阅读说明：本技术 二维免求逆稀疏贝叶斯学习快速稀疏重构方法 (Two-dimensional inversion-free sparse Bayesian learning rapid sparse reconstruction method ) 是由 张双辉 刘永祥 黎湘 霍凯 姜卫东 高勋章 于 2019-09-10 设计创作，主要内容包括：本发明属于信号处理领域,具体涉及一种二维免求逆稀疏贝叶斯学习快速稀疏重构方法,包括以下步骤：S1：对二维稀疏重构问题进行稀疏表示建模；S2：对向量化稀疏信号x以及向量化噪声n进行统计建模；S3：求解向量化稀疏信号x、向量化方差倒数γ与噪声方差倒数α的后验概率；S4：更新辅助变量的矩阵形式Z。与IFSBL方法相比,本发明方法直接对二维信号进行处理,避免了二维信号向量化而产生大矩阵的问题,运算效率明显提升,且显著降低了对计算内存的需求；另一方面,本发明是在统计信号处理框架下实现稀疏重构,与非统计稀疏重构方法相比,具有更易获取全局最优解,对噪声鲁棒性更强,以及算法性能对参数初始化依赖程度不高等优势,工程实用性强。(The invention belongs to the field of signal processing, and particularly relates to a two-dimensional inversion-free sparse Bayesian learning rapid sparse reconstruction method, which comprises the following steps of: s1: carrying out sparse representation modeling on a two-dimensional sparse reconstruction problem; s2: carrying out statistical modeling on a vectorization sparse signal x and vectorization noise n; s3: solving the posterior probability of the vectorized sparse signal x, the vectorized inverse variance gamma and the noise inverse variance alpha; s4: the matrix form Z of the auxiliary variables is updated. Compared with the IFSBL method, the method disclosed by the invention directly processes the two-dimensional signals, avoids the problem that the two-dimensional signals are vectorized to generate a large matrix, obviously improves the operation efficiency and obviously reduces the requirement on a calculation memory; on the other hand, sparse reconstruction is realized under a statistical signal processing framework, compared with a non-statistical sparse reconstruction method, the method has the advantages of being easy to obtain a global optimal solution, strong in noise robustness, low in dependence degree of algorithm performance on parameter initialization and the like, and the engineering practicability is high.)

1. A two-dimensional inversion-free sparse Bayesian learning fast sparse reconstruction method is characterized by comprising the following steps:

s1 sparse representation modeling is carried out on the two-dimensional sparse reconstruction problem:

reconstructing a two-dimensional sparse signal from incomplete observation data, firstly, establishing a relational expression of the two signals, as shown in the following formula:

Y＝AXB^T+N (1)

whereinRespectively representing an observed signal, a left dictionary matrix, a sparse signal, a right dictionary matrix and noise,representing a real number set, P, Q, M, N respectively representing the dimension of the matrix, in the two-dimensional sparse reconstruction problem, the dimension of an observation signal is smaller than that of a sparse signal, namely P is less than M and Q is less than N; because the observation signal and the sparse signal are two-dimensional signals, the sparse signal X cannot be reconstructed from the observation signal Y directly by adopting a traditional sparse reconstruction method, and firstly, the observation signal and the sparse signal need to be vectorized:

y＝Φx+n (2)

wherein y ═ vec (y), x ═ vec (x), n ═ vec (n) respectively represent vectorization of observed signals, sparse signals, and noise, vec (·) represents vectorization of the matrix by column stacking;the representation matrix vectorizes the corresponding dictionary matrix:whereinRepresenting the kronecker product of the matrix;

s2 statistically models the vectorized sparse signal x and the vectorized noise n:

in the statistical modeling process, a Gaussian distribution of a two-layer structure is adopted to carry out statistical modeling on the quantitative sparse signal x: in the first layer, it is assumed that the elements of the vectorized sparse signal x obey a gaussian distribution with a mean value of zero and independent of each other:

where γ represents the inverse variance of the vectorization, x_n、γ_nRespectively indicate toQuantizing the sparse signal x and the nth element of the quantized inverse variance γ;

