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 of graphs and graph signals, we present and discuss a localized form of the graph Fourier transform. To establish analogy with classical signal processing, spectral domain 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 their polynomial approximations and a study of filtering and inversion operations. For rigor, the analysis of energy representation and frames in the localized graph Fourier transform is extended to the energy forms of vertex-frequency distributions, which operate even without the requirement to apply localization windows. Another link with classical signal processing is established through the concept of local smoothness, which is subsequently related to the paradigm of signal smoothness on graphs, a lynchpin which connects the properties of the signals on graphs and graph topology. This all represents a comprehensive account of the relation of general vertex-frequency analysis with classical time-frequency analysis, an important but missing link for more advanced applications of graph signal processing. The theory is supported by illustrative and practically relevant examples.

图形信号处理处理在不规则图形域上观察到的信号。虽然经典图论中已经开发出许多方法来以信号独立的方式对顶点进行聚类和分段，但是基于信号定位的图上数据分析方法代表了一个新的研究方向，这也是图上大数据分析的关键。为此，在概述图和图信号的基本定义之后，我们介绍并讨论图傅立叶变换的局部形式。为了建立与经典信号处理的类比，接下来给出定位窗口的频谱域和顶点域定义。然后将光谱和顶点定位内核与小波变换相关联，其次是它们的多项式逼近以及对滤波和反演运算的研究。为了严格起见，将局部图傅立叶变换中的能量表示和框架分析扩展到顶点频率分布的能量形式，即使不需要应用局部窗口也可以运行。与经典信号处理的另一条联系是通过局部平滑性的概念建立的，局部平滑性随后与图形上信号平滑性的范例有关，这是连接图形和图形拓扑上信号属性的关键。所有这些都全面说明了一般顶点频率分析与经典时频分析之间的关系，而后者对于图形信号处理的更高级应用而言是重要但缺少的环节。