Open Access
This article is
 freely available
 reusable
Sensors 2019, 19(3), 490; https://doi.org/10.3390/s19030490
Article
Accurate Wide Angle SAR Imaging Based on LSCSResidual
^{1}
Key Laboratory of Technology in GeoSpatial Information Processing and Application System, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China
^{2}
School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 101408, China
^{3}
Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China
^{*}
Author to whom correspondence should be addressed.
Received: 4 December 2018 / Accepted: 22 January 2019 / Published: 25 January 2019
Abstract
:Wide angle synthetic aperture radar (WASAR) receives data from a large angle, which causes the problem of aspect dependent scattering. ${L}_{1}$ regularization is a common compressed sensing (CS) model. The ${L}_{1}$ regularization based WASAR imaging method divides the whole aperture into subapertures and reconstructs the subaperture images individually. However, the aspect dependent scattering recovery of it is not accurate. The subaperture images of WASAR can be regarded as the SAR video. The support set among the different frames of SAR video are highly overlapped. Least squares on compressed sensing residuals (LSCSResiduals) can reconstruct the time sequences of sparse signals which change slowly with time. This is to replace CS on the observation by CS on the least squares (LS) residual computed using the prior estimate of the support. In this paper, we introduce LSCSResidual into WASAR imaging. In the iteration of LSCSResidual, the azimuthrange decoupled operators are used to avoid the huge memory cost. Real data processing results show that LSCSResidual can estimate the aspect dependent scatterings of the targets more accurately than CS based methods.
Keywords:
wide angle SAR; compressed sensing; LSCSResidual; aspect dependent1. Introduction
Wide angle synthetic aperture radar (WASAR) receives echo data from a large angle. Advances in synthetic aperture radar (SAR) technology have enabled coherent sensing of WASAR. Circular SAR (CSAR) is a specific case of WASAR whose track is circular. With the increase of the synthetic angle, because of the reflector geometry, shadowing, and coherent scintillation, the problem of aspect dependent scattering [1,2] arises. Traditional imaging methods are based on the isotropic assumption. It means that the scattering is constant in the synthetic aperture angle, which is not valid in WASAR.
To accommodate the aspect dependent scattering, there are mainly two approaches, the subaperture approach and full aperture approach [2]. The subaperture approach [1] divides the whole aperture into the subapertures and assumes that the scatterings are constant in the subaperture. Then, the narrow angle imaging methods such as matched filtering [3] and an ${L}_{1}$ regularization based SAR imaging method [4,5] can be adopted for the subaperture imaging. For the full aperture approach, they can be divided into two kinds. The first one assumes that the scattering during one subaperture is isotropic and reconstructs an imaging model with all subapertures included [6,7,8]. The subaperture images are recovered jointly. The other one is the parametric method [9,10]. It assumes that the scene includes some scattering targets and that their scatterings follow some functions. The scattering functions of the targets are fitting with the whole aperture data included.
In the past decade, compressed sensing (CS) [11,12] has drawn much attention in sparse signal processing, which provides reconstruction guarantees for sparse solutions to linear inverse problems. It is shown that, when the scene is sparse and the measurement matrix satisfies the Restricted Isometry Property (RIP), the signal can be recovered with downsampled data by solving an ${L}_{1}$ minimization problem [12]. An ${L}_{1}$ minimization problem is also known as Basis Pursuit. The theory of Lagrange multipliers indicates that we can solve an unconstrained problem that will yield the same solution, provided that the Lagrange multipler is selected correctly. The unconstrained problem is known as LASSO or ${L}_{1}$ regularization [13]. For the subaperture reconstruction based on ${L}_{1}$ regularization, it mainly has two drawbacks. Firstly, as a common reconstructed model in compressed sensing (CS), ${L}_{1}$ regularization is a biased estimator [14], which means that the amplitude of the targets would be underestimated. Secondly, the support set of the targets is not accurately estimated with the data of one subaperture and there are some missed detections. For the first drawback, it can be solved via debiasedCS proposed in [15,16]. DebiasedCS is a twostep method, which firstly reconstructs the signal with CS and calculates the least squares (LS) estimates on the support set of the signal. For the second drawback, since the support sets of different aspect subaperture images are highly overlapped across the whole aperture [9], this information can be adopted in the subaperture image reconstruction to avoid it.
