Graph Signal Processing: Dualizing GSP Sampling in the Vertex and Spectral Domains

TL;DR

This paper develops a unified graph signal processing sampling theory that bridges vertex and spectral domains using a spectral shift, enabling analogous sampling methods similar to traditional DSP.

Contribution

It introduces a spectral shift operator in GSP, creating a dual framework for sampling in both vertex and spectral domains with clear DSP analogies.

Findings

Unified GSP sampling theory in vertex and spectral domains

Spectral shift operator enables dual sampling methods

GSP sampling reduces to DSP sampling on directed time cycle graph

Abstract

Vertex based and spectral based GSP sampling has been studied recently. The literature recognizes that methods in one domain do not have a counterpart in the other domain. This paper shows that in fact one can develop a unified graph signal sampling theory with analogous interpretations in both domains just like sampling in traditional DSP. To achieve it, we introduce a spectral shift $M$ acting in the spectral domain rather than shift $A$ that acts in the vertex domain. This leads to a GSP theory that starts from the spectral domain, for example, linear shift invariant (LSI) filtering in the spectral domain is with polynomials $P(M)$. We then develop GSP vertex and spectral domain dual versions for each of the four standard sampling steps of subsampling, decimation, upsampling, and interpolation. We show how GSP sampling reduces to DSP sampling when the graph is the directed time cycle graph. Simple examples illustrate the impact of choices that are available in GSP sampling.