论文标题
球形共形参数化的DIRICHLET能量最小化的收敛分析
Convergence Analysis of Dirichlet Energy Minimization for Spherical Conformal Parameterizations
论文作者
论文摘要
在本文中，我们首先在简单连接的封闭表面$ \ MATHCAL {S} $和单位球$ \ Mathbb {s}^2 $之间获得球形完整的参数的理论基础，通过将$ \ overline {\ overline {\ Mathbb {c} c}} $的dirichlet Energy最小化，通过刻有刻录量来减少。可以将Dirichlet能量重写为与南半球和北半球相关的能量的总和，并且可以通过交替求解相应的Laplacian方程来减少等价关系。基于这个理论基础，我们开发了一个修改后的dirichlet能量最小化，并在$ \ mathcal {s} $和$ \ mathbb {s}^2 $之间的球形共形参数化计算中进行了不重要的放气。另外，在某些温和条件下，我们验证了所提出算法的渐近R线性收敛。各种基准的数值实验证实，收敛的假设始终保持并表明开发的修改后的Dirichlet能量最小化的效率，可靠性和鲁棒性。
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.