in the second layer, the vectorized inverse variance γ is assumed to obey the gamma distribution:

wherein a and b respectively represent the shape parameter and the size parameter of the gamma distribution;

the statistical modeling of the quantization noise n is also performed by using a gaussian distribution of a two-layer structure: in the first layer, assuming that the vectorized noise n obeys a zero-mean gaussian distribution, the likelihood function of the vectorized observed signal y also obeys a gaussian distribution:

whereinRepresenting a gaussian distribution, I representing an identity matrix, α representing the inverse of the noise variance;

in the second layer, it is assumed that the inverse α of the noise variance obeys the gamma distribution:

wherein c and d respectively represent the shape parameter and the scale parameter of the gamma distribution;

s3 solves the posterior probabilities of the vectorized sparse signal x, the vectorized inverse variance γ, and the noise inverse variance α:

the IFSBL method is based on a likelihood function of the vectorized observation signal y shown in equation (5) and a prior probability of the vectorized sparse signal x shown in equation (3), and the posterior probability of the vectorized sparse signal x is obtained as follows:

where the desired μ and covariance matrix Σ is given by:

whereinAs an auxiliary variable, the number of variables,<·>expressing an expected operator, wherein T is a constant and the value of T is slightly larger than a Lipschitz constant; diag (·) denotes the generation of a diagonal matrix using vectors; as can be seen from equation (9), the covariance matrix Σ is a diagonal matrix, and thus the inversion thereof can be obtained by inverting each element, and the operation efficiency is significantly improved; however, the calculation of the large matrix Φ is involved in the formula (8), the operation efficiency is low, and the requirement on the memory is high; aiming at the problem, the invention corrects the expected solving process shown in the formula (8) so that the two-dimensional data can be directly processed to avoid large matrix calculation caused by matrix vectorization; specifically, formula (9) can be substituted for formula (8):

whereinRepresenting the division of the elements of the matrix, 1_MN×1A vector having a size of MN × 1 and all elements of 1 is represented; further will beSubstituting formula (10) and utilizing the relevant properties of the kronecker product, one can obtain:

the above equation can be transformed into a matrix form:

whereinIn the form of matrices of the desired mu, the auxiliary variable z and the inverse of the vectorized variance gamma, 1_M×NIs represented by 1_MN×1In the form of a matrix;

the posterior probability of the matrix form Γ of the vectorized inverse variance γ is further calculated, which is obtained by means of the bayesian formula, which also obeys the gamma distribution:

wherein the shape parameterAnd scale parameterAs shown in the following formula:

wherein X_m,nThe m-th row and n-th column elements representing the sparse signal X,is expected toThe calculation expression of (a) is:wherein U is_m,n、Γ_m,nThe m-th row and n-th column elements respectively represent U and gamma;

and finally, calculating the posterior probability of the inverse noise variance alpha, which is known by a Bayesian formula and also obeys gamma distribution:

wherein the shape parameter of the a posteriori probabilityAnd a scale parameterRespectively as follows:

wherein sum (·), "indicates the addition of each element in the matrix and the multiplication of each element in the two matrices, respectively;

the desired < α > and < Γ > involved in equations (12), (18) may be obtained from the posterior probabilities shown in equations (16) and (13), respectively, as shown in the following equations:

<Γ_m,n>matrix form representing vectorized inverse variance γ expectation<Γ>The expectations of the respective elements in (1);

s4 updates the matrix form Z of the auxiliary variable:

after the posterior probabilities of the vectorized sparse signal x, the vectorized inverse variance γ and the noise inverse variance α are calculated, the matrix form Z of the auxiliary variable is further updated by the following formula:

Z＝U (21)

in summary, the process flow of the present invention can be summarized as follows: and (3) circularly iterating the formula (12), the formula (19), the formula (20) and the formula (21) until the relative error between the U obtained by two adjacent iterations reaches a rated threshold, and obtaining the U in the last step, namely the reconstructed sparse signal.

