CN108832934B - Two-dimensional orthogonal matching pursuit optimization algorithm based on singular value decomposition - Google Patents

Two-dimensional orthogonal matching pursuit optimization algorithm based on singular value decomposition Download PDF

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CN108832934B
CN108832934B CN201810553080.1A CN201810553080A CN108832934B CN 108832934 B CN108832934 B CN 108832934B CN 201810553080 A CN201810553080 A CN 201810553080A CN 108832934 B CN108832934 B CN 108832934B
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张�成
陈倩文
王美琴
汪东
韦穗
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Abstract

The invention discloses a two-dimensional orthogonal matching pursuit optimization algorithm based on singular value decomposition, which comprises the steps of obtaining a row measurement matrix, a column measurement matrix, a measured value, a sparse basis and signal sparsity; performing SVD on the two measurement matrixes, updating the measurement matrixes and the measurement values, initializing a residual error and index set and an optimized sensing matrix, finding an index, and calculating a new approximation of a signal; and updating residual errors, continuously iterating, and finally outputting an estimated value and an index set of the signals. The invention realizes the separation of the measurement matrix in the front-end information acquisition and the reconstruction matrix in the back-end reconstruction, and is suitable for a general separable linear system. And carrying out SVD decomposition on the two measurement matrixes to obtain two optimized reconstruction matrixes, effectively eliminating the correlation between the measurement values and obviously improving the reconstruction signal-to-noise ratio and the robustness of the algorithm. And due to the fact that separable operators are used in the design of the measurement matrix, the method can be used for the reconstruction process of the large-scale image.

Description

Two-dimensional orthogonal matching pursuit optimization algorithm based on singular value decomposition
Technical Field
The invention relates to the technical field of image processing and compressed sensing imaging systems, in particular to a two-dimensional orthogonal matching tracking optimization algorithm based on singular value decomposition.
Background
Compressed Sensing (CS), as a new framework in terms of information acquisition and processing, etc., attempts to reduce the collection of redundant data in the information collection step by CS, as compared to a general framework that first collects as much data as possible and then discards the redundant data by digital compression techniques. Data compression is carried out while data are acquired, so that the information acquisition amount is greatly reduced, the information acquisition time is shortened, and the storage space is saved.
The CS theory includes three key technologies: sparse representation of signals, non-coherent measurement and reconstruction algorithms. The fast and efficient reconstruction algorithm is the core of applying the compressed sensing theory to the practical imaging system and is researched in recent yearsOne of the hot spots is characterized in that an original signal is accurately and efficiently recovered from a small amount of linear measurement values, and the original signal can be accurately reconstructed from the small amount of measurement values by utilizing the sparsity or compressibility of the original signal, so that the calculation burden is transferred from a signal acquisition device at the front end to a processor at the rear end. The CS reconstruction algorithms are mainly classified into three categories: the first type is based on0Greedy algorithm of norm; the second type is based on1A convex optimization algorithm of norm; the third category is non-convex optimization algorithms.
Greedy algorithm represented by Orthogonal Matching Pursuit (OMP) algorithm is a commonly used reconstruction algorithm, and has the advantages of high reconstruction speed, low calculation complexity and the like, and is widely applied. The basic idea is to select atoms to form a support set of a signal to be reconstructed based on a certain greedy criterion each time through an iteration method, namely finding out a local optimal solution to gradually approximate an original signal, and finally reconstructing the original signal. By searching for non-zero values in the orthogonal direction, the convergence speed of the algorithm is well improved. However, because the greedy algorithm represented by the OMP selects only one atom for each iteration, only the probability of obtaining approximate reconstruction is given for special measurement matrices (such as gaussian random matrices and bernoulli random matrices), accurate reconstruction is not guaranteed, and the running time of the algorithm is increased due to the fact that the support set is updated for each iteration.
