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Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization

Journal of Scientific Computing volume 58, pages 705–725 (2014)Cite this article

Abstract

Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121–146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed through minimizing the harmonic energy, with two weaknesses: its gradient descent iteration is slow, and its solutions contain undesired parameterization foldings when the underlying surface has long sharp features. In this paper, we propose an algorithm that significantly accelerates the harmonic energy minimization and a method that iteratively removes foldings by taking advantages of the weighted Laplace–Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in Gu and Yau (Commun Inf Syst 2(2):121–146, 2002) and the new results contain no foldings.

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Notes

  1. As our search path is a curve rather than a straight line.

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Acknowledgments

Rongjie Lai’s work is supported by Zumberge Individual Award from USC’s James H. Zumberge Faculty Research and Innovation Fund. Zaiwen Wen’s work is supported in part by NSFC Grant 11101274 and Research Fund (20110073120069) for the Doctoral Program of Higher Education of China. Wotao Yin’s work is supported in part by NSF Grant DMS-0748839 and ONR Grant N00014-08-1-1101. Xianfeng Gu’s work is is supported in part of NSF Nets 1016286, NSF IIS 0916286, NSF CCF 1081424 and ONR N000140910228. Lok Ming Lui’s work is supported in part of the CUHK Direct Grant (Project ID: 2060413) and HKRGC GRF (Project ID: 2130271).

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, USA

    Rongjie Lai

  2. Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiaotong University, Shanghai, China

    Zaiwen Wen

  3. Department of Computational and Applied Mathematics, Rice University, Houston, TX, USA

    Wotao Yin

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

    Xianfeng Gu

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

    Lok Ming Lui

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  1. Rongjie Lai
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  2. Zaiwen Wen
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  3. Wotao Yin
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  4. Xianfeng Gu
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  5. Lok Ming Lui
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Correspondence to Rongjie Lai.

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Lai, R., Wen, Z., Yin, W. et al. Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization. J Sci Comput 58, 705–725 (2014). https://doi.org/10.1007/s10915-013-9752-6

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Keywords

  • Harmonic energy minimization
  • Conformal map
  • Optimization with orthogonality constraints
  • Weighted Laplace–Beltrami eigenfunctions