2. The two-dimensional inversion-free sparse Bayesian learning rapid sparse reconstruction method according to claim 1, characterized in that: the parameters are initialized before the iterative process, in particular: u is initialized toWhereinRepresenting a pseudo-inverse of the matrix; the inverse of the noise variance, α, is initialized to α ═ 0.1var (y)^-1Wherein var (·) represents a variance operator; the parameters a, b, c, and d in the gamma distributions shown in equations (4) and (6) are initialized to a, b, c, d, and 10^-6(ii) a The vectorization inverse variance matrix form gamma is initialized to be 1_M×N。

3. The two-dimensional inversion-free sparse Bayesian learning fast sparse reconstruction method according to claim 1 or 2, characterized by: the specific value of the constant T in S3 is: t2 lambda_max(Φ^TΦ)+10^-10。

4. The two-dimensional inversion-free sparse Bayesian learning fast sparse weight method according to claim 1 orThe method is characterized in that: the rated threshold in S4 is | | | U^(k+1)-U^(k)||₂||U^(k)||₂＜10^-3Wherein U is^(k)、U^(k+1)Respectively representing the U obtained by the k < th > and k +1 < th > iterations, | | · | | luminous flux₂Representing the 2 norm of the matrix.

Technical Field

The invention belongs to the field of signal processing, and particularly relates to a two-dimensional inversion-Free Sparse Bayesian Learning (2D-IFSBL) fast Sparse reconstruction method.

Background

Sparse reconstruction is the core of the compressed sensing technology, and a sparse signal can be accurately reconstructed from incomplete observation data. With continuous development, sparse reconstruction technology has been widely applied in the fields of medical image processing, computer vision, radar imaging, and the like. Classical Sparse reconstruction algorithms include l1 regularization, Basis Pursuit (BP), Orthogonal Matching Pursuit (OMP), Sparse Bayesian Learning (SBL), and the like. The SBL method solves the problem of sparse reconstruction under the framework of a statistical theory, performs statistical prior modeling and posterior solution on an observation signal, a sparse signal and a noise signal, and₁compared with methods such as regularization, BP, OMP and the like, the method has the advantages of being easy to obtain global optimal solution, strong in noise robustness, self-learning of parameters and the like. However, in the solving process of the SBL method, complex matrix inversion operation is involved, and the calculation complexity is high. Literature (Duan H, Yang L, Fang J, and Li H. fast Inverse-Free spark Bayesian Learning via modified evolution [ J.]IEEE Signal Processing Letters,2017,24(6): 774-.

In practical application, a two-dimensional signal sparse reconstruction problem exists, for example, in radar imaging, two-dimensional Fourier transform needs to be carried out on radar echoes, and a two-dimensional sparse reconstruction problem is to reconstruct a radar image from an incomplete radar echo. When the problems are processed, the traditional processing flow is to stack the two-dimensional signals according to columns for vectorization and then process the signals by using the traditional sparse reconstruction method, but a matrix with a larger size is introduced at the moment, so that the operation efficiency is low, the requirement on the memory is high, and the method has the advantages of low operation efficiency and low memory requirement₁The traditional methods such as regularization, BP, OMP, SBL and the like are not satisfiedAnd (5) actual engineering requirements. Although the IFSBL method improves the operation efficiency compared with the SBL method, the two-dimensional signal also needs to be vectorized when the two-dimensional signal is processed, so that large matrix operation cannot be avoided, and the operation efficiency still cannot meet the actual engineering requirements.

Disclosure of Invention

The invention aims to solve the technical problem of how to reconstruct two-dimensional sparse signals from incomplete observation data quickly and efficiently, and reduce the requirement on operation memory so as to improve the engineering practicability.

The invention provides a two-dimensional inversion-free sparse Bayesian learning method aiming at the problem of low operation efficiency of a traditional sparse reconstruction method for processing two-dimensional signals. The method comprises the steps of firstly carrying out sparse representation modeling on a two-dimensional sparse reconstruction problem, and respectively carrying out statistic prior modeling on a sparse signal to be reconstructed and noise; and then calculating the posterior probability of the sparse signal through a Bayes formula based on the prior of the sparse signal to be reconstructed and the likelihood function of the observation signal, thereby realizing the reconstruction of the sparse signal. The two-dimensional signals are directly processed in the calculation process without vectorization, and large matrix processing is avoided, so that the operation efficiency of two-dimensional sparse reconstruction is effectively improved, and the memory requirement is reduced.

