Structured sparsity-driven autofocus algorithm for high-resolution radar imagery

Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Contents lists available at ScienceDirect

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Signal Processing

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journal homepage: www.elsevier.com/locate/sigpro

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Structured sparsity-driven autofocus algorithm for high-resolution radar imagery

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Lifan Zhao a, Lu Wang b, Guoan Bi a,n, Shenghong Li c, Lei Yang a, Haijian Zhang d

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a

School of Electrical and Electronic Engineering, Block S1, 50 Nanyang Avenue, Singapore 639798, Singapore School of Marine Science and Technology, Northwestern Polytechnical University, China Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China d School of Electronic Information, Wuhan University, China b c

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a r t i c l e i n f o

abstract

Article history: Received 22 October 2015 Received in revised form 14 January 2016 Accepted 2 February 2016

Recent development of compressive sensing technology has greatly benefited radar imaging problems. In this paper, we investigate the problem of obtaining enhanced targets of interest such as ships and airplanes, where targets often exhibit structured sparsity in the imaging scene. A novel structured sparsity-driven autofocus algorithm is proposed based on sparse Bayesian framework. The structured sparse prior is imposed on the target scene in a statistical manner. Based on a statistical framework, the proposed algorithm can simultaneously cope with structured sparse recovery and phase error correction problem in an integrated manner. The focused high-resolution radar image can be obtained by iteratively estimating scattering coefficients and phase errors to jointly obtain a structured sparse solution. Due to the structured sparse constraint, the proposed algorithm can desirably preserve the target region and alleviate over-shrinkage problem, compared to previous sparsity-driven auto-focus approaches. Moreover, to accelerate convergence rate of the algorithm, we propose to adaptively eliminate noise-only range cells in phase error estimation stage. The selection is conveniently conducted based on the parameters controlling sparsity degree of the signal in the proposed hierarchical model. The simulated and real data experimental results demonstrate that the proposed algorithm can obtain more concentrated images with much smaller number of iterations, particularly in low Q4 SNR and highly under-sampling scenarios. & 2016 Elsevier B.V. All rights reserved.

Keywords: Radar imagery Compressive sensing High-resolution Structured sparsity Autofocus technique

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1. Introduction

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High-resolution radar imagery has become a popular research area for both civilian and military applications in recent years. The underlying principle of radar imagery is to use synthetic antenna array to achieve 2-dimensional

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Corresponding author. E-mail addresses: [email protected] (L. Zhao), [email protected] (L. Wang), [email protected] (G. Bi), [email protected] (S. Li), [email protected] (L. Yang), [email protected] (H. Zhang).

image [1]. More specifically, high range resolution is obtained by omitting wide-band signals, where linear frequency modulated (LFM) or step LFM signal is chosen to obtain desirable transmitting energy and resolution [1]. The cross-range resolution is determined by width of the synthetic aperture, i.e., coherent processing interval (CPI). Under the assumption of uniform rotation, long CPI can generally result in a higher cross-range resolution. In practical scenarios, however, longer CPI will induce more complicated motion of the target and introduce higherorder Doppler effects, which is contradicted with the assumption of uniform rotation and would undesirably

http://dx.doi.org/10.1016/j.sigpro.2016.02.004 0165-1684/& 2016 Elsevier B.V. All rights reserved.

61 Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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L. Zhao et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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lead to smeared images [1–3]. This problem, however, is conveniently ameliorated by the recent proposed compressive sensing (CS) techniques. In CS based problems, the mathematical model can be formulated as an underdetermined linear equation

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y ¼ ΦΨx

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ð1Þ

where Φ A R with M oN, is the sensing matrix, Ψ A RNN is the sparsifying basis, y A RM is the observation and x A RN is the unknown sparse signal. Since the null space of ΦΨ is non-trivial, least mean square estimation of the problem in (1) can be expressed as, MN

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x^ ¼ arg min‖y  ΦΨx‖22

ð2Þ

x

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which leads to an ill-posed problem and results in infinite number of solutions. Although it is impossible to obtain a unique solution to (1) in general, however, the success of CS technique suggests that under certain conditions, it is possible to recover the high-dimensional signal x from the low-dimensional projections y by algorithms in polynomial time [4–6]. The CS technique has been widely and successfully applied to radar imagery applications to achieve higher resolution with a limited number of pulses and low signalto-noise ratio (SNR) [7–11]. In particular, CS based techniques have shown promising improvements over conventional spectral analysis based methods in synthetic aperture radar (SAR) [6,12], inverse SAR (ISAR) [13,11], passive ISAR [14] and so on. The remarkable advantage of CS based radar imaging approach is that only partial observation is required to obtain image quality comparable to the one using full observation data. The key technique is to properly use sparsity prior to regularize the solution space in a deterministic or probabilistic manner. In the sense of regularization, sparse recovery based approaches are closely related to other regularized methods used in image reconstruction. Recently, to further enhance the performance, structural information, along with sparsity, is introduced in the CS framework to provide robust imagery results in various scenarios [15]. However, it is generally required that pre-processing procedures, such as range compression and alignment, have been successfully conducted before applying CS based technique to achieve high resolution. In practical scenarios, the phase errors still exist in preprocessed data due to the fact that coarse pre-processing cannot eliminate the effects induced by translational motion of the target [16–18]. More specifically, the airborne radar for synthetic aperture radar (SAR) imaging moves to generate the synthetic aperture and the observing terrain is stationary. The SAR may have position errors due to atmosphere turbulence or navigational errors, which may induce phase error causing blurred SAR image. In inverse SAR (ISAR) imaging, radar is stationary on the ground and air-target flies arbitrarily. The translational motion should be compensated since it does not contribute to the image formation. However, due to imprecise compensation of translational motion, the residual phase errors still exist in the model. There is no doubt that performances of the CS algorithm will greatly degrade with the presence of these errors,

