Harmonic Functions for Rotational Symmetry Vector Fields

Representing rotational symmetry vector as a set of vectors is not suitable for design due to lacking of a consistent ordering for measurement. In this paper we introduce a spectral method to find rotation invariant harmonic functions for symmetry vector field design. This method is developed for 3D...

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Veröffentlicht in:Computer graphics forum Jg. 35; H. 7; S. 507 - 516
Hauptverfasser: Shen, Zhongwei, Fang, Xianzhong, Liu, Xinguo, Bao, Hujun, Huang, Jin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Oxford Blackwell Publishing Ltd 01.10.2016
Schlagworte:
Analysis
Categories and Subject Descriptors (according to ACM CCS)
Eigenvalues
Eigenvectors
Fields (mathematics)
Group dynamics
Harmonic functions
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms
Image processing systems
Invariants
Mathematical analysis
Spherical harmonics
Studies
Symmetry
Topological manifolds
ISSN:0167-7055, 1467-8659
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Zusammenfassung:Representing rotational symmetry vector as a set of vectors is not suitable for design due to lacking of a consistent ordering for measurement. In this paper we introduce a spectral method to find rotation invariant harmonic functions for symmetry vector field design. This method is developed for 3D vector fields, but it is applicable in 2D. Given the finite symmetry group G of a symmetry vector field v(x) on a 3D domain Ω, we formulate the harmonic function h(s) as a stationary point of group G. Using the real spherical harmonic (SH) bases, we showed the coefficients of the harmonic functions are an eigenvector of the SH rotation matrices corresponding to group G. Instead of solving eigen problems to obtain the eigenvector, we developed a forward constructive method based on orthogonal group theory. The harmonic function found by our method is not only invariant under G, but also expressive and can distinguish different rotations with respect to G. At last, we demonstrate some vector field design results with tetrahedron‐symmetry, cube‐symmetry and dodecahedron‐symmetry groups.
Bibliographie:istex:73997856868E445D571D2276C1140547C24A0509
ArticleID:CGF13047
ark:/67375/WNG-HG02R8T0-P
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ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.13047