论文标题
持续的同源性与形态动力学之间的某种等效关系
Some equivalence relation between persistent homology and morphological dynamics
论文作者
论文摘要
在数学形态（MM）中，基于动力学的连接过滤器用于过滤图像的极值。同样，持久性是一个来自持续的同源性（pH）和莫尔斯理论（MT）的概念，代表了摩尔斯函数极端的稳定性。由于这两个概念似乎密切相关，因此在本文中，我们研究了它们的关系，我们证明它们在n $ \ ge $ 1 $ 1上是平等的。更重要的是，将最小值与动态的1助剂配对或将相同的1助剂与相同的1助剂配对与持久性的最小值与同一配对的最小值，假设相同的配对，则可以使用相同的配对值，从而具有独特的Morse功能。该结果进一步展示了拓扑数据分析和数学形态有多少相关的，为对这两个研究领域之间关系的更深入研究铺平了道路。
In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n $\ge$ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.
