Multi-windowed graph Fourier frames

Zheng, Xianwei; Tang, Yuan Yan; Zhou, Jiantao; Yuan, Haoliang; Wang, Yulong; Yang, Lina; Pan, Jianjia

doi:10.1109/icmlc.2016.7873023

Cited by 9 publications

(5 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is important to note that, while the concept of window functions for signal localization has been extended to signals defined on graphs [18,66,67,68,69], such extensions are not straightforward, since, owing to inherent properties of graphs as irregular but interconnected domains, even an operation which is very simple in classical time-domain analysis, like the time shift, cannot be straightforwardly generalized to graph signal domain. This has resulted in several approaches to the definition of the graph shift operator, and much ongoing research in this domain [18,66,67,68,69].…”

Section: Vertex-frequency Representationsmentioning

confidence: 99%

Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

Stankovic,

Mandic,

Dakovic

et al. 2019

Preprint

View full text Add to dashboard Cite

Data analytics on graphs deals with information processing of data acquired on irregular but structured graph domains. The focus of Part I of this monograph has been on both the fundamental and higher-order graph properties, graph topologies, and spectral representations of graphs. Part I also establishes rigorous frameworks for vertex clustering and graph segmentation, and illustrates the power of graphs in various data association tasks. Part II embarks on these concepts to address the algorithmic and practical issues centered round data/signal processing on graphs, that is, the focus is on the analysis and estimation of both deterministic and random data on graphs. The fundamental ideas related to graph signals are introduced through a simple and intuitive, yet illustrative and general enough case study of multisensor temperature field estimation. The concept of systems on graph is defined using graph signal shift operators, which generalize the corresponding principles from traditional learning systems. At the core of the spectral domain representation of graph signals and systems is the Graph Discrete Fourier Transform (GDFT), which is defined based on the eigendecomposition of both the adjacency matrix and the graph Laplacian. The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebyshev polynomial approximation series. Ideas related to the sampling of graph signals, and in particular the challenging topic of data dimensionality reduction through graph subsampling, are presented and further linked with compressive sensing. The principles of time-varying signals on graphs and basic definitions related to random graph signals are next reviewed. Localized graph signal analysis in the joint vertex-spectral domain is referred to as the vertex-frequency analysis, since it can be considered as an extension of classical time-frequency analysis to the graph domain of a signal. Important topics related to the local graph Fourier transform (LGFT) are covered, together with its various forms including the graph spectral and vertex domain windows and the inversion conditions and relations. A link between the LGFT with spectral varying window and the spectral graph wavelet transform (SGWT) is also established. Realizations of the LGFT and SGWT using polynomial (Chebyshev) approximations of the spectral functions are further considered and supported by examples. Finally, energy versions of the vertex-frequency representations are introduced, along with their relations with classical time-frequency analysis, including a vertex-frequency distribution that can satisfy the marginal properties. The material is supported by numerous examples.

show abstract

Section: Vertex-frequency Representationsmentioning

confidence: 99%

Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

Stankovic,

Mandic,

Dakovic

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…In graph signal processing (GSP), the vertex-frequency analysis is a hot issue, which is similar to the time-frequency analysis in classical signal processing. This analysis involves various graph-based transforms and the associated frame theory [2,[19][20][21][22][23][24][25][26]. A new framework for constructing Gabor-type frames with sharp bounds for graph signals has been proposed [21].…”

Section: Introductionmentioning

confidence: 99%

A new reconstruction formula and shift frame for WGFRFT

Yan,

2023

Third International Conference on Signal Image Processing and Communication (ICSIPC 2023)

View full text Add to dashboard Cite

The study of signals defined on irregular domains has garnered significant attention, and the field of graph signal processing has emerged as a way to model such signals using underlying graph structures. Vertex frequency analysis is a crucial tool for graph signal analysis and representation. Finding different kinds of frames via the windowed graph fractional Fourier transform (WGFRFT) is necessary to extract the characteristics of graph signals. To overcome this issue, we introduce the concept of the dual of WGFRFT frame and use it to develop a new reconstruction formula for graph signals. Additionally, we propose fractional shift frame. Finally, to highlight the utility of our proposed frames, we present some examples which use our fractional shift frame to extract the spectrum of graph signal.

show abstract

“…Time localization, combined with the modulation by the basis functions, produces kernel functions for classical time-frequency analysis. The classical time-frequency analysis approach has been extended to vertex-frequency analysis for signals defined on graphs [15][16][17][18][19][20][21][22]. This generalization is not straightforward, since graph is a complex and irregular signal domain.…”

Section: Introductionmentioning

confidence: 99%

“…A frame is called Parseval's tight frame if a = b. The LGFT, as given by in (17), represents Parseval's tight frame when…”

mentioning

confidence: 99%

From Time–Frequency to Vertex–Frequency and Back

et al. 2021

View full text Add to dashboard Cite

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.

show abstract

Multi-windowed graph Fourier frames

Cited by 9 publications

References 6 publications

Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

A new reconstruction formula and shift frame for WGFRFT

From Time–Frequency to Vertex–Frequency and Back

Contact Info

Product

Resources

About