RESEARCH Open Access

Resolution enhancement for ISAR imagingvia improved statistical compressivesensingLei Zhang1, Hongxian Wang1 and Zhi-jun Qiao2*

Abstract

Developing compressed sensing (CS) theory reveals that optimal reconstruction of an unknown signal can beachieved from very limited observations by utilizing signal sparsity. For inverse synthetic aperture radar (ISAR), theimage of an interesting target is generally constructed by limited strong scattering centers, representing strongspatial sparsity. Such prior sparsity intrinsically paves a way to improved ISAR imaging performance. In this paper,we develop a super-resolution algorithm for forming ISAR images from limited observations. When the amplitudeof the target scattered field follows an identical Laplace probability distribution, the approach converts super-resolutionimaging into sparsity-driven optimization in the Bayesian statistics sense. We show that improved performance isachievable by taking advantage of the meaningful spatial structure of the scattered field. Further, we use thenonidentical Laplace distribution with small scale on strong signal components and large scale on noise to discriminatestrong scattering centers from noise. A maximum likelihood estimator combined with a bandwidth extrapolationtechnique is also developed to estimate the scale parameters. Real measured data processing indicates the proposalcan reconstruct the high-resolution image though only limited pulses even with low SNR, which shows advantagesover current super-resolution imaging methods.

Keywords: Inverse synthetic aperture radar (ISAR), Super-resolution, Compressive sensing (CS), Statistical compressivesensing, Bayesian, Laplace distribution, Nonidentical distribution

1 IntroductionHigh-resolution radar imaging techniques are widely ap-plied in many military and civilian fields, such as targetclassification and recognition and aircraft traffic control.For success of these applications, sufficient image reso-lution is required to characterize the scattering and geo-metric features of the target. Inverse synthetic apertureradar (ISAR) combines the use of pulse compression,flexible pulse repetition frequency (PRF), and target mo-tions (particularly the rotating motion) to generate two-dimensional high-resolution imagery. In general, rangeresolution is determined by the bandwidth of the trans-mitted signal which is limited by the radar system. Tomitigate this limitation, stepped frequency waveformsare usually employed at a price of longer coherent

processing interval (CPI) [1]. Cross-range resolution isdependent on both the CPI and the target rotationalmotion from the variation of radar viewing angles. It iswell known that CPI should be long enough for high azi-muth resolution, which conflicts with modern radar ac-tivities such as multi-target tracking and searching [2].Due to the CPI limitation, achieving high resolution in

azimuth with only a few pulses through conventionalradar imaging schemes, e.g., range-Doppler (RD) algo-rithm [3], is difficult. This motivates super-resolution(SR) techniques. SR image formation improves the abil-ity to resolve two closely spaced scatterers in compari-son with the Nyquist resolution limit [4], which actuallyinvolves uncertainty as the dimension of the image is re-quired to be higher than that of the measurements. En-hancing both the image contrast enrichment andscattering center localization, SR imaging will also re-duce scintillation effects associated with unresolved scat-terers. A super-resolved image can usually promote the

* Correspondence: leizhang@xidian.edu.cn2School of Mathematical and Statistical Sciences, University of Texas-RioGrande Valley, Edinburg TX 78539, USAFull list of author information is available at the end of the article

EURASIP Journal on Advancesin Signal Processing

© 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 DOI 10.1186/s13634-016-0379-2

