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

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 mathe...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Chuan-Shen, Hu, Yu-Min, Chung
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 03.12.2020
Subjects
Consistency
Data analysis
Digital imaging
Homology
Mathematical analysis
Topology
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422