The technical scheme adopted by the invention for solving the technical problems is as follows: a two-dimensional inversion-free sparse Bayesian learning rapid sparse reconstruction method comprises the following steps:

s1 sparse representation modeling is carried out on the two-dimensional sparse reconstruction problem:

reconstructing a two-dimensional sparse signal from incomplete observation data, firstly, establishing a relational expression of the two signals, as shown in the following formula:

Y＝AXB^T+N (1)

whereinRespectively representing an observed signal, a left dictionary matrix, a sparse signal, a right dictionary matrix and noise,representing a set of real numbers, P, Q, M, N representing the dimensions of the matrix, respectively. In the two-dimensional sparse reconstruction problem, the observed signal dimension is smaller than the sparse signal, i.e. P < M and Q < N. Because the observation signal and the sparse signal are two-dimensional signals, the sparse signal X cannot be reconstructed from the observation signal Y directly by adopting a traditional sparse reconstruction method, and firstly, the observation signal and the sparse signal need to be vectorized:

y＝Φx+n (2)

wherein y ═ vec (y), x ═ vec (x), n ═ vec (n) respectively represent vectorization of observed signals, sparse signals, and noise, vec (·) represents vectorization of the matrix by column stacking;the representation matrix vectorizes the corresponding dictionary matrix:whereinRepresenting the kronecker product of the matrix. The dimension of the dictionary matrix phi is large, so that the problem shown in the formula (2) solved by the traditional sparse reconstruction method is low in operation efficiency, and the requirement on the memory is high, for example, when the sizes of the observation signal Y and the sparse signal X are respectively 64 × 64 and 128 × 128, the size of the dictionary matrix phi is 8192 × 8192, and the efficiency of directly processing the dictionary matrix phi cannot meet the engineering requirement.

S2 statistically models the vectorized sparse signal x and the vectorized noise n:

in the statistical modeling process, the Gaussian distribution of a two-layer structure is adopted to carry out statistical modeling on the quantitative sparse signal x. In the first layer, it is assumed that the elements of the vectorized sparse signal x obey a gaussian distribution with a mean value of zero and independent of each other:

where γ represents the inverse variance of the vectorization, x_n、γ_nSeparately representing vectorized sparse signals x and vectorsThe nth element of the inverse variance γ is normalized.

In the second layer, the vectorized inverse variance γ is assumed to obey the gamma distribution:

wherein a and b represent the shape parameter and the size parameter of the gamma distribution, respectively.

The statistical modeling of the quantization noise n is also performed using a two-layer structure gaussian distribution. In the first layer, assuming that the vectorized noise n obeys a zero-mean gaussian distribution, the likelihood function of the vectorized observed signal y also obeys a gaussian distribution:

whereinDenotes a gaussian distribution, I denotes an identity matrix, and α denotes the inverse of the noise variance.

In the second layer, it is assumed that the inverse α of the noise variance obeys the gamma distribution:

wherein c and d respectively represent the shape parameter and the scale parameter of the gamma distribution.

S3 solves the posterior probabilities of the vectorized sparse signal x, the vectorized inverse variance γ, and the noise inverse variance α:

the process of realizing sparse reconstruction by the SBL method is essentially a process of deducing the posterior probability of the sparse signal by a Bayes formula based on a likelihood function of an observation signal and a prior model of the sparse signal, while the traditional SBL method involves a complex matrix inversion process when solving the vectorized sparse signal x posterior probability, and has low operation efficiency. Matrix inversion is effectively avoided by an inversion-Free Sparse Bayesian learning method (IFSBL) (Duan H, Yang L, Fang J, and Li H. fast Inverse-Free Sparse learning via delayed inversion of evaluation Low Bound approximation [ J ]. IEEE Signal Processing Letters,2017,24(6): 774-. Specifically, the IFSBL method is based on a likelihood function of the vectorized observation signal y shown in equation (5) and a prior probability of the vectorized sparse signal x shown in equation (3), and the posterior probability of the vectorized sparse signal x is obtained as follows:

