Topological Signal Processing over Generalized Cell Complexes

Abstract: 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.