CN110166055B - Multi-channel compressive sensing optimization method and system based on compressive sensing - Google Patents

Multi-channel compressive sensing optimization method and system based on compressive sensing Download PDF

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CN110166055B
CN110166055B CN201910385702.9A CN201910385702A CN110166055B CN 110166055 B CN110166055 B CN 110166055B CN 201910385702 A CN201910385702 A CN 201910385702A CN 110166055 B CN110166055 B CN 110166055B
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张�成
朱园园
汤俊
许海涛
杨佐
潘敏
韦穗
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Abstract

The invention discloses a multi-channel compressive sensing optimization method and system based on compressive sensing, belongs to the technical field of compressive sensing, and comprises the steps of carrying out singular value decomposition on a measurement matrix to obtain an optimized observed value vector, and reconstructing an original signal by utilizing a synchronous orthogonal matching pursuit joint reconstruction algorithm. The invention realizes the separation design of the measurement matrix and the reconstruction matrix by carrying out singular value decomposition on the measurement matrix to obtain the separable reconstruction matrix, reduces the correlation among the measured values due to the mutual orthogonality of the rows of the optimized reconstruction matrix, and reconstructs the original signal by using the measured values, thereby greatly improving the reconstruction precision of the signal.

Description

Multi-channel compressive sensing optimization method and system based on compressive sensing
Technical Field
The invention relates to the technical field of compressed sensing, in particular to a multi-channel compressed sensing optimization method and system based on compressed sensing.
Background
The current information acquisition technology is mainly based on the nyquist sampling theorem, but the signal sampling causes a great deal of data acquisition redundancy and waste of sensor resources. Compressed Sensing (CS) theory shows that if a signal is sparse or compressible, the original signal can be accurately recovered from a small number of measurements below the nyquist sampling rate with a set of incoherent projections, while ensuring the accuracy of signal reconstruction. The theory has been widely applied to wireless sensor networks, image super-resolution reconstruction, seismic exploration and the like.
At present, the CS theory only needs to be applied to the internal signal structure of a single sensor, and it is considered that many application scenarios in real life are networks containing a plurality of sensors. In 2005, baron et al further studied how to sufficiently fuse sparse characteristics of signals and correlation between signals to perform joint processing on distributed signals on the basis of a compressive sensing theory, and further proposed distributed compressive sensing. DCS typically uses Joint Sparse Model (JSM) to characterize the sparsity among multiple signals. In 2005, baron et al studied three simple joint sparse signal models in detail for different application scenarios.
Distributed compressed sensing is mainly aimed at the recovery of multi-channel sparse signals, and the number of measurement values can be further reduced compared with the traditional CS by utilizing the correlation among channels. In a typical Distributed Compressed Sensing (DCS) scenario, the measured signals are sensed for different sensors, each of which is sparse on a set of bases, with the sensors being interrelated.
The DCS is based on compressed sensing, expands the research target from a single signal to distributed sensing of multi-channel signals, if multiple signals are sparse under a certain transform basis and there is a certain correlation between the signals, then at the encoding end, each sparse signal can be compressed and sampled and transmitted independently by using another observation matrix that is not wanted to manage with the transform basis, to obtain the number of measured values that is much smaller than the length of the signal, and then at the decoding end, the accurate reconstruction of all the signals can be realized by using all the measured values to perform joint reconstruction, thereby improving the data compression and the reconstruction accuracy of the signals, and the system block diagram thereof is shown in fig. 1. Aiming at the problem of channel-by-channel compressive sensing in distributed compressive sensing, the invention aims to improve the reconstruction precision of a multi-channel signal and reduce the waste of resources.
Disclosure of Invention
The invention aims to provide a multi-channel compressive sensing optimization method and a multi-channel compressive sensing optimization system based on compressive sensing so as to improve the reconstruction accuracy of multi-channel signals.
In order to achieve the above object, in one aspect, the present invention adopts a compressed sensing-based multi-channel compressed sensing optimization method, including the following steps:
acquiring signals X of each channel, and performing projection measurement on the signals of each channel by using a measurement matrix to obtain an observed value vector YMMV= Φ X + e = a θ + e, wherein: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vector
Figure BDA0002054754370000021
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
optimizing by solving constraints according to the optimized observed value vector
Figure BDA0002054754370000023
The norm reconstructs the original signal.
