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An Efficient Energy Minimization for Conformal Parameterizations

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Abstract

Surface parameterizations have been widely applied to digital geometry processing. In this paper, we propose an efficient conformal energy minimization (CEM) algorithm for computing conformal parameterizations of simply-connected open surfaces with a very small angular distortion and a highly improved computational efficiency. In addition, we generalize the proposed CEM algorithm to computing conformal parameterizations of multiply-connected surfaces. Furthermore, we prove the existence of a nontrivial accumulation point of the proposed CEM algorithm under some mild conditions. Several numerical results show the efficiency and robustness of the CEM algorithm comparing to the existing state-of-the-art algorithms. An application of the CEM on the surface morphing between simply-connected open surfaces is demonstrated thereafter. Thanks to the CEM algorithm, the whole computations for the surface morphing can be performed efficiently and robustly.

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Acknowledgements

The authors want to thank Yen-Jen Cheng, Dr. Wei-Qiang Huang, and Dr. Ching-Sung Liu for useful discussions and thank Prof. So-Hsiang Chou for the help on polishing the manuscript. They also want to thank the anonymous referees for their valuable comments and suggestions. The 3D scanner and camera devices are supported by the ST Yau Center in Taiwan. This work is partially supported by the Ministry of Science and Technology, the National Center for Theoretical Sciences, the Taida Institute for Mathematical Sciences, and the ST Yau Center in Taiwan.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Mei-Heng Yueh, Wen-Wei Lin & Chin-Tien Wu

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

    Shing-Tung Yau

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  1. Mei-Heng Yueh
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  2. Wen-Wei Lin
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  3. Chin-Tien Wu
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  4. Shing-Tung Yau
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Correspondence to Mei-Heng Yueh.

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Yueh, MH., Lin, WW., Wu, CT. et al. An Efficient Energy Minimization for Conformal Parameterizations. J Sci Comput 73, 203–227 (2017). https://doi.org/10.1007/s10915-017-0414-y

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  • DOI: https://doi.org/10.1007/s10915-017-0414-y

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