Solving PDEs on manifolds with global conformal parametrization

Lok Ming Lui, Yalin Wang, Tony F. Chan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

22 Citations (Scopus)

Abstract

In this paper, we propose a method to solve PDEs on surfaces with arbitrary topologies by using the global conformal parametrization. The main idea of this method is to map the surface conformally to 2D rectangular areas and then transform the PDE on the 3D surface into a modified PDE on the 2D parameter domain. Consequently, we can solve the PDE on the parameter domain by using some well-known numerical schemes on ℝ 2. To do this, we have to define a new set of differential operators on the manifold such that they are coordinates invariant. Since the Jacobian of the conformal mapping is simply a multiplication of the conformal factor, the modified PDE on the parameter domain will be very simple and easy to solve. In our experiments, we demonstrated our idea by solving the Navier-Stoke's equation on the surface. We also applied our method to some image processing problems such as segmentation, image denoising and image inpainting on the surfaces.

Original languageEnglish (US)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages307-319
Number of pages13
Volume3752 LNCS
DOIs
StatePublished - 2005
Externally publishedYes
Event3rd International Workshop on Variational, Geometric, and Level Set Methods in Computer Vision, VLSM 2005 - Beijing, China
Duration: Oct 16 2005Oct 16 2005

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume3752 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other3rd International Workshop on Variational, Geometric, and Level Set Methods in Computer Vision, VLSM 2005
CountryChina
CityBeijing
Period10/16/0510/16/05

Fingerprint

Parametrization
Image Inpainting
Conformal mapping
Image denoising
Image Denoising
Conformal Mapping
Numerical Scheme
Navier Stokes equations
Differential operator
Image Processing
Navier-Stokes Equations
Multiplication
Image processing
Segmentation
Topology
Transform
Invariant
Arbitrary
Experiment
Experiments

ASJC Scopus subject areas

  • Computer Science(all)
  • Biochemistry, Genetics and Molecular Biology(all)
  • Theoretical Computer Science

Cite this

Lui, L. M., Wang, Y., & Chan, T. F. (2005). Solving PDEs on manifolds with global conformal parametrization. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 3752 LNCS, pp. 307-319). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3752 LNCS). https://doi.org/10.1007/11567646_26

Solving PDEs on manifolds with global conformal parametrization. / Lui, Lok Ming; Wang, Yalin; Chan, Tony F.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 3752 LNCS 2005. p. 307-319 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3752 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Lui, LM, Wang, Y & Chan, TF 2005, Solving PDEs on manifolds with global conformal parametrization. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 3752 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 3752 LNCS, pp. 307-319, 3rd International Workshop on Variational, Geometric, and Level Set Methods in Computer Vision, VLSM 2005, Beijing, China, 10/16/05. https://doi.org/10.1007/11567646_26
Lui LM, Wang Y, Chan TF. Solving PDEs on manifolds with global conformal parametrization. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 3752 LNCS. 2005. p. 307-319. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/11567646_26
Lui, Lok Ming ; Wang, Yalin ; Chan, Tony F. / Solving PDEs on manifolds with global conformal parametrization. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 3752 LNCS 2005. pp. 307-319 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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