# Block Sparse Bayesian Recovery with Correlated LSM Prior

Authors: Juan Zhao; Xia Bai; Tao Shan; Ran Tao
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

Compressed sensing can recover sparse signals using a much smaller number of samples than the traditional Nyquist sampling theorem. Block sparse signals (BSS) with nonzero coefficients occurring in clusters arise naturally in many practical scenarios. Utilizing the sparse structure can improve the recovery performance. In this paper, we consider recovering arbitrary BSS with a sparse Bayesian learning framework by inducing correlated Laplacian scale mixture (LSM) prior, which can model the dependence of adjacent elements of the block sparse signal, and then a block sparse Bayesian learning algorithm is proposed via variational Bayesian inference. Moreover, we present a fast version of the proposed recovery algorithm, which does not involve the computation of matrix inversion and has robust recovery performance in the low SNR case. The experimental results with simulated data and ISAR imaging show that the proposed algorithms can efficiently reconstruct BSS and have good antinoise ability in noisy environments.


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## Body

## 1. Introduction

Compressed sensing (CS) [1] provides a new sampling and reconstruction paradigm, which can recover sparse signals from linear measurements:
(1)y=Φx+n,where Φ∈RM×NM<N is the measurement matrix, y∈RM is the measurement vector, x∈RN is the sparse signal, and n∈RM is the additive noise. Many recovery algorithms have been presented to reconstruct sparse signals, including orthogonal matching pursuit (OMP) [2] and sparse Bayesian learning (SBL) [3].In some signal processing applications such as ISAR imaging [4] and gene expression levels [5], there are many sparse signals with block structural features; i.e., nonzero elements are often clustered. Inducing structural a priori can largely improve the reconstruction performance. Therefore, to improve the reconstruction effect of block sparse signals (BSS), many algorithms are proposed. For instance, Block-OMP [6] and Block-StOMP [7] are OMP-based approaches. Meanwhile, some block recovery algorithms based on the Bayesian compressed sensing framework are presented, including block sparse Bayesian learning (BSBL) [8], Cluss-MCMC [9], model-based Bayesian CS via local beta process (MBCS-LBP) [10], and pattern-coupled sparse Bayesian learning (PC-SBL) [11]. Among these recovery algorithms, Bayesian algorithms have parameter learning ability and can be applied to recovery arbitrary signals with unknown sparse structures by flexibly imposing different sparse prior models.In [12], Zhang et al. have proposed an expectation-maximization-based variational Bayesian (EM-VB) inference method, which utilizes the Laplacian scale mixture (LSM) model as a sparse prior; i.e., it is assumed that the sparse signal obeys the Laplacian prior because the Laplacian distribution can represent sparseness well. Based on this model, for the BSS with unknown block information, this paper proposes a block Bayesian recovery algorithm by inducing a correlated LSM prior model, which uses the dependence between neighboring elements of the BSS. Furthermore, to improve the computational efficiency of the proposed recovery algorithm, a fast version without matrix inversion is presented, which is suitable for noisy environments, especially in the low SNR case. The experimental results on simulated data and ISAR imaging show that the proposed algorithms have a good reconstruction effect on BSS and can resist noise in noisy environments.The remainder of this paper is organized as follows. In Section2, a correlated LSM prior model for BSS is given. Then, the proposed block Bayesian recovery algorithm and the fast version are derived in Section 3. Simulation experiments are presented in Section 4. Finally, we conclude this paper in Section 5.


