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Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

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Abstract

Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasiconformal mapping (QC-Mapping). Many surface mappings in our physical world are quasiconformal. The angular distortion of a QC mapping can be represented by the Beltrami differentials. According to QC Teichmüller theory, there is a one-to-one correspondence between the set of Beltrami differentials and the set of QC surface mappings under normalization conditions. Therefore, every QC surface mapping can be fully determined by the Beltrami differential and reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a QC mapping associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric (AM) on the domain surface, such that the original QC mapping becomes conformal under the auxiliary metric. The desired QC-mapping can then be obtained by using the conventional conformal mapping method. In this paper, we first formulate a discrete analogue of QC mappings on triangular meshes. Then, we propose an algorithm to compute discrete QC mappings using the discrete Yamabe flow method. To the best of our knowledge, it is the first work to compute the discrete QC mappings for general Riemann surfaces, especially with different topologies. Numerically, the discrete QC mapping converges to the continuous solution as the mesh grid size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

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Authors and Affiliations

  1. Department of Computer Science, Stony Brook University, Stony Brook, NY, 11794, USA

    Wei Zeng & David Xianfeng Gu

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Lok Ming Lui

  3. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Feng Luo

  4. The Hong Kong University of Science and Technology, Kowloon, Hong Kong

    Tony Fan-Cheong Chan

  5. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

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  1. Wei Zeng
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  2. Lok Ming Lui
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  3. Feng Luo
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  4. Tony Fan-Cheong Chan
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  5. Shing-Tung Yau
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  6. David Xianfeng Gu
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Correspondence to David Xianfeng Gu.

Additional information

T. F. Chan is supported by NSF GEO0610079, NSF IIS0914580, NIH U54RR021813 and ONR N000140910105. D. X. Gu is supported by NIH R01EB0075300A1, NSF IIS0916286, NSF CCF0916235, NSF CCF0830550, NSF III0713145, and ONR N000140910228. L.M. Lui is supported by CUHK Direct Grant (Project ID: 2060413).

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Zeng, W., Lui, L.M., Luo, F. et al. Computing quasiconformal maps using an auxiliary metric and discrete curvature flow. Numer. Math. 121, 671–703 (2012). https://doi.org/10.1007/s00211-012-0446-z

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  • DOI: https://doi.org/10.1007/s00211-012-0446-z

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