CN111737639A - A Distributed Repair Method for Time-varying Graph Signals Mixed with 1-norm and 2-norm - Google Patents

A Distributed Repair Method for Time-varying Graph Signals Mixed with 1-norm and 2-norm Download PDF

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CN111737639A
CN111737639A CN202010586812.4A CN202010586812A CN111737639A CN 111737639 A CN111737639 A CN 111737639A CN 202010586812 A CN202010586812 A CN 202010586812A CN 111737639 A CN111737639 A CN 111737639A
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周芳
蒋俊正
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Guilin University of Electronic Technology
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Abstract

The invention discloses a distributed repair method of a 1-norm and 2-norm mixed time-varying graph signal, which aims at the problem of poor repair effect of the existing time-varying graph signal repair method on an edge, and the repair problem of the time-varying graph signal is normalized into an unconstrained optimization problem of 1-norm and 2-norm mixing, wherein the time smoothness of the time-varying graph signal is characterized by adopting a 2-norm, the space smoothness is characterized by adopting a 1-norm with sparse solution, and the optimization problem is solved in a distributed manner by utilizing an alternating direction multiplier method; simulation experiments show that the method has better repairing performance, and particularly has better repairing effect on the edge of the graph signal.

Description

一种1-范数和2-范数混合的时变图信号分布式修复方法A Distributed Repair Method for Time-varying Graph Signals Mixed with 1-norm and 2-norm

技术领域technical field

本发明涉及时变图信号处理技术邻域，具体涉及一种1-范数和2-范数混合的时变图信号分布式修复方法。The invention relates to the technical neighborhood of time-varying graph signal processing, in particular to a time-varying graph signal distributed repair method with a 1-norm and a 2-norm mixed.

背景技术Background technique

时变图信号广泛存在于实际应用中，比如温度传感器每小时采集的各城市温度数据，压力传感器每年采集的各站点海面压力数据等。然而，由于传感器受自身性能及环境因素的影响，采集的数据中会出现部分破坏的现象。因而，需要根据数据间的时空关联性对时变图信号进行修复，尽可能地恢复出原数据。Time-varying graph signals widely exist in practical applications, such as the temperature data of each city collected by temperature sensors every hour, and the sea surface pressure data of each site collected by pressure sensors every year. However, because the sensor is affected by its own performance and environmental factors, the collected data will be partially destroyed. Therefore, it is necessary to repair the time-varying graph signal according to the spatiotemporal correlation between the data, so as to restore the original data as much as possible.

通常情况下，时变图信号涉及的数据量大，需要采取计算复杂度相对低的分布式修复。现有的分布式修复方法中，时变图信号的空间和时间平滑性均采用 2-范数度量，不能很好地表示图信号的空间稀疏性，表现在对图信号的边缘修复效果不好。Usually, the time-varying graph signal involves a large amount of data, which requires distributed repair with relatively low computational complexity. In the existing distributed inpainting methods, both the spatial and temporal smoothness of time-varying graph signals are measured by 2-norm, which cannot well represent the spatial sparsity of graph signals, and the edge inpainting effect of graph signals is not good. .

发明内容SUMMARY OF THE INVENTION

本发明的目的在于针对现有时变图信号修复方法对边缘的修复效果差的问题，而提供一种1-范数和2-范数混合的时变图信号分布式修复方法。The purpose of the present invention is to provide a time-varying graph signal distributed restoration method with a mixture of 1-norm and 2-norm, aiming at the problem that the existing time-varying graph signal restoration method has poor restoration effect on edges.

实现本发明目的的技术方案是：The technical scheme that realizes the object of the present invention is:

一种1-范数和2-范数混合的时变图信号分布式修复方法，包括如下步骤：A 1-norm and 2-norm mixed time-varying graph signal distributed repair method, comprising the following steps:

1)将传感器采样到的时变图信号的修复过程转换为1-范数和2-范数混合的无约束优化问题，利用6-近邻算法构建时变图信号对应的图模型G＝{V,E}，其中V为顶点集，共包括N个顶点，对应于每个传感器节点，E为边集；设D为度矩阵，W为邻接矩阵，I为单位阵，拉普拉斯矩阵定义为