The idea of CS is to compressively sense signals that are sparse in some known domains and then use sparse recovery techniques to recover them. Considering the dynamic CS problem, i.e., the problem of recovering a time sequence of sparse signals, CS recovers each sparse signal in the sequence independently without using any information from other frames. Least squares on compressed sensing residual (LSCSResidual) [17] is to replace CS on the observation by CS on the least squares (LS) residual computed using the prior estimate of the support. It is suitable for dynamic CS problem. It is proved that LSCSResidual can recover the signal better than CS [17].
The subaperture images of WASAR can be regarded as the SAR video. Every frame is the subaperture image indexed by the aspect angle. In WASAR, the backscattering from a complex target at high frequencies can be approximately modeled as a discrete set of the scattering centers [9].The scattering center can be described by the aspectdependent amplitude and position [9]. The supports of the scattering centers overlap across the whole aperture. This information can be adopted in WASAR imaging.
In this paper, we propose a novel subaperture imaging method based on LSCSResidual. The proposed method firstly implements Backprojection (BP) on all of the aperture data. Then, the coarse support set is estimated from the BP image. For every subaperture of WASAR, the least squares estimate on the support set is calculated. Then, the observation residual is calculated. With the residual data, we can solve the residual observation model with ${L}_{1}$ regularization. The accurate supports of subaperture images are estimated from the ${L}_{1}$ regularization image. Finally, the LS estimate on the accurate supports is calculated. Since the structure information and LS estimate on the support set are adopted in the proposed method, it can recover the aspect dependent scattering more accurately than CS and debiasedCS.
In the iteration of LSCSResidual, there are matrixvector products. For large scale scenes, the storage of measurement matrix can cost a huge amount of memory. A common strategy is to adopt the azimuthrange decouple operators in the algorithm. In this paper, the BP based azimuthrange decouple operators are adopted. The memory cost is reduced from $\mathcal{O}\left({n}^{2}\right)$ to $\mathcal{O}\left(n\right)$.
2. WASAR Subaperture Imaging Based on Compressed Sensing
WASAR receives data from a large angle. The configuration of WASAR is depicted in Figure 1. The whole aperture can be divided into subapertures. For the data collected from a little subaperture, its scattering can be regarded as constant. Then, the phase history of the ith ($i=1,2\cdots I$) subaperture is formulated as
where ${r}_{i}$ is the phase history data of the ith subaperture, and m and n are the pixel indexes along x and y. M and N are the pixel numbers along x and y, s is the scattering reflectivity of the ith subaperture which is located at $({x}_{m},{y}_{n})$, ${f}_{p}\phantom{\rule{3.33333pt}{0ex}}(p=1,2\cdots P)$ is the frequency, c is the light velocity, ${\theta}_{q}\phantom{\rule{3.33333pt}{0ex}}(q=1,2\cdots Q)$ is the aspect angle, and ${z}_{i}$ is additive noise.
$$\begin{array}{c}\hfill {r}_{i}({f}_{p},{\theta}_{q})=\sum _{m=1}^{M}\sum _{n=1}^{N}{s}_{i}({x}_{m},{y}_{n})\xb7exp\{j\frac{4\pi {f}_{p}}{c}\xb7({x}_{m}cos\left({\theta}_{q}\right)+{y}_{n}sin\left({\theta}_{q}\right))\}+{z}_{i},\end{array}$$
We vectorize Equation (1) and express it in a compact form
where ${\mathit{r}}_{i}$ is the history data of ith subaperture, ${\mathit{s}}_{i}$ is the backscattering of ith subaperture, and ${\mathit{z}}_{i}$ is the noise, the measurement matrix ${\mathbf{\Phi}}_{i}$ is shown as
where ${\varphi}_{i}(pq,mn)=exp\{j\frac{4\pi {f}_{p}}{c}({x}_{m}cos\left({\theta}_{q}\right)+{y}_{n}sin\left({\theta}_{q}\right))\}$.
