CN111737639A - 1-norm and 2-norm mixed time-varying graph signal distributed restoration method - Google Patents

1-norm and 2-norm mixed time-varying graph signal distributed restoration method 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

1-norm and 2-norm mixed time-varying graph signal distributed restoration method
Technical Field
The invention relates to a time-varying graph signal processing technology neighborhood, in particular to a 1-norm and 2-norm mixed time-varying graph signal distributed repairing method.
Background
The time-varying graph signals are widely used in practical applications, such as urban temperature data acquired by a temperature sensor every hour, sea surface pressure data acquired by a pressure sensor every year, and the like. However, as the sensor is affected by its own performance and environmental factors, some corruption of the collected data may occur. Therefore, it is necessary to restore the time-varying graph signal according to the spatiotemporal correlation between data and restore the original data as much as possible.
Typically, the time-varying graph signals involve large amounts of data, requiring distributed repairs with relatively low computational complexity. In the existing distributed repair method, the spatial and temporal smoothness of the time-varying graph signal are measured by 2-norm, and the spatial sparsity of the graph signal cannot be well represented, which is shown in poor effect of repairing the edge of the graph signal.
Disclosure of Invention
The invention aims to provide a distributed repair method of a 1-norm and 2-norm mixed time-varying graph signal, aiming at the problem that the existing repair method of the time-varying graph signal has poor repair effect on edges.
The technical scheme for realizing the purpose of the invention is as follows:
a1-norm and 2-norm mixed time-varying graph signal distributed restoration method comprises the following steps:
1) converting a repairing process of a time-varying graph signal sampled by a sensor into an unconstrained optimization problem of mixing 1-norm and 2-norm, and constructing a graph model G (V, E) corresponding to the time-varying graph signal by using a 6-nearest neighbor algorithm, wherein V is a vertex set and comprises N vertexes in total, corresponds to each sensor node, and E is an edge set; let D be the degree matrix, W be the adjacency matrix, I be the unit matrix, and the Laplace matrix be defined as
Figure RE-GDA0002578754970000011
2) Setting the dynamic model of the time-varying image signal as
Figure RE-GDA0002578754970000012
Wherein xtIs the graph signal at time t, xt+1Is the graph signal at time t +1, wtIs additive random noise at the time t, tau is a parameter to be estimated, and x is obtained according to the first 10 graph signalstT is 0, …,9, and the parameter τ is estimated to be 0.022;
3) graph signal x for time t in time sequencet＝[x(1),x(2),…,x(N)]TThe restoration is carried out, and the prediction graph signal at the time t is set as
Figure RE-GDA0002578754970000013
Wherein
Figure RE-GDA0002578754970000014
The graph signal at the t-1 th moment after the restoration;
4) according to a certain proportion, randomly destroying the zero-setting image signal to obtain the image signal after zero-setting
Figure RE-GDA0002578754970000021
Wherein xMFor uncorrupted picture signals, xμFor corrupted picture signals, xMThe number of the elements contained in the Chinese character is recorded as | M |;
5) for the destroyed picture signal xμApplying noise to obtain an observed value b of the graph signal, wherein n is additive noise;
6) initialization of original variables for each vertex i ∈ V
Figure RE-GDA0002578754970000022
Auxiliary variable z(0)(i) 0, dual variable w(0)(i) When the iteration time m is 0, the iteration termination threshold value is set;
7) obtaining the original variable by solving the minimum quadratic problem
Figure RE-GDA0002578754970000023
Original variables
Figure RE-GDA0002578754970000024
The expression is as follows:
Figure RE-GDA0002578754970000025
wherein, the symbol (·)-1Representative matrix inversion operation (·)TRepresenting the matrix transpose operation, for each vertex i ∈ V, the repair value of the next graph signal at the m-th iteration
Figure RE-GDA0002578754970000026
Equal to the original variable
Figure RE-GDA0002578754970000027
The value of the ith element in (1), i.e
Figure RE-GDA0002578754970000028
8) For each vertex i ∈ V, auxiliary variables at the mth iteration are calculated
Figure RE-GDA0002578754970000029
Wherein H1(i, j) is the ith row and jth column element value of the high-pass filter corresponding matrix, sigma is the neighborhood radius, B (i, sigma) is the set of vertex i and its sigma order neighborhood nodes, and the function Sβ/γ(. cndot.) is defined as:
Figure RE-GDA00025787549700000210
9) for each vertex i ∈ V, update the dual variable w at the mth iteration(m+1)(i)＝w(m)(i)+s(i)-z(m +1)(i)；
10) For each vertex i ∈ V, judge
Figure RE-GDA00025787549700000211
Whether the result is true or not; if yes, ending the iteration, and then carrying out the m-th iteration to obtain the repair value of the lower graph signal
Figure RE-GDA00025787549700000212
As the t-th timing chart signal
Figure RE-GDA00025787549700000213
The repair result of (2); otherwise, returning the iteration times m +1 to the step 7) to continue the iteration until the iteration times m +1 are reached
Figure RE-GDA00025787549700000214
Until it is established.
