论文标题
PL和离散摩尔斯理论的关键集：对应关系
Critical Sets of PL and Discrete Morse Theory: a Correspondence
论文作者
论文摘要
分段线性（PL）Morse理论和离散的Morse理论用于形状分析任务，以研究离散空间的拓扑特征。尽管它们在平滑的摩尔斯理论中具有共同的起源，但文献中为离散设置提供了各种关键点的概念，从而清楚地了解了它们之间的关系并不明显。本文旨在提供有关两种离散摩尔斯理论的关键点的等效结果。首先，我们证明了PL关键点现有概念的等效性。接下来，在称为相对完美的最佳条件下，我们在PL临界点和组合方法的离散临界简单的集合之间显示了维度不可知的对应关系。最后，我们展示了相对完美的离散梯度向量字段如何构建到维度3。这样，我们保证PL和离散理论中的关键集合之间的正式和操作性连接。
Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical points have been given in the literature for the discrete setting, making a clear understanding of the relationships occurring between them not obvious. This paper aims at providing equivalence results about critical points of the two discretized Morse theories. First of all, we prove the equivalence of the existing notions of PL critical points. Next, under an optimality condition called relative perfectness, we show a dimension agnostic correspondence between the set of PL critical points and that of discrete critical simplices of the combinatorial approach. Finally, we show how a relatively perfect discrete gradient vector field can be algorithmically built up to dimension 3. This way, we guarantee a formal and operative connection between critical sets in the PL and discrete theories.