which will ultimately lead to smeared or blurred radar image. Considerable efforts have been made on developing autofocus techniques for radar imagery applications. Phase gradient autofocus (PGA) [19], image contrast [20] or entropy based autofocus [21] and multi-channel autofocus (MCA) [22] are classical methods to estimate or compensate the phase errors in radar imagery. These algorithms often operate as a post-processing procedure after image formation. More recently, a novel scheme known as sparsity-driven autofocus algorithms is proposed to utilize sparsity criterion to obtain focused image. Alternating regularized approaches [17,23] and sparse Bayesian approach [18] have been developed to obtain focused images in a sparsity-driven manner. Particularly, it is demonstrated that the autofocus sparse Bayesian learning (AFSBL) [18] can achieve better results than l1 regularized method by exploiting uncertainty information during iterations. The underlying idea among these algorithms is to estimate the sparse scattering scene and phase error in an iterative manner. Compared to the conventional technique such as PGA and MEM where autofocus is used as a post-processing procedure, this new scheme provides an integrated framework to simultaneously achieve highresolution image and autofocus. With sparsity constraint, these algorithms can perform much better in low SNR and under-sampled pulse scenarios. Despite the effectiveness of sparsity-driven methods, they generally suffer from over-shrinkage problems, which is mainly due to the process of merely seeking sparsity solution. Another issue inhibiting the performance of these algorithms is that it requires many iterations to converge in low SNR environment, because heavy noise will undoubtedly disturb the estimation process. In this paper, we propose a sparsity-driven autofocus framework to exploit structured sparsity in a statistical way. In our method, a probabilistic model is imposed on the signal hierarchically in a conjugate fashion to enable convenient Bayesian inference. The parameters are inferred from the model based on variational Bayesian expectation maximization framework. In the proposed framework, better estimation of sparse coefficients can be obtained by exploiting structured sparsity, which in turn results in more accurate estimation of phase errors. The remarkable advantage of the proposed algorithm is that it can properly utilize the structured sparsity prior during iterations to ameliorate the over-shrinkage problem. Different from the current state-of-the-art structured sparsity methods, this framework particularly exploits the range consistence of the radar imagery. This treatment has two advantages. The first one is that the proposed framework preserves the conjugacy of the model and allows the utilization of computational inexpensive Bayesian inference method, which is rather simpler than that in [15]. More importantly, in contrast to [15] which requires the whole measurement set for reconstruction, the proposed algorithm can reconstruct each range cell independently and parallelly, which is much more computationally efficient. The other advantage is that it can more robustly determine the parameter controlling sparsity by utilizing adjacent range cells. Moreover, an adaptive procedure is proposed

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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in the phase error correction stage to enable fast convergence, where modified likelihood function is used to be optimized. The rest of the paper is organized as follows. Section 2 presents the motivation of the work reported in this paper. In Section 3, mathematical model for radar imagery is briefly reviewed. The problem is formulated in a statistical manner in Section 4 and solved by tractable inference method in Section 5. In Section 6, experimental results on real data are given to validate the effectiveness of the proposed approach. 1

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T

H

Notation: For any matrix A, A , A and A denote the inverse, transpose and conjugate transpose of A. The i-th row and column of matrix A are represented by Ai and Ai , respectively. The multivariate complex Gaussian distribution with mean vector μξ and covariance matrix Σξ is h
 1 1 exp  ðξ  μξ ÞH Σξ ðξ  defined as CN ξμξ ; Σξ Þ ¼ N π jΣ ξ j i μξ Þ , where ξ is a N-dimensional vector. The Gamma disa  b ξa  1 e  bξ , where a tribution is defined as pðξa; bÞ ¼ Γ ðaÞ and b are shape and rate parameters, respectively.

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2. Motivation The problem of obtaining enhanced images of targets of interest, such as ships and airplanes has been extensively investigated in recent literature. In realistic radar systems, sparsity exists since the targets are always considered to be sparse with respect to the whole imaging scene. In the presence of the ground or sea clutter, pre-processing procedures can be carried out to suppress the ground or sea clutter in order to make use of sparsity, which is a common assumption in many literature concerned with CS based radar imagery [6,7,24–26]. The sparsity-driven autofocus algorithms have provided many desirable characteristics over conventional ones such as PGA and MEM, in terms of simultaneous autofocus, high-resolution and de-noising capabilities. The basic idea of this novel scheme is to iteratively estimate sparse scattering coefficients and phase error to jointly induce sparsity. However, this particular sparsity-inducing scheme may result in undesirable image results, because it merely considers sparsity as the performance measure. More concretely, the weak scatterers cannot be well-preserved and background noise cannot be properly shrunk simply with sparsity constraint. To enhance the performance of this scheme, it is desirable to consider and incorporate more structural information in radar image apart from sparsity. In structured sparsity literature, various structures are intensively investigated in various problems [27]. More recently, the continuity structure is considered in radar imaging problems, where computational intensive sampling method is used to estimate the parameters in [27,15,28,29]. Inspired by the structured sparsity model in [15,28], we intend to exploit the property of range consistence in this paper to simultaneously obtain weak scatterer preservation and noise removal. To facilitate a more convenient inference in this

