Spectral Geometry Processing with Manifold Harmonics

Bruno Vallet 1 Bruno Lévy 1
1 ALICE - Geometry and Lighting
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 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 the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors 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 just 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.
Document type :
Reports
Complete list of metadatas

Cited literature [22 references]  Display  Hide  Download

https://hal.inria.fr/inria-00186931
Contributor : Nicolas Ray <>
Submitted on : Tuesday, November 13, 2007 - 8:49:25 AM
Last modification on : Thursday, January 11, 2018 - 6:20:18 AM
Long-term archiving on : Monday, September 24, 2012 - 3:20:54 PM

File

ManifoldHarmonics.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : inria-00186931, version 1

Collections

Citation

Bruno Vallet, Bruno Lévy. Spectral Geometry Processing with Manifold Harmonics. [Technical Report] 2007. ⟨inria-00186931⟩

Share

Metrics

Record views

1343

Files downloads

1663