Generalized Koebe's method for conformal mapping multiply connected domains

Wei Zeng, Xiaotian Yin, Min Zhang, Feng Luo, Xianfeng Gu

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

18 Scopus citations

Abstract

Surface parameterization refers to the process of mapping the surface to canonical planar domains, which plays crucial roles in texture mapping and shape analysis purposes. Most existing techniques focus on simply connected surfaces. It is a challenging problem for multiply connected genus zero surfaces. This work generalizes conventional Koebe's method for multiply connected planar domains. According to Koebe's uniformization theory, all genus zero multiply connected surfaces can be mapped to a planar disk with multiply circular holes. Furthermore, this kind of mappings are angle preserving and differ by Möbius transformations. We introduce a practical algorithm to explicitly construct such a circular conformal mapping. Our algorithm pipeline is as follows: suppose the input surface has n boundaries, first we choose 2 boundaries, and fill the other n - 2 boundaries to get a topological annulus; then we apply discrete Yamabe flow method to conformally map the topological annulus to a planar annulus; then we remove the filled patches to get a planar multiply connected domain. We repeat this step for the planar domain iteratively. The two chosen boundaries differ from step to step. The iterative construction leads to the desired conformal mapping, such that all the boundaries are mapped to circles. In theory, this method converges quadratically faster than conventional Koebe's method. We give theoretic proof and estimation for the converging rate. In practice, it is much more robust and efficient than conventional non-linear methods based on curvature flow. Experimental results demonstrate the robustness and efficiency of the method.

Original languageEnglish (US)
Title of host publicationProceedings - SPM 2009
Subtitle of host publicationSIAM/ACM Joint Conference on Geometric and Physical Modeling
Pages89-100
Number of pages12
DOIs
StatePublished - Nov 9 2009
EventSPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling - San Francisco, CA, United States
Duration: Oct 5 2009Oct 8 2009

Publication series

NameProceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling

Other

OtherSPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling
CountryUnited States
CitySan Francisco, CA
Period10/5/0910/8/09

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All Science Journal Classification (ASJC) codes

  • Computational Theory and Mathematics
  • Computer Science Applications
  • Computer Vision and Pattern Recognition
  • Mathematics(all)

Keywords

  • Circular
  • Conformal
  • Differential form
  • Holomorphic
  • Multiply connected domain
  • Uniformization

Cite this

Zeng, W., Yin, X., Zhang, M., Luo, F., & Gu, X. (2009). Generalized Koebe's method for conformal mapping multiply connected domains. In Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling (pp. 89-100). [1629267] (Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling). https://doi.org/10.1145/1629255.1629267