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paper, a convenient probabilistic model is proposed to impose range consistency, which can facilitate efficient parameter estimation. The range consistency can be discovered by inspecting into the real radar imagery. For example, the radar image of Mig-25 airplane in Fig. 1(a) is clearly seen that adjacent range cells share similar sparse spectrum profile. For example, Fig. 1(b) shows the Doppler spectra of the Mig-25 data with range cells from 35 to 39. It is observed that the Doppler spectra in these adjacent range cells are highly correlated with each other. The similar phenomenon occurs for Yak-42 airplane in Fig. 1(c) and (d). Therefore, performance improvement can be expected if this desirable structural consistency is exploited in the process of sparse coefficient recovery and autofocus. Remarkably, the structural consistency exhibits in almost all imagery applications for moving targets due to the fact that the target itself is a continuous scattering object. Motivated by this continuity of the radar image, we exploit the structural sparsity in the scattering coefficients during the autofocus process to avoid over-shrinkage. In this novel approach, an integrated framework is proposed to parametrically correct the phase errors and statistically recover the structured sparse coefficients. In this way, more concentrated target region can be obtained, where the weak scatterers can be preserved and isolated strong noise can be eliminated. The better estimation of sparse coefficient will, in turn, lead to more accurate phase error estimation. Another intuition of this paper is that scatterers are clustered in a small portion of the target scene of interest. Taking Fig. 1(c) as an illustrating example, the target only resides from range cell 20 to 100, where there is a number of range cells containing no scatterer. In phase error estimation, the scatter-free range cells contribute little to the estimation and the accuracy of estimation will be degraded due to the influence of noise. However, in the previous approaches, they often blindly use the measurements from all range cells regardless of the existence of scatterer or not. This strategy is simple but not efficient. In this paper, we develop a new strategy to adaptively determine and utilize range cells containing scatterers for better phase error estimation.

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3. Preliminary

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In this section, the mathematic model of radar imagery is briefly reviewed. As discussed in Section 1, the emitting waveform of the radar system is LFM signal. The reason for choosing LFM pulses is to achieve high range resolution after the process of range compression of the radar echoes. Meanwhile, multiple LFM pulses are to be emitted at a repeating frequency fr to obtain high cross-range resolution. Assuming that there exist Ks scatterer centers and the range compressed signal is then expressed as src ðt; t m Þ ¼

Ks X k¼1

     tm 2R ðt m Þ   sinc γ T π t  k c Tr

111 113 115 117 119 121

σ k rect

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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Fig. 1. (a) The Mig-25 dataset and, (b) its Doppler spectra of range cells from 25 to 29. (c) The Yak-42 dataset and, (d) its Doppler spectra of range cells from 76 to 80.

  R ðt m Þ ; exp  j4π k c

101 ð3Þ

where σk is the complex amplitude of the k-th scatterer, c ¼ 3  108 is the speed of light, t denotes fast time, t m ¼ m=f r denotes slow time of the m-th pulse, γ represents chirp rate of LFM signal and T denotes pulse width. Moreover, the approximated distance between radar and the k-th scatter Rk ðt m Þ can be expressed as Rk ðt m Þ  R0 þyk þ xk Δθðt m Þ

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ð4Þ

where R0 is the distance between radar and geometric center of the target, Δθðt m Þ is the rotational angle in m-th pulse, xk and yk are range and azimuth distance to the geometric center of the target, respectively. A matrix, Src ðt; t n Þ, consists of the received data, where range cells and cross-range cells are arranged in horizontal and vertical directions, respectively. For notational brevity, Src is denoted by Y in latter sections of this paper. Since phase error in radar imagery often exhibits range invariant property [17,21], the mathematical model can therefore be

given as the following linear equations: ð5Þ

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where Y A CPM is the received data, X A CNM is the sparse scattering coefficient and N A CPM is zero mean Gaussian noise. The matrix Φ is under-sampled from identity matrix I and Ψ is orthogonal Fourier basis. Therefore, each ~ ~ atom in ΦΨ can be defined as ½e  j2π f i t 1 ; …; e  j2π f i t P T , where t~ i ; i ¼ 1; …; P are under-sampled time instances. In latter sections of the paper, ΦΨ is denoted by A for simplicity.

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Y ¼ EΦΨX þ N

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Remark 1. In CS based radar imagery, the number of under-sampled pulses, P, is set to be much smaller than the number of reconstructed Doppler cells, N, i.e., P⪡N and under-sampled data are obtained by randomly selecting a number of pulses from full aperture data. Notably, apart from random sampling scheme, other sampling schemes can also be designed to accommodate for difference application scenarios, such as achieving higher resolution with limited data or accidentally loss of data. The only difference is to modify matrix A according to specific

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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scheme. The sparsity of the scattering coefficient makes the reconstruction possible. Different from CS based radar imagery reported in [13], we incorporate the phase error in the linear model. The phase error matrix is represented by E ¼ diagðejφ1 ; …; ejφP Þ, which is a diagonal matrix representing fast time invariant and slow time variant errors.

