Graph Signal Processing in a Nutshell

Abstract

Abstract-The framework of graph signal processing was conceived in the last decade with the ambition of generalizing the tools from classical digital signal processing to the case in which the signal is defined over an irregular structure modelled by a graph. Instead of discrete time -what one would call a regular 1-D domain, in which a signal sample is adjacent to only two neighbors and for any pair of contiguous samples the distance is the same -the signals here are defined over graphs and, therefore, the distance and relations between adjacent samples vary along the signal. For instance, one may consider the temperature signal defined from the data of a sensor mesh network. When creating the tools in such a scenario, many challenges arise even with basic concepts of the classical theory. In this paper, the core ideas of graph signal processing are presented, focusing on the two main frameworks developed along the years, and a couple of examples and applications are shown. We conclude drawing attention to a few of the many open opportunities for further studies in the field.

Graph Learning Techniques Using Structured Data for IoT Air Pollution Monitoring Platforms

“…Ribeiro et al [13] analyze, using GSP, data coming from taxis in Manhattan and pluviometry in Brazilian cities. Jablonski [25] analyzes a network of one hundred reference stations monitoring tropospheric ozone (O 3 ) in Poland.…”

“…There is a rich literature on obtaining an optimal Laplacian from structured data. Looking for the application to the content of this paper, [8], [10], [11], [12], [13] give an overview of how Laplacian is used in the calculation of various properties in GSP and [9] gives an overview of how various techniques and algorithms are used to obtain a weight matrix or an optimal Laplacian from the data. Among the most widely used methods, in addition to using a distance-based kernel, is the use of statistical methods and the use of methods based on signal smoothness.…”

“…Therefore, one way to construct the weight matrix, W, specifying the connectivity of the graph G is to use a similarity function and a threshold (T H) to set the weight proportionally to the distance and set some entries to zero. A common choice of the weight of an edge e ij is via a thresholded Gaussian kernel weighting function [8], [13]:…”

“…The construction of graphs for the representation, analysis, processing and visualization of signals on structured data such as those found in a heterogeneous sensor network has had a boost in recent years [8], [9], [10], [11], [12]. The classic method proposed in the literature [8], [13] for this kind of sensor networks is to create the edges that join the nodes of the graph following a function that depends on the distance between the nodes. This method has the disadvantage of not taking into account the similarity of the measurements taken © 2021 IEEE.…”

“…DOI 10.1109/JIOT.2021.3067717 by the sensors. Thus, one of the challenges and key points for the representation of a sensor network using graphs is to estimate the structure of the graph that underlies the measured data [8], [9], [13]. As Dong et al [9] mention, there are several ways to build the graph from the data depending on whether you use i) statistical methods based on graphical probabilistic models such as Markov random fields (MRFs) or Bayesian networks, ii) methods based on graph signal processing (GSP), or iii) methods based on physically motivated models.…”