Computing homology groups of simplicial complexes in R3
Journal of the ACM (JACM)
CGI '00 Proceedings of the International Conference on Computer Graphics
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
Visualization of Seifert Surfaces
IEEE Transactions on Visualization and Computer Graphics
Spectral surface quadrangulation
ACM SIGGRAPH 2006 Papers
Laplace spectra as fingerprints for image recognition
Computer-Aided Design
Designing quadrangulations with discrete harmonic forms
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Discrete laplace operators: no free lunch
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Laplace-Beltrami eigenfunctions for deformation invariant shape representation
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Discrete quadratic curvature energies
Computer Aided Geometric Design
On Linear Variational Surface Deformation Methods
IEEE Transactions on Visualization and Computer Graphics
Mass Density Laplace-Spectra for Image Recognition
CW '07 Proceedings of the 2007 International Conference on Cyberworlds
A Discrete Laplace–Beltrami Operator for Simplicial Surfaces
Discrete & Computational Geometry
ACM SIGGRAPH 2009 papers
Technical Section: Discrete Laplace-Beltrami operators for shape analysis and segmentation
Computers and Graphics
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
Computer-Aided Design
A concise and provably informative multi-scale signature based on heat diffusion
SGP '09 Proceedings of the Symposium on Geometry Processing
Global medical shape analysis using the Laplace-Beltrami spectrum
MICCAI'07 Proceedings of the 10th international conference on Medical image computing and computer-assisted intervention - Volume Part I
Spectral-Driven Isometry-Invariant Matching of 3D Shapes
International Journal of Computer Vision
International Journal of Computer Vision
Convergence, stability, and discrete approximation of Laplace spectra
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Spin transformations of discrete surfaces
ACM SIGGRAPH 2011 papers
Persistent Cohomology and Circular Coordinates
Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
An iterative algorithm for homology computation on simplicial shapes
Computer-Aided Design
Eigenmodes of surface energies for shape analysis
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
SMI 2012: Full Spectral computations on nontrivial line bundles
Computers and Graphics
Complex line bundle Laplacians
The Visual Computer: International Journal of Computer Graphics
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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.