Further, the observation value vector is one measurement vector measured by a single measurement vector model, or J measurement vectors measured by a multi-measurement vector model.
Further, the decomposing the measurement matrix Φ by using the singular value decomposition method to obtain the optimized observation value vector includes:
decomposing the measurement matrix phi by using a singular value decomposition method to obtain a new sensing system:
Figure BDA0002054754370000022
wherein: d1＝diag(σ1,σ2,…,σM) Is M × M dimensional diagonal matrix, phi = UDVTSubmatrix of matrix V
Figure BDA0002054754370000031
As a column orthogonal matrix, i.e.
Figure BDA0002054754370000032
EMAnd EN-MM × M and (N-M) × (N-M) dimensional identity matrices, respectively; d is belonged to RM×NThe matrix is a semi-positive definite diagonal matrix, and elements on non-diagonal lines of the matrix are zero;
left multiplication matrix at both sides of calculation formula of new sensing system
Figure BDA0002054754370000033
Obtaining an optimized observation value vector:
Figure BDA0002054754370000034
wherein:
Figure BDA0002054754370000035
ΦSVD＝V1 T，
Figure BDA0002054754370000036
furthermore, the sparsity among the signals of all the channels is represented by adopting a joint sparse signal model JSM-2.
Further, the optimal constraint is solved according to the optimized observation value vector
Figure BDA0002054754370000038
Reconstructing an original signal by using the norm, specifically:
and processing the optimized observed value vector by adopting a synchronous orthogonal matching pursuit joint reconstruction algorithm to reconstruct an original signal.
In a second aspect, a compressed sensing-based multi-channel compressed sensing optimization system is adopted, and comprises an observed value vector calculation module, an observed value vector optimization module and an original signal reconstruction module;
acquiring signals X of each channel, and performing projection measurement on the signals of each channel by using a measurement matrix to obtain an observed value vector YMMV= Φ X + e = a θ + e, wherein: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
the observation value vector optimization module is used for decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observation value vector
Figure BDA0002054754370000037
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
the original signal reconstruction module is used for solving constraint optimization according to the optimized observed value vector
Figure BDA0002054754370000039
The norm reconstructs the original signal.
Further, the observation vector is one measurement vector measured by a single measurement vector model, or J measurement vectors measured by a multi-measurement vector model.
Further, the observation value vector optimization module comprises a singular value decomposition unit and an optimization unit;
the singular value decomposition unit is used for decomposing the measurement matrix phi by using a singular value decomposition method to obtain a new sensing system:
Figure BDA0002054754370000041
wherein: d1＝diag(σ1,σ2,…,σM) Is M × M dimensional diagonal matrix, phi = UDVTSubmatrix of matrix V
Figure BDA0002054754370000042
As a column orthogonal matrix, i.e. V1 TV1＝EM，
Figure BDA0002054754370000043
EMAnd EN-MM × M and (N-M) × (N-M) dimensional identity matrices, respectively;D∈RM×Nthe matrix is a semi-positive definite diagonal matrix, and elements on non-diagonal lines of the matrix are zero;
the optimization unit is used for calculating the formula YMMVTwo-sided left multiplication matrix
Figure BDA0002054754370000044
Obtaining an optimized observation value vector:
Figure BDA0002054754370000045
wherein:
Figure BDA0002054754370000046
ΦSVD＝V1 T，
Figure BDA0002054754370000047
furthermore, the sparsity among the signals of all the channels is represented by adopting a joint sparse signal model JSM-2;
correspondingly, the optimal constraint optimization is solved according to the optimized observation value vector
Figure BDA0002054754370000048
Reconstructing an original signal by using the norm, specifically:
and processing the optimized observed value vector by adopting a synchronous orthogonal matching pursuit joint reconstruction algorithm to reconstruct an original signal.
In a third aspect, a computer-readable storage medium is employed, comprising a program which, when executed, causes an apparatus comprising the computer-readable storage medium to perform the steps of:
acquiring signals X of each channel, and performing projection measurement on the signals of each channel by using a measurement matrix to obtain an observed value vector YMMV= Φ X + e = a θ + e, where: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vector
Figure BDA0002054754370000051
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
optimizing by solving constraints according to the optimized observation value vector
Figure BDA0002054754370000052
The norm reconstructs the original signal.
