Basic operations in graph signal processing consist in processing signals
indexed on graphs either by filtering them, to extract specific part out of
them, or by changing their domain of representation, using some transformation
or dictionary more adapted to represent the information contained in them. The
aim of this chapter is to review general concepts for the introduction of
filters and representations of graph signals. We first begin by recalling the
general framework to achieve that, which put the emphasis on introducing some
spectral domain that is relevant for graph signals to define a Graph Fourier
Transform. We show how to introduce a notion of frequency analysis for graph
signals by looking at their variations. Then, we move to the introduction of
graph filters, that are defined like the classical equivalent for 1D signals or
2D images, as linear systems which operate on each frequency of a signal. Some
examples of filters and of their implementations are given. Finally, as
alternate representations of graph signals, we focus on multiscale transforms
that are defined from filters. Continuous multiscale transforms such as
spectral wavelets on graphs are reviewed, as well as the versatile approaches
of filterbanks on graphs. Several variants of graph filterbanks are discussed,
for structured as well as arbitrary graphs, with a focus on the central point
of the choice of the decimation or aggregation operators.

本文回顾了图信号处理的一般概念，包括引入谱域、定义谱变换、引入图滤波、多尺度变换和滤波器组等内容。

图滤波器和滤波器组的设计

Design of graph filters and filterbanks

Multiscale transforms designed to process analog and discrete-time signals
and images cannot be directly applied to analyze high-dimensional data residing
on the vertices of a weighted graph, as they do not capture the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
adapt the Laplacian pyramid transform for signals on Euclidean domains so that
it can be used to analyze high-dimensional data residing on the vertices of a
weighted graph. Our approach is to study existing methods and develop new
methods for the four fundamental operations of graph downsampling, graph
reduction, and filtering and interpolation of signals on graphs. Equipped with
appropriate notions of these operations, we leverage the basic multiscale
constructs and intuitions from classical signal processing to generate a
transform that yields both a multiresolution of graphs and an associated
multiresolution of a graph signal on the underlying sequence of graphs.

本文提出了一种新的基于 Laplacian 金字塔变换的图信号多尺度处理方法，能够在加权图上分析高维数据的特征结构与信息多尺度表示，并详细介绍了图下采样、图降维和信号在图上的滤波与插值方法。

用于图信号的多尺度金字塔变换

A Multiscale Pyramid Transform for Graph Signals

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.

本文概述了在图上信号处理的方法，从图谱论、多尺度变换的角度阐述了在处理高维数据的图结构方面的方法和挑战，并强调了考虑到图形数据不规则结构的重要性。

图信号处理新兴领域：将高维数据分析扩展到网络和其他不规则域

The Emerging Field of Signal Processing on Graphs: Extending  High-Dimensional Data Analysis to Networks and Other Irregular Domains

We present in this paper new multiscale transforms on the sphere, namely the
isotropic undecimated wavelet transform, the pyramidal wavelet transform, the
ridgelet transform and the curvelet transform. All of these transforms can be
inverted i.e. we can exactly reconstruct the original data from its
coefficients in either representation. Several applications are described. We
show how these transforms can be used in denoising and especially in a Combined
Filtering Method, which uses both the wavelet and the curvelet transforms, thus
benefiting from the advantages of both transforms. An application to component
separation from multichannel data mapped to the sphere is also described in
which we take advantage of moving to a wavelet representation.

本文介绍了多尺度变换，如各种波变换，其可以用于信号去噪和复原。同时介绍了一种组件分离方法，该方法使用了波变换和曲线变换，从而充分发挥两者的优点。