In 2012, Fang et al proposed an orthogonal matching pursuit algorithm for two-dimensional signals, whose main idea was to represent the two-dimensional measurement Y as a weighted sum of 2D atoms selected from an overcomplete dictionary. In each iteration, the sample matrix is projected onto a two-dimensional atom to select the best matching atom, and then the weights of all selected atoms are updated by the least squares method. The biggest difference compared to conventional algorithms is that the best matching atom selected on iteration is two-dimensional. And demonstrates that 2DOMP is practically equivalent to 1DOMP, but with significantly reduced recovery complexity and memory usage. However, since the reconstruction matrix in the 2DOMP algorithm is the measurement matrix, the requirement of the reconstruction process on the matrix is not considered, which results in poor reconstruction performance in the reconstruction process.
Disclosure of Invention
The invention aims to provide a two-dimensional orthogonal matching pursuit optimization algorithm based on singular value decomposition so as to solve the problems that the 2DOMP algorithm does not have optimal performance and the like.
In order to achieve the above object, the present invention adopts a singular value decomposition-based two-dimensional orthogonal matching pursuit optimization algorithm for reconstructing a compressed two-dimensional sparse signal at a compressed sensing receiving end, including:
obtaining any separable measurement matrix
Figure BDA0001681034680000021
Separable rarefaction base
Figure BDA0001681034680000022
Measured value
Figure BDA0001681034680000023
And a degree of sparsity K, wherein
Figure BDA0001681034680000024
A column measurement matrix is represented which,
Figure BDA0001681034680000025
a matrix of line measurements is represented which,
Figure BDA0001681034680000026
Figure BDA0001681034680000027
which represents the product of the Kronecker reaction,
Figure BDA0001681034680000028
the elements of the representation matrix are real numbers, (g)TRepresenting the transposition operation of the matrix, wherein M is M multiplied by M, N is N multiplied by N, and M is less than or equal to N;
for the measurement matrix phix、ΦyRespectively carrying out singular value decomposition to obtain
Figure BDA0001681034680000029
Figure BDA00016810346800000210
Wherein the matrix UxAnd UyAre respectively a matrix
Figure BDA00016810346800000211
And
Figure BDA00016810346800000212
feature vector of DxAnd DyAre respectively a matrix phixAnd phiyOf singular values of, VxAnd VyAre respectively
Figure BDA00016810346800000213
And
Figure BDA00016810346800000214
the feature vector of (2);
get DxAnd V x1 to m columns of the matrix D are respectively obtained1xAnd matrix V1xTaking DyAnd V y1 to m columns of the matrix D are respectively obtained1yAnd matrix V1yThen has phix＝Ux(D1x O)(V1x V2x)T、Φy＝Uy(D1y O)(V1y V2y)TWherein D is1xAnd D1yDiagonal square matrices that are each m x m, and O represents a matrix of all 0's of size m x (n-m);
by using D1x、Ux、UyAnd D1yUpdating the measured value Y to obtain an updated measured value YSVD；
Using V1xFor the measurement matrix phixOptimizing and utilizing V1yFor the measurement matrix phiyOptimizing to respectively obtain optimized reconstruction matrix phixsAnd phiys；
From the reconstruction matrix phixsOptimized column with sparse basis ΨSensing matrix AxsFrom the reconstruction matrix phiysAnd the sparse basis Ψ to obtain an optimized row-sense matrix
Figure BDA0001681034680000031
Based on 2DOMP algorithm, the updated measured value Y isSVDOptimized column sensing matrix AxsAnd an optimized row sensing matrix
Figure BDA0001681034680000032
And processing is carried out, and an estimated value of the original two-dimensional sparse signal is reconstructed.
Preferably, the calculation model of the measurement value Y is:
Figure BDA0001681034680000033
wherein Z represents a two-dimensional sparse coefficient, AxRepresenting an un-optimized column sensing matrix,
Figure BDA0001681034680000034
representing an unoptimized row-sensing matrix.
