Resolution enhancement for ISAR imaging via improved statistical compressive sensing
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Abstract
Developing compressed sensing (CS) theory reveals that optimal reconstruction of an unknown signal can be achieved from very limited observations by utilizing signal sparsity. For inverse synthetic aperture radar (ISAR), the image of an interesting target is generally constructed by limited strong scattering centers, representing strong spatial sparsity. Such prior sparsity intrinsically paves a way to improved ISAR imaging performance. In this paper, we develop a superresolution algorithm for forming ISAR images from limited observations. When the amplitude of the target scattered field follows an identical Laplace probability distribution, the approach converts superresolution imaging into sparsitydriven optimization in the Bayesian statistics sense. We show that improved performance is achievable by taking advantage of the meaningful spatial structure of the scattered field. Further, we use the nonidentical Laplace distribution with small scale on strong signal components and large scale on noise to discriminate strong scattering centers from noise. A maximum likelihood estimator combined with a bandwidth extrapolation technique is also developed to estimate the scale parameters. Real measured data processing indicates the proposal can reconstruct the highresolution image though only limited pulses even with low SNR, which shows advantages over current superresolution imaging methods.
Keywords
Inverse synthetic aperture radar (ISAR) Superresolution Compressive sensing (CS) Statistical compressive sensing Bayesian Laplace distribution Nonidentical distribution1 Introduction
Highresolution radar imaging techniques are widely applied in many military and civilian fields, such as target classification and recognition and aircraft traffic control. For success of these applications, sufficient image resolution is required to characterize the scattering and geometric features of the target. Inverse synthetic aperture radar (ISAR) combines the use of pulse compression, flexible pulse repetition frequency (PRF), and target motions (particularly the rotating motion) to generate twodimensional highresolution imagery. In general, range resolution is determined by the bandwidth of the transmitted signal which is limited by the radar system. To mitigate this limitation, stepped frequency waveforms are usually employed at a price of longer coherent processing interval (CPI) [1]. Crossrange resolution is dependent on both the CPI and the target rotational motion from the variation of radar viewing angles. It is well known that CPI should be long enough for high azimuth resolution, which conflicts with modern radar activities such as multitarget tracking and searching [2].
Due to the CPI limitation, achieving high resolution in azimuth with only a few pulses through conventional radar imaging schemes, e.g., rangeDoppler (RD) algorithm [3], is difficult. This motivates superresolution (SR) techniques. SR image formation improves the ability to resolve two closely spaced scatterers in comparison with the Nyquist resolution limit [4], which actually involves uncertainty as the dimension of the image is required to be higher than that of the measurements. Enhancing both the image contrast enrichment and scattering center localization, SR imaging will also reduce scintillation effects associated with unresolved scatterers. A superresolved image can usually promote the probability of correct automatic target recognition (ATR) [5, 6]. Consequently, a large fraction of the radar imaging research effort has been taken in developing resolution enhancing algorithms. Generally, there are three categories that conventional SR approaches are often sorted into. The first group is the bandwidth extrapolation (BWE) methods. They are based on the fact that the data is analytic in principle and can be fitted into a linear prediction model [2, 7, 8, 9, 10]. BWE usually applies some modern spectral estimation techniques to determine the coefficients of the prediction model. Burg’s algorithm [2] is one of the most successful estimators. The second general group of SR methods is the parametric spectral estimation techniques. They attempt to estimate parameters of the dominant scattering centers by modern spectral estimation techniques. Probabilistic strategies which model the signal and noise statistically, including maximum a posteriori and maximum entropy [11, 12] schemes have also been developed, yielding good performance. Adaptive spectral estimators [13, 14, 15, 16, 17, 18] utilize sinusoid parametric model to estimate the coefficients of signals by finding a set of filters adaptively. Among this sort, RELAX [13], [14], which estimates the amplitude and frequency of a multicomponent complex sinusoidal signal by minimizing the residual error energy, is one of the most popular algorithms. Imageries generated by these approaches are inherently free from sidelobes, and resolution relies on the precision of spectral estimation. The third group is nonlinear filtering techniques to reduce side lobes while preserving the width of the main lobe. This group includes spacevariant filtering methods such as adaptive sidelobe reduction (ASR) [19] and the related special case of spatially variant apodization (SVA) [20], as well as CLEAN and its modified versions [21, 22]. All the current SR algorithms are more or less sensitive to noise and model error.
