From Time–Frequency to Vertex–Frequency and Back

Abstract

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.

Time-Varying Graph Mode Decomposition

“…Simply put, the graph modes (along the graph geodesics) must be updated in such a way that they have similar values on the pair of nodes having a strong connection. This key requirement is implicitly encoded in the second terms of the TVGMD cost functions in (7) and (9), and the first term of (13) that is part of the augmented Lagrangian function (11). In those terms, Z (k) and its corresponding vector z (k) are directly related to the graph modes g(k) (t) via (1).…”

“…Note that the TVGMD optimization scheme is not strictly ADMM owing to the: i) non-convexity of the original optimization cost function (9); ii) deviation of our optimization formulation from the standard ADMM formulation that requires multiple set of variables having separable objective functions. Therefore, the algorithm is not guaranteed to converge to the global minimum.…”

“…To this end, GSP aims to extend classical signal processing tools to signals supported on irregular (graph) structures. This has led to new concepts and methods extending the harmonic analysis to graph signals e.g., graph Fourier transform [5]; filter bank [6], compressed sensing and reconstruction of graph signals [7]; estimating graph spectrum through related new concepts of stationarity for graphs [8]; and vertex-frequency graph representations [9]. N. Rehman is with the Department of Electrical and Computer Engineering, Aarhus University, 8200 Aarhus N, Denmark e-mail: naveed.rehman@ece.au.dk.…”

New Method for Transient Signal Analysis using the Continuous Wavelet Transform

“…The FT is a fundamental tool in signal processing, but it has limitations when dealing with non-stationary signals such as transients [2]. The Short-Time FT (STFT) addresses some of these limitations by providing timefrequency localization of signals [3]. However, STFT has fixed window parameters that may not be suitable for all transient signals [4].…”

Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2

“…Graph theory and its applications (polyhedra, enumeration, coloring, fullerenes, etc.) has received increasing attention in recent years [1][2][3][4][5], which has paved the way for more directions of research.…”