Discrete Signal Processing on Graphs: Frequency Analysis

TL;DR

This paper extends classical signal processing concepts to data on arbitrary graphs by defining a natural frequency ordering based on total variation, enabling the design of graph filters for applications like sensor fault detection and data classification.

Contribution

It introduces a novel frequency definition for graph signals based on total variation, facilitating the design of graph filters with specified responses.

Findings

Defined a total variation-based frequency ordering for graph signals

Developed methods for designing graph filters with specific frequency responses

Applied the approach to sensor malfunction detection and data classification

Abstract

Signals and datasets that arise in physical and engineering applications, as well as social, genetics, biomolecular, and many other domains, are becoming increasingly larger and more complex. In contrast to traditional time and image signals, data in these domains are supported by arbitrary graphs. Signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs. This paper studies the concepts of low and high frequencies on graphs, and low-, high-, and band-pass graph filters. In traditional signal processing, there concepts are easily defined because of a natural frequency ordering that has a physical interpretation. For signals residing on graphs, in general, there is no obvious frequency ordering. We propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low-, high-, and band-pass graph signals and filters. We study the design of graph filters with specified frequency response, and illustrate our approach with applications to sensor malfunction detection and data classification.