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

• Published in: Journal of scientific computing Vol. 58; no. 3; pp. 705 - 725
• Main Authors: Lai, Rongjie, Wen, Zaiwen, Yin, Wotao, Gu, Xianfeng, Lui, Lok Ming
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-03-2014; Springer Nature B.V
• Subjects: Algorithms; Applications of mathematics; Brain; Computational Mathematics and Numerical Analysis; Computer graphics; Conformal mapping; Energy; Energy conservation; Folding; Fourier transforms; Geometry; Harmonics; Image analysis; Iterative methods; Mapping; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Medical imaging; Methods; Optimization; Optimization techniques; Parameterization; Parametrization; Theoretical; Harmonic energy minimization; Optimization with orthogonality constraints; Weighted Laplace–Beltrami eigenfunctions; Conformal map
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-013-9752-6

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.