Compared with the prior art, the invention has the following technical effects: according to the invention, the singular value decomposition is carried out on the measurement matrix, the separation design of the measurement matrix and the reconstruction matrix is realized, the separable reconstruction matrix is obtained, the correlation among the measured values is reduced because the rows of the optimized reconstruction matrix are mutually orthogonal, the original signal is reconstructed by using the measured values, and the reconstruction precision of the signal is greatly improved.
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The following detailed description of embodiments of the invention refers to the accompanying drawings in which:
FIG. 1 is a block diagram of a distributed compressed sensing system;
FIG. 2 is a schematic flow chart of a compressed sensing optimization method based on compressed sensing;
FIG. 3 is a schematic diagram of a multi-measurement vector model;
FIG. 4 is a schematic structural diagram of a compressed sensing optimization system based on compressed sensing with multiple channels;
FIG. 5 is a graph showing the results of a single experiment of the original signal, observed signal and reconstructed signal;
FIG. 6 is a diagram illustrating the relationship between the success probability and the number of measurements for original signal reconstruction using different methods;
FIG. 7 is a diagram illustrating the relationship between the success probability and the sparsity of original signal reconstruction by different methods;
fig. 8 is a diagram illustrating the relationship between the success probability and the noise change of the original signal reconstruction by using different methods.
Detailed Description
To further illustrate the features of the present invention, refer to the following detailed description of the invention and the accompanying drawings. The drawings are for reference and illustration purposes only and are not intended to limit the scope of the present disclosure.
As shown in fig. 2, the present embodiment adopts a compressed sensing-based multi-channel compressed sensing optimization method, which includes the following steps S1 to S3:
s1, obtaining signals X of each channel, and performing projection measurement on the signals of each channel by using a measurement matrix to obtain an observed value vector YMMV= Φ X + e = a θ + e, where: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
s2, decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vector
Figure BDA0002054754370000061
Wherein: phi (phi) ofSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
s3, according to the optimized observed value vector, optimal constraint is solved
Figure BDA0002054754370000065
The norm reconstructs the original signal.
It should be noted that, in the DCS case, the signals of J channels are all compressible (sparse) or compressible at some sparse basis, that is, any channel signal
Figure BDA0002054754370000062
All can use Nx 1 dimension base vector
Figure BDA0002054754370000063
The linear representation is performed as shown in the following equation:
Figure BDA0002054754370000064
wherein, thetajIs a projection coefficient, and when a signal can be linearly represented by K basis vectors, the signal is said to be K-sparse, i is greater than or equal to 1 and less than or equal to N, and J is greater than or equal to 1 and less than or equal to J.
Further, the observation value vector is one measurement vector measured by a single measurement vector model, or J measurement vectors measured by a multi-measurement vector model. Wherein:
(1) Single Measurement Vector (SMV) problem:
signals with K sparseness on a suitably chosen sparseness basis, such as the orthonormal basis
Figure BDA0002054754370000071
An accurate reconstruction can be obtained by its M linear projections on another set of incoherent bases. Then the single measurement vector problem can be defined as:
Figure BDA0002054754370000072
since X is a column Vector and only a Single Measurement Vector is observed, the above problem is also referred to as a Single Measurement Vector problem (SMV) in CS. Wherein the content of the first and second substances,
Figure BDA0002054754370000073
is the measurement matrix for the jth signal, Ψ is the sparse basis for signal X, herein
Figure BDA0002054754370000074
Taking the identity matrix, in general phijWith xjDifferent from each other. e is the Gaussian noise, and the noise is the noise,
Figure BDA0002054754370000075
representing a real field.
Then there are:
Figure BDA0002054754370000076
wherein the measurement matrix Φ is a block diagonal matrix, each diagonal element
Figure BDA0002054754370000077
Is a random matrix, each sparse signal xjShare a common vector support, J ∈ {1,2, …, J }.
