Design of graph filters and filterbanks

TL;DR

This paper reviews the fundamental concepts of graph signal processing, focusing on graph filters, spectral analysis, and multiscale transforms like wavelets and filterbanks, highlighting their design and implementation for structured and arbitrary graphs.

Contribution

It provides a comprehensive overview of graph filters, spectral domain analysis, and multiscale transforms, including various filterbank variants and their applications to different graph structures.

Findings

Introduction of spectral domain for graph signals

Discussion of various graph filterbank designs

Analysis of multiscale transforms like spectral wavelets

Abstract

Basic operations in graph signal processing consist in processing signals indexed on graphs either by filtering them, to extract specific part out of them, or by changing their domain of representation, using some transformation or dictionary more adapted to represent the information contained in them. The aim of this chapter is to review general concepts for the introduction of filters and representations of graph signals. We first begin by recalling the general framework to achieve that, which put the emphasis on introducing some spectral domain that is relevant for graph signals to define a Graph Fourier Transform. We show how to introduce a notion of frequency analysis for graph signals by looking at their variations. Then, we move to the introduction of graph filters, that are defined like the classical equivalent for 1D signals or 2D images, as linear systems which operate on each frequency of a signal. Some examples of filters and of their implementations are given. Finally, as alternate representations of graph signals, we focus on multiscale transforms that are defined from filters. Continuous multiscale transforms such as spectral wavelets on graphs are reviewed, as well as the versatile approaches of filterbanks on graphs. Several variants of graph filterbanks are discussed, for structured as well as arbitrary graphs, with a focus on the central point of the choice of the decimation or aggregation operators.