Figure RE-GDA0002578754970000011
1) Convert the repair process of the time-varying graph signal sampled by the sensor into an unconstrained optimization problem of a mixture of 1-norm and 2-norm, and use the 6-nearest neighbor algorithm to construct a graph model corresponding to the time-varying graph signal G={V ,E}, where V is the vertex set, including N vertices in total, corresponding to each sensor node, E is the edge set; let D be the degree matrix, W is the adjacency matrix, I is the identity matrix, and the Laplace matrix is defined for
Figure RE-GDA0002578754970000011

2)将时变图信号的动态模型设为

Figure RE-GDA0002578754970000012
其中xt为t时刻的图信号，xt+1为t+1时刻的图信号，wt为t时刻的加性随机噪声，τ为待估计参数，根据前10个图信号xt,t＝0,…,9，估计出参数τ＝0.022；2) Set the dynamic model of the time-varying graph signal as
Figure RE-GDA0002578754970000012
where x t is the graph signal at time t, x t+1 is the graph signal at time t+1, w t is the additive random noise at time t, τ is the parameter to be estimated, according to the first 10 graph signals x t , t =0,...,9, the parameter τ = 0.022 is estimated;

3)按时间先后顺序对t时刻的图信号xt＝[x(1),x(2),…,x(N)]T进行修复，设 t时刻的预测图信号为

Figure RE-GDA0002578754970000013
其中
Figure RE-GDA0002578754970000014
为修复后的第t-1时刻的图信号；3) Repair the graph signal x t =[x(1),x(2),...,x(N)] T at time t in chronological order, and let the predicted graph signal at time t be
Figure RE-GDA0002578754970000013
in
Figure RE-GDA0002578754970000014
is the graph signal at time t-1 after repair;

4)按照一定比例，随机破坏即置零图信号，得到经过置零后的图信号为

Figure RE-GDA0002578754970000021
其中xM为未被破坏的图信号，xμ为被破坏的图信号，xM中包含元素的个数记为|M|；4) According to a certain proportion, random destruction is to zero the graph signal, and the graph signal after zeroing is obtained as
Figure RE-GDA0002578754970000021
where x M is the uncorrupted graph signal, x μ is the corrupted graph signal, and the number of elements contained in x M is recorded as |M|;

5)对破坏后的图信号xμ施加噪声，得到图信号的观测值b＝x+n，其中n 为加性噪声；5) Noise is applied to the damaged graph signal x μ to obtain the observed value of the graph signal b=x+n, where n is additive noise;

6)初始化：对于每个顶点i∈V，原变量

Figure RE-GDA0002578754970000022
辅助变量z(0)(i)＝0，对偶变量w(0)(i)＝0，迭代次数m＝0，迭代终止阈值ε；6) Initialization: For each vertex i∈V, the original variable
Figure RE-GDA0002578754970000022
Auxiliary variable z (0) (i)=0, dual variable w (0) (i)=0, iteration times m=0, iteration termination threshold ε;

7)通过求解最小二次问题，得到原变量

Figure RE-GDA0002578754970000023
原变量
Figure RE-GDA0002578754970000024
表达式为：7) By solving the least quadratic problem, the original variables are obtained
Figure RE-GDA0002578754970000023
original variable
Figure RE-GDA0002578754970000024
The expression is:

Figure RE-GDA0002578754970000025
Figure RE-GDA0002578754970000025

其中，符号(·)-1代表矩阵求逆运算，(·)T代表矩阵转置运算，对于每个顶点 i∈V，第m次迭代下图信号的修复值

Figure RE-GDA0002578754970000026
等于原变量
Figure RE-GDA0002578754970000027
中的第i个元素值，即
Figure RE-GDA0002578754970000028
Among them, the symbol ( ) -1 represents the matrix inversion operation, ( ) T represents the matrix transposition operation, for each vertex i∈V, the mth iteration is the repair value of the signal in the figure below
Figure RE-GDA0002578754970000026
equal to the original variable
Figure RE-GDA0002578754970000027
The i-th element value in , i.e.
Figure RE-GDA0002578754970000028