## 2. Signal Model

In the framework of sparse Bayesian learning, for the measurement model shown in (1), the noise n is generally assumed to obey a Gaussian prior distribution Nn∣0,γ−1IM, and a Gamma distribution for the hyperparameter γ is
(2)pγ;a,b=Gγ;a,b=baΓaγa−1exp−bγ,where Γa=∫0+∞xa−1exp−xdx. The sparse signal x is usually assumed to obey a sparse prior distribution. In the LSM layered prior model [12], the sparse signal x is supposed to follow a Laplacian prior distribution:
(3)px∣λ=∏n=1NLaxn∣0,λn=∏n=1N12λnexp−xnλn,where λnn=1,⋯,N is the scale parameter of the Laplacian distribution for each element in the signal. Since the Inverse-Gamma (IG) distribution is conjugated to Laplacian distribution, the LSM model assumes that the scale parameter λn obeys the IG distribution as follows:
(4)pλ;c,d=∏n=1NIGλn;c,d=∏n=1NdcΓcλn−c−1exp−dλn.In the above model, each hyperparameterλn controls the corresponding signal element xn individually and each signal element is considered to be independent. Considering that nonzero elements of BSS appear in clusters, this requires a more appropriate model for BSS. For the BSS whose structure prior information is unknown, the PC-SBL algorithm [11] assumes that the hyperparameters of adjacent elements have a certain influence on its sparsity. Inspired by the PC-SBL, we assume that the block sparse signal x obeys the following correlated Laplacian prior distribution, i.e.,
(5)px∣λ=∏n=1NLaxn∣0,λn−1,λn,λn+1=∏n=1N1/λn+β1/λn−1+β1/λn+12exp−1λn+β1λn−1+β1λn+1xn,where the parameter β∈0,1 indicates the degree of correlation between adjacent elements in the signal. It can be seen from (5) that the element xn is affected by its own hyperparameter λn and the neighboring ones λn−1 and λn+1. For the elements at both ends x1 and xN, let λ0=0 and λN+1=0. The model (5) makes use of the feature of the block sparse signal, in which the scale parameters λ still obey the IG distribution shown in (4).