Preferably, said utilization D1x、Ux、UyAnd D1yUpdating the measured value Y to obtain an updated measured value YSVDThe method specifically comprises the following steps:
Figure BDA00016810346800000310
preferably, said utilization V1xFor the measurement matrix phixOptimizing and utilizing V1yFor the measurement matrix phiyOptimizing to respectively obtain optimized reconstruction matrix phixsAnd phiysThe method specifically comprises the following steps:
Figure BDA0001681034680000035
Figure BDA0001681034680000036
preferably, the updated measurement value Y is calculated based on the 2DOMP algorithmSVDOptimized column sensing matrix AxsAnd an optimized row sensing matrix
Figure BDA0001681034680000037
Processing is carried out, and an estimated value of an original two-dimensional sparse signal is reconstructed, wherein the method comprises the following steps:
s101, initializing residual R ═ YSVDIndex collection
Figure BDA0001681034680000038
Namely, it is
Figure BDA0001681034680000039
Λ is a set of atomic row-column indices found for storage, ΛrIndex value, Λ, of atomic row found for storagecFor storing the column index value of the found atom, the iteration time t is 1;
s102, searching an atom index value (i, j) when the inner product of the atom and the residual R is maximum;
s103, updating the index set Lambdar＝ΛrUi，Λc＝ΛcUj；
S104, calculating the estimated value of the sparse signal by using a least square method
Figure BDA0001681034680000041
To minimize the Frobenius norm of the residual,
Figure BDA0001681034680000042
Figure BDA0001681034680000043
and
Figure BDA0001681034680000044
is to store per-index set ΛrAnd ΛcThe resulting set of atoms is then selected,
Figure BDA0001681034680000045
a weighting coefficient representing an atom;
s105, residual error is updated
Figure BDA0001681034680000046
S106, judging whether | | R | | non-woven phosphor is met0δ ≦ δ, δ representing a threshold to stop iteration;
s107, if yes, stopping iteration;
s108, if not, t is t +1, and whether t is less than or equal to K is judged;
s109, if yes, executing the step S102;
and S110, if not, stopping the circulation.
Compared with the prior art, the invention has the following technical effects: the invention introduces a matrix separable thought on two dimensions, a separable matrix is separated into a row measurement matrix and a column measurement matrix at a receiving end of compressed sensing, the row measurement matrix and the column measurement matrix are used as the input of a 2DOMP algorithm, Singular Value Decomposition (SVD) is carried out on the row measurement matrix and the column measurement matrix, the row measurement matrix, the column measurement matrix and a measured value are optimized by using the decomposition result of the row measurement matrix and the decomposition result of the column measurement matrix to obtain an optimized reconstruction matrix and a measured value, and then a signal estimation value corresponding to a two-dimensional sparse signal is recovered based on the optimized reconstruction matrix and the measured value in the 2DOMP algorithm. Compared with the existing 2DOMP algorithm, on one hand, the separable matrix is separated into a single matrix as input, the two-dimensional matrix is reduced into two matrices with small dimension, the problem of large calculation amount caused by product operation of a large matrix is solved, the memory space and the calculation complexity required by a storage matrix are obviously reduced, the problems of large matrix storage and calculation amount caused by large dimension in the traditional compression imaging are solved, and the method can be used for the imaging process of large-size images; on the other hand, SVD is introduced into the 2DOMP algorithm, optimization improvement is carried out on the measurement matrix, rows of the optimized measurement matrix are mutually orthogonal, correlation among measurement values is eliminated, selection of a support set is optimized in the reconstruction process, and reconstruction performance of the 2DOMP algorithm 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 flow diagram of separable compressed sensing;
FIG. 2 is a process diagram of a two-dimensional orthogonal matching pursuit optimization method of 2 DOMP-SVD;
FIG. 3 is a schematic diagram of an original two-dimensional sparse signal;
FIG. 4 is a graph of measurements taken when the individual separation matrices are all random Gaussian matrices;
FIG. 5 is a schematic diagram of a reconstructed signal obtained by processing the two-dimensional sparse signal of FIG. 3 using a conventional 2DOMP reconstruction algorithm;
FIG. 6 is a schematic diagram of a reconstructed signal obtained by processing the two-dimensional sparse signal of FIG. 3 using a 2DOMP-SVD reconstruction method;
FIG. 7 is a graph showing a comparison of a reconstructed signal obtained using 2DOMP and a reconstructed signal obtained using 2 DOMP-SVD;
FIG. 8 is a graph showing a comparison of the performance of 2DOMP and 2DOMP-SVD with increasing signal sparsity K for a constant number of measurements M;
FIG. 9 is a graph comparing the performance of 2DOMP and 2DOMP-SVD at different measurement numbers;
FIG. 10 is a graph of reconstructed power comparisons for 2DOMP and 2DOMP-SVD for different input noise interference scenarios.