(2) Multiple Measurement Vector (MMV) problem:
when in use
Figure BDA0002054754370000078
J ∈ {1,2., J } are all the same, i.e., the same measurement matrix is used for each channel, and signal X = [ X ] =1；x2；…；xj；…；xJ]The transmission can be carried out in Multiple channels, and a plurality of Measurement vectors exist after observation, so the problem is also called a Multiple Measurement Vector (MMV) problem. Then the definition is as follows:
Figure BDA0002054754370000081
as shown in figure 3 of the drawings,
wherein, thetaijFor the ith element of the jth channel, J ∈ {1,2. For each individual channel, a measurement vector y can be obtainedj＝Φjxj＝ΦjΨjθj。
The measured values of the J sensing nodes are expressed by the following formula:
Figure BDA0002054754370000082
further, the step S2: decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vector
Figure BDA0002054754370000083
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise. The method specifically comprises the following steps S21 to S22:
s21, decomposing the measurement matrix phi by using a singular value decomposition method to obtain a new sensing system:
Figure BDA0002054754370000084
wherein: d1＝diag(σ1,σ2,…,σM) Is M × M dimensional diagonal matrix, phi = UDVTSubmatrix of matrix V
Figure BDA0002054754370000085
As a column orthogonal matrix, i.e. V1 TV1＝EM，
Figure BDA0002054754370000086
EMAnd EN-MM × M and (N-M) × (N-M) dimensional identity matrices, respectively; d is belonged to RM×NThe matrix is a semi-positive definite diagonal matrix, and elements on non-diagonal lines of the matrix are zero;
s22, calculating formula YMMVTwo-sided left-hand multiplication matrix of
Figure BDA0002054754370000091
Obtaining an optimized observation value vector:
Figure BDA0002054754370000092
wherein:
Figure BDA0002054754370000093
ΦSVD＝V1 T，
Figure BDA0002054754370000094
specifically, singular Value Decomposition (SVD) may decompose any one row full rank measurement matrix Φ as follows:
Φ＝UDVT；
wherein Φ represents a measurement matrix during signal reconstruction, an
Figure BDA0002054754370000095
Are all orthogonal matrices, i.e.:
UUT＝EM，
VVT＝EN，
in the formula: d is belonged to RM×NIs a semi-positive definite diagonal matrix with zero off-diagonal elements, i.e., D = diag (σ)1,σ2,…,σM) K is the rank of the matrix, which is equal to the number of non-zero singular values, σk(k =1, …, M) are all non-zero singular values of matrix Φ, all σkUniquely determined by the decomposition, the formula UDVTKnown as singular value decomposition of Φ.
Taking the decomposition of the measurement matrix in the process of multi-measurement vector model as an example:
measurement matrix phijPerforming SVD operation to obtain formula YMMV＝Φj[x1,x2,...,xJ]+ e is rewritten as the following formula:
Figure BDA0002054754370000101
in the formula: d1＝diag(σ1,σ2,…,σM) Is an M multiplied by M dimension diagonal matrix, 0 is an M multiplied by (N-M) dimension all-zero matrix, a sub-matrix of a matrix V
Figure BDA0002054754370000102
As a column orthogonal matrix, i.e. V1 TV1＝EM，
Figure BDA0002054754370000103
Wherein EMAnd EN-MRespectively, M and (N-M) X (N-M) dimensional identity matrices. By making a pair of YMMVTwo-sided left-hand multiplication matrix
Figure BDA0002054754370000104
It is possible to obtain:
Figure BDA0002054754370000105
a new measurement system is available:
Figure BDA0002054754370000106
in the formula:
Figure BDA0002054754370000107
ΦSVD＝V1 T，
and YMMV= Φ X + e = a θ + e contrast, measurement matrix Φ at this timeSVDIs a row orthogonal matrix, i.e.
Figure BDA0002054754370000108
It should be noted that, in this embodiment, the sparsity between the signals of the channels is represented by using a joint sparse signal model JSM-2. When the JSM-2 model is adopted, all signals share the same vector support, different JSMs correspond to different reconstruction algorithms, in this embodiment, a synchronous Orthogonal Matching Pursuit joint reconstruction algorithm (SOMP) is mainly adopted, that is, the optimized observed value vector is processed by solving the constrained optimal joint reconstruction algorithm
Figure BDA0002054754370000114
The norm reconstructs the original signal.