## 3. Block Bayesian Recovery Algorithms

In Bayesian inference, given observationy, it needs to derive the posterior probability density for all unknown parameters px,λ,γ∣y∝py∣x,γpx∣λpλpγ. Variational Bayesian inference is a widely used method to approximately solve the maximization of a posteriori, which assumes that the variables x,λ,andγ are independent of each other. Let θ=x,λ,γ, and then
(6)pθ∣y≈qθ=∏iqθi=qxqλqγ.For each of these latent variables, the approximate posterior distribution may be computed in an alternating manner as follows:(7)qθi=explnpy,θqθk,k≠i∫explnpy,θqθk,k≠idθi,where ·qθk,k≠i represents the expected operation with respect to the distributions qθk,k≠i. According to (7), the proposed reconstruction algorithm is derived by alternately learning the updating rules of these latent variables.Firstly, the approximate posterior distributionqx is
(8)lnqx∝lnpy∣x,γ+lnpx∣λqλqγ,where py∣x,γ is a Gaussian distribution NΦx,γ−1IM and px∣λ is a Laplacian distribution shown by (5). Since these two distributions are not conjugated, a direct solution is difficult. Similar to [12], let
(9)Lx=−lnpy∣x,γ+lnpx∣λqλqγ∝γ2y−Φx22+∑n=1N1λn+β1λn−1+β1λn+1xn.The maximum a posteriori (MAP) estimate of the signalx can be obtained by x^MAP=argminxLx. The derivative of Lx is
(10)∇xLx=γΦTΦ+Dx−γΦTy,where
(11)D=diag1λn+β1λn−1+β1λn+11xn,and diag⋅ denotes a diagonal matrix with the elements in the bracket. Let the derivative be equal to zero and get the approximate MAP estimate:
(12)x^MAP=γγΦTΦ+D−1ΦTy.So the posterior distribution can be approximated by using the second-order Taylor expansion aroundx^MAP, i.e.,
(13)lnqx≈lnqx^MAP+12x−x∧MAPTHx^MAPx−x^MAP,where Hx^MAP≈−γΦTΦ+D. After similar simplification in [12], qx can be approximated to obey the Gaussian distribution Nx∣μ,Σo with the mean μ=γΣoΦTy and covariance matrix Σo=γΦTΦ+D−1. Due to that, there exists some approximation in the above derivation of the posterior distribution qx; the sparsity of the signal may be underestimated. So a parameter α∈0.5,1 is introduced into the computation of the covariance matrix. Thus, qx is approximated to be the following Gaussian distribution:
(14)qx≈Nx∣μ,Σ,where the mean and covariance matrix, respectively, are
(15)μ=γΣΦTy,Σ=γΦTΦ+αD−1.Secondly, the approximate posterior distributionqλ is
(16)lnqλ∝lnpλ+lnpx∣λqx=lnpλ+lnpx∣λqx.From (4) and (5), we have
(17)lnpλ∝∑n=1Nc+1ln1λn−dλn,(18)lnpx∣λqx∝∑n=1Nln1λn+β1λn−1+β1λn+1−1λn+β1λn−1+β1λn+1xn=∑n=1Nln1λn+β1λn−1+β1λn+1−xn+βxn−1+βxn+11λn≥∑n=1Nln1λn−xn+βxn−1+βxn+11λn.Soqλ can be approximated as
(19)lnqλ∝∑n=1Nc+1+1ln1λn−xn+βxn−1+βxn+1+d1λn.Therefore,qλ obeys an Inverse-Gamma distribution:
(20)pλ;c~,d~=∏n=1NIGλn;c~,d~n,with
(21)c~=c+1,d~n=xn+βxn−1+βxn+1+d.We can also obtain(22)1λn=c~d~n=c+1xn+βxn−1+βxn+1+d,where xn can be computed as follows according to [13]:
(23)xn=2Σn,nπexp−μn22Σn,n+μnerfμn22Σn,n,and erfx=2/π∫0xe−t2dt is the error function.Thirdly, the approximate posterior distribution of the noise parameterqγ is
(24)lnqγ∝lnpγ+lnpy∣x,γqx∝a−1+M2lnγ−y−Φx222+bγ.Soqγ obeys the Gamma distribution:
(25)qγ=Gγ;a~,b~,with
(26)a~=a+M2,b~=y−Φx222+b,and we can obtain
(27)γ=a~b~=a+M/2y−Φx22/2+b,where y−Φx22=y−Φμ22+TrΣΦTΦ and Tr⋅ is the trace of a matrix.Therefore, the whole process of the proposed algorithm is summarized in Algorithm1, where ε is the preset error that can be tolerated. The proposed algorithm can be regarded as an extension of the EM-VB algorithm for the recovery of BSS, which is termed the Block EM-VB algorithm. It has an additional parameter α except for the block parameter β. When α=1andβ=0, the Block EM-VB algorithm reduces to the EM-VB. The parameters α in (15) and β in (22) have great influences on the recovery performance of the Block EM-VB. It is appropriate to set α<1 to void underestimation of the support set if the sparsity of the signal is larger, while larger α can suppress nonzero signal elements and has certain antinoise capacity in noisy environments. For the choice of β, it is similar that larger β may enhance the influence between adjacent elements and could suppress the nonzero values to make the signal more sparse in noisy environments.Algorithm 1: Block EM-VB.