In step 1), the objective function of the unconstrained optimization problem is a weighted sum of data fidelity, a 1-norm airspace non-smooth penalty term and a 2-norm time domain non-smooth penalty term, and the expression is as follows:
Figure RE-GDA0002578754970000031
in formula (1), α is a weighting factor,
Figure RE-GDA0002578754970000032
is an N × N matrix, I|M|Is a unit matrix of | M | × | M |,
Figure RE-GDA0002578754970000033
H1an N × N matrix corresponding to the high-pass filter;
let auxiliary variable z be H1x, then equation (1) becomes:
Figure RE-GDA0002578754970000034
the augmented lagrange function corresponding to equation (2) is:
Figure RE-GDA0002578754970000035
in the formula (3), w is a dual variable, γ is a penalty factor,
based on the augmented Lagrange function, an alternating direction multiplier method is used for iterative solution of a formula (2), and the following results are obtained:
Figure RE-GDA0002578754970000036
in step 4), the ratios are 10%, 20% and 50% of the total signal plot of the destroyed plot signal, respectively.
The invention provides a distributed repair method of a 1-norm and 2-norm mixed time-varying graph signal, which is used for solving the repair problem of the time-varying graph signal into an unconstrained optimization problem of 1-norm and 2-norm mixing, wherein the time smoothness of the time-varying graph signal is described by adopting a 2-norm, the space smoothness is described 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.
Detailed Description
The invention will now be further illustrated with reference to the following examples, but is not intended to be limited thereto.
Example (b):
a1-norm and 2-norm mixed time-varying graph signal distributed restoration method comprises the following steps:
1) the time-varying graph signals selected were temperature data per hour for 218 cities in the united states recorded by day 1/8/2010 with dimensions 218 × 24, X ═ XtT is 0, …,23, which means that a repairing process of a time-varying graph signal sampled by a temperature sensor is converted into an unconstrained optimization problem of mixing 1-norm and 2-norm, and a graph model G corresponding to the time-varying graph signal is constructed by using a 6-neighbor algorithm, wherein V is a vertex set and comprises 128 vertices, corresponding to each sensor node, and E is an edge set; let D be the degree matrix, W be the adjacency matrix, I be the unit matrix, and the Laplace matrix be defined as
Figure RE-GDA0002578754970000049
The unconstrained optimization problem is characterized in that an objective function is a weighted sum of data fidelity, a 1-norm spatial domain unsmooth penalty term and a 2-norm temporal unsmooth penalty term, and an expression is as follows:
Figure RE-GDA0002578754970000041
in formula (1), α is a weighting factor,
Figure RE-GDA0002578754970000042
is an N × N matrix, I|M|Is a unit matrix of | M | × | M |,
Figure RE-GDA0002578754970000043
H1an N × N matrix corresponding to the high-pass filter;
let auxiliary variable z be H1x, then equation (1) becomes:
Figure RE-GDA0002578754970000044
the augmented lagrange function corresponding to equation (2) is:
Figure RE-GDA0002578754970000045
in the formula (3), w is a dual variable, γ is a penalty factor,
based on the augmented Lagrange function, an alternating direction multiplier method is used for iterative solution of a formula (2), and the following results are obtained:
Figure RE-GDA0002578754970000046
2) setting the dynamic model of the time-varying image signal as
Figure RE-GDA0002578754970000047
Wherein xtIs the graph signal at time t, xt+1Is the graph signal at time t +1, wtIs additive random noise at the time t, tau is a parameter to be estimated, and x is obtained according to the first 10 graph signalstT is 0, …,9, and the parameter τ is estimated to be 0.022;
3) graph signal x for time t in time sequencet＝[x(1),x(2),…,x(N)]TRepairing to repair the graph signal at time t