$${\mathit{r}}_{i}={\mathbf{\Phi}}_{i}\xb7{\mathit{s}}_{i}+{\mathit{z}}_{i},$$
$${\mathbf{\Phi}}_{i}=\left[\begin{array}{cccc}{\varphi}_{i}(1,1)& {\varphi}_{i}(1,2)& \dots & {\varphi}_{i}\left(1,MN\right)\\ \vdots & \vdots & \ddots & \vdots \\ {\varphi}_{i}\left(pq,1\right)& {\varphi}_{i}\left(pq,2\right)& \dots & {\varphi}_{i}\left(pq,MN\right)\\ \vdots & \vdots & \ddots & \vdots \\ {\varphi}_{i}\left(PQ,1\right)& {\varphi}_{i}\left(PQ,2\right)& \dots & {\varphi}_{i}\left(PQ,MN\right)\end{array}\right],$$
The subaperture imaging methods for WASAR imaging assume that the scattering of the targets are not relevant to the aspect angle in a narrow angle. Then, a traditional imaging method can be implemented in subaperture image focusing.
CS has been introduced into SAR imaging [4]. When the scene is sparse and the measurement matrix satisfies the restricted isometry property (RIP) condition, Equation (2) can be solved via ${L}_{1}$ regularization [18]
where $\lambda $ is the regularization parameter.
$$\underset{{\mathit{s}}_{i}}{min}\parallel {\mathit{r}}_{i}{\mathbf{\Phi}}_{i}\xb7{\mathit{s}}_{i}{\parallel}_{2}^{2}+\lambda {\parallel {\mathit{s}}_{i}\parallel}_{1}.$$
For Equation (4), the optimality condition is
where ${(\xb7)}^{H}$ is the conjugate transpose and
$$2{\mathbf{\Phi}}_{i}^{H}({\mathbf{\Phi}}_{i}{\mathit{s}}_{i}{\mathit{r}}_{i})+\lambda \mathit{p}=0,$$
$$\mathit{p}=\partial \parallel {\mathit{s}}_{i}{\parallel}_{1}.$$
Suppose the oracle support of the ${\mathit{s}}_{i}$ is T, and then the solution of (4) is
where ${\mathbf{\Phi}}_{\mathit{i}}{}_{T}^{\u2020}={\left({\mathbf{\Phi}}_{\mathit{i}}{}_{T}^{H}{\mathbf{\Phi}}_{\mathit{i}}{}_{T}\right)}^{1}{\mathbf{\Phi}}_{\mathit{i}}{}_{T}^{H}$, ${T}^{C}$ denotes the complement of T. $\mathrm{sign}(\xb7)$ is the signal function formulated as
$${\left({\mathit{s}}_{i}\right)}_{T}={\mathbf{\Phi}}_{\mathit{i}}{}_{T}^{\u2020}{\mathit{r}}_{i}\lambda {\left({\mathbf{\Phi}}_{\mathit{i}}{}_{T}^{H}{\mathbf{\Phi}}_{\mathit{i}T}\right)}^{1}\mathrm{sign}\left({\mathit{s}}_{\mathit{i}T}\right),{\left({\mathit{s}}_{i}\right)}_{{}_{{T}^{C}}}=0,$$
$$\mathrm{sign}\left({\mathit{s}}_{i}\right)=\frac{{\mathit{s}}_{i}}{{\mathit{s}}_{i}}.$$
If the oracle support is accurate, then the first term of (7) is the exact estimate of the signal. The second term of (7) is the bias that is brought by the regularized term of (4).
