@article{hu_sheaf_2020, 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.}, author = {Hu, Chuan-Shen and Chung, Yu-Min}, date = {2020-11-27}, eprint = {2011.13580}, eprinttype = {arxiv}, journaltitle = {{arXiv}:2011.13580 [cs, math]}, keywords = {1 - Computer vision, 1 - Pattern recognition, 2 - Persistent homology, 2 - Sheaf theory, 3 - Images}, title = {A Sheaf and Topology Approach to Generating Local Branch Numbers in Digital Images}, url = {http://arxiv.org/abs/2011.13580}, urldate = {2020-12-02} }