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- Title
Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization.
- Authors
Huang, Tsung-Ming; Liao, Wei-Hung; Lin, Wen-Wei
- 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 핊2 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 핊2. 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 핊2. 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.
- Subjects
SPHERICAL projection; SPHERICAL functions; ENERGY function; CONFORMAL mapping; PARAMETERIZATION
- Publication
Journal of Numerical Mathematics, 2024, Vol 32, Issue 1, p1
- ISSN
1570-2820
- Publication type
Article
- DOI
10.1515/jnma-2022-0072