9 4. Structured sparse Bayesian model 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

The sparse Bayesian learning [30–32] is a promising class of sparse signal recovery algorithms. In [17], the radar imagery problem has been formulated in a Bayesian compressive sensing framework. Because this method cannot exploit higher-order statistical information, error propagation phenomenon exists [18]. In our work, rather than rigidly estimating noise variance and hyperparameter as a pre-processing procedure, a novel approach is developed to adaptively estimate the sparse coefficient and refine other parameters based on a full Bayesian framework. In the following, radar imagery problem will be formulated in a sparse Bayesian framework. Rather than independent modeling of range cells in conventional CS radar imagery, structural consistency is imposed over range cells to obtain better imaging performance. In radar imagery, the noise N is often modeled as an independent identical complex Gaussian distribution among range and Doppler cells. Conditioned on X and noise variance, it is straightforward to see that the received signal Y obeys a complex Gaussian distribution. Therefore, the likelihood function of the received signal Y, conditioned on sparse coefficient X and noise variance α0 1 , can be expressed as pðY i jXi ; α0 ; EÞ ¼ CN ðY i jEAXi ; α0 1 IÞ;

i ¼ 1; …; M

ð6Þ

where α0 is the noise precision (reciprocal of the variance). In sparse Bayesian learning (SBL) based method, sparse signal is hierarchically modeled to achieve sparse prior as well as convenient inference. In conventional radar imagery based on SBL [15,18], it is assumed that each range cell vector Xi follows an independent complex Gaussian distribution, pðXi jαi Þ ¼ CN ðXi j0; Λi Þ;

i ¼ 1; …; M

ð7Þ

where Λi , denoting diagðαi Þ, is the variance vector for Xi . It is noted that when αji approaches zero, its corresponding element in X will be pruned away from the model since it also approaches zero. In order to utilize the spatial aware consistency among adjacent range cells as discussed in Section 2, we propose the following modeling on the ith range cell, pðXi jαÞ ¼

W=2

∏

w ¼  W=2

CN ðXi j0; Λi þ w Þ

ð8Þ

where W is an even number to represent the number of adjacent range cells for range consistent assumption. Fig. 2 shows the consistent structure of the W range cells. This modeling can be interpreted as a statistical way to impose range cell consistency by updating the i-th signal variance conditioned on W adjacent range cells. However, since this

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consistency is intermediately imposed on the signal by manipulating its variance, the proposed modeling should be considered as a flexible way to allow learning from data. Remark 2. The number W can be chosen as an appropriate value to encourage spatial consistency as well as flexible modeling. Intuitively, larger W will induce more interaction among range cells, while smaller W will impose more independence. It is noted this model will correspond to (7) when W¼0, where uncorrelated individual tasks are assumed. The variance αi of the scatterer coefficient Xi , also known as hyper-parameter, obeys an independent Gamma distribution for convenient inference due to its conjugacy to Gaussian distribution [33]. Since it is assumed that adjacent range cells have similar sparse Doppler spectrum, it is justifiable to impose the consistency of adjacent parameter α, pðαi jλÞ ¼

W=2

N

∏ Γ ðαki jη; λi þ w Þ;

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ð9Þ

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where αki is the k-th element in αi . Since λi controls the sparsity degree of the i-th range cell, the above modeling can naturally incorporate this consistency information. It is noted that the marginalized distribution of Xi is a complex Laplace distribution if η ¼ 3=2 with parameter λ. To conveniently estimate the hyper-parameters λ controlling sparsity and noise precision α0, they are modeled into independent Gamma distributions as,

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∏

w ¼  W=2 k ¼ 1

i ¼ 1; …; M

M

pðλjv1 ; v2 Þ ¼ ∏ Γ ðλi jv1 ; v2 Þ;

ð10Þ

pðα0 jv3 ; v4 Þ ¼ Γ ðα0 jv3 ; v4 Þ:

ð11Þ

i¼1

where v1, v2, v3 and v4 are set to be trivial values to induce non-informative prior on these parameters. In this paper, they are all set to be 10  4. The maximum a posterior (MAP) estimation of the above-described probabilistic model suggests a close related convex optimization framework fX; Eg ¼ arg min ‖Y EAX‖2F þ X;E

M X i¼1

  Xi  W=2:i þ W=2 

2;1

:

ð12Þ

This formulation is closely related to the regularized approaches. Because the problem formulation in (12) is non-convex, a straightforward solution is to iteratively estimate X and E, where shallow suboptimal solution rather than the global-optimal one is often obtained. In [17], the formulation can be considered as a special case of (12), with only sparsity constraint. In this paper, rather than utilizing the point estimation obtained in (12), a full Bayesian inference procedure is carried out to make full use of the uncertainty information as well as structured sparsity. In this sense, the obtained solution has smaller possibility of converging to a local minimum, since the posterior estimation can smooth away the shallow local minimum [34]. Remark 3. We have highlighted herein that the success of our proposed formulation fundamentally depends on

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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expressed as ln q ðα0 Þ ¼ 〈ln pðYjX; α0 Þpðα0 jv3 ; v4 Þ〉qðXÞ þc0 : 

3

ð13Þ

where q ðxÞ is the optimal approximated posterior distribution and 〈  〉qðxÞ denotes the expectation with respect to q(x). The updating rule for α0 can be given as the mean of the Gamma distribution, ( " ^ μ‖2 ln q ðα0 Þ p ðMP þv3  1Þln α0  ‖Y  EA

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F

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þ

N X

# )

H ^ Σi þv4 α0 : trace AH E^ EA

ð14Þ

i¼1

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〈α0 〉 ¼

17

Fig. 2. The graphic representation of the proposed correlation structure. (a) The geometry of adjacent range cells. (b) The graphic representation of the signals.

