CN110166055A - A kind of compressed sensing based multichannel compression sensing optimization method and system - Google Patents
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Abstract
The invention discloses a kind of compressed sensing based multichannel compression sensing optimization method and systems, belong to compressed sensing technology field, including carrying out singular value decomposition to calculation matrix, observation vector after being optimized, and combine algorithm for reconstructing using synchronous orthogonal matching pursuit to rebuild original signal.The present invention is by carrying out singular value decomposition to calculation matrix, realize the separate design of calculation matrix and reconstruction matrix, obtain separable restructuring matrix, since the restructuring matrix row of optimization is mutually orthogonal, reduce the correlation between measured value, it uses the measurement to reconstruct original signal, substantially increases the reconstruction precision of signal.
Description
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 using 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 certain 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 YMMVX + e ═ a θ + e, where: x ═ X1,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 vectorWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
optimizing by solving constraints according to the optimized observation value vectorThe 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:
wherein: d1＝diag(σ1,σ2,…,σM) Is an M × M dimensional diagonal matrix, phi ═ UDVTSubmatrix of matrix VAs a column orthogonal matrix, i.e.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;
left multiplication matrix at both sides of calculation formula of new sensing systemObtaining an optimized observation value vector:
wherein:ΦSVD＝V1 T，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 vectorReconstructing 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 YMMVX + e ═ a θ + e, where: x ═ X1,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 vectorWherein: 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 vectorThe 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 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:
wherein: d1＝diag(σ1,σ2,…,σM) Is an M × M dimensional diagonal matrix, phi ═ UDVTSubmatrix of matrix VAs a column orthogonal matrix, i.e. V1 TV1＝EM，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 matrixObtaining an optimized observation value vector:
wherein:ΦSVD＝V1 T，
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 vectorReconstructing 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 YMMVX + e ═ a θ + e, where: x ═ X1,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 vectorWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
optimizing by solving constraints according to the optimized observation value vectorThe norm reconstructs the original signal.
Compared with the prior art, the invention has the following technical effects: 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.
Drawings
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, acquiring the signal X of each channel, and performing projection measurement on the signal of each channel by using a measurement matrix to obtain an observed value vector YMMVX + e ═ a θ + e, where: x ═ X1,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 observation value vectorWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
s3, solving constraint optimization according to the optimized observed value vectorThe norm reconstructs the original signal.
It should be noted that, in the case of DCS, the signals of J channels are all compressible (sparse) or compressible on some sparse basis, i.e. any channel signalAll can use Nx 1 dimension base vectorThe linear representation is performed as shown in the following equation:
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:
with K sparse signals on appropriately chosen sparse basis, such as orthonormal basisAn accurate reconstruction can be obtained by its M linear projections on another set of incoherent basis. The single measurement vector problem can be defined as:
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,is the measurement matrix for the jth signal, Ψ is the sparse basis for signal X, hereinTaking the identity matrix, in general phijWith xjDifferent from each other. e is the Gaussian noise, and the noise is the noise,representing a real field.
Then there are:
wherein the measurement matrix Φ is a block diagonal matrix, each diagonal elementIs a random matrix, each sparse signal xjShare a common vector support, J ∈ {1,2, …, J }.
(2) Multiple Measurement Vector (MMV) problem:
when in useJ ∈ {1, 2..., J } all being the same, i.e., the same measurement matrix is used for each channel, and the signal 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:
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:
further, the above step S2: decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observed value vectorWherein: 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:
wherein: d1＝diag(σ1,σ2,…,σM) Is an M × M dimensional diagonal matrix, phi ═ UDVTSubmatrix of matrix VAs a column orthogonal matrix, i.e. V1 TV1＝EM，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;
s22, calculating formula YMMVTwo-sided left multiplication matrixObtaining an optimized observation value vector:
wherein:ΦSVD＝V1 T，
specifically, Singular Value Decomposition (SVD) may decompose any one row full rank measurement matrix Φ as follows:
Φ＝UDVT；
wherein phi denotes signal reconstructionA measurement matrix in the process, anAre all orthogonal matrices, i.e.:
UUT＝EM，
VVT＝EN，
in the formula: d is belonged to RM×NIs a semi-positive definite diagonal matrix, the off-diagonal elements of which are all zero, 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) is all non-zero singular values of the 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 follows:
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 VAs a column orthogonal matrix, i.e. V1 TV1＝EM，Wherein EMAnd EN-MRespectively, M and (N-M) X (N-M) dimensional identity matrices. By making a pair of YMMVTwo-sided left-hand multiplication matrixIt is possible to obtain:
a new measurement system is available:
in the formula:
ΦSVD＝V1 T，
and YMMVWhen Φ X + e is compared to a θ + e, the measurement matrix ΦSVDIs a row orthogonal matrix, i.e.