8)对于每个顶点i∈V，计算第m次迭代下的辅助变量

Figure RE-GDA0002578754970000029
其中H1(i,j)为高通滤波器对应矩阵的第i行第j列元素值，σ为邻域半径，B(i,σ)为顶点i及其σ阶邻域节点的集合，函数Sβ/γ(·)定义为：8) For each vertex i∈V, calculate the auxiliary variable at the mth iteration
Figure RE-GDA0002578754970000029
where H 1 (i, j) is the element value of the i-th row and j-th column of the matrix corresponding to the high-pass filter, σ is the neighborhood radius, B(i, σ) is the set of vertex i and its σ-order neighborhood nodes, and the function S β/γ (·) is defined as:

Figure RE-GDA00025787549700000210
Figure RE-GDA00025787549700000210

9)对于每个顶点i∈V，更新第m次迭代下的对偶变量 w(m+1)(i)＝w(m)(i)+s(i)-z(m +1)(i)；9) For each vertex i∈V, update the dual variable w (m+1) (i)=w (m) (i)+s(i)-z (m + 1)(i at the mth iteration );

10)对于每个顶点i∈V，判断

Figure RE-GDA00025787549700000211
是否成立；若成立，则结束迭代，则将第m次迭代下图信号的修复值
Figure RE-GDA00025787549700000212
作为第t时刻图信号
Figure RE-GDA00025787549700000213
的修复结果；否则，将迭代次数m+1，返回到步骤7)继续迭代，直到
Figure RE-GDA00025787549700000214
成立为止。10) For each vertex i∈V, judge
Figure RE-GDA00025787549700000211
Whether it is established; if so, the iteration is ended, and the repair value of the signal in the figure below is set for the mth iteration
Figure RE-GDA00025787549700000212
as the graph signal at time t
Figure RE-GDA00025787549700000213
The repair result of
Figure RE-GDA00025787549700000214
until established.

步骤1)中，所述的无约束优化问题，其目标函数为数据保真度、1-范数空域非平滑惩罚项和2-范数时域非平滑惩罚项的加权和，表达式如下：In step 1), the objective function of the unconstrained optimization problem is the weighted sum of data fidelity, the 1-norm space-domain non-smooth penalty term and the 2-norm time-domain non-smooth penalty term, and the expression is as follows:

Figure RE-GDA0002578754970000031
Figure RE-GDA0002578754970000031

公式(1)中，α,β为加权因子，

Figure RE-GDA0002578754970000032
为N×N矩阵，I|M|为|M|×|M| 的单位阵，
Figure RE-GDA0002578754970000033
H1为高通滤波器对应的N×N矩阵；In formula (1), α, β are weighting factors,
Figure RE-GDA0002578754970000032
is an N×N matrix, I |M| is the identity matrix of |M|×|M|,
Figure RE-GDA0002578754970000033
H 1 is the N×N matrix corresponding to the high-pass filter;

令辅助变量z＝H1x，则公式(1)变为：Let auxiliary variable z=H 1 x, then formula (1) becomes:

Figure RE-GDA0002578754970000034
Figure RE-GDA0002578754970000034

公式(2)对应的增广拉格朗日函数为：The augmented Lagrangian function corresponding to formula (2) is:

Figure RE-GDA0002578754970000035
Figure RE-GDA0002578754970000035

公式(3)中，w为对偶变量，γ为惩罚因子，In formula (3), w is the dual variable, γ is the penalty factor,

基于增广拉格朗日函数，利用交替方向乘子法迭代求解公式(2)，得：Based on the augmented Lagrangian function, the alternating direction multiplier method is used to iteratively solve the formula (2), and we get:

Figure RE-GDA0002578754970000036
Figure RE-GDA0002578754970000036

步骤4)中，所述的比例是被破坏的图信号占总信号图的百分数分别为10％、 20％和50％。In step 4), the ratio is that the percentage of the damaged image signal in the total signal image is 10%, 20% and 50%, respectively.