In [14], it is shown that ${L}_{1}$ can reconstruct the targets with the underestimated amplitudes. Some missed detections are also introduced in the results of ${L}_{1}$ regularization. In addition, with less azimuth measurements, the resolution of the subaperture is reduced. The underestimation can be reduced via LS on the support. The missed detections can be reduced when more information is adopted. In the next section, we will propose a novel method for WASAR subaperture imaging.
3. WASAR Imaging Based on LSCSResidual
${L}_{1}$ regularization would cause the errors of the amplitude and support set estimation in WASAR imaging. In this section, we propose a novel WASAR imaging based on LSCSResidual.
In WASAR, the subaperture images can be regarded as the video indexed on subaperture [2], which is a map of reflectivity as a function of viewing angle. The reflectivities of the targets can be described via their amplitudes and positions. Although they vary with aspect angle, the positions are highly overlapped. Some methods for dynamic scene such as video signal processing and dynamic MRI imaging can be introduced to WASAR imaging.
LSCSResidual [17] has been proposed for dynamic CS problems, such as dynamic magnetic resonance imaging (MRI). The idea of LSCSResidual is to perform CS not from the observations, but from the least squares residual computed using the previous support estimation. It is shown that it needs fewer samples and the bounded reconstruction error is smaller than the traditional CS. In the model of LSCSResidual, the information between different frames are used and there is also a debiasing step in the final to reduce the bias caused by ${L}_{1}$ regularization. It can reconstruct the results much more accurately than CS.
Since the support sets of between different WASAR subaperture images are highly overlapped, which means that WASAR imaging can be regarded as a dynamic problem. Thus, LSCSResidual is suitable for WASAR subaperture imaging. In the frame of LSCSResidual, the LS estimate is included, which means that the underestimation of ${L}_{1}$ regularization is avoided. In addition, the support information of different subapertures will be used, which will make the results more accurate.
LSCSResidual mainly has three steps: initial LS estimation, implementing CS on the residual (CSResidual) and final LS estimate.
Initial LS Estimate
For Equation (2), if the support set of ${\mathit{s}}_{{\theta}_{i}}$ is known, we could simply compute the LS estimate on the support while setting all other values to zeros. The previous support can be estimated from the prior information. Suppose the estimated support is T, to compute and initial LS estimate
$${\left({\mathit{s}}_{i,init}\right)}_{T}={\left({\mathbf{\Phi}}_{\mathit{i}}T\right)}^{\u2020}{\mathit{r}}_{i},{\left({\mathit{s}}_{i,init}\right)}_{{T}^{c}}=0.$$
Then, the LS residual is calculated as
$${\mathit{r}}_{i,res}={\mathit{r}}_{i}{\mathbf{\Phi}}_{i}{\mathit{s}}_{i,init}.$$
In WASAR, the scattering of the targets is aspect dependent. However, the support sets of the subaperture images are highly overlapped, which means that a fairly accurate support T can be estimated from the data. T is estimated via
which is the support of the elements whose amplitudes are larger than $\alpha $.
$$T=supp({\mathit{s}}_{0}:{\mathit{s}}_{\mathbf{0}}>\alpha ),$$
In [17], the threshold $\alpha $ is determined by the b%Energy support, which means that T contains at least $b\%$ of the signal energy. In WASAR imaging, we set $b\%=90\%$.
