广义胞复形上的拓扑信号处理
Topological Signal Processing over Generalized Cell Complexes
AI总结：
该研究将拓扑信号处理扩展至广义胞复形以突破单纯复形包含性限制，提出结构推断与分布式滤波方法，在多应用中实现了更优的复杂度-准确性权衡。
摘要：
单纯复形上的拓扑信号处理（Topological Signal Processing, TSP）是近期提出的框架，作为图信号处理（Graph Signal Processing, GSP）的推广，将GSP扩展至分析定义在任意阶集合上的信号（即不仅限于图的顶点），并捕捉数据间任意阶的多元关系。然而，单纯复形需满足所谓的包含性：若一个集合属于该复形，则其所有子集也必须属于该复形。在部分应用中，这是一项严重限制。\n为突破该限制，本文将TSP扩展以处理定义在胞复形上的信号，并推广胞复形的概念以纳入中空胞元。研究表明，尽管代数公式未发生显著变化，但向广义胞复形的扩展大幅拓宽了应用范围。最重要的是，这种新表示在表示复杂度与准确性之间实现了更优的权衡。\n此外，本文提出一种从数据推断胞复形结构的方法，并提出了分布式滤波策略，其中包括一种检索调和分量最稀疏表示的方法。本文针对图像分割、数据流量与交通网络中实测真实流的恢复等不同应用，从复杂度-准确性权衡的角度量化了使用胞复形替代单纯复形的优势。
英文摘要：
Topological Signal Processing (TSP) over simplicial complexes is a framework that has been recently proposed, as a generalization of graph signal processing (GSP), to extend GSP to analyzing signals defined over sets of any order (i.e., not only vertices of a graph) and to capture multiway relations of any order among the data. However, simplicial complexes are required to satisfy the so-called inclusion property, according to which, if a set belongs to the complex, then all its subsets must also belong to the complex. In some applications, this is a severe limitation. To overcome this limit, in this paper we extend TSP to deal with signals defined over cell complexes and we also generalize the concept of cell complexes to include hollow cells. We show that, even if the algebraic formulation does not change significantly, the extension to the generalized cell complexes considerably broadens the number of applications. Most important, the new representation provides a much better trade-off between the complexity of the representation and its accuracy. In addition, we propose a method to infer the structure of the cell complex from data and we propose distributed filtering strategies, including a method to retrieve the sparsest representation of the harmonic component. We quantify the advantages of using cell complexes instead of simplicial complexes, in terms of the complexity/accuracy trade-off, for different applications such image segmentation and recovering of real flows measured on data traffic and transportation networks.
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