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.

翻译：在本文中,我们首先从简单连接的封闭表面$\mathcal{S}$和单位球体$\mathbb{S ⁇ 2$之间得出一个球状符合参数的理论基础,通过星座投影将dirichlet能源在$\ overline_mathbb{C ⁇ C$$中最小化。在一些温和的条件下,我们核查拟议的算法的无症状R线性趋同。各种基准的数值实验证实,趋同的假设总是站不住脚,并表明已开发的Drichlet能源最小化的效率、可靠性和可靠性。