Conventional SR imaging can be regarded as a process that recovers a higher dimensional signal from measurements with much fewer degrees of freedom and is deemed to involve uncertainty mathematically. Developing compressive sensing (CS) theory reveals that an unknown sparse signal can be exactly recovered from very limited measurements with high probability by solving a convex l _{1} optimization problem [23, 24, 25]. Satisfaction of the restricted isometry property (RIP) allows the uncertainty in the l _{1} optimization problem to be overcome to provide an exact or approximate signal recovery. The essence of CS is to exploit prior knowledge of sparsity of the objective signal. This explains the promising performance of radar imaging via regularization techniques. For ISAR imaging, strong scattering centers exhibiting the scattering organization and geometric configuration of the target usually occupy only a fraction of whole bins in the RD plane. In this sense, the ISAR signal is spatially sparse in RD domain. Such sparsity can be exploited by l _{1} optimization to enhance image quality and feature. Many sparsitydriven SR approaches based on l _{1} optimization have been developed for SAR/ISAR imaging. The featureenhanced SAR imaging approach [26, 27] exploits the sparsity of both image and differential image via the l _{ p }(0 < p ≤ 1) optimization to enhance the point and region feature of the radar image. Particularly, the pointenhanced image formation can extrapolate the data bands beyond the observations, yielding SR imaging. It should be noted that the pointenhanced optimization with p = 1 is equivalent to the l _{1} optimization of CS. However, the main problem of the featureenhanced algorithm for radar imaging lies in the difficulty of parameter selection. SR ISAR imaging using the l _{1}FFT, is developed in [28]. Similarly, in [29], a CSbased approach is developed, which can generate a highresolution image with very few pulses, promising promoted performance of ISAR imaging by utilizing spatial sparsity. Based on the assumption that only a few scatterers with different elevations are present in the same rangeazimuth resolution cell, CS is exploited in [30, 31], to provide a new data acquisition scheme and SR imaging using only a few signal samples for tomographic SAR imagery. In [32], the SR capability and robustness of CS for tomographic SAR imaging have been analyzed in detail. All this work indicates that it is possible to substantially enhance the performance of the radar imaging by exploiting target sparsity. Nevertheless, it is usually not an easy task to characterize this sparsity quantitatively due to the uncertainty and complexity of the real target scattered field, together with the inevitable presence of noise and clutter. This makes some regularizationbased methods including the featureenhanced image approach ambiguous in both parameter selection and controlling the sparsity of the recovery.
Generally, it is preferable to represent the sparsity of the target in a statistical manner. In this paper, a SR imaging algorithm is developed wherein the amplitude sparsity of the target scattered field is represented by a Laplace distribution. Based on this, the statistical distribution of the complexvalued scattered field is derived. The SR imaging problem is then converted into solving a sparsityconstraint optimization corresponding to the maximum a posterior (MAP) probability estimation following the Bayesian theory. For simplicity, the proposed algorithm is called Bayesian SR imaging. From Bayesian compressive sensing (BCS) [33] under the identical Laplace distribution assumption, the original Bayesian SR imaging is explicitly coincident with the l _{1}regularization optimization. The sparsity coefficient controls the sparsity of the reconstructed image, which is directly related to the statistics of the target image and noise. We also extend the Bayesian SR imaging by using nonidentical Laplace distributions since the scattered field of a target usually follows the energyassembling and geometric organization of the target. We place Laplace distributions with different scale parameters on each element, by which signal and noise components could be penalized individually in the l _{1} optimization to promote the accuracy of SR imagery reconstruction. A novel scheme to estimate the noise and target statistics parameters are also developed for both the original Bayesian SR and the improved version by combining the constantfalsealarmratio (CFAR) [34] and Burg’s BWE techniques. To achieve a superresolved image stably, we propose an iterative procedure where in each stage the aperture data is extrapolated doubly and the maximum likelihood (ML) estimate of the Laplace scale parameter is updated. The Bayesian SR methods are inherently robust to noise and workable with limited pulses. Both simulated and real data experiments are provided to demonstrate the superiority of Bayesian SR imaging methods over the current SR approaches under different circumstances.
