CN111581852B - Optimal design method of joint time vertex node variable graph filter - Google Patents
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Abstract
The invention discloses an optimal design method of a joint time vertex node variable graph filter, which comprises the steps of firstly providing the joint time vertex node variable graph filter, then providing an effective filter optimal design method, and the method is used for attributing the design of the joint time vertex node variable graph filter into a least square optimization problem, solving the joint time vertex node variable graph filter with larger design freedom and better overall performance, and applying the filter obtained by design to an inverse filtering denoising problem. The filter designed according to the method has higher design freedom degree and better overall characteristic, and has better effect when carrying out inverse filtering denoising on the network measured data.
Description
Technical Field
The invention relates to the technical field of graph signal processing, in particular to an optimal design method of a joint time vertex node graph-changing filter.
Background
The graph signal processing is an important signal processing theoretical framework for processing data in irregular fields such as a sensor network, a temperature network and the like, and extends a plurality of important concepts of traditional signal processing to graph fields, including graph Fourier transform, graph filters, graph filter banks and the like, wherein the graph filters are powerful tools for processing graph signals, and have wide application in aspects of classification, denoising, interpolation and the like of the graph signals. A common form of graph filter is a polynomial filter with respect to the graph laplacian matrix, and the corresponding graph spectral kernel is also a polynomial form. However, the polynomial-diagram filter has a limited degree of freedom in design, and it is difficult to realize a filter having good characteristics both at the diagram vertices and in the frequency domain. Based on the method, the node mapping filter with higher design freedom and better local characteristics is used for processing the problems of network data coding, clustering and the like.
However, the graph filter described above is mainly used for analyzing static signals on a graph, and in an actual network, most of the signals are changed with time axis changes, such as temperature data recorded at successive time points, traffic flow data, and the like. The time-varying signals on the graph need to be processed in the vertex and time domains, such as the correlation of the graph topology and the evolution of the characteristics in the time axis. To analyze the time-varying signals on the graph, a two-dimensional joint-time vertex graph filter, i.e., a two-dimensional polynomial filter, is obtained by extending the product graph model from the graph vertex domain to the joint-time vertex domain. The current two-dimensional polynomial filter is a bivariate function with respect to the matrix polynomial, and the corresponding frequency response is also a bivariate polynomial with respect to the time-frequency domain and the graph-frequency domain. Although the research on the two-dimensional polynomial filter is developed rapidly and plays an important role in practical applications such as continuous time sequence prediction, dynamic image denoising, video restoration and the like, the two-dimensional polynomial filter is mainly designed under the condition of non-node change on the joint time vertex domain, the research is less under the condition of node change, the degree of freedom of node design is low, the design characteristics are not flexible enough, and the joint time vertex graph filter with good characteristics on the joint time vertex domain and the joint frequency domain is difficult to design, so that a more generalized joint time vertex node graph filter is to be proposed under the condition of node change on the joint time vertex domain.
Disclosure of Invention
The invention aims at overcoming the defects of the prior art and provides an optimal design method of a joint time vertex node variable graph filter. The filter designed according to the method has higher design freedom degree and better overall characteristic, and has better effect when carrying out inverse filtering denoising on the network measured data.
