Finite Impulse Response Filters for Simplicial Complexes
This work addresses signal processing challenges in topological data analysis, offering a generalization of graph filters, but it is incremental as it builds on existing Hodge decomposition concepts.
The paper tackles the problem of processing signals on simplicial complexes by proposing a finite impulse response filter based on the Hodge Laplacian, enabling nuanced control over gradient-flow and curl-flow signals for applications like sub-component extraction and denoising.
In this paper, we study linear filters to process signals defined on simplicial complexes, i.e., signals defined on nodes, edges, triangles, etc. of a simplicial complex, thereby generalizing filtering operations for graph signals. We propose a finite impulse response filter based on the Hodge Laplacian, and demonstrate how this filter can be designed to amplify or attenuate certain spectral components of simplicial signals. Specifically, we discuss how, unlike in the case of node signals, the Fourier transform in the context of edge signals can be understood in terms of two orthogonal subspaces corresponding to the gradient-flow signals and curl-flow signals arising from the Hodge decomposition. By assigning different filter coefficients to the associated terms of the Hodge Laplacian, we develop a subspace-varying filter which enables more nuanced control over these signal types. Numerical experiments are conducted to show the potential of simplicial filters for sub-component extraction, denoising and model approximation.