It should be noted that, in this embodiment, singular value decomposition is applied to the measurement matrix, and then an optimized separable reconstruction matrix and an optimized measurement are obtained, since rows of the optimized reconstruction matrix are orthogonal to each other, correlation between measurement values is reduced, and selection of a true support set is optimized, thereby greatly improving reconstruction accuracy of signals.
Further, in the JSM-2 model, the principle of the SOMP algorithm is: the method comprises the following steps of solving the correlation between different measurement values and corresponding compressed sensing measurement matrixes in each greedy iteration process, then summing correlation coefficients corresponding to a signal set support set, selecting the support set with the maximum correlation as a support set for signal reconstruction, wherein the target function is as follows:
Figure BDA0002054754370000111
Figure BDA0002054754370000112
in order to reconstruct the signal(s),
Figure BDA0002054754370000115
for sparse coefficients of signal X
Figure BDA0002054754370000116
Norm, which represents the number of non-zero elements in vector θ.
As shown in fig. 4, the present embodiment discloses a compressed sensing-based multi-channel compressed sensing optimization system, which includes an observation vector calculation module 10, an observation vector optimization module 20, and an original signal reconstruction module 30;
the observation value vector calculation module 10 is configured to obtain a signal X of each channel, and perform projection measurement on the signal of each channel by using a measurement matrix to obtain an observation value vector YMMV= Φ X + e = a θ + e, wherein: x = [ X ]1；x2；…；xj；…；xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
the observed value vector optimization module 20 is configured to decompose the measurement matrix Φ by using a singular value decomposition method to obtain an optimized observed value vector
Figure BDA0002054754370000113
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
the original signal reconstruction module 30 is used for solving the constraint optimization according to the optimized observed value vector
Figure BDA0002054754370000128
The norm reconstructs the original signal.
Further, the observation value vector is one measurement vector measured by a single measurement vector model, or J measurement vectors measured by a multi-measurement vector model.
Further, the observation value vector optimization module 20 includes a singular value decomposition unit 21 and an optimization unit 22;
the singular value decomposition unit 21 is configured to decompose the measurement matrix Φ by using a singular value decomposition method to obtain a new sensing system:
Figure BDA0002054754370000121
wherein: d1＝diag(σ1,σ2,…,σM) Is M × M dimensional diagonal matrix, phi = UDVTSubmatrix of matrix V
Figure BDA0002054754370000122
As a column orthogonal matrix, i.e. V1 TV1＝EM，
Figure BDA0002054754370000123
EMAnd EN-MRespectively, M × M and (N-M) × (N-M) dimensional identity matrices; d is belonged to RM×NThe matrix is a semi-positive definite diagonal matrix, and elements on non-diagonal lines of the matrix are zero;
the optimization unit 22 is used for optimizing the formula YMMVTwo-sided left multiplication matrix
Figure BDA0002054754370000127
Obtaining an optimized observation value vector:
Figure BDA0002054754370000124
wherein:
Figure BDA0002054754370000125
ΦSVD＝V1 T，
Figure BDA0002054754370000126
furthermore, the sparsity among the signals of all the channels is represented by adopting a joint sparse signal model JSM-2; when the JSM-2 model is adopted, all signals share the same vector support, different JSMs correspond to different reconstruction algorithms, in this embodiment, a synchronous Orthogonal Matching Pursuit joint reconstruction algorithm (SOMP) is mainly adopted, that is, the optimized observed value vector is processed by solving the constrained optimal combined reconstruction algorithm
Figure BDA0002054754370000132
The norm reconstructs the original signal.
The present embodiment also discloses a computer-readable storage medium including a program that, when executed, causes an apparatus including the computer-readable storage medium to perform steps S1 to S3 of:
s1, obtaining signals X of each channel, and carrying out projection measurement on the signals of each channel by using a measurement matrix to obtain an observed value vector YMMV= Φ X + e = a θ + e, wherein: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
s2, decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vector
Figure BDA0002054754370000131
Wherein: phi (phi) ofSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
s3, solving constraint optimization according to the optimized observed value vector
Figure BDA0002054754370000133
The norm reconstructs the original signal.
Those of ordinary skill in the art will understand that: all or part of the steps for implementing the method embodiments may be implemented by hardware related to program instructions, and the program may be stored in a computer readable storage medium, and when executed, the program performs the steps including the method embodiments; and the aforementioned storage medium includes: various media that can store program codes, such as ROM, RAM, magnetic or optical disks.