probability of correct automatic target recognition(ATR) [5, 6]. Consequently, a large fraction of the radarimaging research effort has been taken in developingresolution enhancing algorithms. Generally, there arethree categories that conventional SR approaches areoften sorted into. The first group is the bandwidth ex-trapolation (BWE) methods. They are based on the factthat the data is analytic in principle and can be fittedinto a linear prediction model [2, 7–10]. BWE usuallyapplies some modern spectral estimation techniques todetermine the coefficients of the prediction model.Burg’s algorithm [2] is one of the most successful esti-mators. The second general group of SR methods is theparametric spectral estimation techniques. They attemptto estimate parameters of the dominant scattering cen-ters by modern spectral estimation techniques. Probabil-istic strategies which model the signal and noisestatistically, including maximum a posteriori and max-imum entropy [11, 12] schemes have also been devel-oped, yielding good performance. Adaptive spectralestimators [13–18] utilize sinusoid parametric model toestimate the coefficients of signals by finding a set of fil-ters adaptively. Among this sort, RELAX [13], [14],which estimates the amplitude and frequency of a multi-component complex sinusoidal signal by minimizing theresidual error energy, is one of the most popular algo-rithms. Imageries generated by these approaches are in-herently free from sidelobes, and resolution relies on theprecision of spectral estimation. The third group is non-linear filtering techniques to reduce side lobes while pre-serving the width of the main lobe. This group includesspace-variant filtering methods such as adaptive side-lobe reduction (ASR) [19] and the related special case ofspatially variant apodization (SVA) [20], as well asCLEAN and its modified versions [21, 22]. All thecurrent SR algorithms are more or less sensitive to noiseand model error.Conventional SR imaging can be regarded as a process

that recovers a higher dimensional signal from measure-ments with much fewer degrees of freedom and isdeemed to involve uncertainty mathematically. Develop-ing compressive sensing (CS) theory reveals that an un-known sparse signal can be exactly recovered from verylimited measurements with high probability by solving aconvex l1 optimization problem [23–25]. Satisfaction ofthe restricted isometry property (RIP) allows the uncer-tainty in the l1 optimization problem to be overcome toprovide an exact or approximate signal recovery. The es-sence of CS is to exploit prior knowledge of sparsity ofthe objective signal. This explains the promising per-formance of radar imaging via regularization techniques.For ISAR imaging, strong scattering centers exhibitingthe scattering organization and geometric configurationof the target usually occupy only a fraction of whole bins

in the RD plane. In this sense, the ISAR signal isspatially sparse in RD domain. Such sparsity can beexploited by l1 optimization to enhance image qualityand feature. Many sparsity-driven SR approaches basedon l1 optimization have been developed for SAR/ISARimaging. The feature-enhanced SAR imaging approach[26, 27] exploits the sparsity of both image and differen-tial image via the lp(0 < p ≤ 1) optimization to enhancethe point and region feature of the radar image. Particu-larly, the point-enhanced image formation can extrapo-late the data bands beyond the observations, yielding SRimaging. It should be noted that the point-enhancedoptimization with p = 1 is equivalent to the l1optimization of CS. However, the main problem of thefeature-enhanced algorithm for radar imaging lies in thedifficulty of parameter selection. SR ISAR imaging usingthe l1-FFT, is developed in [28]. Similarly, in [29], a CS-based approach is developed, which can generate a high-resolution image with very few pulses, promisingpromoted performance of ISAR imaging by utilizingspatial sparsity. Based on the assumption that only a fewscatterers with different elevations are present in thesame range-azimuth resolution cell, CS is exploited in[30, 31], to provide a new data acquisition scheme andSR imaging using only a few signal samples for tomo-graphic SAR imagery. In [32], the SR capability and ro-bustness of CS for tomographic SAR imaging have beenanalyzed in detail. All this work indicates that it is pos-sible to substantially enhance the performance of theradar imaging by exploiting target sparsity. Nevertheless,it is usually not an easy task to characterize this sparsityquantitatively due to the uncertainty and complexity ofthe real target scattered field, together with the inevit-able presence of noise and clutter. This makes someregularization-based methods including the feature-enhanced image approach ambiguous in both parameterselection 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 im-aging algorithm is developed wherein the amplitudesparsity of the target scattered field is represented by aLaplace distribution. Based on this, the statistical distri-bution of the complex-valued scattered field is derived.The SR imaging problem is then converted into solvinga sparsity-constraint optimization corresponding to themaximum a posterior (MAP) probability estimation fol-lowing the Bayesian theory. For simplicity, the proposedalgorithm is called Bayesian SR imaging. From Bayesiancompressive sensing (BCS) [33] under the identical La-place distribution assumption, the original Bayesian SRimaging is explicitly coincident with the l1-regularizationoptimization. The sparsity coefficient controls the spars-ity of the reconstructed image, which is directly relatedto the statistics of the target image and noise. We also

