Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow

Tsz Ching Ng The Chinese University of Hong Kong Xianfeng Gu Stony Brook University Ronald Lok Ming Lui The Chinese University of Hong Kong

Computational Geometry mathscidoc:1609.09021

Journal of Scientific Computing, 60, (2), 249–275, 2013.10
A numerical method for computing the extremal Teichmüller map between multiply-connected domains is presented. Given two multiply-connected domains, there exists a unique Teichmüller map (T-Map) between them minimizing the conformality distortion. The extremal T-Map can be considered as the ‘most conformal’ map between multiply-connected domains. In this paper, we propose an iterative algorithm to compute the extremal T-Map using the Beltrami holomorphic flow (BHF). The BHF procedure iteratively adjusts the initial map based on a sequence of Beltrami coefficients, which are complex-valued functions defined on the source domain. It produces a sequence of quasi-conformal maps, which converges to the T-Map minimizing the conformality distortion. We test our method on synthetic data together with real human face data. Results show that our algorithm computes the extremal T-Map between two multiply-connected domains of the same topology accurately and efficiently.
Teichmüller map, Extremal map, Multiply-connected, Beltrami holomorphic flow, Beltrami coefficient, Quasiconformal map
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@inproceedings{tsz2013computing,
  title={Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow},
  author={Tsz Ching Ng, Xianfeng Gu, and Ronald Lok Ming Lui},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160905204703338485633},
  booktitle={Journal of Scientific Computing},
  volume={60},
  number={2},
  pages={249–275},
  year={2013},
}
Tsz Ching Ng, Xianfeng Gu, and Ronald Lok Ming Lui. Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow. 2013. Vol. 60. In Journal of Scientific Computing. pp.249–275. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160905204703338485633.
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