  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.