Input: y,Φ,ε, Max_iter,β,α.Initialize: μ0,λ,γ,a,b,c,dWhileμt+1−μt2>ε or t≤Max_iterdoUpdate:(1) Compute the meanμ and the covariance matrix Σ by (15) and (11).(2) Computeλ−1 according to (22) and (23).(3) Computeγ via (27).Output: x^=μRemark 1.
From (23), it can be seen that the computation of xn is related to the erfx involving integral operation. In practice, to reduce the complexity, erfx can be calculated by utilizing some approximation equations such as
(28)erfx≈1−e−4x2/π1+8x4π13−1π1/2.
From the process of the Block EM-VB algorithm, it can be seen that its complexity is almost the same as that of the PC-SBL.The proposed Block EM-VB algorithm involves the matrix inversion shown in (15), which is the main computational complexity. It is better to consider the fast version of the proposed algorithm. To void computation of the inverse of the matrix, we can use the following approximate posterior distribution, which is expressed as
(29)pθ∣y≈qθ=∏i=1Nqxiqλqγ.That is to say, it assumes the independence on the posterior of each coefficient element of the signal. Similarly, by using (7), we can alternately learn the updating rules of these latent variables.Firstly, the approximate posterior distributionqxii=1,2,⋯,N is
(30)lnqxi∝lnpy∣x,γ+lnpxi∣λi−1,λi,λi+1qγqλqxjj≠i,where py∣x,γ=NΦx,γ−1IM and pxi∣λi−1,λi,λi+1=1/λi+β1/λi−1+β1/λi+1/2exp−1/λi+β1/λi−1+β1/λi+1xi. Let
(31)lxi=−lnpy∣x,γ+lnpxi∣λi−1,λi,λi+1qγqλqxjj≠i∝γ2y−Φx22qxjj≠i+1λi+β1λi−1+β1λi+1xi∝γ2−2yTΦx+xTΦTΦxqxjj≠i+1λi+β1λi−1+β1λi+1xi∝γ2−2yTφixi+φiTφixi2+2xiφiT∑j≠iφjxj+1λi+β1λi−1+β1λi+1xi,where φi is the i-th column of Φ. The derivative of lxi is
(32)∇xilxi=γ2−2yTφi+2φiTφixi+2φiT∑j≠iφjxj+1λi+β1λi−1+β1λi+1xixi.Let∇xilxi=0 and obtain the following approximate MAP estimate:
(33)x~i=γγφiTφi+1λi+β1λi−1+β1λi+11xi−1y−∑j≠iφjxjTφi.Similarly,qxi is approximated to obey the following Gaussian distribution:
(34)qxi≈Nxi∣μi,σi2,where
(35)μi=γσi2y−∑j≠iφjxjTφi,σi2=γφiTφi+1λi+β1λi−1+β1λi+1αxi−1,in which a parameter α is also introduced to avoid underestimation of the sparsity of the signal. Thus, xi=μii=1,2,⋯,N can be computed in a sequential manner.Secondly, the approximate posterior distributionqλ still obeys an Inverse-Gamma distribution shown in (20) and the computation of λ−1 is the same as (22).Thirdly, the approximate posterior distributionqγ obeys the Gamma distribution Gγ;a~,b~ and we can obtain γ=a~/b~=a+M/2/y−Φx22/2+b, where
(36)y−Φx22=y−Φμ22+TrΣΦTΦ=y−Φμ22+∑i=1Nσi2φiTφi,and μ=μ1,μ2,⋯,μNT and Σ≜diagσi2.Compared with the Block EM-VB algorithm, the above algorithm has low computational complexity due to without matrix inversion in each iteration. It can be called Fast Block EM-VB, and its process is summarized in Algorithm2. In noisy environments, the correlations between adjacent signal elements may be weakened, especially in the low SNR case, so it is appropriate to assume independence on the posterior of each coefficient element, which implies that the Fast Block EM-VB is suitable to recover the BSS under the low SNR case.Algorithm 2: Fast Block EM-VB.
Input: y,Φ,ε, Max_iter,β,αInitialize: μ0,λ,γ,a,b,c,dWhileμt+1−μt2>ε or t≤Max_iterdoUpdate:(1) Compute the meanμi and the variance σi2i=1,2,⋯,N, sequentially by (35).(2) Computeλ−1 according to (22) and (23).(3) Computeγ via (27) and (36).Output: x^=μ


