A Discrete Extrinsic and Intrinsic Dirac Operator

Hoffmann, Tim; Ye, Zi

doi:10.1080/10586458.2020.1727798

Cited by 4 publications

(8 citation statements)

References 15 publications

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“…3 Decomposition of Z 2 into graphs with black and white vertices family. In [8] it was shown that the family of minimal surfaces can be interpreted by the face-edge-constraint minimal surfaces. Specifically, a discrete surface f : V(G) → R 3 with normal defined on faces n :…”

Section: Remarkmentioning

confidence: 99%

“…However, this definition did not cover the whole associated families of minimal surfaces or CMC surfaces either as discrete isothermic triangulated surfaces. In [8], Ye and Hoffmann reformulated these minimal surfaces with a discrete Dirac operator and gave a hint of more general isothermic surfaces. Finally this paper will fill the gap and obtain the isothermicity that covers all discrete surfaces of the associated families of a minimal surface and a CMC surface, see Sects.…”

Section: Introductionmentioning

confidence: 99%

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Discrete constant mean curvature surfaces on general graphs

Hoffmann

Kobayashi

2022

Geom Dedicata

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The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Discrete constant mean curvature surfaces on general graphs

Hoffmann

Kobayashi

2022

Geom Dedicata

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“…Thus conjugation by E −1 ij sends any vector lying on face i to face j, and −n i to n j . Intuitively speaking, E ij carries information about a connection structure between the affine spaces of faces j and i [12].…”

Section: A Geometric Interpretation Of Hyperedgesmentioning

confidence: 99%

Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing

Folgoc

Castro

Tan

et al. 2019

Lecture Notes in Computer Science

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We investigate discrete spin transformations, a geometric framework to manipulate surface meshes by controlling mean curvature. Applications include surface fairing -flowing a mesh onto say, a reference sphere -and mesh extrusion -e.g., rebuilding a complex shape from a reference sphere and curvature specification. Because they operate in curvature space, these operations can be conducted very stably across large deformations with no need for remeshing. Spin transformations add to the algorithmic toolbox for pose-invariant shape analysis. Mathematically speaking, mean curvature is a shape invariant and in general fully characterizes closed shapes (together with the metric). Computationally speaking, spin transformations make that relationship explicit. Our work expands on a discrete formulation of spin transformations. Like their smooth counterpart, discrete spin transformations are naturally close to conformal (angle-preserving). This quasi-conformality can nevertheless be relaxed to satisfy the desired tradeoff between area distortion and angle preservation. We derive such constraints and propose a formulation in which they can be efficiently incorporated. The approach is showcased on subcortical structures.

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“…Our discrete surface representation via nets relates to [HSFW17], where however the normals were discretized differently. For a more systematic and detailed analysis of this discretization see [HY18]. Let X = (V, F, E) be a discrete surface, where V , F and E denote the set of vertices, faces and edges.…”

Section: Discretizationmentioning

confidence: 99%

“…without knowing the vertex positions. Since all of the applications in this paper are based on immersed surfaces, we will not present the details on this process but refer the readers to [HY18]. Remarkably, in parallel to our work, a closely related spinor connection structure has been developed by Chern et al [CKPS18] for the purposes of isometric shape embeddings.…”

Section: Discrete Dirac Operatorsmentioning

confidence: 99%

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Diamanti

Tang

et al. 2018

Computer Graphics Forum

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Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace‐Beltrami operator, which has the often‐cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.

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A Discrete Extrinsic and Intrinsic Dirac Operator

Cited by 4 publications

References 15 publications

Discrete constant mean curvature surfaces on general graphs

Discrete constant mean curvature surfaces on general graphs

Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

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