Details
| Original language | English |
|---|---|
| Pages (from-to) | 398-409 |
| Number of pages | 12 |
| Journal | Computers and Graphics (Pergamon) |
| Volume | 36 |
| Issue number | 5 |
| Early online date | 27 Mar 2012 |
| Publication status | Published - Aug 2012 |
Abstract
Computing the spectral decomposition of the Laplace-Beltrami operator on a manifold M has proven useful for applications such as shape retrieval and geometry processing. The standard operator acts on scalar functions which can be identified with sections of the trivial line bundle M×R. In this work we propose to extend the discussion to Laplacians on nontrivial real line bundles. These line bundles are in one-to-one correspondence with elements of the first cohomology group of the manifold with Z2 coefficients. While we focus on the case of two-dimensional closed surfaces, we show that our method also applies to surfaces with boundaries. Denoting by β the rank of the first cohomology group, there are 2β different line bundles to consider and each of these has a naturally associated Laplacian that possesses a spectral decomposition. Using our new method it is possible for the first time to compute the spectra of these Laplacians by a simple modification of the finite element basis functions used in the standard trivial bundle case. Our method is robust and efficient. We illustrate some properties of the modified spectra and eigenfunctions and indicate possible applications for shape processing. As an example, using our method, we are able to create spectral shape descriptors with increased sensitivity in the eigenvalues with respect to geometric deformations and to compute cycles aligned to object symmetries in a chosen homology class.
Keywords
- Computational topology, Finite elements, Laplace operator, Spectral geometry processing, Vector bundles
ASJC Scopus subject areas
- Computer Science(all)
- Software
- Computer Science(all)
- Signal Processing
- Engineering(all)
- General Engineering
- Computer Science(all)
- Human-Computer Interaction
- Computer Science(all)
- Computer Vision and Pattern Recognition
- Computer Science(all)
- Computer Graphics and Computer-Aided Design
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In: Computers and Graphics (Pergamon), Vol. 36, No. 5, 08.2012, p. 398-409.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Spectral computations on nontrivial line bundles
AU - Vais, Alexander
AU - Berger, Benjamin
AU - Wolter, Franz Erich
PY - 2012/8
Y1 - 2012/8
N2 - Computing the spectral decomposition of the Laplace-Beltrami operator on a manifold M has proven useful for applications such as shape retrieval and geometry processing. The standard operator acts on scalar functions which can be identified with sections of the trivial line bundle M×R. In this work we propose to extend the discussion to Laplacians on nontrivial real line bundles. These line bundles are in one-to-one correspondence with elements of the first cohomology group of the manifold with Z2 coefficients. While we focus on the case of two-dimensional closed surfaces, we show that our method also applies to surfaces with boundaries. Denoting by β the rank of the first cohomology group, there are 2β different line bundles to consider and each of these has a naturally associated Laplacian that possesses a spectral decomposition. Using our new method it is possible for the first time to compute the spectra of these Laplacians by a simple modification of the finite element basis functions used in the standard trivial bundle case. Our method is robust and efficient. We illustrate some properties of the modified spectra and eigenfunctions and indicate possible applications for shape processing. As an example, using our method, we are able to create spectral shape descriptors with increased sensitivity in the eigenvalues with respect to geometric deformations and to compute cycles aligned to object symmetries in a chosen homology class.
AB - Computing the spectral decomposition of the Laplace-Beltrami operator on a manifold M has proven useful for applications such as shape retrieval and geometry processing. The standard operator acts on scalar functions which can be identified with sections of the trivial line bundle M×R. In this work we propose to extend the discussion to Laplacians on nontrivial real line bundles. These line bundles are in one-to-one correspondence with elements of the first cohomology group of the manifold with Z2 coefficients. While we focus on the case of two-dimensional closed surfaces, we show that our method also applies to surfaces with boundaries. Denoting by β the rank of the first cohomology group, there are 2β different line bundles to consider and each of these has a naturally associated Laplacian that possesses a spectral decomposition. Using our new method it is possible for the first time to compute the spectra of these Laplacians by a simple modification of the finite element basis functions used in the standard trivial bundle case. Our method is robust and efficient. We illustrate some properties of the modified spectra and eigenfunctions and indicate possible applications for shape processing. As an example, using our method, we are able to create spectral shape descriptors with increased sensitivity in the eigenvalues with respect to geometric deformations and to compute cycles aligned to object symmetries in a chosen homology class.
KW - Computational topology
KW - Finite elements
KW - Laplace operator
KW - Spectral geometry processing
KW - Vector bundles
UR - http://www.scopus.com/inward/record.url?scp=84860723540&partnerID=8YFLogxK
U2 - 10.1016/j.cag.2012.03.027
DO - 10.1016/j.cag.2012.03.027
M3 - Article
AN - SCOPUS:84860723540
VL - 36
SP - 398
EP - 409
JO - Computers and Graphics (Pergamon)
JF - Computers and Graphics (Pergamon)
SN - 0097-8493
IS - 5
ER -