## 4. Simulation Experiments

In this section, some simulation experiments are carried out to demonstrate the performances of the proposed Block EM-VB algorithm and its fast version. A comparison with other algorithms such as EM-VB [12] and PC-SBL [11] is also given.


### 4.1. Performance Analysis via Simulated Data

In the following simulation, let the length of the sparse signal beN=100, and an arbitrary block Gaussian sparse signal is randomly generated with its nonzero entries randomly distributed in B=3 blocks, and the measurement matrix is a random Gaussian matrix. The number of Monte Carlo simulations is 200. The parameters in EM-VB and in the Block EM-VB and its fast version are set as a=b=c=d=10−6, Max_iter=200, and ε=10−5. The parameters α and β in the Block EM-VB and its fast version will be set adaptively according to the noiseless and noisy cases because their performances are sensitive to the choice of α and β. The parameters of PC-SBL are set the same as those in [11].First, we discuss the influences of two parametersα and β on the performance of the proposed Block EM-VB and make a comparison with the EM-VB, which is a special case of the Block EM-VB when α=1andβ=0. To demonstrate the effect of these two parameters, the performances of the Block EM-VB with α=0.7andβ=0 and α=1andβ=0.5 are also given. The support recovery rate is used to evaluate the performance of the Block EM-VB algorithm with different parameters. The recovered support of the sparse signal is defined as suppx^=i,∣x^i∣>0.001, and then the support recovery rate is defined by ∣suppx^∩suppx∣/∣suppx^∪suppx∣, where ⋅ denotes the number of elements in a set. If the overlap between the estimated support and the true support is more, the recovery rate is closer to 1. Figure 1 plots the support recovery rates of different algorithms versus the sparsity level K when the number of measurements is M=30 and M=40, respectively. It can be seen that the parameters of the Block EM-VB algorithm have an important influence on the recovery performance. The appropriate parameter α<1 can avoid the underestimation of the support of the sparse signals, and the block parameter β>0 is helpful to recover the block sparse signals. Thus, the proposed Block EM-VB algorithm with appropriate parameters has better performance than the EM-VB.Figure 1
The support recovery rate versus the sparsity in the noiseless case when (a)M=30 and (b) M=40.
(a)(b)Then, we make a comparison between the Block EM-VB and its fast version with the PC-SBL. In the noiseless case, the success rate is used to evaluate the performances of these different algorithms. When a trial satisfiesx^‐x22/x22<10−6, it is regarded as a successful trial. The success rate is defined as the percentage of successful trials in the total of independent trials. In the noisy case, the reconstruction performance of each algorithm is evaluated by the normalized mean square error (NMSE), where NMSE=x^‐x22/x22.In the noiseless case, letα=0.7andβ=0.5 in the Block EM-VB and α=0.5andβ=0.5 in the Fast Block EM-VB. Figure 2 plots the success rate of individual recovery algorithm versus the sparsity level K when the number of measurement is M=30 and M=40, respectively. Then, let the sparsity K=20 and K=25, and the success rate of each algorithm versus the number of measurements M is shown in Figure 3. From these results, it is observed that the Block EM-VB is superior to the PC-SBL when the number of measurements is less or the sparsity is smaller. The Fast Block EM-VB is inferior to these two algorithms due to the independent assumption.Figure 2
The success rate vs. the sparsity level in the noiseless case when (a)M=30 and (b) M=40.
(a)(b)Figure 3
The success rate vs. the number of measurements in the noiseless case when (a)K=20 and (b) K=25.
(a)(b)Then, we consider the noisy case where the Gaussian noise is added to the measurements with the signal-to-noise ratio (SNR) defined as20log10Φx2/n2. It should be noted that the setting of parameter α in the proposed algorithms needs to consider the trade-off between removing the noise and maintaining the nonzero elements of the signal. A larger value of α tends to suppress noise while losing the nonzero values of the signal. Compared with the Block EM-VB, the fast version needs to select smaller α to ensure the recovery of nonzero elements of the sparse signal because of its inherent denoising ability. Let SNR=10dB and α=0.85andβ=0.5 in the Block EM-VB and α=0.5andβ=0.5 in the Fast Block EM-VB. The reconstruction results of different algorithms when K=25 and M=50 are given in Figure 4. It can be seen that the Block EM-VB algorithm and its fast version have better reconstruction performance than the PC-SBL in the low SNR environment.Figure 4
The reconstructed signal by different algorithms whenN=100, M=50, K=25, and SNR=10dB. (a) Original signal, (b) reconstructed signal by PC-SBL with NMSE=0.2362, (c) reconstructed signal by Block EM-VB with NMSE=0.2184, and (d) reconstructed signal by Fast Block EM-VB with NMSE=0.1092.
(a)(b)(c)(d)The NMSE of each algorithm versus the sparsityK in the case of SNR=10dB is shown in Figure 5, where the numbers of measurements are given as M=40 and M=50, respectively. Figure 6 plots the NMSE of individual recovery algorithm versus the number of measurements M in the case of SNR=10dB, where K=20 and K=25, respectively. It can be found that the Block EM-VB algorithm is superior to the PC-SBL when the number of the measurements is larger or the sparsity is less, and the latter is better than the former when the number of the measurements is less and the sparsity is larger. It is also observed that the Fast Block EM-VB is better than the Block EM-VB when the number of measurements is less or the sparsity is larger. In addition, it is shown that the Fast Block EM-VB outperforms the PC-SBL when the sparsity is less or the number of the measurements is larger.Figure 5
The NMSE vs. the sparsity level whenSNR=10dB and (a) M=40 and (b) M=50.
(a)(b)Figure 6
The NMSE vs. the number of measurements whenSNR=10dB and (a) K=20 and (b) K=25.
(a)(b)The performances of each algorithm in different SNR cases are shown in Figure7, where K=20 and M=50. For the Block EM-VB, the parameters β=0.5 and α is set to vary with the SNR, i.e., α=0.85 when the SNR varies from 0 dB to 15 dB, α=0.75 when the SNR changes from 20 dB to 25 dB, and α=0.73 for the case of 30 dB. The parameters in the Fast Block EM-VB algorithm are still set as α=0.5 and β=0.5. From Figure 7, it can be seen that the Block EM-VB algorithm and the fast version have good reconstruction performance in the case of low SNR when compared with the PC-SBL. The Fast Block EM-VB especially has robust recovery performance in respect of noise immunity.Figure 7
The comparison of different algorithms under different SNR whenN=100, K=20, and M=50.Finally, the average runtimes of these algorithms versus the length of signalsN by 5 independent trials are given in Figure 8, where M=N/2, K=N/10, SNR=10dB, and the number of nonzero blocks B=K/5. It validates that the Block EM-VB almost has considerable computational complexity as the PC-SBL and the Fast Block EM-VB has the highest computational efficiency compared with other recovery algorithms, which makes it have a potential advantage in practical application.Figure 8
The average runtimes vs. the length of signalN.