Figure RE-GDA0002578754970000048
For example, the prediction map signal at time t is
Figure RE-GDA0002578754970000051
Wherein
Figure RE-GDA0002578754970000052
The graph signal at the t-1 th moment after the restoration;
4) randomly destroying or zeroing the graph signals to obtain graph signals according to the proportion of 10%, 20% and 50% of the total graph
Figure RE-GDA0002578754970000053
Wherein xMFor uncorrupted picture signals, xμFor corrupted picture signals, xMThe number of the elements contained in the Chinese character is recorded as | M |;
5) for the destroyed picture signal xμApplying noise to obtain an observed value b of the graph signal, wherein n is additive noise;
6) initialization of original variables for each vertex i ∈ V
Figure RE-GDA0002578754970000054
Auxiliary variable z(0)(i) 0, dual variable w(0)(i) When the iteration time m is 0, the iteration termination threshold value is set;
7) by solving the least quadratic problem, equation (4) above, the original variables are obtained
Figure RE-GDA0002578754970000055
Original variables
Figure RE-GDA0002578754970000056
The expression of (a) is:
Figure RE-GDA0002578754970000057
wherein, the symbol (·)-1Representative matrix inversion operation (·)TRepresenting the matrix transpose operation, for each vertex i ∈ V, the repair value of the next graph signal at the m-th iteration
Figure RE-GDA0002578754970000058
Equal to the original variable
Figure RE-GDA0002578754970000059
The value of the ith element in (1), i.e
Figure RE-GDA00025787549700000510
8) For each vertex i ∈ V, calculate the m-th timeAuxiliary variables under iteration
Figure RE-GDA00025787549700000511
Wherein H1(i, j) is the ith row and jth column element value of the high-pass filter corresponding matrix, sigma is the neighborhood radius, B (i, sigma) is the set of vertex i and its sigma order neighborhood nodes, and the function Sβ/γ(. cndot.) is defined as:
Figure RE-GDA00025787549700000512
9) for each vertex i ∈ V, update the dual variable w at the mth iteration(m+1)(i)＝w(m)(i)+s(i)-z(m +1)(i)；
10) For each vertex i ∈ V, judge
Figure RE-GDA00025787549700000513
Whether the result is true or not; if yes, ending the iteration, and then carrying out the m-th iteration to obtain the repair value of the lower graph signal
Figure RE-GDA00025787549700000514
As the t-th timing chart signal
Figure RE-GDA00025787549700000515
The repair result of (2); otherwise, returning the iteration times m +1 to the step 7) to continue the iteration until the iteration times m +1 are reached
Figure RE-GDA0002578754970000061
Until it is established.
In order to verify the effectiveness of the method, the method is subjected to a simulation experiment and compared with the existing method; the evaluation indexes of the repairing method are as follows:
(1)
Figure RE-GDA0002578754970000062
(2)
Figure RE-GDA0002578754970000063
wherein x is0And x are the actual and repaired values of the graph signal, respectively, and the performance comparison shown in table 1 below was obtained by testing, and the average values of 24 RMSE and MRE were taken in table 1.
TABLE 1 repair Performance of American City temperature data
Figure RE-GDA0002578754970000064
Further verifying the effect of the method on map signal edge repair, constructing a Minnesota traffic map signal with smooth segments, wherein the values of the map signal are respectively 1 and-1, and the map signal comprises an edge. According to the method, graph signals are damaged according to the proportion of 10%, the graph signals are repaired by the method and the existing method, the penalty of spatial smoothness is respectively 1-norm and 2-norm, and after the graph signals are repaired by the method and the existing method, RMSE values are respectively calculated to be 0.044 and 0.0625.
From the simulation results, the method of the present invention has better repairing performance than the existing method, especially in the aspect of repairing the graph signal edge.