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sparsity constraint, since the CS based high-resolution imaging problem is itself an under-determined problem. We argue that only imposing structural information without sparsity, such as total variation (TV) norm or Markov random filed (MRF), cannot give any reasonable results in our problem.

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5. Structured sparsity-driven autofocus algorithm

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Because the intractability of the marginal distribution, pðYÞ, results in the unattainability of the full posterior, we employ variational Bayesian expectation maximization (VBEM) in structured sparsity-driven autofocus framework. The VBEM algorithm is briefly reviewed in Appendix. The noise variance and structured sparse coefficient are modeled as random variables, while phase error matrix E is modeled as a deterministic parameter. Therefore, estimation of these parameters will be carried out in variational expectation stage and maximization stage, respectively.

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5.1. Noise estimation

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Since the noise is assumed to be independent and identically (i.i.d.) distributed in all resolution and Doppler cells, full measurement data are used to estimate the noise variance. Due to conjugacy of Gamma and Gaussian distribution, the posterior of noise variance parameter can be

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5.2. Structured sparse coefficient estimation

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MP þ v3 ^ μ‖2 þ PM traceðAH E^ H EA ^ Σi Þ þ v4 : ‖Y  EA i¼1 F

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α0 as,

Different from [24] where noise-only range cells are used for noise estimation, this method utilizes all the range cells to obtain estimation which is refined iteratively.

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In this stage, hierarchical parameters in sparse model are updated individually to allow partially sharing information in a spatial consistent manner. We will show in the updating rule that information sharing is naturally incorporated by the statistical model given above. (i) The estimation of X: Based on the Markov blanket, the approximated posterior can be given as, * + W=2 ^ ln q ðXi Þ ¼ ln pðY i jXi ; EÞ ∏ CN ðXi j0; Λi þ w Þ þ c0 w ¼  W=2

ð16Þ where c0 is a constant with respect to Xi . By substituting (6) and (8) into (16), we obtain that each Xi obeys a complex Gaussian distribution as h i 1 ð17Þ ln q ðXi Þ p ðXi  μi ÞH Σi ðXi  μi Þ where μi and Σi are given by H

ð18Þ 0*

H ^ þ diag@ Σi ¼ 4hα0 iAH E^ EA

iþ W=2 X

+13  1 1 ./ αi A5 :

ln q ðαi Þ p ln

∏

w ¼  W=2

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Based on the above equations, it is concluded that αi determines sparsity profile of the signal Xi . However, it is worthwhile to point out that the updating rule for the i-th signal does not directly incorporate information of adjacent W sparse signals. This validates our argument that spatial range consistency is not rigidly imposed on the signal. (ii) The estimation of αi : Based on the Markov blanket, approximated posterior can be given as, * + W=2

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ð19Þ

i  W=2



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μi ¼ hα0 iΣi AH E^ Yi 2

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pðXi þ w jαi Þpðαi jλi Þ

:

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qðXi Þqðλi Þ

ð20Þ

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Substituting (8) and (9) into (20), the approximated posterior of αji can be transformed into a canonical generalized inverse Gaussian (GIG) distribution as, 2 3 * + W=2 X  q ðαji Þ p ðη  1Þln αji 4  2λi αji  2 jxjði þ wÞ j2 αji 1 5: w ¼  W=2

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ð21Þ

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Therefore, the k-th moment of the GIG distribution [35] is given in (20) in next page, where κa is the modified Bessel function of the second kind:

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E1 k 0D PW=2 2 D E 2 w ¼  W=2 jxjði þ wÞ j2 A   αkji ¼ @ 2λi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Effi  D PW=2 κη  1  W þ k 2λi 2 m ¼  W=2 jxjði þ wÞ j2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Effi  D PW=2 κη  1  W 2λi 2 w ¼  W=2 jxjði þ wÞ j2

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M

i¼1

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:

ð23Þ

∏M qðαi Þ i ¼ 1

Substituting (9) and (10) into (23), it is seen that the approximated posterior for λi obeys a Gamma distribution due to the prior conjugacy, 2 0 1 3 W=2 N X X ηN þ v1  1 qðλi Þ p λi exp 4  @ αkði þ wÞ þ þv2 Aλi 5:

43

ð24Þ

45

Since the updating for hyper-parameter only involves firstorder moment of the distribution, the mean of the Gamma distribution can be given by

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ηWN þ v1 : PN k ¼ 1 αkði þ wÞ þv2 w ¼  W=2



λi ¼ PW=2

ð25Þ

5.3. Phase error estimation The solution to E can be formulated by minimizing the negative expected log-likelihood function as, E^ ¼ arg min〈  lnpðY; X; α; λ; EÞ〉qðXÞqðαÞqðλÞ

ð27Þ

ð26Þ

It is noted that the above problem is a convex optimization with a closed-form solution [36]. In order to properly

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ð28Þ

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ð29Þ 81

However, this updating scheme is inefficient in low SNRs scenarios due to the influence of strong noise. To motivate a computational efficient algorithm with desirable convergence rate, we argue that it is not suitable to use full measurement likelihood to obtain estimation of phase errors, since some of noise-only range cells cannot contribute to better phase error estimation. Therefore, eliminating noise-only cells can help to alleviate the impact of heavy noise on sparse recovery process. To accelerate convergence rate of the algorithm, we propose to adaptively select range cells that contain strong scatterers. An intuitive method is to measure the power of Xi to determine which range cells to be selected. However, directly determining from the estimation of sparse coefficient requires further design of detection strategy. Rather than explicitly determining range cells from sparse coefficients, selection procedure can be simply conducted in the λ parameter space. This criterion is convenient and effective since λi itself is a measure on sparsity of signal, i.e. lower λi indicates stronger signal. Therefore, by arranging vector λ in an ascent order, denoted as λ~ ¼ ½λ~ I1 ; …; λ~ IM T , the selection rule can be expressed as,

