Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

Tsung-Ming Huang; Wei-Hung Liao; Wen-Wei Lin

doi:10.1515/jnma-2022-0072

Published by De Gruyter August 24, 2023

Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

Tsung-Ming Huang , Wei-Hung Liao and Wen-Wei Lin

From the journal Journal of Numerical Mathematics

https://doi.org/10.1515/jnma-2022-0072

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Abstract

Here, we extend the finite distortion problem from bounded domains in ℝ² to closed genus-zero surfaces in ℝ³ by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊² by minimizing the total area distortion energy on ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊². The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊². In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

Keywords: spherical equiareal parameterization; stretch energy minimization; R-linear convergence

JEL Classification: 68U05; 65D18; 65𝔼10

Funding statement: This work was partially supported by the Ministry of Science and Technology (MoST), the National Center for Theoretical Sciences, and the ST Yau Center in Taiwan. W.-W. Lin and T.-M. Huang were partially supported by MoST 110-2115-M-A49-004- and 110-2115-M-003-012-MY3, respectively.

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Received: 2022-08-18

Revised: 2023-07-31

Accepted: 2023-08-10

Published Online: 2023-08-24

Published in Print: 2024-03-25

Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

Abstract

References

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