Signal Processing on Cell Complexes

TL;DR

This paper introduces a framework for signal processing on regular cell complexes, unifying various non-Euclidean structures like graphs and meshes, and develops Hodge Laplacians for filtering applications.

Contribution

It presents a novel approach to signal processing on regular cell complexes, deriving Hodge Laplacians and enabling convolutional filtering on these structures.

Findings

Hodge Laplacians can be derived for regular cell complexes.

Convolutional filters can be constructed using these Laplacians.

Applicable to linear and neural network-based filtering.

Abstract

The processing of signals supported on non-Euclidean domains has attracted large interest recently. Thus far, such non-Euclidean domains have been abstracted primarily as graphs with signals supported on the nodes, though the processing of signals on more general structures such as simplicial complexes has also been considered. In this paper, we give an introduction to signal processing on (abstract) regular cell complexes, which provide a unifying framework encompassing graphs, simplicial complexes, cubical complexes and various meshes as special cases. We discuss how appropriate Hodge Laplacians for these cell complexes can be derived. These Hodge Laplacians enable the construction of convolutional filters, which can be employed in linear filtering and non-linear filtering via neural networks defined on cell complexes.