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I ¼ fI i : i ¼ 1; …; kg

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ð30Þ

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where k can be set to be a number of range cells containing scatters. The value of k can be conveniently set by a preprocessing procedure, which is not discussed further for brevity. Based on the selected range cell set I , the modified likelihood can be defined as,

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pðY~ i ; XI i ; αI i ; λi ; EÞ ¼ CN ðY~ i jEAI XI i ; α0 1 IÞ

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ð31Þ

where AI represents the column under-sampled matrix of A and XI i is the under-sampled vector. The modified measurement Y~ can be given as, Y~ ¼ Y  E^

E

¼ arg min〈‖Y EAX‖22 〉qðXÞ : E

traceðμ

Re½Yi ðAi μÞH  P H μÞ þ M k ¼ 1 traceðAi Ai Σk Þ

H AH A i i

Ei ¼ ai þ jbi :

w ¼  W=2 k ¼ 1

47

ai ¼

where i ¼ 1; …; P. The estimation of E is given by,

It is remarkable to note that in the updating equation for αi , the W adjacent sparse coefficients from Xi  W=2 to Xi þ W=2 are incorporated. This incorporation of information naturally imposes estimation of adjacent range cells into the estimation of the current range cell, where structural consistency can be statistically obtained. Rather than explicitly imposing this structural information on the signal, the prior is implicitly imposed on the variance to enable a flexible modeling process. (iii) The estimation of λ: The approximated posterior of λ can be obtained by * + q ðλÞ ¼ exp ln ∏ pðαi jλi Þpðλi jv1 ; v2 Þ

utilize uncertainty information, we incorporate Σ to enhance estimation accuracy by modeling real and imaginary parts of the error as ai and bi, respectively [18]. By introducing these two parameters instead of the angle φi, uncertainty information can be naturally incorporated into the estimation process to achieve enhanced estimation of E in each iteration. Taking derivative with respect to ai and bi and setting them to zeros, we can obtain

Im½Y i ðAi μÞH  : bi ¼ P H H traceðμH Ai Ai μÞ þ M k ¼ 1 traceðAi Ai Σk Þ

ð22Þ

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AI c XI c 

ð32Þ

n1

is the estimation of phase error in the (n-1)-th where E^ iteration and I c is the complement set of I . This manipulation can be considered as eliminating the influence of non-scattering cells. Therefore, estimation of phase error

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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16: Update Eði; iÞ by (33) and (34). 17: end for 18: end while 19: Output: X.

in the n-th iteration can be expressed as,   n E^ ¼ arg min Y  E^ n  1 AI c XI c   EAI XI  2 qðXÞ 2

E

Similarly, taking derivative with respect to ai and bi, we can obtain Re½Y~ i ðAiI μI i ÞH  P H μI i Þ þ M k ¼ 1 traceðAiI AiI Σk ðI ; I ÞÞ

ð33Þ

Im½Y~ i ðAiI μI i ÞH  bi ¼ PM H H A A μ traceðμH k ¼ 1 traceðA iI A iI Σk ðI ; I ÞÞ I i iI iI I i Þ þ

ð34Þ

ai ¼

traceðμ

H AH A I i iI iI

where Σk ðI ; I Þ represents a matrix with columns and rows under-sampled from Σ. In summary, the proposed structured autofocus sparse Bayesian learning (SAFSBL) can be given by iteratively updating X, α, λ, α0 and E as summarized in Algorithm 1. Algorithm learning.

1. Structured

autofocus

sparse

Bayesian

1: Input: Y, A, E, α, v1, v2, v3, v4. 2: while  Converge do 3: % I: NoiseEstimation

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Remarkably, by imposing the structured sparsity on the sparse coefficient X and manipulating Ei;i as a þ jb, improved estimation can be obtained during the iterative process. Fast convergence can also be facilitated by the adaptive selection of range cells in each iteration. Remark 4. In our problem, both phase error parameters and sparse coefficients are required to be estimated. An iterative procedure is necessary to be carried out, where the error propagation issue inevitably existing in regularization problems which have been solved by MAP estimation. It is particularly important to encode the structured sparsity in a hierarchical Bayesian model as in our formulation. Moreover, the Bayesian formulation of the structured sparsity can conveniently preserve the estimation uncertainty information during iterations, which can alleviate effect of error propagations.