### 4.2. Application in ISAR Imaging

The inverse synthetic aperture radar (ISAR) imaging is appropriately implemented under the framework of sparse signal recovery due to the sparse characteristic of the target [14]. In this experiment, the “Yak-42” dataset is used, in which the number of range cells is 256 and the number of pulses is 256. 128 pulses are randomly sampled to simulate the sparse aperture data. Here, we use the MATLAB code provided in [14], where the PC-SBL adopts a pruning operation. For a fair comparison, the proposed Block EM-VB and the fast version also use a similar pruning operation and the parameters in these two algorithms are set as Max_iter=200 and ε=10−6. Image entropy is usually used to measure image quality in ISAR imaging. The smaller image entropy means better reconstruction performance. The image entropy is defined as
(37)Entropy=−∑i∑jxi,j2Elogxi,j2E,where E=∑i∑jxi,j2 is the energy of the radar image x.Here, the image is reconstructed by each range cell. Figures9(a), 9(c), and 9(e) give the reconstruction results of these algorithms in the noiseless case, where we set α=1andβ=1 for the Block EM-VB and α=0.8andβ=1 for the Fast Block EM-VB. In the noisy case, the data are corrupted by additive Gaussian noise and let α=0.85andβ=1 for the Block EM-VB and α=0.6andβ=1 for the Fast Block EM-VB. Figures 9(b), 9(d), and 9(f) demonstrate the results in the case of SNR≈3dB (the noise variance γ−1=0.0025). The entropy values of these algorithms are shown in Table 1. From these reconstruction results, it can be seen that the image obtained by the Block EM-VB and its fast version has better quality in the noisy case, which implies that the Block EM-VB and its fast version have strong noise immunity ability. Table 2 gives the corresponding runtimes of these algorithms, which demonstrates that the Fast Block EM-VB has the highest computational efficiency and can be used in real-time processing.Figure 9
Reconstructed images with different algorithms (the left column is the noiseless case, and the right column is the noisy case).
(a)(b)(c)(d)(e)(f)Table 1
The PSNR results of different algorithms.
PC-SBLBlock EM-VBFast Block EM-VBNoiseless5.63845.32755.4201Noisy5.54575.06895.0059Table 2
Comparison of the runtimes (s) of different algorithms.
PC-SBLBlock EM-VBFast Block EM-VBNoiseless27.71307.83913.3582Noisy30.394031.64836.8828


