We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we compare the Wrap complex, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the lexicographic optimal homologous chains, considered by Cohen-Steiner, Lieutier, and Vuillamy. We show that given any cycle in a Delaunay complex at some radius threshold, the lexicographically optimal homologous cycle is supported on the Wrap complex at the same threshold, thereby establishing a close connection between the two methods. This result is obtained as a consequence of a general connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory, which is of independent interest.

翻译：具体地说,我们比较了Edelsbrunner作为Delaunay综合体的子复合体引入的环状综合体,以及Cohen-Steiner、Lieutier和Vuillamy考虑的词典最佳同系链。我们发现,在某个半径阈值的Delaunay综合体中,如果存在任何循环,那么在同一个临界点的环状综合体上支持了法律上的最佳同系循环,从而在这两种方法之间建立了密切的联系。 取得这一结果的原因是,独立感兴趣的离心摩斯理论在计算持续同质学中的周期减少与代数理论的变异性流之间的一般联系。