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4: Update α0 by (15) 5: % II: Structured Sparse Coefficient Estimation 6: for i ¼ 1: M do 7: Update μi and Σi by (18) and (19). 8: Update αi by (22) 9: Update λi by (25) 10: end for 11: % III: Phase Error Correction 12: for i ¼ 1: P do 13: % Without Adaptive Selection 14: Update Eði; iÞ by (27) and (28) 15: % With Adaptive Selection

6. Experimental results

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The performance comparison of the proposed SAFSBL algorithm with other recent reported ones are presented in this section. In the following experiments, phase error obeys N ð0; 1Þ. The MSE of phase error estimation is given by,

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1 MSEe ¼ ‖E^  E‖22 P

Table 1 Radar system parameters for Mig-25 dataset. Centroid frequency fc Band width B Repetition frequency fr Number of range cells M Number of pulses N

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9 GHz 512 MHz 15 Hz 64 64

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ð35Þ

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where E^ is the estimated phase error and E is true phase error. The measures used to evaluate the quality of radar image are correlation and entropy. The correlation is defined as   vecðXr ÞT vecðXtrue Þ    ð36Þ Corr ¼  vecðXr Þ vecðXtrue Þ 2 2

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Fig. 3. (a) True phase error. The radar image of Mig-25 dataset (SNR¼ 5 dB) obtained by (b) RD algorithm, (c) l1, (d) SAFSBL and (e) SAFSBL (Adaptive). The phase errors in the first row obeys uniform distribution with ½  π: π and in the second row obeys uniform distribution with ½  π=2: π=2.

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Fig. 4. (a) True phase error. The radar image of Mig-25 dataset (SNR ¼5 dB) obtained by (b) AFSBL2, (c) SAFSBL and (d) SAFSBL (Adaptive). The sampling scheme in the first row is random sampling and in the second row is uniform sampling.

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Number of iteration Fig. 5. The convergence curves of AFSBL2, SAFSBL and SAFSBL(Adaptive) in terms of (a) MSEe , (b) Image Entropy and (c) Image Correlation.

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61 Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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Table 2 Corr and Entropy of the reconstructed radar image. SNR

5

5 db

7

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Corr Entropy Corr Entropy Corr Entropy

AFSBL

SAFSBL

SAFSBL(Adaptive)

0.7321 2.9132 0.5689 3.1273 0.4756 3.3008

0.7716 2.5251 0.6211 2.8655 0.5624 3.1386

0.8418 1.8744 0.7771 2.3502 0.6541 2.7950

11 13 15 17

Table 3 Radar system parameters for Yak-42 dataset. Centroid frequency fc Band width B Repetition frequency fr Number of range cells M Number of pulses N

10 GHz 400 MHz 25 Hz 128 128

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and entropy of the image is defined as X Entropy ¼  p  log 2ðpÞ

ð37Þ

where p is histogram of the gray level image. The correlation metric measures similarity of the recovered image with the true one, while the entropy measures concentration quality of the image. In the following experiments, phase errors are uniformly distributed within ½  π : π  unless specified. The l1 minimization problem is solved by alternating direction method of multipliers (ADMM) [37] and mixed TV-l1 minimization is also solved by ADMM [38]. The regularization parameter for l1 minimization is set to be 1:6 J AH Y J 1 for Mig-25 dataset and 7 J AH Y J 1 for Yak-42 dataset.

37

6.1. Synthetic data experiment

39

The radar system parameters of Mig-25 synthetic dataset are given in Table 1. In the following experiments, we will focus on the performance comparison of the AFSBL2 [18], SAFSBL and SAFSBL (Adaptive). In the following sections, number of iterations is 80 unless specified. Firstly, in order to show the difference of the proposed schemes, both non-adaptive and adaptive ones, we use 30 iterations in this experiment. It is shown in Fig. 3 that the target is not well focused using l1 minimization method because it makes use of merely sparsity. Due to utilization of structured sparsity, both the SAFSBL and SAFSBL (Adaptive) can achieve more concentrated target imaging result. The reason that this structured information can greatly enhance phase error correction is because this particular priori can preserve the target region. In particular, the SAFSBL (Adaptive) algorithm can obtain the most concentrated target region, because its inherent adaptive strategy in selecting range cells can alleviate noise effects during recovery process. Fig. 4 is to show performances of the proposed method with different sampling scheme. The first row of the figures is generated by randomly selecting 32 pulses from full

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data, while the second row of figures is generated by selecting the first 32 pulses of the full data. As seen from these figures, results of the proposed methods are rather similar under different sampling schemes. It can be concluded that the proposed algorithm can be rather straightforwardly applied to different sampling schemes and can still achieve desirable results. To further demonstrate convergence of the algorithms, Fig. 5 shows convergence curves of these three algorithms in terms of MSEe , Corr and Entropy. In order to demonstrate that the proposed adaptive scheme is particularly useful in low SNR environment, SNR is set to be 0 dB. We have carried out 40 iterations to observe full convergence path. It can be observed that the estimation of E for SAFSBL (Adaptive) can converge within 15 iterations only, while SAFSBL requires 30 and AFSBL2 requires 40. It seems that image entropy takes more iterations to converge due to the fact that shrinkage of heavy noise cells will continue even when phase error estimation has converged. Therefore, it can be concluded that with adaptive selection procedure, the proposed SAFSBL can converge faster. Notably, upon convergence, the proposed SAFSBL and adaptive SAFSBL can achieve similar results. The performances of the algorithms are also evaluated in different SNR scenarios, where low SNRs ranging from  5 dB to 5 dB are highlighted. As given in Table 2, the proposed SAFSBL (Adaptive) algorithm can obtain lower entropy, higher correlation and lower MSEe . When these algorithms converge, the SAFSBL (Adaptive) can obtain the best performance for all performance measures.