Claims (3)

1. A distributed repair method for a 1-norm and 2-norm mixed time-varying graph signal is characterized by comprising the following steps:
1) converting a repairing process of a time-varying graph signal sampled by a sensor into an unconstrained optimization problem of mixing 1-norm and 2-norm, and constructing a graph model G (V, E) corresponding to the time-varying graph signal by using a 6-nearest neighbor algorithm, wherein V is a vertex set and comprises N vertexes in total, corresponds to each sensor node, and E is an edge set; let D be the degree matrix, W be the adjacency matrix, I be the unit matrix, and the Laplace matrix be defined as
Figure FDA0002554101360000011
2) Setting the dynamic model of the time-varying image signal as
Figure FDA0002554101360000012
Wherein xtIs the graph signal at time t, xt+1Is the graph signal at time t +1, wtIs additive random noise at the time t, tau is a parameter to be estimated, and x is obtained according to the first 10 graph signalstT is 0, …,9, and the parameter τ is estimated to be 0.022;
3) graph signal x for time t in time sequencet＝[x(1),x(2),…,x(N)]TRepairing, wherein the prediction graph signal at the time t is
Figure FDA0002554101360000013
Wherein
Figure FDA0002554101360000014
The graph signal at the t-1 th moment after the restoration;
4) according to a certain proportion, randomly destroying the zero-setting image signal to obtain the image signal after zero-setting
Figure FDA0002554101360000015
Wherein xΜFor uncorrupted picture signals, xμFor corrupted picture signals, xΜThe number of the elements contained in the Chinese character is recorded as | M |;
5) for the destroyed picture signal xμApplying noise to obtain an observed value b of the graph signal, wherein n is additive noise;
6) initialization of original variables for each vertex i ∈ V
Figure FDA0002554101360000016
Auxiliary variable z(0)(i) 0, dual variable w(0)(i) When the iteration time m is 0, the iteration termination threshold value is set;
7) obtaining the original variable by solving the minimum quadratic problem
Figure FDA0002554101360000017
Original variables
Figure FDA0002554101360000018
The expression of (a) is:
Figure FDA0002554101360000019
wherein, the symbol (·)-1Representative matrix inversion operation (·)TRepresenting the matrix transpose operation, for each vertex i ∈ V, the repair value of the next graph signal at the m-th iteration
Figure FDA00025541013600000110
Equal to the original variable
Figure FDA00025541013600000111
The value of the ith element in (1), i.e
Figure FDA00025541013600000112
8) For each vertex i ∈ V, auxiliary variables at the mth iteration are calculated
Figure FDA00025541013600000113
Wherein H1(i, j) is the ith row and jth column element value of the high-pass filter corresponding matrix, sigma is the neighborhood radius, B (i, sigma) is the set of vertex i and its sigma order neighborhood nodes, and the function Sβ/γ(. cndot.) is defined as:
Figure FDA0002554101360000021
9) for each vertex i ∈ V, update the dual variable w at the mth iteration(m+1)(i)＝w(m)(i)+s(i)-z(m+1)(i)；
10) For each vertex i ∈ V, judge
Figure FDA0002554101360000022
Whether the result is true or not; if yes, ending the iteration, and then carrying out the m-th iteration to obtain the repair value of the lower graph signal
Figure FDA0002554101360000023
As the t-th timing chart signal
Figure FDA0002554101360000024
The repair result of (2); otherwise, returning the iteration times m +1 to the step 7) to continue the iteration until the iteration times m +1 are reached
Figure FDA0002554101360000025
Until it is established.
2. The distributed restoration method for 1-norm and 2-norm mixed time-varying graph signals according to claim 1, wherein in step 1), the objective function of the unconstrained optimization problem is a weighted sum of data fidelity, a 1-norm spatial non-smooth penalty term and a 2-norm temporal non-smooth penalty term, and the expression is as follows:
Figure FDA0002554101360000026
in formula (1), α is a weighting factor,
Figure FDA0002554101360000027
is an N × N matrix, IΜIs a unit matrix of | M | × | M |,
Figure FDA0002554101360000028
H1an N × N matrix corresponding to the high-pass filter;
let auxiliary variable z be H1x, then equation (1) becomes:
Figure FDA0002554101360000029
the augmented lagrange function corresponding to equation (2) is:
Figure FDA00025541013600000210
in the formula (3), w is a dual variable, γ is a penalty factor,
based on the augmented Lagrange function, an alternating direction multiplier method is used for iterative solution of a formula (2), and the following results are obtained:
Figure FDA0002554101360000031
3. the distributed repair method for the 1-norm and 2-norm mixed time-varying graph signals according to claim 1, wherein in the step 4), the proportion of the corrupted graph signals is 10%, 20% and 50% of the total graph signal.
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