Abstract
Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain.
Highlights
Graph signal processing (GSP) has emerged as an effective solution to handle data with an irregular support
Instrumental to GSP are the notions of the graph shift operator (GSO), which is a matrix that accounts for the topology of the graph, and the graph Fourier transform (GFT), which transforms a graph signal to the so-called graph frequency domain leading to a graph frequency signal
We propose a support of a graph frequency signal by means of a graph, which we denominate as dual graph,1 and its corresponding dual GSO
Summary
Journal of Fourier Analysis and Applications (2021) 27:49 graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain. Keywords Graph signal processing · Dual graph shift operator · Frequency support · Graph Fourier transform
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