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

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.