“…Besides ∥W ∥ 1 , also other functions of the principal curvatures obey inequality (3) and yield a sensible curvature measure. Bergmann et al [2019] define total variation as ∥W ∥ 2 , where ∥W ∥ 2 = (κ 2 1 + κ 2 2 ) 1/2 is the Frobenius norm of the shape operator. We use ∥W ∥ 1 because of its relation to the discrete mesh energy E.…”

Section: Total Absolute Curvature Of Surfacesmentioning

confidence: 99%

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Visual smoothness of polyhedral surfaces

et al. 2019

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Representing smooth geometric shapes by polyhedral meshes can be quite difficult in situations where the variation of edges and face normals is prominently visible. Especially problematic are saddle-shaped areas of the mesh, where typical vertices with six incident edges are ill suited to emulate the more symmetric smooth situation. The importance of a faithful discrete representation is apparent for certain special applications like freeform architecture, but is also relevant for simulation and geometric computing. In this paper we discuss what exactly is meant by a good representation of saddle points, and how this requirement is stronger than a good approximation of a surface plus its normals. We characterize good saddles in terms of the normal pyramid in a vertex. We show several ways to design meshes whose normals enjoy small variation (implying good saddle points). For this purpose we define a discrete energy of polyhedral surfaces, which is related to a certain total absolute curvature of smooth surfaces. We discuss the minimizers of both functionals and in particular show that the discrete energy is minimal not for triangle meshes, but for principal quad meshes. We demonstrate our procedures for optimization and interactive design by means of meshes intended for architectural design.

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Section: Previous Workmentioning

confidence: 99%

Section: Total Absolute Curvature Of Surfacesmentioning

confidence: 99%

Visual smoothness of polyhedral surfaces

et al. 2019

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“…Section 4 is devoted to the formulation of an ADMM method which generalizes the split Bregman algorithm to the manifold-valued problem (1.3). This paper is accompanied by a companion paper Bergmann, Herrmann, et al, 2019 where we introduce a discrete counterpart of (1.2) on simplicial meshes, as they are frequently used in finite element discretizations of PDEs. On these piecewise flat surfaces, the normal vector n jumps across edges and the definition (1.2) needs to be generalized.…”

Section: Introductionmentioning

confidence: 99%

“…This derivative can be obtained by standard shape calculus techniques, which are not our concern here. A concrete example is considered in the companion paper Bergmann, Herrmann, et al, 2019. Next we consider the second term in (4.3). Due to the chosen splitting, the vector fields d i are merely transformed along with Ω according to the perturbation (3.1) and thus we define their perturbed counterparts as…”

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Total variation of the normal vector field as shape prior

et al. 2020

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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.

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Geometry Processing Problems Using the Total Variation of the Normal Vector Field

Bergmann

Herrmann

Herzog

et al. 2019

Proc Appl Math and Mech

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This paper presents a novel approach to the formulation and solution of discrete geometry processing problems including mesh denoising. The main quantity of interest is the piecewise constant unit normal vector field, which we consider on piecewise flat, triangulated surfaces. In a similar fashion as one does for images defined on 'flat' domains, our goal is to remove noise while preserving shape features such as sharp edges. To this end, we model a denoising problem via a quadratic vertex tracking term and a regularizer based on the total variation of the normal vector field. Since the latter has values on the unit sphere, its total variation reduces to a finite sum of the geodesic distances of neighboring normals. We solve the model numerically by applying split-Bregmann iterations and provide results for the 'fandisk' benchmark.

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

Cited by 5 publications

References 41 publications

Visual smoothness of polyhedral surfaces

Visual smoothness of polyhedral surfaces

Total variation of the normal vector field as shape prior

Geometry Processing Problems Using the Total Variation of the Normal Vector Field

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