The technical scheme for realizing the aim of the invention is as follows:
the optimization design method of the joint time vertex node variable graph filter is different from the prior art in that the method comprises the following steps:
1) Defining a joint time vertex node variogram filter: first, a two-dimensional polynomial graph filter is defined as follows:
in the middle of
Is a directed circulation diagram->
Laplacian matrix in time domain, wherein +.>
Is a unitary matrix->
Is an adjacency matrix;
Is a common figure->
Normalized Laplace matrix on vertex domain, wherein +.>
Is a unitary matrix->
For the degree matrix->
Is a weighted adjacency matrix; k (K) t And K g For the filter length in the joint temporal vertex domain, a k,l For the filter coefficients +.>
For the Cronecker product, from equation (1), it can be derived that each node on the product graph of the two-dimensional polynomial filter is given the same weight coefficient a k,l I.e. filter coefficient a k,l The effect is the same for all nodesExpressed as a non-node-dependent characteristic of a two-dimensional polynomial filter, the degree of freedom of design is only (K t +1)×(K g +1), the flexible design characteristics of the filter are limited, in order to improve the design freedom of the joint time vertex graph filter and improve the overall performance of the filter, a more general joint time vertex graph filter is provided, and the joint time vertex node graph filter is defined as follows:
in the method, in the process of the invention,
is a vector of NT×1, N and T each represent a normal map +.>
And directed cyclic graph->
Is of the size of diag (a) (k,l) ) Representing a diagonal matrix, the diagonal elements being represented by the vector a (k,l) In the composition, in the formula (2), a joint time vertex node change graph filter can be obtained, that is, the degree of freedom of the two-dimensional node change graph filter is NT× (K t +1)(K g +1), far greater than the degree of freedom of the two-dimensional polynomial filter, in addition, it can be seen that the two-dimensional node-change filter weights different +.>
Endowed on each node of the product graph, which ensures flexibility and performance of the filter design;
2) Definition of two-dimensional node-change-map filter H 2D,NV (((i-1) n+j)) behavior:
where T represents a transpose of the vector or matrix, and the ranges of i and j are i=1, …, T, j=1, …, N, respectively;
{e i } T×1 sum { e } j } N×1 The standard basis vector is represented, that is, the elements other than the ith and jth elements in the vector are all 0, and for convenience in designing the filter, the following formula is defined:
based on the above formula (4), formula (5), formula (6), formula (3) can be expressed as:
wherein I is NT Is an identity matrix with the size of NT x NT;
3) Design of two-dimensional node change-map filter H 2D,NV : design of two-dimensional node-varying filter H 2D,NV Approximating an ideal filter operator
The following optimization problems are summarized:
wherein F represents the F norm of the matrix, a (k,l) Representing a filter to be solvedThe coefficients, according to equation (3) and equation (7) in step 2), equation (8) can be equivalently the least squares optimization problem as follows:
by solving the optimization problem, equation (9), the filter coefficients can be obtained as:
in the method, in the process of the invention,
thereby obtaining the approximate ideal filter operator +.>
Is a two-dimensional node-change filter;
4) Application of two-dimensional node variable filter in inverse filtering denoising: assuming that a noise-containing mapping signal with size n×t is y=x+n, where X and N are an original signal and a noise signal, respectively, when the original signal and the noise signal are vectorized, y=vec (Y) =x+n, the dimensions of X and N are nt×1, and the inverse filtering denoising problem is expressed as the following unconstrained optimization problem:
in the method, in the process of the invention,
for the polynomial of the joint Laplace matrix, μ is a weight factor, the relation between the first term and the second term in the optimization problem can be adjusted, and the inverse filtering problem, namely formula (11), is solved to obtain the optimal solution as follows:
wherein I is NT Is an identity matrix with the size of NT x NT,
is an ideal inverse filter operator;
5) Inverse filtering denoising: in combination with the optimization problem (9) in step 3) and the equation (10) for solving the filter coefficients, an ideal inverse filter operator approximating step 4) can be designed
Is a two-dimensional node-change filter H 2D,NV Finally, the method is used for solving the problem of denoising of the inverse filter.
The optimization problem formula (9) in the step 3) is a least square optimization problem, the formula (9) is solved by adopting a method for solving the least square problem, and a solution of the filter coefficient, namely the formula (10), is obtained, so that the two-dimensional node mapping filter is obtained.
In the step 4) of the method,
where q represents the joint Laplace matrix +.>
To the power q of (2).
In step 4), an ideal inverse filter operator
I NT Represents an identity matrix of size NT.times.NT, mu is the weighting factor,/is the weight factor>
Compared with the prior art, the technical scheme adopts an optimization method to design the joint time vertex node graph changing filter, the design problem of the filter is reduced to be a least square problem, the objective function is the approximation of the two-dimensional node graph changing filter to an ideal inverse filter operator, and the optimization problem is split into the approximation of each row of the two-dimensional node graph changing filter and each row of the ideal inverse filter operator to obtain the solution of the filter coefficient, thereby obtaining the two-dimensional node graph changing filter.
The two-dimensional node variable filter designed according to the method has larger design freedom, more flexible design characteristics and better overall performance, and has better effect and is closer to the denoising level of an ideal inverse filter operator when the ideal inverse filter operator is approximately denoised, namely, the synthesized time-varying data or the actually measured time-varying network signal is denoised, compared with the existing method.