The organization of this paper is as follows. Section 2 introduces the Bayesian SR imaging approaches based on identical and nonidentical statistics assumptions. In Section 3, we focus on three issues: statistics estimation, imaging procedure, and a modified QuasiNewton solver. Finally, in Section 4, simulated and realdata experimental results are given to show the effectiveness of the proposed approaches.
2 Resolution enhancement with statistical compressive sensing
 A.
Signal model
 1)
Complete motion compensation. In (2), only the additive noise is taken into account while other model errors such as range shift and residual phase error are assumed removed. There are many precise motion compensation algorithms [35, 36, 37, 38, 39, 40, 41] applicable making this assumption rigid.
 2)
Stationarity assumption. The signal amplitude and Doppler frequency are assumed to be invariant. In this paper, we focus on the SR imaging with short CPI, during which the signal can be regarded as stationary and involves no time variance of both the reflectivity and Doppler for a scatterer.
 B.
Bayesian SR (BSR) imaging via independent and identical (IID) statistics assumption
ISAR imagery demonstrates the spatial distribution of the target scattered field in the RD plane. Dominant scattering centers usually occupy only a fraction of the whole RD bins even though they contribute a majority of the energy, while signals from weak scattering centers contribute little to image formation. This sparse characteristic of ISAR images can be exploited to achieve SR. In this section, we develop a SR approach by combining Bayesian statistics and compressive sensing theory. According to sparse Bayesian learning (SBL) [42] and BCS [33], the sparsity can be formulized by placing a sparsenesspromoting prior on A. In general, a Laplace probability distribution can be used to enforce the sparsity prior on the recovered objective signal when the signal is real valued [42, 43]; however, it is not straightforward to extend the Laplace distribution to the complexvalued model. In the following content, an extension of the twolayer SBL model [42] is presented and a Laplace distribution for complex signal is developed to encourage sparse representation of A.
 C.
Improved Bayesian SR (IBSR) imaging based on nonidentical statistics assumption
where s _{ m } = S(:, m) denotes the mth column of S corresponding to a _{ m }. In contrast to BSR in (15), the image is modeled by the nonidentical Laplace distribution in (20). We designate it improved BSR (IBSR). IBSR imaging can discriminate pixels containing the strong scattering centers from the weak ones, which emphasizes the spatial configuration and strong components in image formation. To this end, each element of the image is treated individually to favor the recovery of the structural organization of the target scattered field. To encourage the probability of scattering center reconstruction, and enforce the noise be near zero, small scale parameters are preferable for scattering centers, while large scale parameters for noise components. Furthermore, similar to the optimization in weighted compressive sensing [50], the weights in the l _{1}norm penalty term in (20) will influence the contributions of different components to the penalty function improving reconstruction accuracy and efficiency. Therefore, if proper scale parameters are utilized in (20), different distributions of signal and noise can be obtained, leading to more precise signal recovery and more effective noise suppression.
3 Statistics estimation and imaging solver
 A.
Statistics estimation and imaging procedure
 1)
We apply Fourier transform (FT) to the rangecompressed data to generate the coarse RD image and estimate the noise variance by using the pure noise samples. Those noise samples are also used to develop a CFAR threshold to remove noise in the background of the coarse image. It should be emphasized that the noise within the target region cannot be removed; however, due to the high SNR gain from twodimensional coherent integration, this noise does not affect much. The denoised image is then transformed back into the time domain by an inverse Fourier transform (IFT). This is followed by Burg’s BWE to extrapolate the aperture twice, and then the scale parameter is estimated from the BWE image based on the ML rule. This step functions as the initialization.
 2)
The Bayesian optimization for an image with a SR factor of two is developed with the statistics from the last step and then solved by a QuasiNewton solver (this solver will be introduced in the following section). The reconstructed SR image is transformed back into the time domain by IFT, after which, Burg’s BWE with double extrapolation is employed to obtain an image with SR at a factor of four and to estimate the scale parameter for the next stage.
 3)
We repeat the second step until a desired SR is reached.
This shows that the ML estimator of γ _{1} is the reciprocal of the amplitude mean of Ā _{1}. Without loss of generality, in the initialized step, all variables are indicated by the subscript g = 1, and the initialization step is embedded in the first stage of the whole procedure.
