Vertex-frequency graph signal processing: A comprehensive review

Stanković, Ljubiša; Mandic, Danilo P.; Daković, Miloš; Scalzo, Bruno; Brajović, Miloš; Sejdić, Ervin; Constantinides, A.G.

doi:10.1016/j.dsp.2020.102802

Cited by 32 publications

(16 citation statements)

References 36 publications

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“…A recent trend of particular interest in neuroimaging data analysis is the application of methods from the field of graph signal processing [5][6][7][8][9][10]. In these applications, anatomical information obtained from DTI and structural MRI is used to construct the connectome graph [11], and combined with functional imaging data such as BOLD-fMRI or EEG/MEG to investigate structure-function relationships in the human brain (see [12,13] for reviews).…”

Section: Introductionmentioning

confidence: 99%

Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome

et al. 2021

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Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed “connectome harmonics”, have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI). As an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome graph constructed from diffusion tensor imaging (DTI) and structural MRI data. We show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of resting-state fMRI data, and predict its functional connectivity, with a high level of detail. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.

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Section: Introductionmentioning

confidence: 99%

Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome

et al. 2021

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“…Classical short-time Fourier transform (STFT) admits time-frequency localization of the analyzed signal using the Fourier transform of the windowed and shifted versions of the signal. This principle is possible in the graph signal processing [29,31]. However, since this approach requires sophisticated approaches to the vertex shift operation on the signals, the spectral domain localization is more commonly used in vertex-frequency analysis.…”

Section: Spectral Domain Localized Graph Fourier Transform (Lgft)mentioning

confidence: 99%

“…• Sine window is obtained as the square root of the raised cosine window in (28). Obviously, this window will satisfy (31). Its form is…”

Section: Frame Decomposition-wola Conditionmentioning

confidence: 99%

“…, K − 1, in the form given by ( 28) or ( 33) should be realized using their approximations by polynomials whose order is (M − 1). Although the approximation based on the Chebyshev polynomial is most commonly used for this purpose [25,31], we will revisit alternative approaches [38] as well.…”

Section: Polynomial Lgft Approximationmentioning

confidence: 99%

“…The LGFT defined by ( 17) is a tight frame if the condition in (31) or ( 49) holds. This is the condition that is used to define transfer functions shown in Figure 5b,c.…”

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From Time–Frequency to Vertex–Frequency and Back

et al. 2021

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The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

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