Fourier could be a data scientist: From graph Fourier transform to signal processing on graphs (original) (raw)
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Graph Signal Processing - Part II: Processing and Analyzing Signals on Graphs
2019
The focus of Part I of this monograph has been on both the fundamental properties, graph topologies, and spectral representations of graphs. 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). The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebys...
IEEE Signal Processing Magazine, 2000
In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogues to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting, and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.
A SYSTEMATIC REVIEW OF GRAPH SIGNAL PROCESSING
Graph signal processing(GSP) is a representation of data in graphical format with directed or undirected vertices. In many applications such as big data networks, economic and social networks analysis signals with graph is relevant. Harmonic analysis for processing the signals with spectral and algebric graphical thereotical concepts are merged and analyzed with respect to signal processing schemes on graphs. In this work, main challenges of GSP are discussed with Graph Spectral Domains (GSD) and when processing the signals on graph. The information is extracted efficiently from the highdimensional data by using operators of signals on graph and transformation of graph on signal are highlighted in this work. Finally, a brief discussion of open issues of GSP are reviewed.
To further understand graph signals
arXiv (Cornell University), 2022
Graph signal processing (GSP) is a framework to analyze and process graph-structured data. Many research works focus on developing tools such as Graph Fourier transforms (GFT), filters, and neural network models to handle graph signals. Such approaches have successfully taken care of "signal processing" in many circumstances. In this paper, we want to put emphasis on "graph signals" themselves. Although there are characterizations of graph signals using the notion of bandwidth derived from GFT, we want to argue here that graph signals may contain hidden geometric information of the network, independent of (graph) Fourier theories. We shall provide a framework to understand such information, and demonstrate how new knowledge on "graph signals" can help with "signal processing".
Vertex-Frequency Graph Signal Processing
ArXiv, 2019
Graph signal processing deals with signals which are observed on an irregular graph domain. While many approaches have been developed in classical graph theory to cluster vertices and segment large graphs in a signal independent way, signal localization based approaches to the analysis of data on graph represent a new research direction which is also a key to big data analytics on graphs. To this end, after an overview of the basic definitions in graphs and graph signals, we present and discuss a localized form of the graph Fourier transform. To establish an analogy with classical signal processing, spectral- and vertex-domain definitions of the localization window are given next. The spectral and vertex localization kernels are then related to the wavelet transform, followed by a study of filtering and inversion of the localized graph Fourier transform. For rigour, the analysis of energy representation and frames in the localized graph Fourier transform is extended to the energy fo...
Extended Adjacency and Scale-Dependent Graph Fourier Transform via Diffusion Distances
IEEE Transactions on Signal and Information Processing over Networks
This paper proposes the augmentation of the adjacency model of networks for graph signal processing. It is assumed that no information about the network is available, apart from the initial adjacency matrix. In the proposed model, additional edges are created according to a Markov relation imposed between nodes. This information is incorporated into the extended-adjacency matrix as a function of the diffusion distance between nodes. The diffusion distance measures similarities between nodes at a certain diffusion scale or time, and is a metric adopted from diffusion maps. Similarly, the proposed extendedadjacency matrix depends on the diffusion scale, which enables the definition of a scale-dependent graph Fourier transform. We conduct theoretical analyses of both the extended adjacency and the corresponding graph Fourier transform and show that different diffusion scales lead to different graph-frequency perspectives. At different scales, the transform discriminates shifted ranges of signal variations across the graph, revealing more information on the graph signal when compared to traditional approaches. The scale-dependent graph Fourier transform is applied for anomaly detection and is shown to outperform the conventional graph Fourier transform.
On the Graph Fourier Transform for Directed Graphs
IEEE Journal of Selected Topics in Signal Processing
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Created by The Institute of Electrical and Electronics Engineers (IEEE) for the benefit of humanity.
Complex Basis For Spectral Analysis of Graph Signals
International Journal of Mathematics Trends and Technology, 2020
The Signal Processing on Graph (SPG) is an emerging field of research aiming to develop accurate methods for big data analysis by combining graph theory and classical signal processing methods. One key method in signal processing on graph is the so-called Graph Fourier Transform (GFT) which is a generalization of the Classical Fourier Transform (defined for data lying on regular domains :1D for times series or 2D for images) to data lying on networks. Those network data are viewed like a set of interrelated data points lying on a graph whose graph vertices map the data points and graph links encode the relationship between data. In the classical framework, the Fourier transform is a linear operator that performs the mapping of a vector from its initial representation domain to the frequency domain through the Fourier matrix which is an orthonormal basis formed by complex exponential vectors constructed from powers of the complex number. Those vectors are of a key importance in the properties of the transform and its applications. However, for each graph Fourier transform proposed in the literature, although its graph Fourier matrix is orthonormal, its vectors are not complex as in the classical framework, limiting the extension and the use of some useful properties of the classical Fourier transform to the graph signals framework. In this work, we present a method to define a complex orthonormal basis for the graph Fourier transform that allows to perform spectral analysis for graph signals in the frequency domain. The graph Fourier basis we defined is identical to the Fourier basis when applied to graph signals defined on a regular domain. We applied the proposed method successfully to signal detection on an irregularly sampled sensor network.