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. 1 to fig. 2, the present embodiment discloses a two-dimensional orthogonal matching pursuit optimization algorithm based on singular value decomposition, which is used for performing signal reconstruction on a compressed two-dimensional sparse signal at a receiving end of compressed sensing. The basic idea is to combine SVD and 2DOMP algorithm to form a two-dimensional orthogonal matching pursuit Optimization algorithm (2DOMP Optimization Based on Singular Value Decomposition,2DOMP-SVD), which is to introduce a two-dimensional separable matrix, separate the two-dimensional separable matrix into a column measurement matrix and a row measurement matrix, operate the original image, execute SVD Decomposition on the two separated measurement matrices in the 2DOMP algorithm, update the measurement Value by using the Decomposition result, update the measurement matrix to obtain a reconstruction matrix at the rear end, thereby separate the rear end reconstruction matrix from the front end measurement matrix, optimize the rear end reconstruction matrix by SVD, so that the rows of the reconstruction matrix are orthogonal to each other, and meet the requirement of the rear end reconstruction process on the orthogonalization of the reconstruction matrix. The method specifically comprises the following steps:
s1, obtaining any separable measuring matrix
Figure BDA0001681034680000061
Separable rarefaction base
Figure BDA0001681034680000062
Measured value
Figure BDA0001681034680000063
And a degree of sparsity K, wherein
Figure BDA0001681034680000064
A column measurement matrix is represented which,
Figure BDA0001681034680000065
a matrix of line measurements is represented which,
Figure BDA0001681034680000066
Figure BDA0001681034680000067
which represents the product of the Kronecker reaction,
Figure BDA0001681034680000068
the elements of the representation matrix are real numbers, (g)TDenotes the transpose operation of a matrix, M x M, Nn is multiplied by n, m is less than or equal to n, and m and n are positive integers;
s2, measuring matrix phix、ΦyRespectively carrying out singular value decomposition to obtain
Figure BDA0001681034680000069
Figure BDA00016810346800000610
Wherein the matrix UxAnd UyAre respectively a matrix
Figure BDA00016810346800000611
And
Figure BDA00016810346800000612
feature vector of DxAnd DyAre respectively a matrix phixAnd phiyOf singular values of, VxAnd VyAre respectively
Figure BDA00016810346800000613
And
Figure BDA00016810346800000614
the feature vector of (2);
s3, getting DxAnd V x1 to m columns of the matrix D are respectively obtained1xAnd matrix V1xTaking DyAnd V y1 to m columns of the matrix D are respectively obtained1yAnd matrix V1yThen has phix＝Ux(D1x O)(V1x V2x)T、Φy＝Uy(D1y O)(V1y V2y)TWherein D is1xAnd D1yDiagonal square matrices that are each m x m, and O represents a matrix of all 0's of size m x (n-m);
s4, use of D1x、Ux、UyAnd D1yUpdating the measured value Y to obtain an updated measured value YSVD；
S5, use of V1xFor the measurement matrix phixOptimizing and utilizing V1yFor the measurement matrix phiyOptimizing to respectively obtain optimized reconstruction matrix phixsAnd phiys；
S6, according to the reconstruction matrix phixsOptimized column sense matrix A with sparse basis ΨxsFrom the reconstruction matrix phiysAnd the sparse basis Ψ to obtain an optimized row-sense matrix
Figure BDA0001681034680000071
S7, based on 2DOMP algorithm, the updated measured value YSVDOptimized column sensing matrix AxsAnd an optimized row sensing matrix
Figure BDA0001681034680000072
And processing is carried out, and an estimated value of the original two-dimensional sparse signal is reconstructed.
Further, as shown in fig. 1, in the above step S1, a two-dimensional signal is assumed
Figure BDA0001681034680000073
Is sparse on the sparse basis Ψ, with a sparsity K (K ═ n)2) I.e. only K nonzero values, the sparse coefficient is Z epsilon Cn×nI.e. X ═ Ψ Z ΨTC denotes the matrix element as a constant, measured value
Figure BDA0001681034680000074
m denotes the number of rows and n denotes the number of columns.
