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.

翻译：我们研究了在形状重建方法背景下离散莫尔斯理论与持续同调之间的联系。具体而言，我们考察了Edelsbrunner引入的作为Delaunay复形子复形的Wrap复形构造，以及Cohen-Steiner、Lieutier和Vuillamy在类似背景下考虑的字典序最优同调循环构造。我们证明：对于给定半径参数的Delaunay复形中的任意循环，其字典序最优同调循环支撑于相同参数的Wrap复形上，从而建立了这两种方法之间的密切联系。这一结果是通过建立持续同调计算中循环约简与离散莫尔斯理论代数推广中梯度流之间的基本联系而获得的。