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 Scopus citations

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 publicationVariational, Geometric, and Level Set Methods in Computer Vision - Third International Workshop, VLSM 2005, Proceedings
Pages307-319
Number of pages13
DOIs
StatePublished - Dec 1 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)0302-9743
ISSN (Electronic)1611-3349

Other

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

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Lui, L. M., Wang, Y., & Chan, T. F. (2005). Solving PDEs on manifolds with global conformal parametrization. In Variational, Geometric, and Level Set Methods in Computer Vision - Third International Workshop, VLSM 2005, Proceedings (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