While branching network structures abound in nature, their objective analysis is more difficult than expected because existing quantitative methods often rely on the subjective judgment of branch structures. This problem is particularly pronounced when dealing with images comprising discrete particles. Here we propose an objective framework for quantitative analysis of branching networks by introducing the mathematical definitions for internal and external structures based on topological data analysis, specifically, persistent homology. We compare persistence diagrams constructed from images with and without plots on the convex hull. The unchanged points in the two diagrams are the internal structures and the difference between the two diagrams is the external structures. We construct a mathematical theory for our method and show that the internal structures have a monotonicity relationship with respect to the plots on the convex hull, while the external structures do not. This is the phenomenon related to the resolution of the image. Our method can be applied to a wide range of branch structures in biology, enabling objective analysis of numbers, spatial distributions, sizes, and more. Additionally, our method has the potential to be combined with other tools in topological data analysis, such as the generalized persistence landscape.

翻译：自然界中广泛存在着枝状网络结构，然而其客观分析远比预期困难，因为现有定量方法往往依赖于对枝状结构的主观判断。当处理包含离散粒子的图像时，这一问题尤为突出。本文提出一个用于枝状网络定量分析的客观框架，通过引入基于拓扑数据分析（特别是持久同调）的内部结构与外部结构的数学定义。我们比较带凸包图与不带凸包图构建的持久图，两图中保持不变的点为内部结构，差异部分为外部结构。我们为该方法构建了数学理论，证明内部结构相对于凸包图存在单调性关系，而外部结构则不具备此性质。这一现象与图像分辨率相关。我们的方法可应用于生物学中广泛的枝状结构，实现数量、空间分布、尺寸等特征的客观分析。此外，该方法有望与拓扑数据分析中的其他工具（如广义持久景观）相结合。