本发明提供的一种1-范数和2-范数混合的时变图信号分布式修复方法，该方法将时变图信号的修复问题归结为1-范数和2-范数混合的无约束优化问题，其中时变图信号的时间平滑性采用2-范数刻画，空间平滑性采用具有稀疏解的 1-范数刻画，利用交替方向乘子法分布式求解优化问题；经过仿真实验，实验结果表明，本发明方法具有更优的修复性能，特别是对图信号边缘的修复效果更好。The invention provides a time-varying graph signal distributed restoration method with a 1-norm and a 2-norm mixture. Constrained optimization problem, in which the time smoothness of the time-varying graph signal is characterized by 2-norm, and the spatial smoothness is characterized by 1-norm with sparse solution, and the optimization problem is solved by the alternating direction multiplier method. The experimental results show that the method of the present invention has better repair performance, especially the repair effect on the edge of the graph signal.

具体实施方式Detailed ways

下面结合实施例对发明内容做进一步阐述，但不是对本发明的限定。The content of the invention will be further described below in conjunction with the examples, but it is not intended to limit the invention.

实施例：Example:

一种1-范数和2-范数混合的时变图信号分布式修复方法，包括如下步骤：A 1-norm and 2-norm mixed time-varying graph signal distributed repair method, comprising the following steps:

1)选择的时变图信号为2010年8月1日传感器记录下的美国218个城市的每小时温度数据，维数为218×24，用X＝{xt,t＝0,…,23}表示，将温度传感器采样到的时变图信号的修复过程转换为1-范数和2-范数混合的无约束优化问题，利用6-近邻算法构建时变图信号对应的图模型G＝{V,E}，其中V为顶点集，共包括128个顶点，对应于每个传感器节点，E为边集；设D为度矩阵，W为邻接矩阵，I为单位阵，拉普拉斯矩阵定义为

Figure RE-GDA0002578754970000049
1) The selected time-varying graph signal is the hourly temperature data of 218 cities in the United States recorded by the sensor on August 1, 2010, with a dimension of 218×24, using X={x t ,t=0,…,23 } indicates that the repair process of the time-varying graph signal sampled by the temperature sensor is converted into an unconstrained optimization problem of a mixture of 1-norm and 2-norm, and the 6-nearest neighbor algorithm is used to construct the graph model corresponding to the time-varying graph signal G= {V,E}, where V is the vertex set, including a total of 128 vertices, corresponding to each sensor node, E is the edge set; let D be the degree matrix, W is the adjacency matrix, I is the identity matrix, Laplace The matrix is defined as
Figure RE-GDA0002578754970000049

无约束优化问题，其目标函数为数据保真度、1-范数空域非平滑惩罚项和 2-范数时域非平滑惩罚项的加权和，表达式如下：For unconstrained optimization problem, the objective function is the weighted sum of data fidelity, 1-norm space-domain non-smooth penalty term and 2-norm time-domain non-smooth penalty term, and the expression is as follows:

Figure RE-GDA0002578754970000041
Figure RE-GDA0002578754970000041

公式(1)中，α,β为加权因子，

Figure RE-GDA0002578754970000042
为N×N矩阵，I|M|为|M|×|M| 的单位阵，
Figure RE-GDA0002578754970000043
H1为高通滤波器对应的N×N矩阵；In formula (1), α, β are weighting factors,
Figure RE-GDA0002578754970000042
is an N×N matrix, I |M| is the identity matrix of |M|×|M|,
Figure RE-GDA0002578754970000043
H 1 is the N×N matrix corresponding to the high-pass filter;

令辅助变量z＝H1x，则公式(1)变为：Let auxiliary variable z=H 1 x, then formula (1) becomes:

Figure RE-GDA0002578754970000044
Figure RE-GDA0002578754970000044

公式(2)对应的增广拉格朗日函数为：The augmented Lagrangian function corresponding to formula (2) is:

Figure RE-GDA0002578754970000045
Figure RE-GDA0002578754970000045

公式(3)中，w为对偶变量，γ为惩罚因子，In formula (3), w is the dual variable, γ is the penalty factor,

基于增广拉格朗日函数，利用交替方向乘子法迭代求解公式(2)，得：Based on the augmented Lagrangian function, the alternating direction multiplier method is used to iteratively solve the formula (2), and we get:

Figure RE-GDA0002578754970000046
Figure RE-GDA0002578754970000046

2)将时变图信号的动态模型设为

Figure RE-GDA0002578754970000047
其中xt为t时刻的图信号，xt+1为t+1时刻的图信号，wt为t时刻的加性随机噪声，τ为待估计参数，根据前10个图信号xt,t＝0,…,9，估计出参数τ＝0.022；2) Set the dynamic model of the time-varying graph signal as
Figure RE-GDA0002578754970000047
where x t is the graph signal at time t, x t+1 is the graph signal at time t+1, w t is the additive random noise at time t, τ is the parameter to be estimated, according to the first 10 graph signals x t , t =0,...,9, the parameter τ = 0.022 is estimated;

3)按时间先后顺序对t时刻的图信号xt＝[x(1),x(2),…,x(N)]T进行修复，以修复t时刻的图信号

Figure RE-GDA0002578754970000048
为例说明修复过程，t时刻的预测图信号为
Figure RE-GDA0002578754970000051
其中
Figure RE-GDA0002578754970000052
为修复后的第t-1时刻的图信号；3) Repair the graph signal x t =[x(1),x(2),...,x(N)] T at time t in chronological order to restore the graph signal at time t
Figure RE-GDA0002578754970000048
As an example to illustrate the repair process, the predicted graph signal at time t is
Figure RE-GDA0002578754970000051
in
Figure RE-GDA0002578754970000052
is the graph signal at time t-1 after repair;

4)按照被破坏的图信号占总信号图的百分数分别为10％、20％和50％比例，随机破坏即置零图信号，得到图信号

Figure RE-GDA0002578754970000053
其中xM为未被破坏的图信号，xμ为被破坏的图信号，xM中包含元素的个数记为|M|；4) According to the percentage of the damaged graph signal in the total signal graph, the percentages are 10%, 20% and 50% respectively, and the graph signal is randomly destroyed, that is, the graph signal is set to zero, and the graph signal is obtained.
Figure RE-GDA0002578754970000053
where x M is the uncorrupted graph signal, x μ is the corrupted graph signal, and the number of elements contained in x M is recorded as |M|;

5)对破坏后的图信号xμ施加噪声，得到图信号的观测值b＝x+n，其中n 为加性噪声；5) Noise is applied to the damaged graph signal x μ to obtain the observed value of the graph signal b=x+n, where n is additive noise;

6)初始化：对于每个顶点i∈V，原变量

Figure RE-GDA0002578754970000054
辅助变量z(0)(i)＝0，对偶变量w(0)(i)＝0，迭代次数m＝0，迭代终止阈值ε；6) Initialization: For each vertex i∈V, the original variable
Figure RE-GDA0002578754970000054
Auxiliary variable z (0) (i)=0, dual variable w (0) (i)=0, iteration times m=0, iteration termination threshold ε;

7)通过求解最小二次问题，即上述公式(4)，得到原变量

Figure RE-GDA0002578754970000055
原变量
Figure RE-GDA0002578754970000056
的表达式为：7) By solving the least quadratic problem, that is, the above formula (4), the original variable is obtained
Figure RE-GDA0002578754970000055
original variable
Figure RE-GDA0002578754970000056
The expression is:

Figure RE-GDA0002578754970000057
Figure RE-GDA0002578754970000057

其中，符号(·)-1代表矩阵求逆运算，(·)T代表矩阵转置运算，对于每个顶点 i∈V，第m次迭代下图信号的修复值

Figure RE-GDA0002578754970000058
等于原变量
Figure RE-GDA0002578754970000059
中的第i个元素值，即
Figure RE-GDA00025787549700000510
Among them, the symbol ( ) -1 represents the matrix inversion operation, ( ) T represents the matrix transposition operation, for each vertex i∈V, the mth iteration is the repair value of the signal in the figure below
Figure RE-GDA0002578754970000058
equal to the original variable
Figure RE-GDA0002578754970000059
The i-th element value in , i.e.
Figure RE-GDA00025787549700000510

