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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.

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 [

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 [

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 [

Another important class of vertex-frequency representations, called the vertex-frequency energy distributions, were recently introduced in [

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

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 [

A weighted undirected graph with

The definition of the graph Laplacian, using the weight matrix

The Laplacian matrix, like any other quadratic matrix, can be written using its eigenvectors and eigenvalues as

Graph signal samples,

The graph discrete Fourier transform (GDFT) of a signal

The inverse graph discrete Fourier transform (IGDFT) follows from the property

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

In classic signal analysis, when the signal domain is time, the signal

From the Laplacian eigendecomposition, we have

Since the eigenvalues of the Laplacian matrix are equal to the quadratic form

The graph signal

An example of the time domain signals and graph signals, with various values of the global smoothness

Signals in the time domain (left) and in the graph domain (right). The global signal smoothness

Constant signal

Slow varying signal

Fast varying signal

Now consider the signal whose form is given by a weighted sum of the eigenvectors,

The smoothness of graph signals is used in graph topology learning [

The local graph signal smoothness is introduced next. Its properties are analyzed in the second part of this section.

Assume the simplest case, when the analyzed signal is proportional to the

The smoothness index can be related to the frequency in the time domain signal analysis [

For the time domain signals with a time-varying spectrum, the concept of instantaneous frequency is introduced. Several approaches to the instantaneous frequency exist [

The discrete-time definition of the squared instantaneous frequency is

In the previous section, we show that the second-order difference of a time domain signal corresponds to the elements of

An example of the signal with varying local smoothness in the time domain.

In analogy with (

We have assumed that

Some of the properties of the local smoothness are described next.

Consider a monocomponent signal

Its local smoothness

In the time domain signal analysis, this property means that the instantaneous frequency of a sinusoidal signal is equal to its frequency.

Assume a piecewise monocomponent signal

where

For each interior vertex

An example of a piecewise monocomponent graph signal is presented in Figure

For subset

The local smoothness of the piecewise monocomponent graph signal from Figure

An ideal vertex-frequency distribution can be defined as

It has been assumed that the local smoothness is rounded to the nearest eigenvalue.

For the graph and the signal presented in Figure

This distribution can be used as a local smoothness estimator since, for each vertex

Piecewise monocomponent signal on the graph. The signal is composed of three parts. The support sets

The local smoothness values for the graph signal shown in Figure

The ideal vertex-frequency distribution of a graph signal.

and the estimated local smoothness is

For a multicomponent graph signal with

the local smoothness, calculated by definition, is

From classic time domain analysis, we know that a multicomponent signal cannot be analyzed directly; that is, the instantaneous frequency is not defined as a single value in this case. We can only estimate the instantaneous frequencies of the individual components. The same holds for graph signals, where we should decompose multicomponent signals to the individual components and then calculate the local smoothness for each component.

A vertex-frequency distribution

The ideal vertex-frequency distribution

For a vertex-frequency distribution

The energy vertex-frequency distributions follow the concept of the time-frequency energy distributions in classic signal analysis. The estimation of the local smoothness can be obtained by using the vertex-frequency representations that localize the graph signal energy on the local smoothness. Here we will present the vertex-frequency energy distribution, a reduced interference vertex-frequency distribution, and the graph signal spectrogram, as the tools for local smoothness estimation.

The energy of a signal

A vertex-frequency distribution

We will show that the vertex-frequency distribution defined by (

For the vertex-frequency distribution defined by (

Vertex-frequency energy distribution with its marginal values.

In order to reduce the cross-terms interferences and to preserve the marginal properties, a general class of reduced interference time-frequency distributions is extended to the graph signals [

The reduced interference vertex-frequency distribution is presented in Figure

Reduced interference vertex-frequency distribution obtained using the exponential kernel.

Now we will consider a general case and review the conditions that the distribution kernel should satisfy in order to preserve the marginal properties.