It should be noted that the compressed sensing-based multi-channel compressive sensing optimization system and the computer-readable storage medium disclosed in this embodiment correspond to the disclosed steps of the compressed sensing-based multi-channel compressive sensing optimization method, and are not described herein again.
To verify the effectiveness of the method presented herein, this section designed 4 sets of experiments for the same sparse signal.
Experiment 1: single reconstruction experiment
For the single multi-channel signal recovery problem, Y is setSMV＝ΦX+eAnd YMMV＝Φj[x1,x2,…,xJ]The signal recovery in the + e equation is named as SMV and MMV models, and after SVD is introduced, it is named as SMV + SVD and MMV + SVD, respectively. A single reconstitution experiment was performed for the SMV, MMV, SMV + SVD and MMV + SVD problems as shown in figure 5: fig. 5 (a) and 5 (b) show the original signal and the observed signal, respectively, and fig. 5 (c), 5 (d), 5 (e), and 5 (f) show the reconstructed results of the original signals of the SMV, MMV, SMV + SVD, and MMV + SVD methods, respectively. In the experiment herein, the length of the signal N =150, the number of measurements M =100, and the signal sparsity K =40. The OMP algorithm and the SOMP algorithm are used for signal reconstruction in the SMV and MMV problems, respectively.
The image reconstruction effect is quantified by using the Signal to Noise Ratio (SNR), and as can be seen from fig. 5, the SNRs in the SMV, SMV + SVD, MMV, and MMV + SVD problems are 64dB, 83dB, 352dB, and 337dB, respectively. In a single experiment, the MMV method is obviously superior to the SMV problem, particularly, when SVD is used as a preprocessing step of an observed signal, the reconstruction effect of the signal is obviously superior to the SMV problem, and the reconstruction precision is obviously improved.
On the basis of a single reconstruction experiment, in order to further analyze the performance of the method provided by the scheme, the scheme respectively carries out the following analysis on three variables in the signal reconstruction process: the signal length N, the number of measurements M and the added noise N were repeated 1000 times, and the following experiments were designed considering that the two parameters were kept constant and the reconstruction success probability under various methods was changed for the other variable.
Experiment 2: reconstruction experiment with different measurement times
The experiment was designed in consideration of how the reconstruction probability changes under various methods of changing the measurement times M under the condition that the signal length N and the signal sparsity K are constant. In the experiment, the signal length N =150 and the sparsity K =40 of the signal are kept unchanged, the value of the measurement times M is changed from 40 to 160, the step length is set to be 5, and the sparse signal and the measurement matrix are independently and randomly generated under the setting of each group of parameters (N, M and K). Under the same group of sparse signals, respectively introducing SVD (singular value decomposition) and not introducing SVD into the measurement matrix, and finally reconstructing by using different methods, wherein the experiment is independently repeated for 1000 times, and the probability of success in reconstruction of the 1000 experimental results is counted, as shown in FIG. 6.
Experiment 3: different sparsity reconstruction experiment
Similarly, the experiment was designed in consideration of how the reconstruction probability changes under various methods by changing the sparsity K of the signal under the condition that the signal length N and the measurement frequency M are fixed. In the experiment, the signal length N =150 and the measurement frequency M =100 of the signal are kept unchanged, the value of the sparsity K is changed from 10 to 100, the step length is set to be 5, and the sparse signal and the measurement matrix are independently and randomly generated under the setting of each group of parameters (N, M and K). Under the same group of sparse signals, respectively introducing SVD (singular value decomposition) and not introducing SVD into the measurement matrix, and finally reconstructing by using different methods, wherein the experiment is independently repeated for 1000 times, and the probability of success in reconstruction of 1000 experiments is counted, as shown in FIG. 7.
Experiment 4: robustness testing
The experiment was designed taking into account the noise immunity under noise of the method proposed herein. In this experiment, the signal length N =150, the number of times of signal measurement M =100, and the signal sparsity K =40 were kept constant, the setting parameter σ represents the noise standard deviation, the value of σ was changed from 0 to 0.5, and the step size was set to 0.035. The sparse signals and measurement matrices are independently randomly generated at each set of parameter (N, M, K) settings. Under the same group of sparse signals, respectively introducing SVD (singular value decomposition) and not introducing SVD into the measurement matrix, and finally reconstructing by using different methods, wherein the experiment is independently repeated for 1000 times, and the probability of success in reconstruction of 1000 experiments is counted, as shown in FIG. 8.