Variable description in the stagebystage procedure
g  Subscript of the stage number 

\( {\overline{N}}_g={2}^g\cdot N \)  Azimuth dimension of SR image in the gth stage 
A _{ g }  SR image \( \left({\overline{N}}_g\times M\right) \) 
Â _{ g }  Estimation of A _{ g } (\( {\overline{N}}_g\times M \)) 
Ā _{ g }  BWE SR image \( \left({\overline{N}}_g\times M\right) \) in the gth stage 
F _{ g }  Partial Fourier matrix \( \left({\overline{N}}_g\times N\right) \) 
It should be noted that the Burg’s BWE plays an essential role in the estimation of the Laplace scale parameter. Implemented in a recursive way, Burg’s BWE is stable in aperture extrapolation. In real data processing, the order of the linear prediction model in the BWE is typically set high, such as one third of the data length, to ensure estimation precision of the largest coefficients without bringing spurious scattering centers [10]. Additionally, in the Burg’s BWE, the retention of primary data and phase coherency of the extrapolated signal are meaningful for the precise estimation of the scale parameter. However, in [53], detailed theoretic analysis reveals that Burg’s BWE has limited SR ability at a factor of 2.6 for two identical pointscatterers, and beyond that, the negative effect becomes evident, which is the main reason why the extrapolation factor of two in each stage is set.
 B.
An effective solver to the IBSR optimization
where \( {\widehat{\mathbf{a}}}_m^{(h)} \) is the estimator of A(:, m) in the hth iteration. To accelerate the iteration, the conjugate gradient algorithm (CGA) [59] can be applied to avoid the Hessian matrix inversion in (34). A fixed threshold δ _{CG} is used for a complete run of the CGA. Without prior information about A, it is initialized as Â _{0} = F ^{ H } S. We iterate h until \( {\left{\widehat{\mathbf{a}}}_m^{\left(h+1\right)}{\widehat{\mathbf{a}}}_m^{(h)}\right}_2/{\left{\widehat{\mathbf{a}}}_m^{(h)}\right}_2\le \rho \), where ρ is a small parameter for the predetermined threshold, or h reaches a predetermined maximum iteration number.
In the following computational cost analysis of the QuasiNewton solver, we keep track of the multiplications. The main computational cost of the solver for IBSR optimization is in using CGA to iterate (34). One may note that the term F ^{ H } F corresponds to the partial Fourier matrix F, allowing us to use a FFT to efficiently calculate F ^{ H } Fa _{ m } in (31). We perform an inverse fast Fourier transform (IFFT) to a _{ m }, set the components corresponding to the vacant aperture to zero, and then apply a FFT dramatically reducing the computational load. Let the number of the CGA iterations to solve (34) be N _{CGA}. The computational cost of the FFT and IFFT in solving (34) is \( 2{N}_{\mathrm{CGA}}\overline{N}{ \log}_2\overline{N} \) flops. If there are N _{QN} iterations in the QuasiNewton solver, the total computation cost is approximately \( 2{N}_{\mathrm{QN}}{N}_{\mathrm{CGA}}\overline{N}{ \log}_2\overline{N} \) flops, while the computational load of calculating matrix inversion through the Cholesky factorization [60] is up to \( {N}_{\mathrm{QN}}{\overline{N}}^3/3 \) flops. Since N _{CGA} is usually on the order of several tens, the efficiency improvement via FFT and CGA is obvious. In practice, we can start with a relaxed CGA tolerance δ _{CG} and reduce it as we iterate the QuasiNewton solver. For further improvements, there are optimization algorithms such as the preconditioned conjugate gradient algorithms [60], which can reduce the number of iterations of the standard CGA.
4 Performance analysis
 A.
Simulation for SR analysis
where a and â is the ideal and the reconstructed signal, respectively, and \( \overline{N} \) denotes the length of the reconstruction. We repeat the SR reconstruction 1000 times with different noise levels and use the average MSE as an evaluating metric.

H _{0}: There is at most one scatterer present in the reconstruction signal;

H _{1}: There are exactly two scatterers inside the given Doppler bins of the reconstruction signal.