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 2 of 19

extend the Bayesian SR imaging by using nonidenticalLaplace distributions since the scattered field of a targetusually follows the energy-assembling and geometricorganization of the target. We place Laplace distribu-tions with different scale parameters on each element,by which signal and noise components could be penal-ized individually in the l1 optimization to promote theaccuracy of SR imagery reconstruction. A novel schemeto estimate the noise and target statistics parameters arealso developed for both the original Bayesian SR and theimproved version by combining the constant-false-alarm-ratio (CFAR) [34] and Burg’s BWE techniques. Toachieve a super-resolved image stably, we propose an it-erative procedure where in each stage the aperture datais extrapolated doubly and the maximum likelihood(ML) estimate of the Laplace scale parameter is updated.The Bayesian SR methods are inherently robust to noiseand workable with limited pulses. Both simulated andreal data experiments are provided to demonstrate thesuperiority of Bayesian SR imaging methods over thecurrent SR approaches under different circumstances.The organization of this paper is as follows. Section 2

introduces the Bayesian SR imaging approaches basedon identical and nonidentical statistics assumptions. InSection 3, we focus on three issues: statistics estimation,imaging procedure, and a modified Quasi-Newtonsolver. Finally, in Section 4, simulated and real-data ex-perimental results are given to show the effectiveness ofthe proposed approaches.

2 Resolution enhancement with statisticalcompressive sensing

A. Signal model

Considering that an ISAR system observes a maneu-vering target and collects a number of pulses coherently,our aim is to achieve a high-resolution image of the tar-get. At first, conventional motion compensation is ap-plied to the raw data followed by range compression andsignal alignment to produce the signal S. We assumethat the data contains N pulses with index from 0 to N−1 and each pulse composes M range bins. Then, therange-compressed data set is given by S = [snm]N ×M.Based on the short CPI assumption, the data can beregarded as a measured patch of the Fourier transformof the target scattered field relevant to some aspect an-gles and so the echoed signals can be rewritten as

S ¼ FAþε ð1Þ

where A¼ a�nm½ � �N�M is the �N �M �N > Nð Þ super-resolved ISAR image whose pixel values correspond tothe amplitude of the scattering centers, ε is the additive

noise matrix with the same size as S, and F is a �N �Mpartial Fourier matrix given by

F ¼1 1 ⋯ 11 ω ⋯ ω N−1ð Þ

⋮ ⋮ ⋱ ⋮1 ω N−1ð Þ ⋯ 1

2664

3775N�N

and ω ¼ exp −j2π�N

� �

ð2ÞNotably, matrix F can be regarded as an over-completed

dictionary, which bridges the recorded data and the SRimage. It should be emphasized that the above signalmodel is based on two significant assumptions.

1) Complete motion compensation. In (2), only theadditive noise is taken into account while othermodel errors such as range shift and residual phaseerror are assumed removed. There are many precisemotion compensation algorithms [35–41] applicablemaking this assumption rigid.

2) Stationarity assumption. The signal amplitude andDoppler frequency are assumed to be invariant. Inthis paper, we focus on the SR imaging with shortCPI, during which the signal can be regarded asstationary and involves no time variance of both thereflectivity and Doppler for a scatterer.

