TY - JOUR

T1 - Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

AU - Huang, Tsung Ming

AU - Liao, Wei Hung

AU - Lin, Wen Wei

N1 - Publisher Copyright:
© 2024 Walter de Gruyter GmbH. All rights reserved.

PY - 2024/3/1

Y1 - 2024/3/1

N2 - 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.

AB - 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.

KW - R-linear convergence

KW - spherical equiareal parameterization

KW - stretch energy minimization

UR - http://www.scopus.com/inward/record.url?scp=85170279743&partnerID=8YFLogxK

U2 - 10.1515/jnma-2022-0072

DO - 10.1515/jnma-2022-0072

M3 - Article

AN - SCOPUS:85170279743

SN - 1570-2820

VL - 32

SP - 1

EP - 25

JO - Journal of Numerical Mathematics

JF - Journal of Numerical Mathematics

IS - 1

ER -