Note that we take it as a failure when artifacts occur in the reconstruction, which means that the definition of P _{ D } is slightly different from that in [32]. In general, the number of scatterers in the reconstructed signal can be converted into the problem of model selection by some information theory rules before the SR reconstruction [32]. In the manner of signal detection, we utilize the CFAR technique by a predefined threshold to determine the number of signal components. The threshold is calculated according to the added noise variance before the SR reconstruction. The false alarm rate is adjusted experientially according to the given noise variance to maintain the threshold value constant at 0.3, which is selected optimally based on the performance of IBSR. In the following experiments, bins of the reconstructed signal exceeding the threshold are determined to be signal components and the total number of components is outputted for the hypothesis test.
First, we apply the methods to onedimensional synthetic signals covering the Doppler range −255 to 256 Hz. All signals contain N = 32 regularly sampled acquisitions with sample interval 1/512 s, providing a Rayleigh resolution of 16 Hz. Each signal contains two scatterers corresponding to two closely located Doppler points whose amplitudes are equally set to unity and have zero phases. We consider four cases of different distances between the two scatterers to analyze the SR power of the methods. In the four cases, the two scatterers are located at Doppler frequencies of 0/8 Hz, 0/4 Hz, 0/2 Hz, and 0/1 Hz, respectively. To separate the two scatterers in all cases with the given central 32 samples within the dwell time [−31/512, 32/512] (s), we perform the SR methods with SR factors 2, 4, 8, and 16, corresponding to the reconstructed signal lengths 64, 128, 256, and 512, respectively. Complexvalued Gaussian noise is added in to the synthetic signals to generate SNRs from 0 to 20 dB, wherein SNR is defined as the energy ratio of a single point and the added noise.
 1)
The performance of BSR can approach cvxbased CS imaging, especially in the cases of relative high resolution. Therefore, although BSR never performs better than CSbased imaging, BSR is preferable in some ideal applications due to its high efficiency.
 2)
Based on the nonidentical statistical assumption and accommodation of statistical parameters estimation implemented by CFAR and Burg’s BWE, IBSR is more robust to noise and limitation of measurements.
 B.
Real data set description and evaluation metrics
In the following, real groundbased data is used for a performance analysis of BSR and IBSR for SR. Real data experimental comparisons of the Burg’s BWE, CS, ICS, BSR, and IBSR are given. The performance analysis in the following is carried out by considering mainly two factors: the pulse amount and the noise interference.
where A and \( \widehat{\mathbf{A}}={\left[{\widehat{a}}_{nm}\right]}_{\overline{N}\times M} \) denote the original and reconstructed SR image, respectively, ⊙ denotes the Hadamard product, and < · >, the operator for summing up all components of a matrix. Notably, the coherence is defined with respect to the magnitude. Since the phases of the image are random, they usually provide few contributions to feature extraction for ATR [62]. Therefore, we calculate the coherence only with the magnitude of image in (36).
 C.
Performance versus pulses amount
 D.
Performance versus SNR
5 Conclusions
In this paper, we present SR imaging algorithms based on BCS. By combining the statistics estimation and CS, they are implemented by solving optimizations with l _{1}norm optimization, derived from the MAP estimations. The sparsity level of the reconstructed image is determined by extracting the statistics from data stage by stage. Both BSR and IBSR are robust to strong noise and can reconstruct an image while suppressing strong noise. Based on the fact that the scattered field usually has distinctive spatial configuration, the nonidentical Laplace distribution is introduced in IBSR to discriminate prominent scattering centers from weak ones and noise in the l _{1} penalty term. The nonidentical Laplace distribution promotes performance in terms of noise tolerance and limited pulses. The contribution of this paper illuminates the possibility of substantially improving the performance of SR imaging by leveraging statistical models coinciding with the image optimally. Surely, there are other novel statistics models that can also be used to represent the sparsity and local feature of radar image ideally. Recent studies show that Gaussian mixture models [64] are optimal candidates to describe the structural dependence and sparsity of a target. This will be surveyed in future work. The improved statistical compressive sensing would be also beneficial for multiantenna setting for SAR [65] and ISAR imaging for maneuvering targets [66, 67].
Notes
Acknowledgements
The authors thank the anonymous reviewers for their valuable comments. This work was supported by the National Natural Science Foundation of China (No. 61301280 and 61301293).
Competing interests
The authors declare that they have no competing interests.
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