Figure BDA0001681034680000075
And
Figure BDA0001681034680000076
respectively a single separation matrix in the column and row directions. Similarly, assume a separable sparse basis ΨSCan be expressed as
Figure BDA0001681034680000077
Wherein
Figure BDA0001681034680000078
Figure BDA0001681034680000079
In this case, the measurement values can be written as:
Figure BDA00016810346800000710
in the formula, AxAnd
Figure BDA00016810346800000711
respectively column and row sensing matrices, operating on columns and rows, respectively, of a two-dimensional sparse signal, i.e. ax＝ΦxΨ，
Figure BDA00016810346800000712
Then, the 2DOMP algorithm or the 2DOMP-SVD algorithm is adopted to reconstruct the original signal from the measured value Y, and the process of reconstructing the original signal from the measured value Y by solving the following optimization problem is as follows:
Figure BDA00016810346800000713
subject to||vec(Z)||0≤K
wherein the content of the first and second substances,
Figure BDA00016810346800000714
representing the estimated value of the sparse signal obtained using the reconstruction algorithm, vec representing the vectorization operation, pulling a matrix into column vectors,
Figure BDA00016810346800000715
represents a value corresponding to when the value of the expression is minimized, | g | | non-calculation2Representing Frobenius norm, | | g | | purple0Represents a 0-norm, α ═ vec (z).
As shown in fig. 2, the process of reconstructing the original signal from the measured value Y by using the 2DOMP-SVD algorithm is as follows:
(1) respectively combining two measurement matrixes
Figure BDA0001681034680000081
Sparse base
Figure BDA0001681034680000082
Measured value
Figure BDA0001681034680000083
And the signal sparsity K is used as the input of the 2DOMP-SVD algorithm;
(2) for single measurement matrix phi respectivelyxPerforming singular value decomposition to obtain Ux，Dx，VxFor a single measurement matrix phiyPerforming singular value decomposition to obtain Uy，Dy，Vy. Wherein:
Figure BDA0001681034680000084
Figure BDA0001681034680000085
(3) respectively take DxM columns of (1) < M > form a matrix D1xGet Vx1: m columns of (A) form (V)1xRespectively take DyM columns of (1) form D1yGet Vy1: m columns of (A) form (V)1y. Wherein:
Figure BDA0001681034680000086
Figure BDA0001681034680000087
in the formula (I), the compound is shown in the specification,
Figure BDA0001681034680000088
Figure BDA0001681034680000089
are all orthogonal matrices and are all provided with a matrix,
Figure BDA00016810346800000810
Imand In-mAre all identity matrices with dimensions of m x m and (n-m) x (n-m), respectively.
Figure BDA00016810346800000811
Are all semi-positive definite diagonal square matrices, and O represents a full 0 matrix with dimension m x (n-m).
(4) Updating the measured values
Figure BDA00016810346800000812
Obtaining an optimized reconstruction matrix phixsAnd phiys. Wherein:
Figure BDA00016810346800000813
by left multiplication of the above formula
Figure BDA00016810346800000814
Right riding Uy(D1y)-1To obtain
Figure BDA00016810346800000815
The optimized measured values and the separation reconstruction matrix are finally obtained as follows:
Figure BDA00016810346800000816
Figure BDA00016810346800000817
Figure BDA0001681034680000091
at this time,. phixsAnd phiysAre all row orthogonal matrices, i.e.
Figure BDA0001681034680000092
ImRepresenting an identity matrix with dimensions m x m.
(5) Optimizing a single sensing matrix Axs＝ΦxsΨ，Ays＝ΦysΨ。
(6) Based on 2DOMP algorithm, for measured value YSVDArray sensing matrix AxsAnd a row sensing matrix
Figure BDA0001681034680000093
And processing is carried out, and an estimated value of the original two-dimensional coefficient signal is reconstructed.