where the desired μ and covariance matrix Σ is given by:

whereinAs an auxiliary variable, the number of variables,<·>expressing the calculation of an expected operator, wherein T is a constant and takes a value slightly larger than a Lipschitz constant: t2 lambda_max(Φ^TΦ)+10^-10Wherein λ is_max(Φ^TPhi) represents phi^TMaximum eigenvalue of Φ. diag (·) denotes the generation of a diagonal matrix using vectors. As can be seen from equation (9), the covariance matrix Σ is a diagonal matrix, and thus the inversion thereof can be obtained by inverting each element, and the operation efficiency is significantly improved. However, the calculation of the large matrix Φ in equation (8) is inefficient and requires a high memory. Aiming at the problem, the invention corrects the expectation solving process shown in the formula (8) so that the expectation solving process can directly process two-dimensional data to avoid large matrix calculation caused by matrix vectorization. Specifically, formula (9) can be substituted for formula (8):

whereinRepresenting the division of the elements of the matrix, 1_MN×1A vector having a size of MN × 1 and all elements of 1 is represented. Further will beSubstituting formula (10) and utilizing the relevant properties of the kronecker product, one can obtain:

the above equation can be transformed into a matrix form:

whereinIn the form of matrices of the desired mu, the auxiliary variable z and the inverse of the vectorized variance gamma, 1_M×NIs represented by 1_MN×1In the form of a matrix. As shown in equation (12), the calculation process of U only involves matrix operation with the maximum size of M × N, so the calculation efficiency is significantly higher than that of equation (8), and the requirement on the memory is reduced.

The posterior probability of the matrix form Γ of the vectorized inverse variance γ is further calculated, which is obtained by means of the bayesian formula, which also obeys the gamma distribution:

wherein the shape parameterAnd scale parameterAs shown in the following formula:

wherein X_m,nThe m-th row and n-th column elements representing the sparse signal X,is expected toThe calculation expression of (a) is:wherein U is_m,n、Γ_m,nThe m-th row and n-th column elements of U and Γ are shown, respectively.

And finally, calculating the posterior probability of the inverse noise variance alpha, which is known by a Bayesian formula and also obeys gamma distribution:

wherein the shape parameter of the a posteriori probabilityAnd a scale parameterRespectively as follows:

wherein sum (·),. fwdarw respectively denote the addition of elements in the matrix and the multiplication of each element of the two matrices.

The desired < α > and < Γ > values referred to in equations (12), (18) may be obtained from the posterior probabilities shown in equations (16) and (13), respectively, as shown in the following equations:

<Γ_m,nmatrix form representing vectorized inverse variance γ expectation<Expectation of each element in gamma;

s4 updates the matrix form Z of the auxiliary variable:

after the posterior probabilities of the vectorized sparse signal x, the vectorized inverse variance γ and the noise inverse variance α are calculated, the matrix form Z of the auxiliary variable is further updated by the following formula:

Z＝U (21)

in summary, the process flow of the present invention can be summarized as follows: and (3) circularly iterating the formula (12), the formula (19), the formula (20) and the formula (21) until the relative error between the U obtained by two adjacent iterations reaches a rated threshold (for example | | U:)^(k+1)-U^(k)||₂/||U^(k)||₂＜10^-3Wherein U is^(k)、U^(k+1)Respectively representing the U obtained by the k < th > and k +1 < th > iterations, | | · | | luminous flux₂Representing 2 norm of the matrix), and the U obtained in the last step is the reconstructed sparse signal. Furthermore, the parameters need to be initialized before the iterative process, in particular: u is initialized toWhereinRepresenting a pseudo-inverse of the matrix; the inverse of the noise variance, α, is initialized to α ═ 0.1var (y)^-1Wherein var (·) represents a variance operator; the parameters a, b, c, and d in the gamma distributions shown in equations (4) and (6) are initialized to a, b, c, d, and 10^-6(ii) a The vectorization inverse variance matrix form gamma is initialized to be 1_M×N。

