Details
| Original language | English |
|---|---|
| Pages (from-to) | 718-729 |
| Number of pages | 12 |
| Journal | Computers and Graphics (Pergamon) |
| Volume | 37 |
| Issue number | 6 |
| Early online date | 12 Jun 2013 |
| Publication status | Published - Oct 2013 |
Abstract
The well-known Laplace-Beltrami operator, established as a basic tool in shape processing, builds on a long history of mathematical investigations that have induced several numerical models for computational purposes. However, the Laplace-Beltrami operator is only one special case of many possible generalizations that have been researched theoretically. Thereby it is natural to supplement some of those extensions with concrete computational frameworks. In this work we study a particularly interesting class of extended Laplacians acting on sections of flat line bundles over compact Riemannian manifolds. Numerical computations for these operators have recently been accomplished on two-dimensional surfaces. Using the notions of line bundles and differential forms, we follow up on that work giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to three-dimensional Riemannian manifolds, yielding a method that is able to deal robustly with volumetric objects of intricate shape and topology. We investigate and visualize the two-dimensional zero sets of the first eigenfunctions of the modified Laplacians, yielding an approach for constructing characteristic well-behaving, particularly robust homology generators invariant under isometric deformation. The latter include nicely embedded Seifert surfaces and their non-orientable counterparts for knot complements.
Keywords
- Computational topology, FEM, Knots, Laplace operator, Seifert surfaces, Spectral geometry, Vector bundles
ASJC Scopus subject areas
- Computer Science(all)
- Software
- Computer Science(all)
- Signal Processing
- Engineering(all)
- 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. 37, No. 6, 10.2013, p. 718-729.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Laplacians on flat line bundles over 3-manifolds
AU - Vais, Alexander
AU - Brandes, Daniel
AU - Thielhelm, Hannes
AU - Wolter, Franz Erich
PY - 2013/10
Y1 - 2013/10
N2 - The well-known Laplace-Beltrami operator, established as a basic tool in shape processing, builds on a long history of mathematical investigations that have induced several numerical models for computational purposes. However, the Laplace-Beltrami operator is only one special case of many possible generalizations that have been researched theoretically. Thereby it is natural to supplement some of those extensions with concrete computational frameworks. In this work we study a particularly interesting class of extended Laplacians acting on sections of flat line bundles over compact Riemannian manifolds. Numerical computations for these operators have recently been accomplished on two-dimensional surfaces. Using the notions of line bundles and differential forms, we follow up on that work giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to three-dimensional Riemannian manifolds, yielding a method that is able to deal robustly with volumetric objects of intricate shape and topology. We investigate and visualize the two-dimensional zero sets of the first eigenfunctions of the modified Laplacians, yielding an approach for constructing characteristic well-behaving, particularly robust homology generators invariant under isometric deformation. The latter include nicely embedded Seifert surfaces and their non-orientable counterparts for knot complements.
AB - The well-known Laplace-Beltrami operator, established as a basic tool in shape processing, builds on a long history of mathematical investigations that have induced several numerical models for computational purposes. However, the Laplace-Beltrami operator is only one special case of many possible generalizations that have been researched theoretically. Thereby it is natural to supplement some of those extensions with concrete computational frameworks. In this work we study a particularly interesting class of extended Laplacians acting on sections of flat line bundles over compact Riemannian manifolds. Numerical computations for these operators have recently been accomplished on two-dimensional surfaces. Using the notions of line bundles and differential forms, we follow up on that work giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to three-dimensional Riemannian manifolds, yielding a method that is able to deal robustly with volumetric objects of intricate shape and topology. We investigate and visualize the two-dimensional zero sets of the first eigenfunctions of the modified Laplacians, yielding an approach for constructing characteristic well-behaving, particularly robust homology generators invariant under isometric deformation. The latter include nicely embedded Seifert surfaces and their non-orientable counterparts for knot complements.
KW - Computational topology
KW - FEM
KW - Knots
KW - Laplace operator
KW - Seifert surfaces
KW - Spectral geometry
KW - Vector bundles
UR - http://www.scopus.com/inward/record.url?scp=84880062116&partnerID=8YFLogxK
U2 - 10.1016/j.cag.2013.05.013
DO - 10.1016/j.cag.2013.05.013
M3 - Article
AN - SCOPUS:84880062116
VL - 37
SP - 718
EP - 729
JO - Computers and Graphics (Pergamon)
JF - Computers and Graphics (Pergamon)
SN - 0097-8493
IS - 6
ER -