Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Equiareal parameterizations

TL;DR

This paper develops a theoretical foundation and a convergent algorithm for spherical equiareal parameterizations of genus-zero surfaces, extending finite distortion problems to 3D surfaces and demonstrating R-linear convergence.

Contribution

It introduces a novel theoretical framework for equiareal parameterizations via energy minimization and proposes a modified stretch energy minimization algorithm with proven convergence.

Findings

Algorithm exhibits asymptotic R-linear convergence.

Numerical experiments confirm efficiency and robustness.

Theoretical foundation extends finite distortion problems to 3D surfaces.

Abstract

In this paper, we first extend the finite distortion problem from the bounded domains in $\mathbb{R}^2$ to the closed genus-zero surfaces in $\mathbb{R}^3$ by the stereographic projection. Then we derive a theoretical foundation for spherical equiareal parameterizations between a simply connected closed surface $\mathcal{M}$ and a unit sphere $\mathbb{S}^2$ via minimizing the total area distortion energy on $\overline{\mathbb{C}}$. Provided we determine the minimizer of the total area distortion energy, the minimizer composed with the initial conformal map determines the equiareal map between the extended planes. Taking the inverse stereographic projection, we can derive the equiareal map between $\mathcal{M}$ and $\mathbb{S}^2$. The total area distortion energy can be rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres, respectively, and can be decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization for the computation of the spherical equiareal parameterization between $\mathcal{M}$ and $\mathbb{S}^2$. In addition, under some mild conditions, we verify that our proposed algorithm has asymptotically R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate the assumptions for convergence always hold and indicate the efficiency, reliability and robustness of the developed modified stretch energy minimization.