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.

通过引入基于拓扑数据分析的数学定义，我们提出了一种定量分析分支网络的客观框架，比较在具有和不具有凸包绘图的图像中构建的持续图，通过判断两个图之间的不变点以及区别，我们构建了一个数学理论，并表明内部结构与凸包绘图之间存在单调性关系，而外部结构则不存在该关系，该方法可以应用于生物学中的各种分支结构的客观分析，实现对数量、空间分布、大小等的分析，另外，该方法还有与拓扑数据分析中的其他工具相结合的潜力，比如广义持续景观。