Critical points with discrete Morse theory

Peihong Guo, E. Akleman, Ying He, Xiaoning Wang, W. Liu
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Abstract

In this work, we present some of the unexpected observations resulted from our recent research. We, recently, needed to identify a small number of important critical points, i.e. minimum, maximum and saddle points, on a given manifold mesh surface. All critical points on a manifold triangular mesh can be identified using discrete Gaussian curvature, which is given as ki = 2π − Σj θi,j where ki is vertex defect (the discrete Gaussian curvature) of the vertex i and θi,j is the corner of the vertex in the triangle j. A very useful property coming with vertex defect is the discrete version of Gauss-Bonnet theorem: the sum of all vertex defects is always constant as Σi ki = 2π(2−2g) where g is the genus of the mesh. Any vertex with a non-zero vertex defect is really an critical point of the surface. However, identification of interesting critical points is hard with vertex defect alone. As it can be seen in Figure 1(a), even we ignore vertex defects that are small, too many vertices are still chosen and this information is not really useful to make any conclusion of the shape of the surface.
离散莫尔斯理论的临界点
在这项工作中,我们提出了一些意想不到的观察结果,从我们最近的研究。最近,我们需要在给定的流形网格表面上识别少量重要的临界点,即最小点、最大值和鞍点。所有临界点流形三角网格可以使用离散高斯曲率,确定这是作为ki = 2π−Σjθ,j, ki顶点缺陷(离散高斯曲率)的顶点我和θ,j是三角形顶点的角落j。一个非常有用的属性与顶点的缺陷是离散的版本Gauss-Bonnet定理:顶点缺陷总是不断的总和作为Σ我ki = 2π2−2 g, g网的一个属。任何有非零顶点缺陷的顶点都是曲面的一个临界点。然而,仅用顶点缺陷很难识别出感兴趣的临界点。从图1(a)中可以看出,即使我们忽略了较小的顶点缺陷,仍然选择了太多的顶点,这些信息对于得出曲面形状的任何结论都没有真正的用处。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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