Convergence Analysis of Dirichlet Energy Minimization for Spherical Conformal Parameterizations

TL;DR

This paper provides a theoretical foundation and an efficient algorithm for spherical conformal parameterizations of surfaces, demonstrating convergence and robustness through numerical experiments.

Contribution

It introduces a modified Dirichlet energy minimization method with convergence analysis for spherical conformal parameterizations.

Findings

Theoretical basis for spherical conformal parameterizations via Dirichlet energy.

Development of a modified energy minimization algorithm with convergence guarantees.

Numerical experiments confirm efficiency, reliability, and robustness.

Abstract

In this paper, we first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface $\mathcal{S}$ and a unit sphere $\mathbb{S}^2$ by minimizing the Dirichlet energy on $\overline{\mathbb{C}}$ by stereographic projection. The Dirichlet energy can be rewritten as the sum of the energies associated with the southern and northern hemispheres and can be decreased under an equivalence relation by alternatingly solving the corresponding Laplacian equations. Based on this theoretical foundation, we develop a modified Dirichlet energy minimization with nonequivalence deflation for the computation of the spherical conformal parameterization between $\mathcal{S}$ and $\mathbb{S}^2$. In addition, under some mild conditions, we verify the asymptotically R-linear convergence of the proposed algorithm. Numerical experiments on various benchmarks confirm that the assumptions for convergence always hold and indicate the efficiency, reliability and robustness of the developed modified Dirichlet energy minimization.