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

arXiv: https://arxiv.org/abs/2402.07436 · HTML: https://mlsanity.com/paper/2402.07436/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 5 of 5)

- [The persistence landscape and some of its properties](https://mlsanity.com/paper/1810.04963/) (2018)
- [A persistence landscapes toolbox for topological statistics](https://mlsanity.com/paper/1501.00179/) (2014)
- [Topological analysis of the connectome of digital reconstructions of neural microcircuits](https://mlsanity.com/paper/1601.01580/) (2016)
- [Persistent homology machine learning for fingerprint classification](https://mlsanity.com/paper/1711.09158/) (2017)
- [From trees to barcodes and back again: theoretical and statistical perspectives](https://mlsanity.com/paper/2010.11620/) (2020)

## Built on by (most cited first, 1 of 1)

- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)

## Abstract (arXiv)

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.