It should be noted that, based on the 2DOMP algorithm, the measured value Y is measuredSVDArray sensing matrix AxsAnd a row sensing matrix
Figure BDA0001681034680000094
Processing is carried out, and an estimated value of an original two-dimensional coefficient signal is reconstructed, and the method specifically comprises the following steps:
s101, initializing residual R ═ YSVDIndex collection
Figure BDA0001681034680000095
Namely, it is
Figure BDA0001681034680000096
Λ is a set of atomic row-column indices found for storage, ΛrIndex value, Λ, of atomic row found for storagecFor storing the column index value of the found atom, the iteration time t is 1;
s102, searching an atom index value (i, j) when the inner product of the atom and the residual R is maximum:
Figure BDA0001681034680000097
wherein the content of the first and second substances,
Figure BDA0001681034680000098
represents the (i, j) th atom,
Figure BDA0001681034680000099
aiand ajRespectively, the column sensing matrix A after singular value decompositionxsAnd row sensing matrix
Figure BDA00016810346800000910
Wherein (i ', j') represents an index value of each atom, and a row number and a column number corresponding to the atom whose value is the largest after the inner product of each atom and the residual is calculated are assigned to (i, j).
S103, updating index set lambdar＝ΛrUi，Λc＝ΛcUj；
S104, calculating the estimated value of the sparse signal by using a least square method
Figure BDA00016810346800000911
To minimize the Frobenius norm of the residual, resulting in a new signal approximation
Figure BDA00016810346800000912
Figure BDA00016810346800000913
And
Figure BDA00016810346800000914
is to store per-index set ΛrAnd ΛcThe resulting set of atoms is then selected,
Figure BDA00016810346800000915
a weighting coefficient representing an atom;
s105, residual error is updated
Figure BDA00016810346800000916
S106, judging whether | | R | | non-woven phosphor is met0δ ≦ δ, δ representing a threshold to stop iteration;
s107, if yes, stopping iteration and executing the step S102;
s108, if not, t is t +1, and whether t is less than or equal to K is judged;
s109, if yes, executing the step S102;
and S110, if not, stopping the circulation.
According to the scheme, a separable matrix is separated into a measurement matrix in the row direction and a measurement matrix in the column direction, the dimension of the two-dimensional large matrix is reduced into two matrixes with smaller dimensions, the problem of large calculated amount caused by product operation of the large matrix is solved, the problems of matrix storage, calculated amount and the like caused by larger dimensions in the traditional compression imaging method are further solved, and the complexity involved in implementation, storage and calculation of the imaging matrix is obviously reduced by the separable CS design. Meanwhile, a singular value decomposition method is introduced into the 2DOMP algorithm, optimization improvement is carried out on the measurement matrix, rows of the optimized measurement matrix are mutually orthogonal, correlation among measurement values is eliminated, selection of a support set is optimized in the reconstruction process, and reconstruction performance of the 2DOMP algorithm is greatly improved.
As shown in fig. 3 to 7, to compare the reconstruction performance of the 2DOMP algorithm with the 2DOMP-SVD algorithm, a single reconstruction experiment was performed to test the reconstruction performance when the same measurement matrix and measurement values were used with 2DOMP and 2DOMP-SVD, respectively:
fig. 3 shows an original two-dimensional sparse signal, which has a dimension of 32 × 32, the sparse basis is an orthonormal basis, and the signal sparsity K is 150. For the 2DOMP algorithm, if high-precision reconstruction is required, the measurement number is generally 3-5 times of sparsity, but in the design of a measurement matrix, the complexity is reduced due to the introduction of a separable operator, but more measurement values are required. In the experiment, in order to better compare the influence of singular value decomposition in the reconstruction algorithm on the experiment result, the measurement number is selected as the measurement with higher reconstruction difficulty in the 2DOMP algorithm. Measured value 20X 20, single separate measurement matrix phixAnd phiyIs a 20 x 32 gaussian matrix,its elements meet the standard normal distribution.
Fig. 4 is a measured value obtained when the single separation matrix is a random gaussian matrix, fig. 5 is a reconstructed signal obtained when 2DOMP is used as a reconstruction algorithm, and fig. 6 is a reconstructed signal obtained when a 2DOMP-SVD optimization algorithm is used. FIGS. 7- (e) and 7- (f) are comparisons of 2DOMP and 2DOMP-SVD reconstructed signals.