Notice that the LS residual, ${\mathit{r}}_{i,res}$, can be rewritten as
$${\mathit{r}}_{i,res}={\mathbf{\Phi}}_{i}{\mathit{\beta}}_{i},{\mathit{\beta}}_{i}={\mathit{s}}_{i}{\mathit{s}}_{i,init}.$$
CSResidual
In this step, CS is implemented on the LS residual, i.e., solve (12) with CS in the following model
$$\underset{{\mathit{\beta}}_{i}}{min}\parallel {\mathit{r}}_{i,res}{\mathbf{\Phi}}_{i}{\mathit{\beta}}_{i}{\parallel}_{2}^{2}+\lambda {\parallel {\mathit{\beta}}_{i}\parallel}_{1}.$$
Iterative shrinkage thresholding algorithm (ISTA) [19] can be used to solve (13). In the iteration of ISTA, there are no matrix inversions involved. ISTA is preferred for its simplicity in implementation for distributed or parallel recovery due to nature of the involved matrixvector multiplications [20,21]. The iteration is formulated as
where $\mu \in (0,\parallel \mathit{A}{\parallel}_{2}^{2})$ is the step size controlling the convergence, $\lambda $ is the regularization parameter, and f is the iterative function of ISTA. In the iteration, the value of $\lambda $ is
where $\widehat{{\mathit{\beta}}_{i}^{t}}{}_{K+1}$ is the $(K+1)$th largest element of $\widehat{{\mathit{\beta}}_{i}^{t}}$ and $K=\parallel \widehat{{\mathit{\beta}}_{i}^{t}}{\parallel}_{0}$.
$$\widehat{{\mathit{\beta}}_{i}^{t}}={\mathit{\beta}}_{i}^{t}+\mu \left[{\mathbf{\Phi}}_{i}^{H}({\mathit{r}}_{i}{\mathbf{\Phi}}_{i}{\mathit{\beta}}_{i}^{t})\right],$$
$${\mathit{\beta}}_{i}^{t+1}={f}_{\lambda \mu}\left(\widehat{{\mathit{\beta}}_{i}^{t}}\right)=\left\{\begin{array}{cc}\hfill \mathrm{sgn}\left(\widehat{{\mathit{\beta}}_{i}^{t}}\right)\left(\right\widehat{{\mathit{\beta}}_{i}^{t}}\lambda \mu ),\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}\widehat{{\mathit{\beta}}_{i}^{t}}>\lambda \mu \hfill \\ \hfill 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$
$$\lambda =\widehat{{\mathit{\beta}}_{i}^{t}}{}_{K+1}/\mu ,$$
The final estimation is
$$\widehat{{\mathit{s}}_{i}}={\mathit{\beta}}_{i}+{\mathit{s}}_{i,\mathrm{init}}.$$
Final LS Estimation
It is shown that ${\mathit{\beta}}_{i}$ is obtained after ${L}_{1}$ regularization, and the estimation will be biased towards zeros. Thus, a debiasing step is needed
$${T}^{\prime}=\mathrm{supp}\left(\widehat{{\mathit{s}}_{i}}\right),$$
$${\mathit{s}}_{i}{}_{T}={\left({\mathbf{\Phi}}_{i}{}_{{T}^{\prime}}\right)}^{\u2020}{\mathit{r}}_{i},{\mathit{s}}_{i}{}_{{{T}^{\prime}}^{C}}=0.$$
After the construction of the subaperture images, the generalized likelihood ratio test (GLRT) [1] can be implemented for the final composite image. GLRT is defined as
where ${s}^{i}(x,y)$ is the scattering at pixel $(x,y)$ of ith subaperture.
$$s(x,y)=\underset{i}{max}\left{s}^{i}(x,y)\right,$$
The algorithm is summarized in Algorithm 1.
Algorithm 1 LSCSResidual based WASAR imaging. 

In WASAR imaging, it will cost huge amount of memory to store the measurement matrix. The azimuthrange decouple operators can be used to reduce the memory cost [5]. In this paper, we take BP based operators to substitute the measurement matrix and its conjugate transpose in real WASAR imaging. With the BP based operators, the memory cost can be reduced dramatically. If we reconstruct the measurement matrix, the memory cost is $\mathcal{O}\left(PQMN\right)$. With the BP based operators, the memory cost is $\mathcal{O}\left(MN\right)$. It means that, with the measurement matrix, the memory cost is reduced from $\mathcal{O}\left({n}^{2}\right)$ to $\mathcal{O}\left(n\right)$.