## 4.1. Performance Analysis via Simulated Data

In the following simulation, let the length of the sparse signal beN=100, and an arbitrary block Gaussian sparse signal is randomly generated with its nonzero entries randomly distributed in B=3 blocks, and the measurement matrix is a random Gaussian matrix. The number of Monte Carlo simulations is 200. The parameters in EM-VB and in the Block EM-VB and its fast version are set as a=b=c=d=10−6, Max_iter=200, and ε=10−5. The parameters α and β in the Block EM-VB and its fast version will be set adaptively according to the noiseless and noisy cases because their performances are sensitive to the choice of α and β. The parameters of PC-SBL are set the same as those in [11].First, we discuss the influences of two parametersα and β on the performance of the proposed Block EM-VB and make a comparison with the EM-VB, which is a special case of the Block EM-VB when α=1andβ=0. To demonstrate the effect of these two parameters, the performances of the Block EM-VB with α=0.7andβ=0 and α=1andβ=0.5 are also given. The support recovery rate is used to evaluate the performance of the Block EM-VB algorithm with different parameters. The recovered support of the sparse signal is defined as suppx^=i,∣x^i∣>0.001, and then the support recovery rate is defined by ∣suppx^∩suppx∣/∣suppx^∪suppx∣, where ⋅ denotes the number of elements in a set. If the overlap between the estimated support and the true support is more, the recovery rate is closer to 1. Figure 1 plots the support recovery rates of different algorithms versus the sparsity level K when the number of measurements is M=30 and M=40, respectively. It can be seen that the parameters of the Block EM-VB algorithm have an important influence on the recovery performance. The appropriate parameter α<1 can avoid the underestimation of the support of the sparse signals, and the block parameter β>0 is helpful to recover the block sparse signals. Thus, the proposed Block EM-VB algorithm with appropriate parameters has better performance than the EM-VB.Figure 1
The support recovery rate versus the sparsity in the noiseless case when (a)M=30 and (b) M=40.
(a)(b)Then, we make a comparison between the Block EM-VB and its fast version with the PC-SBL. In the noiseless case, the success rate is used to evaluate the performances of these different algorithms. When a trial satisfiesx^‐x22/x22<10−6, it is regarded as a successful trial. The success rate is defined as the percentage of successful trials in the total of independent trials. In the noisy case, the reconstruction performance of each algorithm is evaluated by the normalized mean square error (NMSE), where NMSE=x^‐x22/x22.In the noiseless case, letα=0.7andβ=0.5 in the Block EM-VB and α=0.5andβ=0.5 in the Fast Block EM-VB. Figure 2 plots the success rate of individual recovery algorithm versus the sparsity level K when the number of measurement is M=30 and M=40, respectively. Then, let the sparsity K=20 and K=25, and the success rate of each algorithm versus the number of measurements M is shown in Figure 3. From these results, it is observed that the Block EM-VB is superior to the PC-SBL when the number of measurements is less or the sparsity is smaller. The Fast Block EM-VB is inferior to these two algorithms due to the independent assumption.Figure 2
The success rate vs. the sparsity level in the noiseless case when (a)M=30 and (b) M=40.
(a)(b)Figure 3
The success rate vs. the number of measurements in the noiseless case when (a)K=20 and (b) K=25.
(a)(b)Then, we consider the noisy case where the Gaussian noise is added to the measurements with the signal-to-noise ratio (SNR) defined as20log10Φx2/n2. It should be noted that the setting of parameter α in the proposed algorithms needs to consider the trade-off between removing the noise and maintaining the nonzero elements of the signal. A larger value of α tends to suppress noise while losing the nonzero values of the signal. Compared with the Block EM-VB, the fast version needs to select smaller α to ensure the recovery of nonzero elements of the sparse signal because of its inherent denoising ability. Let SNR=10dB and α=0.85andβ=0.5 in the Block EM-VB and α=0.5andβ=0.5 in the Fast Block EM-VB. The reconstruction results of different algorithms when K=25 and M=50 are given in Figure 4. It can be seen that the Block EM-VB algorithm and its fast version have better reconstruction performance than the PC-SBL in the low SNR environment.Figure 4
The reconstructed signal by different algorithms whenN=100, M=50, K=25, and SNR=10dB. (a) Original signal, (b) reconstructed signal by PC-SBL with NMSE=0.2362, (c) reconstructed signal by Block EM-VB with NMSE=0.2184, and (d) reconstructed signal by Fast Block EM-VB with NMSE=0.1092.
(a)(b)(c)(d)The NMSE of each algorithm versus the sparsityK in the case of SNR=10dB is shown in Figure 5, where the numbers of measurements are given as M=40 and M=50, respectively. Figure 6 plots the NMSE of individual recovery algorithm versus the number of measurements M in the case of SNR=10dB, where K=20 and K=25, respectively. It can be found that the Block EM-VB algorithm is superior to the PC-SBL when the number of the measurements is larger or the sparsity is less, and the latter is better than the former when the number of the measurements is less and the sparsity is larger. It is also observed that the Fast Block EM-VB is better than the Block EM-VB when the number of measurements is less or the sparsity is larger. In addition, it is shown that the Fast Block EM-VB outperforms the PC-SBL when the sparsity is less or the number of the measurements is larger.Figure 5
The NMSE vs. the sparsity level whenSNR=10dB and (a) M=40 and (b) M=50.
(a)(b)Figure 6
The NMSE vs. the number of measurements whenSNR=10dB and (a) K=20 and (b) K=25.
(a)(b)The performances of each algorithm in different SNR cases are shown in Figure7, where K=20 and M=50. For the Block EM-VB, the parameters β=0.5 and α is set to vary with the SNR, i.e., α=0.85 when the SNR varies from 0 dB to 15 dB, α=0.75 when the SNR changes from 20 dB to 25 dB, and α=0.73 for the case of 30 dB. The parameters in the Fast Block EM-VB algorithm are still set as α=0.5 and β=0.5. From Figure 7, it can be seen that the Block EM-VB algorithm and the fast version have good reconstruction performance in the case of low SNR when compared with the PC-SBL. The Fast Block EM-VB especially has robust recovery performance in respect of noise immunity.Figure 7
The comparison of different algorithms under different SNR whenN=100, K=20, and M=50.Finally, the average runtimes of these algorithms versus the length of signalsN by 5 independent trials are given in Figure 8, where M=N/2, K=N/10, SNR=10dB, and the number of nonzero blocks B=K/5. It validates that the Block EM-VB almost has considerable computational complexity as the PC-SBL and the Fast Block EM-VB has the highest computational efficiency compared with other recovery algorithms, which makes it have a potential advantage in practical application.Figure 8
The average runtimes vs. the length of signalN.


