Novel definition and quantitative analysis of branch structure with topological data analysis

TL;DR

This work defines a mathematically objective way to analyze branching networks in discrete images using persistent homology by separating internal (network) and external (protruding) structures through $PD_1(X)$ and $PD_1(X \cup U)$, where $U$ are hull points. It proves a monotonicity property for internal structures with respect to hull plots while external structures lack such monotonicity, enabling robust, geometry-aware quantitative analyses. The authors apply the framework to lymph vessels, neurons, and blood vessels, showing how counts and persistence landscapes characterize internal complexity and spatial distributions, and they discuss how this approach compares to non-TDA and other TDA methods. They also introduce a generalized persistence landscape to extend invertibility and differentiability, paving the way for future integration with multiparameter persistence and classification tasks. The framework provides a principled, objective tool for analyzing branch-like biological structures and can be combined with other TDA tools for enhanced insight.

Abstract

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.

Figures (7)

• Figure 1: (a,b) Examples of the images of lymph vessels that we will analyze in this paper
• Figure 2: (a) The image of lymph vessels. We use the white points as a point cloud. (b) The persistence diagram calculated from the original image of lymph vessels. (c) The points on the convex hull of the image and the original points. (d) The persistence diagram calculated from the points on the convex hull and the points of the original image
• Figure 3: An example to make clear the meaning of the expression: a loop's equivalence class can serve as a basis of the homology group at the birth of a bar
• Figure 4: (a) The original image and its barcode. (b) The original image plus the point on the left opening and the resulting barcode. (c) The image in (b) plus the points on the lower opening and the resulting barcode. The large loop which is a detectable external structure in (b) cannot be a detectable external structure in (c)
• Figure 5: The result of detecting external/internal structures in the images (red/blue points)

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3: detectable internal structure
• Proposition 1
• Definition 4: birth and death for loops
• Definition 5
• Lemma 2
• proof
• Lemma 3
• proof