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“Spectral Geometry Processing with Manifold Harmonics”
Bruno Vallet and Bruno Lévy
Computer Graphics Forum (Proceedings Eurographics), 2008

Abstract: 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.

BibTex reference

@ARTICLE{vallet:MH:2008,
   AUTHOR     = "Bruno Vallet and Bruno Lévy",
   TITLE      = "Spectral Geometry Processing with Manifold Harmonics",
   JOURNAL    = "Computer Graphics Forum (Proceedings Eurographics)",
   YEAR       = "2008",
}

Supplemental material, links, hindsight ...

See also our book Polygon Mesh Processing , Botsch, Kobbelt, Pauly, Alliez and Levy, AK Peters. Chapter 3 (Differential Geometry) and Chapter 4 (Smoothing, Fourrier transform and Manifold Harmonics, diffusion flow, fairing ...).

Resources

[Download Presentation]

[Download Demo] (Windows executable)

See also the implementation in Graphite (Manifold Harmonics plugin).

See also our SMI 2006 paper .

This paper previously appeared as a tech report , with the derivations in FEM (Finite Elements Modeling) instead of DEC (Discrete Exterior Calculus). This gives the same result in the end. Both are interesting: DEC derivations are shorter, FEM derivations are more self-contained.

Resources on Laplace-Belatrami and DEC

(some of them suggested by Ramsay Dyer)

Further reading:

Manifold learning

Spectral geometry processing

Other references