## 4.2. Application in ISAR Imaging

The inverse synthetic aperture radar (ISAR) imaging is appropriately implemented under the framework of sparse signal recovery due to the sparse characteristic of the target [14]. In this experiment, the “Yak-42” dataset is used, in which the number of range cells is 256 and the number of pulses is 256. 128 pulses are randomly sampled to simulate the sparse aperture data. Here, we use the MATLAB code provided in [14], where the PC-SBL adopts a pruning operation. For a fair comparison, the proposed Block EM-VB and the fast version also use a similar pruning operation and the parameters in these two algorithms are set as Max_iter=200 and ε=10−6. Image entropy is usually used to measure image quality in ISAR imaging. The smaller image entropy means better reconstruction performance. The image entropy is defined as
(37)Entropy=−∑i∑jxi,j2Elogxi,j2E,where E=∑i∑jxi,j2 is the energy of the radar image x.Here, the image is reconstructed by each range cell. Figures9(a), 9(c), and 9(e) give the reconstruction results of these algorithms in the noiseless case, where we set α=1andβ=1 for the Block EM-VB and α=0.8andβ=1 for the Fast Block EM-VB. In the noisy case, the data are corrupted by additive Gaussian noise and let α=0.85andβ=1 for the Block EM-VB and α=0.6andβ=1 for the Fast Block EM-VB. Figures 9(b), 9(d), and 9(f) demonstrate the results in the case of SNR≈3dB (the noise variance γ−1=0.0025). The entropy values of these algorithms are shown in Table 1. From these reconstruction results, it can be seen that the image obtained by the Block EM-VB and its fast version has better quality in the noisy case, which implies that the Block EM-VB and its fast version have strong noise immunity ability. Table 2 gives the corresponding runtimes of these algorithms, which demonstrates that the Fast Block EM-VB has the highest computational efficiency and can be used in real-time processing.Figure 9
Reconstructed images with different algorithms (the left column is the noiseless case, and the right column is the noisy case).
(a)(b)(c)(d)(e)(f)Table 1
The PSNR results of different algorithms.
PC-SBLBlock EM-VBFast Block EM-VBNoiseless5.63845.32755.4201Noisy5.54575.06895.0059Table 2
Comparison of the runtimes (s) of different algorithms.
PC-SBLBlock EM-VBFast Block EM-VBNoiseless27.71307.83913.3582Noisy30.394031.64836.8828


## 5. Conclusions

Considering the clustered structural features of nonzero elements of block sparse signals, this paper proposes the Block EM-VB algorithm for signal recovery, which is based on a correlated LSM model. Furthermore, a fast version of the Block EM-VB is presented, which can recover the block sparse signals with lower computational complexity because of no inversion in each iteration. Experimental results with simulation data and ISAR imaging demonstrate that the Block EM-VB and its fast version have good BSS reconstruction performance and noise tolerance capability, especially in the low SNR scenarios, which implies that the proposed algorithms can be potentially applied in various signal processing fields.



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*Source: 9942694-2021-10-06.xml*