Detailed Description
The following describes the invention in further detail with reference to examples, but is not intended to limit the invention.
Examples:
an optimization design method of a joint time vertex node variable graph filter comprises the following steps:
1) Defining a joint time vertex node variogram filter: first, a two-dimensional polynomial graph filter is defined as follows:
in the middle of
Is a directed circulation diagram->
Laplacian matrix in time domain, wherein +.>
Is a unitary matrix->
Is an adjacency matrix;
Is a common figure->
Normalized Laplace matrix on vertex domain, wherein +.>
Is a unitary matrix->
For the degree matrix->
Is a weighted adjacency matrix; k (K) t And K g For the filter length in the joint temporal vertex domain, a k,l For the filter coefficients +.>
For the Cronecker product, from equation (1), it can be derived that each node on the product graph of the two-dimensional polynomial filter is given the same weight coefficient a k,l I.e. filter coefficient a k,l The effect is the same for all nodes, the non-node-varying characteristic expressed as a two-dimensional polynomial filter, the degree of freedom of design is only (K t +1)×(K g +1), the flexible design characteristics of the filter are limited, in order to improve the design freedom of the joint time vertex graph filter and improve the overall performance of the filter, a more general joint time vertex graph filter is provided, and the joint time vertex node graph filter is defined as follows:
in the method, in the process of the invention,
is a vector of NT×1, N and T each represent a normal map +.>
And directed cyclic graph->
Is of the size of diag (a) (k,l) ) Representing a diagonal matrix, the diagonal elements being represented by the vector a (k,l) In the composition, in the formula (2), a joint time vertex node change graph filter can be obtained, that is, the degree of freedom of the two-dimensional node change graph filter is NT× (K t +1)(K g +1), far greater than the degree of freedom of the two-dimensional polynomial filter, in addition, it can be seen that the two-dimensional node-change filter weights different +.>
The method has the advantages that the flexibility and the performance of the filter design are guaranteed by the method on each node of the product graph, compared with a two-dimensional polynomial filter, the two-dimensional node variable filter has higher degree of freedom and the like, but the difficulty is also increased for the design of the filter;
2) Definition of two-dimensional node-change-map filter H 2D,NV (((i-1) n+j)) behavior:
where T represents a transpose of the vector or matrix, and the ranges of i and j are i=1, …, T, j=1, …, N, respectively;
{e i } T×1 sum { e } j } N×1 The standard basis vector is represented, that is, the elements other than the ith and jth elements in the vector are all 0, and for convenience in designing the filter, the following formula is defined:
based on the above formula (4), formula (5), formula (6), formula (3) can be expressed as:
wherein I is NT Is an identity matrix with the size of NT x NT;
3) Design of two-dimensional node change-map filter H 2D,NV : design of two-dimensional node-varying filter H 2D,NV Approximating an ideal filter operator
The following optimization problems are summarized:
wherein F represents the F norm of the matrix, a (k,l) Representing the filter coefficients to be solved, according to equations (3) and (7) in step 2), equation (8) may be equivalently the following least squares optimization problem:
by solving the optimization problem, equation (9), the filter coefficients can be obtained as:
in the method, in the process of the invention,
thereby obtaining the approximate ideal filter operator +.>
Is a two-dimensional node-change filter;
4) Application of two-dimensional node variable filter in inverse filtering denoising: assuming that a noise-containing mapping signal with size n×t is y=x+n, where X and N are an original signal and a noise signal, respectively, when the original signal and the noise signal are vectorized, y=vec (Y) =x+n, the dimensions of X and N are nt×1, and the inverse filtering denoising problem is expressed as the following unconstrained optimization problem:
in the method, in the process of the invention,
for the polynomial of the joint Laplace matrix, μ is a weight factor, the relation between the first term and the second term in the optimization problem can be adjusted, and the inverse filtering problem, namely formula (11), is solved to obtain the optimal solution as follows:
wherein I is NT Is an identity matrix with the size of NT x NT,
is an ideal inverse filter operator;
5) Inverse filtering denoising: in combination with the optimization problem (9) in step 3) and the equation (10) for solving the filter coefficients, an ideal inverse filter operator approximating step 4) can be designed
Is a two-dimensional node-change filter H 2D,NV Finally, the method is used for solving the problem of denoising of the inverse filter.
