  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.