BP mainly includes two operations: range Fourier transform and azimuth coherent addition. The imaging and raw data generation procedures are formulated as
where $\mathcal{F}$ and ${\mathcal{F}}^{1}$ are the the Fourier transform pairs, $\mathcal{H}$ is azimuth coherent addition operator and ${\left(\mathcal{H}\right)}^{1}$ is its inverse operation, $\mathcal{R}$ reshapes the vector into matrix and ${\mathcal{R}}^{1}$ reshapes the matrix into a vector.
$$\mathcal{I}\left\{\xb7\right\}\cong {\mathcal{R}}^{1}\left\{\mathcal{H}\left\{{\mathcal{F}}^{1}\left\{\mathcal{R}\{\xb7\}\right\}\right\}\right\},$$
$$\mathcal{G}\left\{\xb7\right\}\cong \mathcal{R}\left\{\mathcal{F}\left\{{\mathcal{H}}^{1}\left\{{\mathcal{R}}^{1}\{\xb7\}\right\}\right\}\right\},$$
4. Real Data Experiment
In this section, we will use two datasets to show the effectiveness of the proposed method.
4.1. Turntable Data
The turntable data collected by the Institute of Electronics, Chinese Academy of Sciences will be used to show the effectiveness of the proposed method. The real data of a metal tank model are measured in an anechoic chamber on a turntable, which is in uniform circular motion. The radar is a stepped frequency type and has a center frequency of 15 GHz and bandwidth 6 GHz. The turntable plane and its center are set as the imaging ground plane and the coordinate origin, respectively. The radius of equivalent circular passes is 8.54 m. The ${360}^{\circ}$ whole aperture is divided into 36 subapertures. The pixel size of the SAR image is 0.25 cm × 0.38 cm.
We reconstruct the subaperture images with BP, CS, debiasedCS and LSCSResidual. The results of the three methods are shown in Figure 2. Figure 2a is the result of BP, which is used as the referenced image. Compared with the result of the three method, the result of LSCSResidual remains less artifects as shown in the white circle.
To compare the performance of the three methods in the reconstruction of aspect dependent scattering, we select an aspect dependent scattering target P and plot its aspect dependent amplitude curve Figure 3. The result of BP is used as the reference. In Figure 3, we select Area 1 to show the performance of LSCSResidual to reduce the underestimation. Area 2 in Figure 3 is selected to show the performance of LSCSResidual to reduce the missed detections. As shown in Figure 3 Area 1, CS underestimates the amplitude of the target. The results of DebiasedCS and LSCSResidual highly overlap the result of BP. So DebiasedCS and LSCSResidual avoid the underestimation caused by $CS$. In Figure 3 Area 2, CS and debiasedCS fail in reconstructing the weak scattering. Since the support information of the other subapertures is adopted in LSCSResidual, the support of weak scattering target is preserved in the subapertures. So with the prior support information and the final debiasing step, LSCSResidual can reconstruct the aspect dependent scatterings of the targets more accurately than CS and debiasedCS.
The time taken by the three algorithms is given in the Table 1. DebiasedCS takes more time because of the debias step compared with CS. Compared with the former two algorithms, LSCSResidual takes similar amount time.
4.2. Gotcha Volumetric SAR DATA
Gotcha volumetric SAR dataset [22] is Xband circular SAR data that consists of CSAR phase history data collected at the Xband with a 640MHz bandwidth. The spotlighted scene is a parking lot in an urban environment. The scene consists of numerous civilian vehicles and reflectors.
In this experiment, the HH polarization data are used. The whole aperture of ${360}^{\circ}$ is divided into 180 subapertures. Every subaperture is ${4}^{\circ}$. The apertures overlap every ${2}^{\circ}$. The pixel size is 0.2 m × 0.2 m. The area of reflectors is chosen. We reconstruct the scene with BP, CS, debiasedCS and LSCSResidual. The GLRT results of the four methods are shown in Figure 4.