The optimization problem formula (9) in the step 3) is a least square optimization problem, the formula (9) is solved by adopting a method for solving the least square problem, and a solution of the filter coefficient, namely the formula (10), is obtained, so that the two-dimensional node mapping filter is obtained.
In the step 4) of the method,
where q represents the joint Laplace matrix +.>
To the power q of (2).
In step 4), an ideal inverse filter operator
I NT Represents an identity matrix of size NT.times.NT, mu is the weighting factor,/is the weight factor>
Specifically:
simulation case 1:
using two-dimensional node-change filters H 2D,NV Approximate ideal inverse filter operator
And (3) carrying out inverse filtering denoising experiments on the synthesized time-varying graph signals: in the simulation, a random sensor network diagram with a node n=100 is firstly constructed by using a nearest distance algorithm, and then a time-varying diagram signal x= [ X ] of 100×30 is generated 1 ,x 2 ,…,x 30 ]Its picture signal at time t is set to +.>
Wherein the initial signal x 1 Is a low frequency signal f t Vectorizing X to a white Gaussian noise signalWith x=vec (X), the signal size range is [ -19.39,38.31]The noise signal n is [ -sigma, sigma]Obeying uniform distribution, sigma is a standard noise factor, the weight factor is set to mu=0.6, and the filter length is set to K t ＝K g =2 and let->
Table 1 shows the filter design of the method of the present example and the prior method 1 (two-dimensional polynomial graph filter) in approximately ideal inverse filter operator +.>
Performance comparison of inverse filtering denoising is carried out on the time-varying graph signals, wherein Input (containing noise) represents Input signal-to-noise ratio, and Explicit (denoising) represents the noise between ideal inverse filtering operator +.>
The denoising signal-to-noise ratio obtained below.
TABLE 1
| σ | 1 | 2 | 5 | 10 | 15 |
| Input (with noise) | 24.85 | 18.83 | 10.87 | 4.85 | 1.33 |
| Explicit (denoising) | 26.21 | 20.48 | 12.61 | 6.60 | 3.08 |
| Existing method 1 (denoising) | 21.73 | 18.80 | 12.41 | 6.67 | 3.20 |
| This example method (denoising) | 24.02 | 19.91 | 12.66 | 6.74 | 3.24 |
As can be seen from Table 1, the two-dimensional node variable filter designed by the method of the present example approximates to an ideal inverse filter operator
When the noise factor sigma is smaller, the denoising performance of the time-varying graph signal is better than that of a two-dimensional polynomial filter, and is closer to an ideal inverse filtering operator +.>
This also verifies the two-dimensional node transformation graph filtering designed by the methodThe wave filter has a larger design degree and is superior to a two-dimensional polynomial filter in the overall denoising performance.
Simulation case 2:
in the simulation, actual measurement data, namely global sea level pressure data, is used for verifying the validity of the designed two-dimensional node variable filter in processing real data. The global sea level pressure data includes sea level pressure data from 1984 to 2010, and the recorded pressure data for each time period is average pressure data every five days. In this simulation, a graph model is constructed on sea level pressure data of node n=500 by a nearest distance algorithm, then average sea level data of last 100 days in 2010 is selected as an original time-varying graph signal x, that is, average pressure data of every five days in 100 days is taken as pressure data at one moment, that is, t=20, x ranges from 95.2762kPa to 106.9132kPa, a weighting factor μ=0.4 is set, and other parameters such as filter length and L are set p Etc. are set as in simulation case 1.