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 combined reconstruction algorithmThe 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: in order to reconstruct the signal(s),for sparse coefficients of signal XNorm, 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 YMMVX + e ═ a θ + e, where: x ═ X1；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 valueVector of posterior observationsWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
the original signal reconstruction module 30 is used for solving the constrained optimal condition according to the optimized observed value vectorThe 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:
wherein: d1＝diag(σ1,σ2,…,σM) Is an M × M dimensional diagonal matrix, phi ═ UDVTSubmatrix of matrix VAs a column orthogonal matrix, i.e. V1 TV1＝EM，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 matrixObtaining an optimized observation value vector:
wherein:ΦSVD＝V1 T，
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 algorithmThe 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 as follows:
s1, acquiring the signal X of each channel, and performing projection measurement on the signal of each channel by using a measurement matrix to obtain an observed value vector YMMVX + e ═ a θ + e, where: x ═ X1,x2,…,xj,…,xJ]Phi is the measurement matrix, A is the perception matrix, and theta is the projectionCoefficient, e is gaussian noise;
s2, decomposing the measurement matrix phi by using a singular value decomposition method to obtain an optimized observation value vectorWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
s3, solving constraint optimization according to the optimized observed value vectorThe 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 formula is named SMV and MMV models, and after SVD is introduced, they are named SMV + SVD and MMV + SVD, respectively. Performing single reconstruction experiment on SMV, MMV, SMV + SVD and MMV + SVD problems, such as graphAnd 5, as follows: 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, the length N of the signal is 150, the measurement number M is 100, and the signal sparsity K is 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, and 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 steps 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 is 150, the sparsity K of the signal is 40, the value of the number of times of measurement M is changed from 40 to 160, the step length is set to 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 is 150, the measurement times M of the signal is 100, the value of the sparsity K is changed from 10 to 100, the step length is set to 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 was 150, the number of times of signal measurement M was 100, and the degree of sparsity K of the signal was 40, and the setting parameter σ represents the standard deviation of noise, and the magnitude of the value 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 is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (10)
1. A multi-channel compressive sensing optimization method based on compressive 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 YMMVX + e ═ a θ + e, where: x ═ X1,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 measurement matrix phiVector of observationsWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
according to the optimized observed value vector, the optimal l is constrained by solving0The norm reconstructs the original signal.
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 or 2, wherein the decomposing the measurement matrix Φ by using a singular value decomposition method to obtain an optimized observation value vector comprises:
decomposing the measurement matrix phi by using a singular value decomposition method to obtain a new sensing system:
wherein: d1＝diag(σ1,σ2,…,σM) Is an M × M dimensional diagonal matrix, phi ═ UDVTSubmatrix of matrix VAs a column orthogonal matrix, i.e. V1 TV1＝EM，EMAnd EN-MRespectively, M × M and (N-M) × (N-M) dimensional identity matrices; d is belonged to RM×NIs a semi-positive definite diagonal matrix with off-diagonal elementsAll elements are zero;
in said YMMVTwo-sided left-hand multiplication matrixObtaining an optimized observation value vector:
wherein:ΦSVD＝V1 T，
4. the compressed sensing-based multi-channel compressed sensing optimization method according to claim 3, wherein sparsity among signals of each channel is characterized by adopting a joint sparse signal model JSM-2.
5. The compressed sensing-based multi-channel compressed sensing optimization method of claim 4, wherein the optimal l is solved by solving a constraint according to the optimized observation vector0Reconstructing 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.
6. 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 YMMVX + e ═ a θ + e, where: x ═ X1,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 vectorWherein: 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 vector0The norm reconstructs the original signal.
7. The compressed sensing-based multi-channel compressed sensing optimization system of claim 6, 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.
8. The compressed sensing-based multi-channel compressed sensing optimization system according to claim 6 or 7, wherein the observation 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:
wherein: d1＝diag(σ1,σ2,…,σM) Is an M × M dimensional diagonal matrix, A ═ UDVTSubmatrix of matrix VAs a column orthogonal matrix, i.e. V1 TV1＝EM，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 matrixObtaining an optimized observation value vector:
wherein:ΦSVD＝V1 T，
9. the compressed sensing-based multi-channel compressed sensing optimization system according to claim 8, wherein sparsity among signals of each channel is characterized by adopting a joint sparse signal model JSM-2;
correspondingly, the optimal l is constrained by solving the optimized observation value vector0Reconstructing 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.
10. A computer-readable storage medium comprising a program that, 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 YMMVX + e ═ a θ + e, where: x ═ X1,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 vectorWherein: phiSVDIs a row orthogonal matrix, and e' is optimized Gaussian noise;
according to the optimized observed value vector, the optimal l is constrained by solving0The norm reconstructs the original signal.
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