B. Bayesian SR (BSR) imaging via independent andidentical (IID) statistics assumption

Generally, the components of ε are approximated as azero-mean complex Gaussian noise, namely, its imaginaryand real parts (denoted by εr and εi) are subject to inde-pendent zero-mean Gaussian distribution. The complex-valued normal probability density function (PDF) of ε withunknown variance σ2 is given by

pðε σ2�� � ¼ πσ2

� �−N⋅Mexp −

1σ2

εk k2F� �

ð3Þ

where ⋅k k2F denotes the square of the Frobenius norm of a

matrix (the Mk k2F . Notation for a matrix M = [bnm]N ×M

denotes Mk k2F¼X

n;mbnmj j2 ). Therefore, we have the

Gaussian likelihood function of the observation, which is

pðS A; σ2�� � ¼ πσ2

� �−N⋅Mexp −

1σ2

S‐FAk k2F� �

ð4Þ

ISAR imagery demonstrates the spatial distribution of thetarget scattered field in the RD plane. Dominant scatteringcenters usually occupy only a fraction of the whole RD binseven though they contribute a majority of the energy, whilesignals from weak scattering centers contribute little toimage formation. This sparse characteristic of ISAR imagescan be exploited to achieve SR. In this section, we developa SR approach by combining Bayesian statistics and

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 3 of 19

compressive sensing theory. According to sparse Bayesianlearning (SBL) [42] and BCS [33], the sparsity can be for-mulized by placing a sparseness-promoting prior on A. Ingeneral, a Laplace probability distribution can be used toenforce the sparsity prior on the recovered objective signalwhen the signal is real valued [42, 43]; however, it is notstraightforward to extend the Laplace distribution to thecomplex-valued model. In the following content, an exten-sion of the two-layer SBL model [42] is presented and a La-place distribution for complex signal is developed toencourage sparse representation of A.Let anm = rnm + j ⋅ inm, where rnm and inm denote the real

and imagery part of anm, respectively. Then the PDF ofanm can be expressed by the joint PDF of rnm and inm. Fol-lowing the two-layer SBL probabilistic framework [42], weassume both the real and imaginary parts independentlyfollow the same zero-mean Gaussian distribution withvariance β. Therefore, the PDF of anm has a complexGaussian form as

pðanm βj Þ ¼ pðrnm; inm βj Þ¼ 1

πβexp −

r2nm þ i2nmβ

� �ð5Þ

In the two-layer hierarchical prior model of SBL, ahyperprior PDF for β is introduced. The gamma distri-bution is usually used to induce the sparsity propertiesof the reconstruction [42], that is

pðβ a; bj Þ ¼ Gammaðβ a; bj Þ ¼ ba

Γ að Þ βa−1 exp −b⋅βð Þ ð6Þ

where Gamma(· |a, b) denotes a gamma PDF with shapeparameter a and rate parameter b. A detailed analysis onparameter selection for this PDF can be found in [44].Integrating out β, we have

pðanm a; bj Þ ¼Z ∞

0pðanm βj Þpðδ2 γ2

�� Þdβ

¼Z ∞

0

1πβ

exp −r2nm þ i2nm

β

� �⋅ba

Γ að Þ βa−1 exp −b⋅βð Þdβ

ð7ÞFollowing the two-layer SBL probabilistic framework

[33, 42, 43], we set a ¼ 32

p

�anm

32; b

�����

¼Z ∞

0

1πβ

exp −r2nm þ i2nm

2β

� �⋅

b32

Γ32

� � β12 exp −b⋅βð Þdβ

¼ b32

π⋅Γ32

� �Z ∞

0β−12 exp −

r2nm þ i2nm2β

−b⋅β� �

⋅dβ

¼ b32

π⋅Γ32

� �Z ∞

0exp −

r2nm þ i2nm2β

−b⋅β� �

dβ12

ð8Þ

By using the relationZ ∞

0exp −

12

h2u2 þ k2u−2� �� �

du

¼ffiffiffiffiffiπ2h2

qexp − hkj jð Þ [45], the PDF of anm is developed as

a function of b, which is given by

pðanm bj Þ ¼ bπexp −

ffiffiffiffiffi2b

panmj j

�ð9Þ

Setting γ ¼ ffiffiffiffiffi2b

p, (9) becomes

pðanm γj Þ ¼ γ2

2πexp −γ anmj jð Þ ð10Þ

Supposing the image pixels are subject to the IID PDFin (10), we have the likelihood function of A as

pðA γj Þ ¼YN

�−1;M−1

n¼0;m¼0

γ2

2π

� �exp −γ anmk k1� �

¼ γ2

2π

� �N�⋅M

⋅ exp −γ Ak k1� � ð11Þ

where Ak k1 ¼Xn¼1

N� XM

m¼1

anmk k1 . Then, reconstructing SR

image is shifted into a classical problem of estimating Afrom noisy observation S, which has the MAP estimator

