基于拓扑数据分析的分支结构新定义与定量分析
Novel definition and quantitative analysis of branch structure with topological data analysis
AI总结：
针对分支网络结构客观分析困难的问题，提出基于拓扑数据分析（持久同调）的内部与外部结构数学定义及定量分析框架，实现对生物学分支结构的客观分析并具备与其他拓扑工具结合的潜力。
摘要：
尽管分支网络结构在自然界中普遍存在，但其客观分析比预期的更为困难，因为现有的定量方法通常依赖于对分支结构的主观判断。当处理由离散粒子组成的图像时，这一问题尤为突出。在此，我们提出了一种用于定量分析分支网络的客观框架，通过引入基于拓扑数据分析（特别是持久同调）的内部和外部结构的数学定义来实现。我们比较了在凸包上带有和不带有节点的图像所构建的持久性图。两图中保持不变的点构成内部结构，两者之间的差异构成外部结构。我们为该方法构建了数学理论，并证明了内部结构相对于凸包上的节点具有单调性关系，而外部结构则没有。这是与图像分辨率相关的现象。我们的方法可应用于生物学中广泛的分支结构，能够对数量、空间分布、大小等进行客观分析。此外，该方法还有潜力与拓扑数据分析中的其他工具相结合，例如广义持久性景观。
英文摘要：
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.
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