基于层和拓扑的数字图像局部分支数生成方法
A Sheaf and Topology Approach to Generating Local Branch Numbers in Digital Images
摘要：
本文提出了一种结合拓扑数据分析和层理论的理论方法。拓扑数据分析是数学和计算机科学中的一个新兴领域，它关注数据的形状，并已在许多科学学科中被证明是有效的。层理论是代数几何中的一门数学学科，它为描述几何对象的局部一致性提供了一个框架。持续同源性（PH）是TDA的主要驱动力之一，其思想是在不同尺度上跟踪几何对象的变化。持久性图（PD）以多集合的形式总结了PH的信息。虽然PD提供了关于底层对象的有用信息，但它缺乏关于PD中特定生成元对的局部一致性的精细关系，层结构为描述PH中局部对象的合并关系提供了一个新的视角，本文的目的是建立一个理论框架，利用层理论揭示更精细的信息，从PH值。我们还表明，所提出的理论可以应用于识别数字图像中的局部对象的分支数。
英文摘要：
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.
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