8)对于每个顶点i∈V，计算第m次迭代下的辅助变量

Figure RE-GDA00025787549700000511
其中H1(i,j)为高通滤波器对应矩阵的第i行第j列元素值，σ为邻域半径，B(i,σ)为顶点i及其σ阶邻域节点的集合，函数Sβ/γ(·)定义为：8) For each vertex i∈V, calculate the auxiliary variable at the mth iteration
Figure RE-GDA00025787549700000511
where H 1 (i, j) is the element value of the i-th row and j-th column of the matrix corresponding to the high-pass filter, σ is the neighborhood radius, B(i, σ) is the set of vertex i and its σ-order neighborhood nodes, and the function S β/γ (·) is defined as:

Figure RE-GDA00025787549700000512
Figure RE-GDA00025787549700000512

9)对于每个顶点i∈V，更新第m次迭代下的对偶变量 w(m+1)(i)＝w(m)(i)+s(i)-z(m +1)(i)；9) For each vertex i∈V, update the dual variable w (m+1) (i)=w (m) (i)+s(i)-z (m + 1)(i at the mth iteration );

10)对于每个顶点i∈V，判断

Figure RE-GDA00025787549700000513
是否成立；若成立，则结束迭代，则将第m次迭代下图信号的修复值
Figure RE-GDA00025787549700000514
作为第t时刻图信号
Figure RE-GDA00025787549700000515
的修复结果；否则，将迭代次数m+1，返回到步骤7)继续迭代，直到
Figure RE-GDA0002578754970000061
成立为止。10) For each vertex i∈V, judge
Figure RE-GDA00025787549700000513
Whether it is established; if so, the iteration is ended, and the repair value of the signal in the figure below is set for the mth iteration
Figure RE-GDA00025787549700000514
as the graph signal at time t
Figure RE-GDA00025787549700000515
The repair result of
Figure RE-GDA0002578754970000061
until established.

为验证上述方法的有效性，对本发明的方法进行仿真实验，并与现有的方法进行比较；修复方法的评价指标为：In order to verify the effectiveness of the above method, the method of the present invention is simulated and compared with the existing method; the evaluation index of the repair method is:

(1)

Figure RE-GDA0002578754970000062
(1)
Figure RE-GDA0002578754970000062

(2)

Figure RE-GDA0002578754970000063
(2)
Figure RE-GDA0002578754970000063

其中，x0和x分别为图信号的真实值和修复值，经测试，得到如下表1所示的性能对比，表1中取24个RMSE和MRE平均值。Among them, x 0 and x are the real value and repair value of the graph signal, respectively. After testing, the performance comparison shown in Table 1 below is obtained. In Table 1, the average value of 24 RMSE and MRE is taken.

表1美国城市温度数据的修复性能Table 1. Restoration performance for temperature data in U.S. cities

Figure RE-GDA0002578754970000064
Figure RE-GDA0002578754970000064

进一步验证本发明方法在图信号边缘修复上的效果，构造分段平滑的明尼苏达州交通图信号，图信号取值分别为1和-1，包含一条边缘。按照10％比例破坏图信号，利用本发明方法和现有方法对其进行修复，空间平滑性惩罚分别采取 1-范数和2-范数，利用本发明方法和现有方法修复后图信号后，计算出RMSE值分别为0.044和0.0625。To further verify the effect of the method of the present invention on the edge restoration of the map signal, a segmented smooth Minnesota traffic map signal is constructed, and the map signal values are 1 and -1 respectively, including an edge. The graph signal is destroyed according to the proportion of 10%, and it is repaired by the method of the present invention and the existing method. The spatial smoothness penalty adopts 1-norm and 2-norm respectively. After the graph signal is repaired by the method of the present invention and the existing method , the RMSE values were calculated to be 0.044 and 0.0625, respectively.

从上述仿真结果可以看出，本发明方法比现有方法具有更好的修复性能，尤其表现在对图信号边缘的修复方面。It can be seen from the above simulation results that the method of the present invention has better repair performance than the existing method, especially in repairing the edge of the graph signal.