The invention has the following beneficial effects: compared with the IFSBL method, the method can directly process the two-dimensional signals, avoids the problem that the two-dimensional signals are vectorized to generate a large matrix, obviously improves the operation efficiency and obviously reduces the requirement on a calculation memory; on the other hand, the 2D-IFSBL method provided by the invention realizes sparse reconstruction under a statistical signal processing framework, and has the advantages of being easier to obtain a global optimal solution, stronger in noise robustness, lower in degree of dependence of algorithm performance on parameter initialization and the like compared with a non-statistical sparse reconstruction method, and the engineering practicability is strong.

Drawings

FIG. 1 is a flow chart of an embodiment of the present invention;

FIG. 2 shows sparse reconstruction results under different observation signal sizes;

FIG. 3 shows sparse reconstruction results under different signal-to-noise ratio (SNR) conditions;

FIG. 4 is a comparison of relative estimation errors of sparse signals reconstructed by different methods under different observation signal sizes;

FIG. 5 is a comparison of relative estimation errors of sparse signals reconstructed by different methods under different SNR conditions;

FIG. 6 shows a time comparison calculated by different methods for different observed signal sizes.

Detailed Description

The invention is further illustrated with reference to the accompanying drawings:

FIG. 1 is a general process flow of the present invention.

The invention relates to a two-dimensional inversion-free sparse Bayesian learning rapid sparse reconstruction method, which comprises the following four steps of:

s1: carrying out sparse representation modeling on a two-dimensional sparse reconstruction problem;

s2: carrying out statistical modeling on a vectorization sparse signal x and vectorization noise n;

s3: solving the posterior probability of the vectorized sparse signal x, the vectorized inverse variance gamma and the noise inverse variance alpha;

s4: the matrix form Z of the auxiliary variables is updated.

Experiments were conducted using simulation data to compare the performance of the method of the present invention with IFSBL method (Duan H, Yang L, Fang J, and Li H. fast Inverse-Free spark Bayesian Learning via delayed evolution lower Bound maximum knowledge [ J ]. IEEE Signal Processing Letters,2017,24(6): 774-) 778) and two-dimensional smooth 0-norm method (2D-SL0) (Aboozar G, Massoud B Z, Christian J. spark analysis of two-dimensional signals [ C ]. IEEE International Conference on Acoustics, Speech and Signal Processing 2009, Taipei 2009: 3157-. In the experimental process, firstly, a two-dimensional sparse signal X with the size of M multiplied by N is generated in a simulation mode, and the signal comprises K non-zero points; the sizes of the left dictionary matrix A and the right dictionary matrix B are respectively P multiplied by M, Q multiplied by N, and all elements of the left dictionary matrix A and the right dictionary matrix B are subjected to standard normal distribution; the noise N is white Gaussian noise; finally, the observation signal Y is generated by equation (1).

In the first experiment, it is assumed that the two-dimensional sparse signal X includes K — 9 non-zero points, each of which takes a value of 1 and is uniformly distributed in the center of the image, so as to facilitate comparison. The size of the two-dimensional sparse signal X is set to 80 × 80, i.e., M ═ N ═ 80. The signal-to-noise ratio (SNR) of the observation signal Y is set to 20 dB. Changing the size of the observed signal Y: let P-Q25, 40, 55 to compare the performance of the three methods 2D-SL0, IFSBL and the 2D-IFSBL proposed by the present invention under different observed signal size conditions. Fig. 2 shows the sparse signals reconstructed by the three methods, where the first, second, and third rows correspond to the sparse reconstruction results under the conditions of P25, 40, and 55, respectively. As can be seen from the figure, the sparse signals obtained by the 2D-IFSBL and IFSBL methods provided by the present invention are completely consistent, and when P is 25, the 2D-SL0 reconstruction result is better, and when P is 55, the results obtained by the 2D-IFSBL and IFSBL methods provided by the present invention are less affected by noise, and the reconstructed sparse signal is better than the 2D-SL0 method.

