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

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Abstract

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.

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Acknowledgments

Lok Ming Lui is supported by RGC GRF (Grant No: CUHK401811). Xianfeng Gu is supported in part of NSF Nets 1016286, NSF IIS 0916286, NSF CCF 1081424 and ONR N000140910228.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    Tsz Ching Ng & Lok Ming Lui

  2. Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY, USA

    Xianfeng Gu

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  1. Tsz Ching Ng
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Correspondence to Lok Ming Lui.

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Ng, T.C., Gu, X. & Lui, L.M. Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow. J Sci Comput 60, 249–275 (2014). https://doi.org/10.1007/s10915-013-9791-z

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  • DOI: https://doi.org/10.1007/s10915-013-9791-z

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