As shown in fig. 7, by using a signal-to-noise ratio (SNR) as an objective metric, in fig. 7, only a part of the 2DOMP algorithm is accurately reconstructed, a large part of the reconstruction results have a large error, the SNR of the reconstruction results is only 8.601dB, and the reconstruction SNR of the same group of sparse signals and measurement matrices using the 2DOMP-SVD optimization algorithm reaches 20.3042dB, which is significantly improved.
The performance tests of the 2DOMP algorithm and the 2DOMP-SVD algorithm are respectively carried out under the conditions that the measured value is unchanged and the signal sparsity K is gradually increased, and the test comparison result is shown in FIG. 8. Gradually increasing the value of the sparsity K from 30 to 180, and setting the step size to 10; the dimension of the two-dimensional signal is 32 multiplied by 32; the sparse base is an orthonormal base; the two-dimensional measurement size is 20 × 20. And randomly generating a sparse signal and a measurement matrix under each group of parameters. And respectively reconstructing the same group of sparse signals and the measurement matrix by adopting 2DOMP and 2DOMP-SVD, and calculating an SNR value between a reconstruction result and an original signal. Under each set of parameter settings, the experiment was performed 1000 times independently, and the reconstructed SNR for each time was counted. The SNR threshold is set to 25dB, i.e. if the reconstruction SNR is greater than 25dB, the signal is deemed to be successfully reconstructed, otherwise, the signal fails. The reconstructed power under each set of parameters is counted as shown in fig. 4. It can be seen that under the same conditions, 2DOMP-SVD has a significant advantage over 2DOMP that is not optimized.
Fig. 9 is a graph showing the comparison of the performance of the 2DOMP and 2DOMP-SVD reconstruction algorithms tested under different numbers of measurements. The signal dimension is 32 × 32; sparsity K is 50; the sparse base is an orthonormal base; the measurement number is set to M × M, M is set to 10 to 24, and the step size is 1. Similarly, under each group of parameters, a sparse signal and a measurement matrix are randomly generated, and then the SNR value between the reconstruction result of the 2DOMP and the 2DOMP-SVD and the original signal is adopted for comparing the same sparse signal and the same measurement matrix. Under the setting of each group of parameters, the method is independently executed for 1000 times, the SNR threshold value is set to be 25dB, namely if the SNR is more than 25dB, a signal is successfully reconstructed, otherwise, the method fails. The reconstructed power curve plotted from 1000 replicates is shown in fig. 5. As can be seen from fig. 9, the same set of measurement matrices and measurements, using the 2DOMP-SVD algorithm, significantly improved the reconstruction.
And respectively carrying out robustness testing experiments of 2DOMP and 2DOMP-SVD at the same testing times. The result is shown in fig. 10, where the signal dimension is 32 × 32; the number of measurements was 20 × 20; sparsity K40; the magnitude of the input noise value is set to 12dB to 30dB with a step size of 1 dB. Then the same set of coefficient signals is calculated and the measurement matrix uses 2DOMP and 2DOMP-SVD as SNR values between the reconstruction result and the original signal at the time of the reconstruction algorithm. Under each set of parameter setting, the method is independently executed for 1000 times, the SNR threshold is set to be 25dB, namely if the SNR is more than 25dB, the signal is successfully reconstructed, otherwise, the method fails. The reconstructed power curve drawn by 1000 repeated experiments is shown in fig. 6, and it is obvious that the optimized 2DOMP-SVD can obviously improve the robustness of the system.
