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

Authors:
Rongjie Lai;Zaiwen Wen;Wotao Yin;Xianfeng Gu;Lok Ming Lui
Affiliations:
Department of Mathematics, University of Southern California, Los Angeles, USA;Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiaotong University, Shanghai, China;Department of Computational and Applied Mathematics, Rice University, Houston, USA;Department of Computer Science, State University of New York at Stony Brook, Stony Brook, USA;Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
Venue:
Journal of Scientific Computing
Year:
2014

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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.