论文标题
在德劳尼建筑群中包装周期：桥接持续的同源和离散的摩尔斯理论
Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory
论文作者
论文摘要
我们在形状重建方法的背景下研究离散的摩尔斯理论与持久同源性之间的联系。具体而言，我们将Edelsbrunner引入的包裹络合物的构建为Delaunay Complex的子复合物，以及在类似环境中的Cohen-Steiner，Lieutier和Vuillamy考虑的词典最佳同源周期的词素最佳同源周期。我们表明，对于给定半径参数的Delaunay复合物中的任何循环，在包装复合体上支持相同参数的词典最佳同源周期，从而在两种方法之间建立了密切的连接。我们通过在持续的同源性计算中的循环减少与离散莫尔斯理论的代数概括中的梯度流中的减少之间建立基本联系来获得这一结果。
We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen-Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.
