Computing Quasiconformal Maps Using an Auxiliary Metric and Discrete Curvature Flow

Wei Zeng Stony Brook University Ronald Lok Ming Lui The Chinese University of Hong Kong Feng Luo Rutgers University Tony F. Chan The Hong Kong University of Science and Technology Shing-Tung Yau Harvard University Xianfeng Gu Stony Brook University

Computational Geometry mathscidoc:1609.09028

Numerische Mathematik, 121, (4), 671-703, 2012.8
Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasi-conformal map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichmüller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of quasi-conformal surface maps. Therefore, every quasi-conformal surface map can be fully determined by the Beltrami differential and can be 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 quasi-conformal map associated with the prescribed Beltrami differential. We firstly formulate a discrete analog of quasi-conformal maps on triangular meshes. Then, we propose an algorithm to compute discrete quasi-conformal maps. The main strategy is to define a discrete auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the newly defined discrete metric. The associated map can then be obtained by using the discrete Yamabe flow method. Numerically, the discrete quasi-conformal map converges to the continuous real solution as the mesh 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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@inproceedings{wei2012computing,
  title={Computing Quasiconformal Maps Using an Auxiliary Metric and Discrete Curvature Flow},
  author={Wei Zeng, Ronald Lok Ming Lui, Feng Luo, Tony F. Chan, Shing-Tung Yau, and Xianfeng Gu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160906102503780379640},
  booktitle={Numerische Mathematik},
  volume={121},
  number={4},
  pages={671-703},
  year={2012},
}
Wei Zeng, Ronald Lok Ming Lui, Feng Luo, Tony F. Chan, Shing-Tung Yau, and Xianfeng Gu. Computing Quasiconformal Maps Using an Auxiliary Metric and Discrete Curvature Flow. 2012. Vol. 121. In Numerische Mathematik. pp.671-703. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160906102503780379640.
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