As shown in fig. 5-8: in experiment 1, the signal reconstruction success probability increases with the number of measurements M for the SMV, SMV + SVD, MMV and MMV + SVD methods. In experiment 2, the signal reconstruction success probability decreases as the signal sparsity K increases. In experiment 3, in the presence of noise, the reconstruction success probability of the signal is reduced along with the increase of the standard deviation of the added noise, but the reconstruction effect of the MMV problem is always better than that of the SMV problem; after the SVD is applied to the perception matrix as a preprocessing step for signal reconstruction, the reconstruction effect is obviously better than the case without the SVD. In summary, their reconstruction performance is in turn: the SMV method is the worst, the SMV + SVD method is the next, the MMV method is the next, and the MMV + SVD method provided by the text has the best reconstruction effect.
It should be noted that, singular value decomposition is applied to the measurement matrix, and then an optimized separable reconstruction matrix and an optimized measurement are obtained, since rows of the optimized reconstruction matrix are mutually orthogonal, correlation between measurement values is reduced, and selection of a real support set is optimized, thereby greatly improving reconstruction accuracy of signals. And the numerical experiment result shows that for the recovery of the multi-channel signal, the complexity of system design is reduced, the reconstruction precision of the signal is greatly improved, and in addition, the reconstruction success probability under the noise interference condition is also obviously improved.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and should not be taken as limiting the scope of the present invention, which is intended to cover any modifications, equivalents, improvements, etc. within the spirit and scope of the present invention.

Claims (8)

1. A multichannel compressed sensing optimization method based on compressed sensing is characterized by comprising the following steps:
acquiring signals X of each channel, and performing projection measurement on the signals of each channel by using a measurement matrix to obtain an observed value vector YMMV= Φ X + e = a θ + e, wherein: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vector
Figure FDA0003830026690000011
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
according to the optimized observed value vector, the optimal l is constrained by solving0Reconstructing an original signal by using the norm;
the method for decomposing the measurement matrix phi by using the singular value decomposition method to obtain the optimized observation value vector comprises the following steps:
decomposing the measurement matrix phi by using a singular value decomposition method to obtain a new sensing system:
Figure FDA0003830026690000012
wherein: d1＝diag(σ1,σ2,…,σM) Is M × M dimensional diagonal matrix, phi = UDVTSubmatrix of matrix V
Figure FDA0003830026690000013
Is a matrix of orthogonal columns,i.e. V1 TV1＝EM，
Figure FDA0003830026690000014
EMAnd EN-MRespectively, M × M and (N-M) × (N-M) dimensional identity matrices; d belongs to RM×NThe matrix is a semi-positive definite diagonal matrix, and elements on non-diagonal lines of the matrix are zero;
in said YMMVTwo-sided left-hand multiplication matrix
Figure FDA0003830026690000018
Obtaining an optimized observation value vector:
Figure FDA0003830026690000015
wherein:
Figure FDA0003830026690000016
ΦSVD＝V1 T，
Figure FDA0003830026690000017
2. the compressed sensing-based multi-channel compressed sensing optimization method of claim 1, wherein the observation vector is one measurement vector measured by a single measurement vector model or J measurement vectors measured by a multi-measurement vector model.
3. The compressed sensing-based multi-channel compressed sensing optimization method according to claim 1, wherein sparsity among signals of each channel is characterized by adopting a joint sparse signal model JSM-2.
4. The compressed sensing-based multi-channel compressed sensing optimization method of claim 3, wherein the basis-optimal isThe normalized observed value vector is optimized by solving the constraint0Reconstructing an original signal by using the norm, specifically:
and processing the optimized observed value vector by adopting a synchronous orthogonal matching pursuit combined reconstruction algorithm to reconstruct an original signal.
