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“Manifold Harmonics”
Bruno Vallet and Bruno Lévy
Tech report, 2007

Abstract: (TR) We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvector of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, 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. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.

BibTex reference

@TECHREPORT{vallet:MH:2007,
   AUTHOR     = "Bruno Vallet and Bruno Lévy",
   TITLE      = "Manifold Harmonics",
   INSTITUTION = "INRIA - ALICE Project Team",
   YEAR       = "2007",
}

Supplemental material, links, hindsight ...

Further reading

  • 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 ...).
  • This tech-report was published later as an article in Eurographics.
  • See also our SMI 2006 paper .