  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.