5. A multi-channel compressed sensing optimization system based on compressed sensing is characterized by comprising an observed value vector calculation module, an observed value vector optimization module and an original signal reconstruction module;
the observation value vector calculation module is used for acquiring the signals X of each channel and performing projection measurement on the signals of each channel by using a measurement matrix to obtain an observation value vector YMMV= Φ X + e = a θ + e, wherein: x = [ X ]1,x2,…,xj,…,xJ]Phi is a measurement matrix, A is a perception matrix, theta is a projection coefficient, and e is Gaussian noise;
the observation value vector optimization module is used for decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observation value vector
Figure FDA0003830026690000021
Wherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
the original signal reconstruction module is used for solving the constraint optimal l according to the optimized observed value vector0Reconstructing an original signal by using the norm;
the observation value vector optimization module comprises a singular value decomposition unit and an optimization unit;
the singular value decomposition unit is used for decomposing the measurement matrix phi by using a singular value decomposition method to obtain a new sensing system:
Figure FDA0003830026690000031
wherein: d1＝diag(σ1,σ2,…,σM) Is an M multiplied by M dimensional diagonal matrix, A =UDVTSubmatrix of matrix V
Figure FDA0003830026690000032
As a column orthogonal matrix, i.e. V1 TV1＝EM，
Figure FDA0003830026690000033
EMAnd EN-MRespectively, M × M and (N-M) × (N-M) dimensional identity matrices; d is belonged to RM×NThe matrix is a semi-positive definite diagonal matrix, and elements on non-diagonal lines of the matrix are zero;
the optimization unit is used for calculating the formula YMMVTwo-sided left multiplication matrix D1 -1UTAnd obtaining an optimized observed value vector:
Figure FDA0003830026690000034
wherein:
Figure FDA0003830026690000035
ΦSVD＝V1 T，
Figure FDA0003830026690000036
6. the compressed sensing-based multi-channel compressed sensing optimization system of claim 5, wherein the observation vector is one measurement vector measured by a single measurement vector model or J measurement vectors measured by a multi-measurement vector model.
7. The compressed sensing-based multi-channel compressed sensing optimization system according to claim 6, wherein sparsity among signals of each channel is characterized by adopting a joint sparse signal model JSM-2;
accordingly, the optimized observation value is usedVector, by solving for constrained optimal l0Reconstructing an original signal by using the norm, specifically:
and processing the optimized observed value vector by adopting a synchronous orthogonal matching pursuit joint reconstruction algorithm to reconstruct an original signal.
8. A computer-readable storage medium, storing a computer program which, when executed by a processor, causes the processor to carry out the steps of the method according to any one of claims 1 to 4.
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Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN105490682A (en) * 2015-11-27 2016-04-13 安庆师范学院 Compressive sensing signal recovery algorithm based on SVD (singular value decomposition) and OMP (orthogonal matching pursuit)
CN107516301A (en) * 2017-08-30 2017-12-26 中国科学院光电技术研究所 It is a kind of based on compressed sensing in image reconstruction calculation matrix constitution optimization method
CN108832934A (en) * 2018-05-31 2018-11-16 安徽大学 A kind of two-dimensional quadrature match tracing optimization algorithm based on singular value decomposition
CN109523486A (en) * 2018-11-22 2019-03-26 合肥工业大学 Based on the multichannel brain electric signal reconfiguring method of robust compressed sensing under noise circumstance

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20120259590A1 (en) * 2011-04-11 2012-10-11 Jong Chul Ye Method and apparatus for compressed sensing with joint sparsity

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN105490682A (en) * 2015-11-27 2016-04-13 安庆师范学院 Compressive sensing signal recovery algorithm based on SVD (singular value decomposition) and OMP (orthogonal matching pursuit)
CN107516301A (en) * 2017-08-30 2017-12-26 中国科学院光电技术研究所 It is a kind of based on compressed sensing in image reconstruction calculation matrix constitution optimization method
CN108832934A (en) * 2018-05-31 2018-11-16 安徽大学 A kind of two-dimensional quadrature match tracing optimization algorithm based on singular value decomposition
CN109523486A (en) * 2018-11-22 2019-03-26 合肥工业大学 Based on the multichannel brain electric signal reconfiguring method of robust compressed sensing under noise circumstance

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
Data Compression in Smart Distribution Systems via Singular Value Decomposition;Julio Cesar Stacchini de Souza;《IEEE TRANSACTIONS ON SMART GRID》;20170131;第8卷(第1期);275-284 *

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