To evaluate the aspect dependent reconstruction performance of different methods, we select an aspect dependent scattering target and plot its reconstructed aspect dependent scattering. The selected target is a reflector that distributes across several pixels. We reconstruct the subaperture images with BP, CS, debiasedCS and LSCSResidual. To compare the aspect dependent scattering reconstruction performance of the three methods, we add the intensities of these pixels together and plot the results in Figure 5. Figure 5 is the main valid scattering area of the reflector. BP result is used as the reference. It is shown that the result of LSCSResidual is highly overlapped with the results of BP. The intensities of CS and debiasedCS are less than BP and LSCSResidual. The underestimation of debiasedCS is mainly caused by the missed detections. Since there are some missed detections in the result of debiasedCS, the intensities of the debiasedCS is less than BP. CS causes bias and missed detections because of the regularizer term. With the bias and missed detections, the peak of CS is lower than the other three methods.
5. Conclusions
In this paper, an accurate WASAR imaging algorithm based on LSCSResidual is proposed. The traditional regularized subaperture imaging method based on ${L}_{1}$ regularization introduces the bias and missed detections which will cause inaccurate aspect dependent scattering estimates. To overcome this problem, LSCSResidual has been introduced into WASAR imaging. LSCSResidual mainly has three steps: initial LS estimate, CS on the residual and final LS estimate. The LS estimate step can be used to reduce the bias. The missed detections are reduced because the support information is adopted in the process of the LSCSResidual. The proposed method accommodates aspect dependent scattering better than CS and debiasedCS. The experiment results demonstrate its validity.
Author Contributions
Conceptualization, Z.W.; Data Curation, B.Z.; Methodology, Z.W.; Project Administration, B.Z.; Supervision, B.Z. and Y.W.; Validation, Z.W.; Writing—Original Draft, Z.W.; Writing—Review and Editing, B.Z.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No. 61571419.
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
SAR  Synthetic aperture radar 
WASAR  Wide angle synthetic aperture radar 
CSAR  Circular synthetic aperture radar 
LS  Least squares 
CS  Compressed sensing 
LSCSResidual  Least squares on compressed sensing residual 
BP  Backprojection 
CSResidual  CS on the residual 
ISTA  Iterative shrinkage thresholding algorithm 
GLRT  Generalized likelihood ratio test 
References
 Moses, R.; Lee, P.; Mujdat, C. Wideangle SAR imaging. Proc. SPIE 2004, 5427, 164–175. [Google Scholar]
 Ash, J.; Emre, E.; Lee, P.; Edmund, Z. WideAngle Synthetic Aperture Radar Imaging: Models and algorithms for anisotropic scattering. IEEE Signal Process. Mag. 2014, 31, 16–26. [Google Scholar] [CrossRef]
 Soumekh, M. Synthetic Aperture Radar Signal Processing; Wiley: New York, NY, USA, 1999. [Google Scholar]
 Baraniuk, R.; Steeghs, P. Compressive Radar Imaging. In Proceedings of the IEEE Radar Conference, Boston, MA, USA, 17–20 April 2007; pp. 17–20. [Google Scholar]
 Zhang, B.; Hong, W.; Wu, Y. Sparse microwave imaging: Principles and applications. Sci. China Inf. Sci. 2012, 55, 1722–1754. [Google Scholar] [CrossRef]
 Stojanovic, I.; Mujdat, C.; William, K. Joint space aspect reconstruction of wideangle SAR exploiting sparsity. Proc. SPIE 2008, 6970. [Google Scholar] [CrossRef]
