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A Dirac Operator for Extrinsic Shape Analysis
Author(s) -
Liu HsuehTi Derek,
Jacobson Alec,
Crane Keenan
Publication year - 2017
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.13252
Subject(s) - operator (biology) , eigenfunction , laplace–beltrami operator , mathematics , dirac operator , eigenvalues and eigenvectors , spectrum (functional analysis) , laplace operator , surface (topology) , metric (unit) , mathematical analysis , differential operator , geometry , boundary value problem , physics , p laplacian , biochemistry , chemistry , operations management , repressor , quantum mechanics , transcription factor , economics , gene
The eigenfunctions and eigenvalues of the Laplace‐Beltrami operator have proven to be a powerful tool for digital geometry processing, providing a description of geometry that is essentially independent of coordinates or the choice of discretization. However, since Laplace‐Beltrami is purely intrinsic it struggles to capture important phenomena such as extrinsic bending, sharp edges, and fine surface texture. We introduce a new extrinsic differential operator called the relative Dirac operator , leading to a family of operators with a continuous trade‐off between intrinsic and extrinsic features. Previous operators are either fully or partially intrinsic. In contrast, the proposed family spans the entire spectrum: from completely intrinsic (depending only on the metric) to completely extrinsic (depending only on the Gauss map). By adding an infinite potential well to this (or any) operator we can also robustly handle surface patches with irregular boundary. We explore use of these operators for a variety of shape analysis tasks, and study their performance relative to operators previously found in the geometry processing literature.