A fast algorithm for vertex-frequency representations of signals on graphs

Abstract

The windowed Fourier transform (short time Fourier transform) and the S-transform are widely used signal processing tools for extracting frequency information from non-stationary signals. Previously, the windowed Fourier transform had been adopted for signals on graphs and has been shown to be very useful for extracting vertex-frequency information from graphs. However, high computational complexity makes these algorithms impractical. We sought to develop a fast windowed graph Fourier transform and a fast graph S-transform requiring significantly shorter computation time. The proposed schemes have been tested with synthetic test graph signals and real graph signals derived from electroencephalography recordings made during swallowing. The results showed that the proposed schemes provide significantly lower computation time in comparison with the standard windowed graph Fourier transform and the fast graph S-transform. Also, the results showed that noise has no effect on the results of the algorithm for the fast windowed graph Fourier transform or on the graph S-transform. Finally, we showed that graphs can be reconstructed from the vertex-frequency representations obtained with the proposed algorithms.

Differences in brain networks during consecutive swallows detected using an optimized vertex–frequency algorithm

“…According to the previous study, the α domain can be represented as multiplication of the signal spectra shifted by each frequency point with the spectra of the window function (Jestrović et al, 2017). We can write alpha domain as: $$\alpha (l^{\prime },l)=\widehat{f}(l^{\prime }+l)·\widehat{w}(l),$$ where f̂ is the spectra of the graph signal and ŵ is the spectra of the window function, while l and l ′ refer to the graph signal frequency.…”

“…Thus, signal processing on graphs is a valuable clinical tool to detect anomalies in graphs (Sun et al, 2005; Noble and Cook, 2003; Eberle and Holder, 2007). In our previous study, we developed an algorithm for calculating a fast windowed graph Fourier transform ( FWGFT ) and a fast graph S-transform ( FGST ) (Jestrović et al, 2017). In the same study, we showed that vertex-frequency representations (graph signal processing equivalents of time-frequency representations (Sejdić et al, 2009)) of the brain network during healthy swallowing has a distinctive pattern.…”

Fractional Spectral Graph Wavelets and Their Applications

“…On the other hand, Fourier transform and wavelet transform were extended to graph domain, obtaining graph Fourier transform (GFT) [33][34][35][36][37][38][39][40] and graph wavelet transform (GWT) [41][42][43][44] to handle signal defined on the vertices of weighted graphs. Two basic approaches to signal processing on graphs have been considered: The first is rooted in the spectral graph theory [45] and builds upon the graph Laplacian matrix [33].…”

Local Smoothness of Graph Signals

“…The theory and methods for processing the graph signals are introduced and presented in [1][2][3][4][5]. Graph signal processing applications in biomedical systems [6,7] and analysis of big data [8] provide insight into the graph framework advantages and real-world potential.…”

“…Signals with varying local vertex behaviors are a class of signals called nonstationary graph signals. One approach to the analysis of nonstationary graph signals is vertex-frequency analysis [7,[9][10][11][12][13][14][15], which is a counterpart of time-frequency analysis [16][17][18] in classic signal processing.…”