Claims (3)

1.一种1-范数和2-范数混合的时变图信号分布式修复方法，其特征在于，包括如下步骤：1. a time-varying graph signal distributed repair method of 1-norm and 2-norm mixing, is characterized in that, comprises the steps: 1)将传感器采样到的时变图信号的修复过程转换为1-范数和2-范数混合的无约束优化问题，利用6-近邻算法构建时变图信号对应的图模型G＝{V,E}，其中V为顶点集，共包括N个顶点，对应于每个传感器节点，E为边集；设D为度矩阵，W为邻接矩阵，I为单位阵，拉普拉斯矩阵定义为
Figure FDA0002554101360000011
1) Convert the repair process of the time-varying graph signal sampled by the sensor into an unconstrained optimization problem of a mixture of 1-norm and 2-norm, and use the 6-nearest neighbor algorithm to construct a graph model corresponding to the time-varying graph signal G={V ,E}, where V is the vertex set, including N vertices in total, corresponding to each sensor node, E is the edge set; let D be the degree matrix, W is the adjacency matrix, I is the identity matrix, and the Laplace matrix is defined for
Figure FDA0002554101360000011
 2)将时变图信号的动态模型设为
Figure FDA0002554101360000012
其中xt为t时刻的图信号，xt+1为t+1时刻的图信号，wt为t时刻的加性随机噪声，τ为待估计参数，根据前10个图信号xt,t＝0,…,9，估计出参数τ＝0.022；2) Set the dynamic model of the time-varying graph signal as
Figure FDA0002554101360000012
where x t is the graph signal at time t, x t+1 is the graph signal at time t+1, w t is the additive random noise at time t, τ is the parameter to be estimated, according to the first 10 graph signals x t , t =0,...,9, the parameter τ = 0.022 is estimated; 3)按时间先后顺序对t时刻的图信号xt＝[x(1),x(2),…,x(N)]T进行修复，其中，t时刻的预测图信号为
Figure FDA0002554101360000013
其中
Figure FDA0002554101360000014
为修复后的第t-1时刻的图信号；3) Repair the graph signal x t =[x(1),x(2),...,x(N)] T at time t in chronological order, where the predicted graph signal at time t is
Figure FDA0002554101360000013
in
Figure FDA0002554101360000014
is the graph signal at time t-1 after repair; 4)按照一定比例，随机破坏即置零图信号，得到经过置零后的图信号为
Figure FDA0002554101360000015
其中xΜ为未被破坏的图信号，xμ为被破坏的图信号，xΜ中包含元素的个数记为|Μ|；4) According to a certain proportion, random destruction is to zero the graph signal, and the graph signal after zeroing is obtained as
Figure FDA0002554101360000015
Wherein x M is the image signal that is not destroyed, x μ is the image signal that is destroyed, and the number of elements contained in x M is denoted as |M|; 5)对破坏后的图信号xμ施加噪声，得到图信号的观测值b＝x+n，其中n为加性噪声；5) Noise is applied to the damaged graph signal x μ to obtain the observed value of the graph signal b=x+n, where n is additive noise; 6)初始化：对于每个顶点i∈V，原变量
Figure FDA0002554101360000016
辅助变量z(0)(i)＝0，对偶变量w(0)(i)＝0，迭代次数m＝0，迭代终止阈值ε；6) Initialization: For each vertex i∈V, the original variable
Figure FDA0002554101360000016
Auxiliary variable z (0) (i)=0, dual variable w (0) (i)=0, iteration times m=0, iteration termination threshold ε; 7)通过求解最小二次问题，得到原变量
Figure FDA0002554101360000017
原变量
Figure FDA0002554101360000018
的表达式为：7) By solving the least quadratic problem, the original variables are obtained
Figure FDA0002554101360000017
original variable
Figure FDA0002554101360000018
The expression is:
Figure FDA0002554101360000019
Figure FDA0002554101360000019
 其中，符号(·)-1代表矩阵求逆运算，(·)T代表矩阵转置运算，对于每个顶点i∈V，第m次迭代下图信号的修复值
Figure FDA00025541013600000110
等于原变量
Figure FDA00025541013600000111
中的第i个元素值，即
Figure FDA00025541013600000112
Among them, the symbol ( ) -1 represents the matrix inversion operation, ( ) T represents the matrix transpose operation, for each vertex i∈V, the repair value of the signal in the following image is the mth iteration
