Mathematical Problems in Engineering

Volume 2019, Article ID 3208569, 14 pages

https://doi.org/10.1155/2019/3208569

## Local Smoothness of Graph Signals

^{1}University of Montenegro, Podgorica, Montenegro^{2}University of Pittsburgh, Pittsburgh, PA, USA

Correspondence should be addressed to Miloš Daković; em.ca@solim

Received 31 August 2018; Accepted 19 March 2019; Published 8 April 2019

Academic Editor: Luigi Rodino

Copyright © 2019 Miloš Daković et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Analysis of vertex-varying spectral content of signals on graphs challenges the assumption of vertex invariance and requires the introduction of vertex-frequency representations as a new tool for graph signal analysis. Local smoothness, an important parameter of vertex-varying graph signals, is introduced and defined in this paper. Basic properties of this parameter are given. By using the local smoothness, an ideal vertex-frequency distribution is introduced. The local smoothness estimation is performed based on several forms of the vertex-frequency distributions, including the graph spectrogram, the graph Rihaczek distribution, and a vertex-frequency distribution with reduced interferences. The presented theory is illustrated through numerical examples.

#### 1. Introduction

Graph signal processing is a new and quickly developing field. Many practical signals can be considered as graph signals. The theory and methods for processing the graph signals are introduced and presented in [1–5]. Graph signal processing applications in biomedical systems [6, 7] and analysis of big data [8] provide insight into the graph framework advantages and real-world potential.

In the case of large graphs, we may not be interested in the analysis of the entire graph signal, but rather interested in its local behavior. Signals with varying local vertex behaviors are a class of signals called nonstationary graph signals. One approach to the analysis of nonstationary graph signals is vertex-frequency analysis [7, 9–15], which is a counterpart of time-frequency analysis [16–18] in classic signal processing.

The main representatives of the vertex-frequency representations are local vertex spectrum and its energetic version, graph spectrogram. Window functions are used to localize graph signals in a neighborhood of the considered vertex [9, 12, 15].

Another important class of vertex-frequency representations, called the vertex-frequency energy distributions, were recently introduced in [13, 14]. This class is a counterpart to the class of quadratic time-frequency distributions in classic signal analysis. It has been shown that the graph version of the Rihaczek distribution is of special interest for graph signals since it does not require a localization window. The reduced interference distributions can be derived from the Rihaczek distribution by using appropriate kernel functions. This class of representations, under certain conditions, satisfies marginal properties in both the vertex domain and the spectral domain.

An important concept that is used in classic time domain signal analysis for the description of local signal behavior around a time instant is the instantaneous frequency. The local smoothness is introduced in this paper as an extension of the instantaneous frequency concept to graph signal analysis. The local smoothness is defined by using the graph signal Laplacian matrix. The vertex-frequency representations can be highly concentrated along the local spectral index, corresponding to the local signal smoothness. This property is used to define local smoothness estimators based on the vertex-frequency representations.

After an introduction, we will review the fundamental theory of graph signal processing. This review will include the graph Fourier transform and the global signal smoothness in Section 2. Then, the local signal smoothness will be introduced and its properties derived within Section 3. The vertex-frequency representations, along with their connections to the local signal smoothness, will be presented in Section 4. The theory will be illustrated through a nonstationary graph signal example.

#### 2. Graph Signals

A graph is defined as a set of vertices and a set of edges connecting these vertices. In signal processing, such a structure can be considered as the domain of a signal. The signal values are defined at the graph vertices. The graph Fourier transform (graph spectrum) is defined through the eigenvalue decomposition of the graph Laplacian matrix. Here, we will present a review of the graph spectrum and the global signal smoothness index calculated using the Laplacian matrix [10].

##### 2.1. Graph Signal and Spectrum

A weighted undirected graph with vertices will be considered. The edge weights are nonzero if there is an edge between the vertices and . If there is no edge between the vertices and , the corresponding weight is equal to zero, . The weight matrix is a matrix whose elements are . It is a symmetric matrix (since the underlying graph is undirected), with zeros on the main diagonal.

The definition of the graph Laplacian, using the weight matrix and its elements , is given bywhere is a diagonal matrix, called the degree matrix. Its diagonal elements are obtained from as , while for .

The Laplacian matrix, like any other quadratic matrix, can be written using its eigenvectors and eigenvalues asIn this decomposition, the matrix consists of the matrix eigenvectors, denoted by , as its columns. The diagonal matrix of eigenvalues , , is denoted by . The eigenvectors and eigenvalues of are calculated from . Here we will consider the case with simple eigenvalues, whose multiplicity is one.

Graph signal samples, , , are sensed/defined at each graph vertex . These signal samples can be written in vector form as an vector:

The graph discrete Fourier transform (GDFT) of a signal is defined by [10]The coefficients in the GDFT are calculated as the projections of the considered graph signal to the eigenvectors

The inverse graph discrete Fourier transform (IGDFT) follows from the property that holds for the Laplacian matrix eigenvectors, where is an identity matrix. The IGDFT relation is , with

The GDFT concept can be extended to the directed graphs. The cases of repeated eigenvalues can also easily be included in the analysis [19–21].

##### 2.2. Global Graph Signal Smoothness

In classic signal analysis, when the signal domain is time, the signal smoothness can be defined through a second-order difference . Since classic time domain signal processing can be considered as graph signal processing on a circular graph, the second-order difference can be written as , where is the Laplacian of the circular graph. The signal smoothness can be measured as cumulative energy of the signal changes . It can also be calculated as . In matrix notation, we get the quadratic form . This approach can be extended to general (non-circular) graphs.

From the Laplacian eigendecomposition, we haveorsince for an eigenvector holds . For an arbitrary eigenvector and the corresponding eigenvalue , here we omitted index for notation simplicity. The quadratic form of an eigenvector is equal to the corresponding eigenvalue. This quadratic form can be used as a measure of the signal smoothness. We can write the quadratic form as Since , the last relation can also be rewritten asThe sum of the previous two relations producesObviously, a small value of corresponds to slow eigenvector variations , within the neighboring/connected vertices. This means that the eigenvectors calculated with a small represent a low-pass (slow-varying) part of the graph signal.

Since the eigenvalues of the Laplacian matrix are equal to the quadratic form , they are nonnegative. It is known that at least one eigenvalue of the Laplacian is zero. The corresponding eigenvector is constant, i.e., maximally smooth signal.

The graph signal smoothness is defined, in general, using a full analogy with (8). By normalizing the quadratic form,With the signal energy, the smoothness index definition is obtained as

An example of the time domain signals and graph signals, with various values of the global smoothness , is presented in Figure 1. It is obvious that small values correspond to the smooth (slow-varying) signals and that large values of indicate fast-varying signals.