Spectral Geometry Processing with Manifold Harmonics

• Published in: Computer graphics forum Vol. 27; no. 2; pp. 251 - 260
• Main Authors: Vallet, B., Lévy, B.
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 01-04-2008; Wiley
• Subjects: 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
• DOI: 10.1111/j.1467-8659.2008.01122.x

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.