单纯形上的拓扑信号处理
Topological Signal Processing over Simplicial Complexes
摘要：
本文的目标是建立基本工具来分析定义在拓扑空间上的信号，即一组点以及一组邻域关系。这种设置不需要度量的定义，因此对于处理在非度量空间上定义的信号特别有用。我们专注于定义在单纯复形上的信号。图信号处理（GSP）表示拓扑信号处理（TSP）的特殊情况，指的是信号仅与图的顶点相关联的情况。即使该理论可以应用于任何顺序的信号，我们专注于定义在一个图形的边缘的信号，并显示如何建立一个单纯复杂的二阶，即包括三角形，在边缘信号的分析中产生的好处。在回顾了代数拓扑的基本原理之后，我们导出了任意阶信号的采样理论，并强调了不同阶信号之间的相互作用。然后，我们提出了一种方法来推断一个单纯复形的拓扑数据。最后，我们将应用于真实边缘信号和离散矢量场分析，以说明所提出的方法的优点。
英文摘要：
The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.
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