 Cong, X.; Guan, G.; Yong, J.; Li, X.; Wen, G.; Huang, X.; Qun, W. A novel adaptive wideangle SAR imaging algorithm based on Boltzmann machine model. Muldimens. Syst. Signal Process. 2016, 29, 119–135. [Google Scholar] [CrossRef]
 Wei, Z.; Jiang, C.; Zhang, B.; Bi, H.; Hong, W.; Wu, Y. WASAR imaging with backprojection based group complex approximate message passing. Electron. Lett. 2016, 52, 1950–1952. [Google Scholar] [CrossRef]
 Trintinalia, L.C.; Rajan, B.; Ling, H. Scattering center parameterization of wideangle backscattered data using adaptive Gaussian representation. IEEE Trans. Antennas Propag. 1997, 45, 1664–1668. [Google Scholar] [CrossRef]
 Gerry, M.J.; Potter, L.C.; Gupta, I.J.; Van Der Merwe, A. A parametric model for synthetic aperture radar measurements. IEEE Trans. Antennas Propag. 1999, 47, 1179–1188. [Google Scholar] [CrossRef]
 Donoho, D. Compressed Sensing. IEEE Trans. Inf. Theory 2006, 52, 1289–1306. [Google Scholar] [CrossRef]
 Candes, E.; Wakin, M. An Introduction to Compressive Sampling. IEEE Signal Process. Mag. 2008, 25, 21–30. [Google Scholar] [CrossRef]
 Tibshirani, R. Regression Shrinkage and Selection via the Lasso. J. R. Stat. Soc. Ser. B 1996, 58, 267–288. [Google Scholar] [CrossRef]
 Osher, S.; Feng, R.; Jiechao, X.; Yuan, Y.; Wotao, Y. Sparse recovery via differential inclusions. Appl. Comput. Harmon. Anal. 2016, 41, 436–469. [Google Scholar] [CrossRef][Green Version]
 Candes, E.; Tao, T. The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Stat. 2007, 35, 2313–2351. [Google Scholar] [CrossRef][Green Version]
 Figueiredo, M.A.T.; Nowak, R.D.; Wright, S.J. Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems. IEEE J. Sel. Top. Signal Process. 2007, 1, 586–597. [Google Scholar] [CrossRef][Green Version]
 Vaswani, N. LSCSResidual (LSCS): Compressive Sensing on Least Squares Residual. IEEE Trans. Signal Process. 2010, 58, 4108–4120. [Google Scholar] [CrossRef][Green Version]
 Candes, E.; Romberg, J.; Tao, T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 2006, 52, 489–509. [Google Scholar] [CrossRef]
 Daubechies, I.; Defrise, M.; De Mol, C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 2004, 57, 1413–1457. [Google Scholar] [CrossRef][Green Version]
 Ravazzi, C.; Fosson, S.; Magli, E. Distributed Iterative Thresholding for ℓ_{0}/ ℓ_{1}Regularized Linear Inverse Problems. IEEE Trans. Inf. Theory 2015, 61, 2081–2100. [Google Scholar] [CrossRef]
 Fiandrotti, A.; Fosson, S.; Ravazzi, C.; Magli, E. PISTA: Parallel Iterative Soft Thresholding Algorithm for Sparse Image Recovery. In Proceedings of the Picture Coding Symposium (PCS), San Jose, CA, USA, 8–11 December 2013; pp. 325–328. [Google Scholar]
 Casteel, C.; Gorham, L.; Minardi, M.; Scarborough, S.; Naidu, K.; Majumder, U. A challenge problem for 2D/3D imaging of targets from a volumetric data set in an urban environment. Proc. SPIE 2007, 6568. [Google Scholar] [CrossRef]
Figure 2.
Results of the four methods. (a) GLRT result of BP; (b) GLRT result of CS; (c) GLRT result of debiasedCS; (d) GLRT result of LSCSResidual.
Figure 4.
Results of the four methods. (a) GLRT result of BP; (b) GLRT result of CS; (c) GLRT result of debiasedCS; (d) GLRT result of LSCSResidual.
CS  DebiasedCS  LSCSResidual 

22.31  23.43  20.08 
CS  DebiasedCS  LSCSResidual 

154.94  162.69  139.45 
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).