A Sheaf and Topology Approach to Generating Local Branch Numbers in Digital Images

TL;DR

This paper introduces a novel theoretical framework combining sheaf theory and topological data analysis to extract detailed local information, such as branch numbers, from digital images.

Contribution

It develops a sheaf-based approach to analyze persistent homology, revealing finer local relations and branch structures in digital images.

Findings

Sheaf theory enhances the analysis of persistent homology.

The approach successfully identifies local branch numbers in digital images.

The framework provides new insights into local geometric consistency.

Abstract

This paper concerns a theoretical approach that combines topological data analysis (TDA) and sheaf theory. Topological data analysis, a rising field in mathematics and computer science, concerns the shape of the data and has been proven effective in many scientific disciplines. Sheaf theory, a mathematics subject in algebraic geometry, provides a framework for describing the local consistency in geometric objects. Persistent homology (PH) is one of the main driving forces in TDA, and the idea is to track changes of geometric objects at different scales. The persistence diagram (PD) summarizes the information of PH in the form of a multi-set. While PD provides useful information about the underlying objects, it lacks fine relations about the local consistency of specific pairs of generators in PD, such as the merging relation between two connected components in the PH. The sheaf structure provides a novel point of view for describing the merging relation of local objects in PH. It is the goal of this paper to establish a theoretic framework that utilizes the sheaf theory to uncover finer information from the PH. We also show that the proposed theory can be applied to identify the branch numbers of local objects in digital images.