Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI

TL;DR

This paper introduces a novel sampling framework called Graph-FRI for perfect reconstruction of sparse signals on circulant graphs, extending finite rate of innovation theory to the graph domain with potential for arbitrary graph approximation.

Contribution

It extends FRI theory to graph signals, proposing a sampling scheme for sparse signals on circulant graphs and a method for graph coarsening that preserves key properties.

Findings

Perfect reconstruction from spectral samples at size 2K

Decomposition of spectral transform using e-spline wavelets

Framework applicable to arbitrary graphs via approximation

Abstract

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, and subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.