TABLE 2
| σ | 5 | 10 | 1 | 20 | 25 |
| Input (with noise) | 30.84 | 24.82 | 21.30 | 18.80 | 16.86 |
| Explicit (denoising) | 32.21 | 26.20 | 22.67 | 20.18 | 18.24 |
| Existing method 1 (denoising) | 27.17 | 24.33 | 21.78 | 19.68 | 17.94 |
| This example method (denoising) | 29.28 | 25.33 | 22.32 | 20.01 | 18.18 |
Table 2 reflects the comparison of the present example method with the prior art method 1 (two-dimensional polynomial graph filter)
And when the sea level pressure data is similar, performing inverse filtering denoising performance comparison on the actually measured sea level pressure data. As can be seen from Table 2, the two-dimensional node variable filter is closer to the ideal inverse filter operator when denoising the measured data>
Is to be subjected to denoisingThe performance of the two-dimensional node variable filter is better than that of a two-dimensional polynomial filter, and the effect of denoising the measured network data by the two-dimensional node variable filter designed by the example is also shown. The two simulation examples show that the filter designed by the method has better denoising performance than a two-dimensional polynomial filter in the aspect of processing the synthesized time-varying diagram signals and the actual network data, and can be closer to the denoising effect of an ideal filtering operator, and the two-dimensional node variable filter designed by the method is verified from the side to have larger design freedom, more flexible design characteristics and better overall performance. />
Claims (2)
1. The optimal design method of the joint time vertex node variable graph filter is characterized by comprising the following steps of:
1) Defining a joint time vertex node variogram filter: first, a two-dimensional polynomial graph filter is defined as follows:
in the middle of
Is a directed circulation diagram->
Laplacian matrix in time domain, wherein +.>
Is a matrix of units which is a matrix of units,
is an adjacency matrix;
Is a common figure->
Normalized Laplace matrix on vertex domain, wherein +.>
Is a unitary matrix->
For the degree matrix->
Is a weighted adjacency matrix; k (K) t And K g For the filter length in the joint temporal vertex domain, a k,l For the filter coefficients +.>
For the Cronecker product, from equation (1), it can be derived that each node on the product graph of the two-dimensional polynomial filter is given the same weight coefficient a k,l I.e. filter coefficient a k,l The effect is the same for all nodes, the non-node-varying characteristic expressed as a two-dimensional polynomial filter, the degree of freedom of design is only (K t +1)×(K g +1) then defining a joint temporal vertex node variogram filter as:
in the method, in the process of the invention,
is a vector of NT×1, N and T each represent a normal map +.>
And directed cyclic graph->
Is of the size of diag (a) (k,l) ) Representing a diagonal matrix, the diagonal elements being represented by the vector a (k,l) Composition;
2) Definition of two-dimensional node-change-map filter H 2D,NV (((i-1) n+j)) behavior:
where T represents a transpose of the vector or matrix, and the ranges of i and j are i=1, …, T, j=1, …, N, respectively;
{e i } T×1 sum { e } j } N×1 The standard basis vector, i.e. the vector has 0 for all elements except the i and j elements except 1, defines the following formula:
based on the above formula (4), formula (5), formula (6), formula (3) can be expressed as:
wherein I is NT Is an identity matrix with the size of NT x NT;
3) Design of two-dimensional node change-map filter H 2D,NV : design of two-dimensional node-varying filter H 2D,NV Approximating an ideal filter operator
The following optimization problems are summarized:
wherein F represents the F norm of the matrix, a (k,l) Representing the filter coefficients to be solved, according to equations (3) and (7) in step 2), equation (8) may be equivalently the following least squares optimization problem:
by solving the optimization problem, equation (9), the filter coefficients can be obtained as:
in the method, in the process of the invention,
thereby obtaining the approximate ideal filter operator
Is a two-dimensional node-change filter;
4) Application of two-dimensional node variable filter in inverse filtering denoising: assuming that a noise-containing mapping signal with size n×t is y=x+n, where X and N are an original signal and a noise signal, respectively, when the original signal and the noise signal are vectorized, y=vec (Y) =x+n, the dimensions of X and N are nt×1, and the inverse filtering denoising problem is expressed as the following unconstrained optimization problem:
in the method, in the process of the invention,
for the polynomial of the joint Laplace matrix, μ is a weight factor, the relation between the first term and the second term in the optimization problem can be adjusted, and the inverse filtering problem, namely formula (11), is solved to obtain the optimal solution as follows:
wherein I is NT Is an identity matrix with the size of NT x NT,
is an ideal inverse filter operator;
5) Inverse filtering denoising: combining the optimization problem (9) in step 3) and the equation (10) for solving the filter coefficients, the ideal inverse filter operator approximating step 4) can be obtained
Is a two-dimensional node-change filter H 2D,NV Finally, the method is used for solving the inverse filtering denoising.
2. The method for optimizing design of joint-time vertex node variogram filter as in claim 1, wherein in step 4), ideal inverse filter operator
I NT Represents an identity matrix of size NT.times.NT, mu is the weighting factor,/is the weight factor>
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