The second experiment compares the performance of the three methods under different SNR conditions. The experimental parameters were set as follows: P-Q-55, M-N-80, K-9, SNR 15dB, 7.5dB, and 0dB respectively, and the sparse signals obtained by the three methods under different SNR conditions are shown in fig. 3. As can be seen from the figure, the sparse signals reconstructed by the 2D-IFSBL and IFSBL methods provided by the invention are obviously affected by noise to a lower extent than the 2D-SL0 method, especially under the condition that the SNR is 0dB, the 2D-SL0 method basically fails, and the 2D-IFSBL and IFSBL methods provided by the invention can still accurately reconstruct the sparse signals, which shows that the robustness of the sparse signals to noise is better than that of the 2D-SL0 method.

Algorithm performance was further compared by monte carlo experiments. Firstly, the relative errors of the reconstructed sparse signals by the three methods under different observation signal sizes are compared, and experimental parameters are set as follows: m80, K10, SNR 20dB, P, Q, 10 to 60, and 5 steps. Under the condition of each different observed signal size, 100 Monte Carlo experiments are repeatedly carried out, A, B, X and N are randomly valued in each experiment, wherein K-10 non-zero values in the sparse signal X obey standard normal distribution, and the positions in X are completely random. The average sparse reconstruction relative error obtained by the three methods in 100 monte carlo experiments is shown in fig. 4. As can be seen, the performance of the 2D-IFSBL method and the IFSBL method provided by the invention is completely consistent, when P is less than 40, the 2D-SL0 method is better, and when P is more than 40, the relative error obtained by the 2D-IFSBL method and the IFSBL method is lower than that obtained by the 2D-SL0 method.

Further comparing the performances of the three methods under different SNR conditions by Monte Carlo experiments. The experimental parameters were set as follows: M-N-80, P-Q-55, K-10, SNR ranging from-12 dB to 16dB, with a step size of 2 dB. Under each SNR condition, 100 monte carlo experiments were performed, and the average sparse reconstruction relative error of the three methods was recorded, and the results are shown in fig. 5. As can be seen from the figure, the 2D-IFSBL method and the IFSBL method provided by the invention are comprehensively superior to the 2D-SL0 method, and the low relative error of sparse reconstruction is obtained under any SNR condition, so that the strong robustness is verified.

And finally, comparing the operation efficiency of the three methods. Also using the monte carlo experiment, the experimental parameters were set as follows: SNR 20dB, M N. Changing the value of N, wherein the value range is 50-250, the step length is 10, carrying out 100 Monte Carlo experiments on each N condition, and recording the average calculation time of the three algorithms (the calculation platform: Intel (R) Corei7-8550U @1.8GHz), wherein the result is shown in FIG. 6. As can be seen from the figure, the 2D-SL0 method has the highest operation efficiency, and the second method of the 2D-IFSBL method provided by the invention is the lowest IFSBL method. The IFSBL method has high requirement on the calculation memory because of the large matrix calculation, and when N is more than 150, the calculation memory is overrun and the algorithm is invalid. In contrast, on the premise of keeping the performance consistent with the IFSBL, the 2D-IFSBL method provided by the invention improves the highest operation efficiency by more than 100 times, reduces the requirement on the calculation memory, and still only needs about 5 seconds to reconstruct a sparse signal when N is 250, thereby meeting the actual engineering requirements.

The experimental results show that the invention corrects the traditional IFSBL, so that the traditional IFSBL can directly process two-dimensional data. In the problem of two-dimensional sparse reconstruction, the method provided by the invention has the advantages that on the premise of keeping the performance of the traditional IFSBL method, the operation efficiency is improved by more than 100 times, and the requirement on the calculation memory is obviously reduced. Compared with the 2D-SL0 method, the method disclosed by the invention belongs to a statistical sparse reconstruction method and has the advantage of stronger robustness. Although the algorithm performance is lower than that of the 2D-SL0 method under the condition that the observed data size is small, the algorithm performance is obviously better than that of the 2D-SL0 method under the condition of low SNR, and the method is suitable for application scenes with low SNR.