论文标题
在数字图像中生成本地分支机构的捆绑和拓扑方法
A Sheaf and Topology Approach to Generating Local Branch Numbers in Digital Images
论文作者
论文摘要
本文涉及一种结合拓扑数据分析（TDA）和捆绑理论的理论方法。拓扑数据分析是数学和计算机科学领域不断上升的领域，涉及数据的形状，并已被证明在许多科学学科中有效。代数几何形状中的数学主题束缚理论为描述几何对象中局部一致性提供了一个框架。持续的同源性（PH）是TDA的主要驱动力之一，其想法是在不同尺度上跟踪几何对象的变化。持久图（PD）以多集的形式总结了pH的信息。尽管PD提供了有关基础对象的有用信息，但它缺乏关于PD中特定发电机的局部一致性的良好关系，例如pH中两个连接的组件之间的合并关系。捆结构为描述pH中局部对象的合并关系提供了一种新颖的观点。本文的目的是建立一个理论框架，该框架利用束缚理论从pH中揭示了更精细的信息。我们还表明，提出的理论可以应用于数字图像中本地对象的分支数字。
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.