Figure FDA00025541013600000110
equal to the original variable
Figure FDA00025541013600000111
The i-th element value in , i.e.
Figure FDA00025541013600000112
 8)对于每个顶点i∈V，计算第m次迭代下的辅助变量
Figure FDA00025541013600000113
其中H1(i,j)为高通滤波器对应矩阵的第i行第j列元素值，σ为邻域半径，B(i,σ)为顶点i及其σ阶邻域节点的集合，函数Sβ/γ(·)定义为：8) For each vertex i∈V, calculate the auxiliary variable at the mth iteration
Figure FDA00025541013600000113
where H 1 (i, j) is the element value of the i-th row and j-th column of the matrix corresponding to the high-pass filter, σ is the neighborhood radius, B(i, σ) is the set of vertex i and its σ-order neighborhood nodes, and the function S β/γ (·) is defined as:
Figure FDA0002554101360000021
Figure FDA0002554101360000021
 9)对于每个顶点i∈V，更新第m次迭代下的对偶变量w(m+1)(i)＝w(m)(i)+s(i)-z(m+1)(i)；9) For each vertex i∈V, update the dual variable w (m+1) (i)=w (m) (i)+s(i)-z (m+1) (i at the mth iteration ); 10)对于每个顶点i∈V，判断
Figure FDA0002554101360000022
是否成立；若成立，则结束迭代，则将第m次迭代下图信号的修复值
Figure FDA0002554101360000023
作为第t时刻图信号
Figure FDA0002554101360000024
的修复结果；否则，将迭代次数m+1，返回到步骤7)继续迭代，直到
Figure FDA0002554101360000025
成立为止。10) For each vertex i∈V, judge
Figure FDA0002554101360000022
Whether it is established; if so, the iteration is ended, and the repair value of the signal in the figure below is set for the mth iteration
Figure FDA0002554101360000023
as the graph signal at time t
Figure FDA0002554101360000024
The repair result of
Figure FDA0002554101360000025
until established. 2.根据权利要求1所述的一种1-范数和2-范数混合的时变图信号分布式修复方法，其特征在于，步骤1)中，所述的无约束优化问题，其目标函数为数据保真度、1-范数空域非平滑惩罚项和2-范数时域非平滑惩罚项的加权和，表达式如下：2. a kind of 1-norm and 2-norm mixed time-varying graph signal distributed repair method according to claim 1, is characterized in that, in step 1), described unconstrained optimization problem, its target The function is the weighted sum of the data fidelity, the 1-norm spatial non-smooth penalty term and the 2-norm temporal non-smooth penalty term, and the expression is as follows:
Figure FDA0002554101360000026
Figure FDA0002554101360000026
 公式(1)中，α,β为加权因子，
Figure FDA0002554101360000027
为N×N矩阵，IΜ为|Μ|×|Μ|的单位阵，
Figure FDA0002554101360000028
H1为高通滤波器对应的N×N矩阵；In formula (1), α, β are weighting factors,
Figure FDA0002554101360000027
is an N×N matrix, I M is the identity matrix of |M|×|M|,
Figure FDA0002554101360000028
H 1 is the N×N matrix corresponding to the high-pass filter; 令辅助变量z＝H1x，则公式(1)变为：Let auxiliary variable z=H 1 x, then formula (1) becomes:
Figure FDA0002554101360000029
Figure FDA0002554101360000029
 公式(2)对应的增广拉格朗日函数为：The augmented Lagrangian function corresponding to formula (2) is:
Figure FDA00025541013600000210
Figure FDA00025541013600000210
 公式(3)中，w为对偶变量，γ为惩罚因子，In formula (3), w is the dual variable, γ is the penalty factor, 基于增广拉格朗日函数，利用交替方向乘子法迭代求解公式(2)，得：Based on the augmented Lagrangian function, the alternating direction multiplier method is used to iteratively solve the formula (2), and we get:
Figure FDA0002554101360000031
Figure FDA0002554101360000031
 3.根据权利要求1所述的一种1-范数和2-范数混合的时变图信号分布式修复方法，其特征在于，步骤4)中，所述的比例是被破坏的图信号占总信号图的百分数分别为10％、20％和50％。3. a kind of 1-norm and 2-norm mixed time-varying graph signal distributed repair method according to claim 1, is characterized in that, in step 4), described ratio is the graph signal that is destroyed The percentages of the total signal map were 10%, 20% and 50%, respectively.
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