Graph Spectral Image Processing

1Graph Spectral Filtering

Yuichi TANAKA

Tokyo University of Agriculture and Technology, Japan

1.1. Introduction

The filtering of time- and spatial-domain signals is one of the fundamental techniques for image processing and has been studied extensively to date. GSP can treat signals with irregular structures that are mathematically represented as graphs. Theories and methodologies for the filtering of graph signals are studied using spectral graph theory. In image processing, graphs are strong tools for representing structures formed by pixels, like edges and textures.

The filtering of graph signals is not only an extension of that for standard time- and spatial-domain signals, but it also has its own interesting properties. For example, GSP can represent traditional pixel-dependent image filtering methods as graph spectral domain filters. Furthermore, theory and design methods for wavelets and filter banks, which are studied extensively in signal and image processing, are also updated to treat graph signals.

In this chapter, the spectral-domain filtering of graph signals is introduced. In section 1.2, the filtering of time-domain signals is briefly described as a starting point. The filtering of graph signals, both in the vertex and spectral domains, is detailed in section 1.3, in addition to its relationship with classical filtering. Edge-preserving image smoothing is represented as a graph filter in section 1.4. Furthermore, a framework of filtering by multiple graph filters, ...