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 at 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.

翻译：我们研究了形状重建方法背景下离散莫尔斯理论与持久同调之间的联系。具体而言，我们考察了由Edelsbrunner引入的作为德尔尼复形子复形的Wrap复形的构造，以及由Cohen-Steiner、Lieutier和Vuillamy在类似设定中研究的词典序最优同调环的构造。我们证明，在给定半径参数下，德尔尼复形中的任意环所对应的词典序最优同调环均支撑于同一参数的Wrap复形上，从而建立了这两种方法之间的紧密联系。通过建立持久同调计算中环缩减与离散莫尔斯理论代数推广中梯度流之间的基本联系，我们得到了这一结果。