Convergence of Dirichlet Energy Minimization for Spherical Conformal Parameterizations

Wei Hung Liao, Tsung Ming Huang*, Wen Wei Lin, Mei Heng Yueh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


In this paper, we first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface S and a unit sphere S2 by minimizing the Dirichlet energy on C¯ with stereographic projection. The Dirichlet energy can be rewritten as the sum of the energies associated with the southern and northern hemispheres and can be decreased under an equivalence relation by alternatingly solving the corresponding Laplacian equations. Based on this theoretical foundation, we develop a modified Dirichlet energy minimization with nonequivalence deflation for the computation of the spherical conformal parameterization between S and S2 . In addition, under some mild conditions, we verify the asymptotically R-linear convergence of the proposed algorithm. Numerical experiments on various benchmarks confirm that the assumptions for convergence always hold and demonstrate the efficiency, reliability and robustness of the developed modified Dirichlet energy minimization.

Original languageEnglish
Article number29
JournalJournal of Scientific Computing
Issue number1
Publication statusPublished - 2024 Jan


  • Asymptotically R-linear convergence
  • Dirichlet energy minimization
  • Nonequivalence deflation
  • Spherical conformal parameterization

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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