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6.2. Real data experiment 95 The Yak-42 dataset is tested with the proposed and other popular algorithms in various scenarios. The radar system parameters of this dataset are given in Table 3. In these experiments, we particularly demonstrate both high-resolution and autofocus results achieved by the proposed SAFSBL algorithm compared with those obtained by l1 based method and recently proposed AFSBL method. In all the following figures, all the radar images are normalized and presented in contour style for better visualization purpose. The image size is 128  128 and the noiseless RD imagery result with full measurement is shown in Fig. 1(c). As observed, the yak-42 image obtained by the full measurements is well concentrated. In this case, we make use of this dataset to be a reference and generate the experimental data by adding noise on this dataset. Therefore, SNR of the generated dataset is actually below the predefined one. In the following experimental results, we particularly compare our results with sparsity-driven autofocus approach, i.e., l1 regularized approach [17], mixed TV-l1 and AFSBL2. To provide a fair comparison with other approaches who require more iterations to converge, the number of iterations is set to be 80 in the following experiment unless specified. In Fig. 6, the imaging results are compared with different phase error levels. It is seen that the SAFSBL can give very concentrated target region compared with the sparsity-driven AFSBL2. In fact, with increase in phase error level, the SAFSBL did not obviously degrade.

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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However, the AFSBL2 degrades significantly. From this experiment, it can be validated that by exploiting structured sparsity, concentrated imaging performance can be achieved. In Fig. 7, the autofocus performance in terms of different SNRs is provided to demonstrate the effectiveness of the proposed algorithm. In general, all of these compared algorithms can achieve reasonable results when SNR is

8 dB, where obtained target image is well concentrated. However, when SNR decreases, degraded imagery results appear, particularly those obtained by l1 and mixed TV-l1 based regularized approaches. The AFSBL2 can achieve better results although the target is not well concentrated. In contrast, the proposed SAFSBL can achieve much more concentrated target region due to utilization of range cell consistency. When SNR is as low as 3 dB, the l1 approach

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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almost fails to obtain a meaningful image, the mixed TV-l1 appears to have too many artifacts and the AFSBL2 suffers from loss of scattering coefficients. However, the SAFSBL demonstrates promising imagery results with a much more concentrated target region and less artifacts around the region. The improvement in the imagery results obtained by SAFSBL can validate the effectiveness in exploiting structural information during estimating of sparse coefficients. In Fig. 8, the autofocus performance in terms of different under-sampling ratios is provided to show the robustness of the proposed algorithm. More measurements generally give better imagery results for all these algorithms. Compared with l1 approach, mixed TV-l1 and AFSBL2, the proposed SAFSBL can achieve better concentrated image with more coefficients recovered in the target region under different under-sampling ratios, where a much more concentrated image can be observed. The performance of SAFSBL is competing with that of RD imagery with full measurement data given in Fig. 1. Therefore, it can be concluded that the SAFSBL exploiting structure sparsity can substantially benefit the autofocus performances.

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7. Conclusion

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In this paper, an adaptive phase error correction technique for high-resolution radar imagery is proposed based on sparse Bayesian method. In the proposed algorithm, a Bayesian learning framework is formulated, where noise variance, structured sparse scattering coefficients and

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phase errors are updated iteratively by using variational Bayesian expectation maximization technique. The main idea is to jointly achieve structural sparsity by this particular parametric learning process. In this way, the autofocus can be more effectively and robustly performed by enforcing structured sparse estimation of the scattering coefficients. More importantly, an adaptive procedure is proposed to achieve fast convergence. Numerical experimental results demonstrate the effectiveness of the proposed SAFSBL algorithm in providing more concentrated target image in various scenarios.

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Acknowledgment 107 This work is partially supported by project funds MOE2014-T2-1-079 and AcRF Tier-1 RG103/14, Singapore.

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Appendix A. Review on variational Bayesian expectation maximization The intuitive idea of this method is to use a particular class of distribution to approximate the true posterior. Based on the factorisable assumption of the posterior in VBEM method, we can have J

pðΘjY; θÞ  ∏ qðΘi Þ

113 115 117 119

ðA:1Þ

i¼1

where Θ denotes set of the unknown random parameters and θ denotes deterministic parameters. Based on the

Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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mean-field assumption [39], the VB procedure can be obtained by minimizing KL divergence of the true distribution and approximated one, where Kullback Leibler (KL) divergence between distribution p(x) and q(x) is defined as, Z  pðxÞ dx: ðA:2Þ DKL ð pðxÞqðxÞÞ ¼ pðxÞ ln qðxÞ Therefore, optimal approximated distribution can be derived by solving the following optimization problem,



  q Θ ¼ arg minDKL q Θ J p ΘY; θ qðΘÞ

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¼ arg min qðΘÞ

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Z


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qðΘÞ dΘ: pðΘjY; θÞ

The best approximated distribution for can be thus expressed as [39], n o q ðΘi Þ ¼ exp 〈ln pðΘ; YÞ〉qðΘ⧹Θi Þ

ðA:3Þ

Θi given q ðΘj a i Þ ðA:4Þ

where 〈  〉qðÞ represents the expectation with respect to qðÞ. The maximization procedure is to maximize the loglikelihood with respect to the deterministic parameter,

θ ¼ arg min〈  log pðY; θÞ〉qðΘÞ θ

[14]

[15]

[16] [17]

[18]

[19]

[20]

[21]

[22] [23]

ðA:5Þ

The VBEM algorithm can be summarized as a process of iteratively updating the random and deterministic parameters.

[24]

[25]

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Please cite this article as: L. Zhao, et al., Structured sparsity-driven autofocus algorithm for high-resolution radar imagery, Signal Processing (2016), http://dx.doi.org/10.1016/j.sigpro.2016.02.004i

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