Spectral Geometry Processing with Manifold Harmonics

We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out o...

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Vydáno v:Computer graphics forum Ročník 27; číslo 2; s. 251 - 260
Hlavní autoři: Vallet, B., Lévy, B.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford, UK Blackwell Publishing Ltd 01.04.2008
Wiley
Témata:
Algorithms
Computational Geometry and Object Modeling
Computational Geometry and Object Modeling, Hierarchy and geometric transformations
Computer graphics
Computer programming
Computer Science
Eigenvectors
Fourier transforms
Geometry
Graphics
Hierarchy and geometric transformations
I.3.5 [Computer Graphics]
Studies
Topological manifolds
ISSN:0167-7055, 1467-8659
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Shrnutí:We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band‐by‐band spectrum computation algorithm and an out‐of‐core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited‐memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.
Bibliografie:ark:/67375/WNG-5V99VKTZ-R
ArticleID:CGF1122
istex:B73CBA9B303CAA04753104D06B669D3CAF272870
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ISSN:0167-7055
1467-8659
DOI:10.1111/j.1467-8659.2008.01122.x