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Convergence of Dirichlet Energy Minimization for Spherical Conformal Parameterizations

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Abstract

In this paper, we first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface \(\mathcal {S}\) and a unit sphere \(\mathbb {S}^2\) by minimizing the Dirichlet energy on \(\overline{\mathbb {C}}\) 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 \(\mathcal {S}\) and \(\mathbb {S}^2\). 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.

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Acknowledgements

This work was partially supported by the Ministry of Science and Technology (MoST), the National Center for Theoretical Sciences, the Nanjing Center for Applied Mathematics, and the ST Yau Center in Taiwan. W.-W. Lin, T.-M. Huang, and M.-H. Yueh were partially supported by NSTC 110-2115-M-A49-004-, 110-2115-M-003-012-MY3, 109-2115-M-003-010-MY2 and 110-2115-M-003-014-.

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Authors and Affiliations

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

    Wei-Hung Liao & Wen-Wei Lin

  2. Department of Mathematics, National Taiwan Normal University, Taipei, 116, Taiwan

    Tsung-Ming Huang & Mei-Heng Yueh

  3. Nanjing Center for Applied Mathematics, Nanjing, China

    Wen-Wei Lin

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  1. Wei-Hung Liao
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Correspondence to Tsung-Ming Huang.

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Liao, WH., Huang, TM., Lin, WW. et al. Convergence of Dirichlet Energy Minimization for Spherical Conformal Parameterizations. J Sci Comput 98, 29 (2024). https://doi.org/10.1007/s10915-023-02424-x

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