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 (3)

1. The utility model provides a two-dimentional orthogonal matching pursuit optimization algorithm based on singular value decomposition which characterized in that, the receiving end that is used for compressed sensing carries out the reconsitution to the two-dimentional sparse signal after compressing, includes:
obtaining any separable measurement matrix
Figure FDA0003305632770000011
Separable rarefaction base
Figure FDA0003305632770000012
Measured value
Figure FDA0003305632770000013
And a degree of sparsity K, wherein
Figure FDA0003305632770000014
A column measurement matrix is represented which,
Figure FDA0003305632770000015
a matrix of line measurements is represented which,
Figure FDA0003305632770000016
Figure FDA0003305632770000017
which represents the product of the Kronecker reaction,
Figure FDA0003305632770000018
the elements of the representation matrix are real numbers, (g)TDenotes the transpose operation of the matrix, M × M, N × N, M ≦ N, M denotes the measurement value Y ∈ Rm×mAnd the number of rows and columns, and the like, n represents the two-dimensional signal X ∈ Rn×nThe number of rows and columns is equal, and M and N are dimensions of the separable measurement matrix;
for the measurement matrix phix、ΦyRespectively carrying out singular value decomposition to obtain
Figure FDA0003305632770000019
Figure FDA00033056327700000110
Wherein the matrix UxAnd UyAre respectively a matrix
Figure FDA00033056327700000111
And
Figure FDA00033056327700000112
feature vector of DxAnd DyAre respectively a matrix phixAnd phiyOf singular values of, VxAnd VyAre respectively
Figure FDA00033056327700000113
And
Figure FDA00033056327700000114
the feature vector of (2);
get DxAnd Vx1 to m columns of the matrix D are respectively obtained1xAnd matrix V1xTaking DyAnd Vy1 to m columns of the matrix D are respectively obtained1yAnd matrix V1yThen has phix＝Ux(D1x O)(V1x V2x)T、Φy＝Uy(D1y O)(V1y V2y)TWherein D is1xAnd D1yDiagonal square matrices that are each m x m, and O represents a matrix of all 0's of size m x (n-m);
by using D1x、Ux、UyAnd D1yUpdating the measured value Y to obtain an updated measured value YSVD；
Using V1xFor the measurement matrix phixOptimizing and utilizing V1yFor the measurement matrix phiyOptimizing to respectively obtain optimized reconstruction matrix phixsAnd phiysThe method specifically comprises the following steps:
Figure FDA00033056327700000115
Figure FDA00033056327700000116
from the reconstruction matrix phixsOptimized column sense matrix A with sparse basis ΨxsFrom the reconstruction matrix phiysAnd the sparse basis Ψ to obtain an optimized row-sense matrix
Figure FDA00033056327700000117
Based on 2DOMP algorithm, the updated measured value Y isSVDOptimized column sensing matrix AxsAnd an optimized row sensing matrix
Figure FDA00033056327700000118
Processing is carried out, and an estimated value of an original two-dimensional sparse signal is reconstructed, wherein the method comprises the following steps:
s101, initializing residual R ═ YSVDIndex collection
Figure FDA0003305632770000021
Namely, it is
Figure FDA0003305632770000022
Λ is a set of atomic row-column indices found for storage, ΛrIndex value, Λ, of atomic row found for storagecFor storing the column index value of the found atom, the iteration time t is 1;
s102, searching an atom index value (i, j) when the inner product of the atom and the residual R is maximum;
s103, updating the index set Lambdar＝ΛrUi，Λc＝ΛcUj；
S104, calculating the estimated value of the sparse signal by using a least square method
Figure FDA0003305632770000023
To minimize the Frobenius norm of the residual, resulting in a new signal approximation
Figure FDA0003305632770000024
Figure FDA0003305632770000025
And
Figure FDA0003305632770000026
is to storeSet Λ by indexrAnd ΛcThe resulting set of atoms is then selected,
Figure FDA0003305632770000027
a weighting coefficient representing an atom;
s105, residual error is updated
Figure FDA0003305632770000028
S106, judging whether | | R | | non-woven phosphor is met0δ ≦ δ, δ representing a threshold to stop iteration;
s107, if yes, stopping iteration;
s108, if not, t is t +1, and whether t is less than or equal to K is judged, wherein K represents the signal sparsity;
s109, if yes, executing the step S102;
and S110, if not, stopping the circulation.
2. The singular value decomposition-based two-dimensional orthogonal matching pursuit optimization algorithm of claim 1, wherein the computational model of the measurement Y is:
Figure FDA0003305632770000029
wherein Z represents a sparse coefficient, AxRepresenting an un-optimized column sensing matrix,
Figure FDA00033056327700000210
representing an unoptimized row-sensing matrix.
3. The singular value decomposition-based two-dimensional orthogonal matching pursuit optimization algorithm of claim 1, wherein the utilizing D is1x、Ux、UyAnd D1yUpdating the measured value Y to obtain an updated measured value YSVDThe method specifically comprises the following steps:
Figure FDA00033056327700000211
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