A sum of all

The vertex marginal property of the distribution

Moreover, if this condition is satisfied, then the vertex moment property holds

The frequency marginal property holds if

If the frequency marginal property holds, then the frequency moment property holds as well,

The local smoothness property (

The reduced interference distributions can be used as estimators of the local smoothness. The local smoothness is estimated as the eigenvalue that corresponds to the position of the maximum in

The reduced interference distribution

In the classic time-frequency analysis, the short-time Fourier transform and the spectrogram are well-developed tools for analysis of nonstationary signals. Their extension to the graph signals leads to the vertex-frequency spectrogram. It can be calculated as the spectrum of a signal

In a special case, when

The spectrogram of a graph signal is defined as

The vertex marginal of the spectrogram

A summation over

The localization windows

The localization window, defined in the spectral domain, is equal to [

where

where

The localization window

where

Here we will review a method for obtaining the localization window functions, at each vertex, in matrix form [

The vertices whose distance is

We have used the following notation:

When

The matrix for the graph localization windows is formed as

The graph signal weighted by the localization window is calculated by using the previous matrix as

An example of the vertex-frequency spectrogram calculated by the vertex domain localization window is presented in Figure

Local vertex-frequency spectrum calculated by using the localization windows defined in the spectral domain.

Local vertex-frequency spectrum calculated using the vertex neighborhood windows.

The relation that would connect the vertex domain spectrogram (

The kernel that corresponds to the vertex domain spectrogram (

Here, the local smoothness property cannot be satisfied. However, since the graph spectrogram is concentrated along the local smoothness, we can still use the maximum-based estimator of the local smoothness

Now we will calculate the local signal smoothness by using the Laplacian applied to the graph signal (

The obtained results are presented in Figure

The number of outliers and the mean squared error of the local smoothness for the considered estimators.

Laplacian | Rih. dist. | RID | LVS spec. | LVS vert. | ||||||
---|---|---|---|---|---|---|---|---|---|---|

SNR | NO | MSE | NO | MSE | NO | MSE | NO | MSE | NO | MSE |

| 4 | 0.027 | 0 | 0.000 | 0 | 0.000 | 4 | 0.161 | 4 | 0.161 |

50 | 4 | 0.027 | 0 | 0.000 | 0 | 0.000 | 4 | 0.161 | 4 | 0.161 |

40 | 8 | 0.027 | 0 | 0.000 | 0 | 0.000 | 4 | 0.161 | 5 | 0.165 |

30 | 20 | 0.032 | 0 | 0.000 | 0 | 0.000 | 4 | 0.161 | 5 | 0.165 |

20 | 32 | 0.057 | 0 | 0.000 | 0 | 0.000 | 4 | 0.161 | 5 | 0.165 |

10 | 36 | 0.261 | 0 | 0.000 | 1 | 0.027 | 4 | 0.172 | 5 | 0.036 |

5 | 38 | 0.848 | 3 | 0.177 | 1 | 0.027 | 4 | 0.172 | 5 | 0.042 |

0 | 38 | 1.557 | 5 | 0.290 | 3 | 0.095 | 6 | 0.278 | 6 | 0.270 |

-2.5 | 38 | 2.294 | 11 | 0.648 | 8 | 0.693 | 6 | 0.278 | 7 | 0.282 |

-5 | 38 | 2.194 | 14 | 1.127 | 16 | 1.814 | 9 | 0.581 | 9 | 0.555 |

-7.5 | 38 | 2.701 | 21 | 2.115 | 19 | 2.293 | 11 | 0.667 | 12 | 0.826 |

-10 | 38 | 3.052 | 25 | 2.486 | 23 | 2.474 | 19 | 1.089 | 15 | 1.177 |

-15 | 38 | 3.469 | 33 | 3.005 | 33 | 3.477 | 27 | 1.683 | 27 | 2.503 |

Local signal smoothness for the graph signal presented in Figure

Next, we will consider a noisy signal. The signal is corrupted by a Gaussian noise added to the signal samples. Signal-to-noise ratio (SNR) is varied from

The ratio of the concentration measures is calculated using the

In this paper, the local smoothness of graph signals is introduced and analyzed. Methods for local smoothness estimation, based on the signal Laplacian and the vertex-frequency representations, are given and applied to examples with graph signals. It has been shown that the local smoothness is a counterpart of the instantaneous frequency in classic signal analysis and can be estimated using vertex-frequency distributions. Finally, the vertex-frequency energy distributions, including a reduced interference distribution, and the local vertex spectrogram with two windowing techniques, are considered as the local smoothness estimators.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.