A Sð Þ ¼ arg maxA∈CN

��M

p Að jS½ Þ� ð12Þ

By Bayes rule, we get

A Sð Þ ¼ arg maxA∈CN

��M

p S A; σ2�� �

⋅ p A γj Þð ��� ð13Þ

Which is equivalent to

A Sð Þ ¼ arg maxA∈CN

��M

log p S A; σ2�� �� þ log p A γj Þð �½ g��

ð14ÞSubstituting (4) and (11) into (14), the MAP estimator

becomes

A Sð Þ ¼ arg maxA∈C �N�N

−1σ2

S‐FAk k2F−γ Ak k1� �

¼ arg minA∈C �N�N

S‐FAk k2Fþμ Ak k1� � ð15Þ

where μ = σ2γ is defined as the sparsity coefficient whichis directly related to the unknown statistics of noise andthe target signal. One can note that the optimizationcomposes two different terms: the l2-norm preserves thedata fidelity of the solution and the l1-norm imposesmost elements of A to be small with a few large ones inaccordance with its spatial sparsity. The complex-valuedLaplace distribution model in (11) can be also applied inother complex-valued imaging problems, such as tera-hertz imaging [46]. Clearly, MAP estimator of a SR

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 4 of 19

image corresponds to a l1-regularized optimizationwhich is often referred to as the basis pursuit de-noising(BPDN) problem [47]. It is also widely accepted that theproblem in (15) is also equivalent to the l1-constrainedoptimization in CS [48]; however, BSR imaging hasevident differences with them. Compared to the point-enhanced algorithm [26], the l1-norm weighting coeffi-cient in (15) is explicitly associated with the noise andtarget statistics in a Bayesian sense. Furthermore, differ-ent from the CS-based SR imaging in [29], BSR is closelyconnected to convex quadratic programming [49] mak-ing it very efficient and accurate.

C. Improved Bayesian SR (IBSR) imaging based onnonidentical statistics assumption

In the BSR optimization (15), all the components of Aare treated equally as they are assumed to follow thesame Laplace distribution. In reality, the scattered fieldof a distributed target usually has the spatial-assemblingfeature corresponding to the phenomenon that most sig-nificant scattering centers are concentrated around asmall region in the RD image. Neighboring pixels tendto be large or small simultaneously, characterizing them-selves with a specific spatial organization. In this sense,placing the identical Laplace distribution on all compo-nents of the image would ignore both the spatial config-uration and the energy-assembling feature of the targetscattering filed. Nevertheless, it is of great potential tosubstantially improve the performance of BSR imagingby leveraging statistics models that better suit the targetimage. In order to respect the local spatial organizationand discriminate different elements of the image statisti-cally, we introduce nonidentical Laplace distributions foreach component of the ISAR image. Following the defin-ition in (10), for the specific component anm =A(n,m),the Laplace density function is rewritten as

pðanm γnm�� � ¼ γ2nm

2π

� �⋅ exp −γnm anmj j� � ð16Þ

Supposing the image pixels are independent, we havePDF of the SR image as

pA γ00; γ01⋯; γ �N−1ð Þ M−1ð Þ��� �

¼YM−1

m¼0

Y�N−1

n¼0

γ2nm2π

� �exp −γnm anmj j� �

¼YM−1

m¼0

Y�N−1

n¼0

γ2nm2π

� �" #exp − Wmamk k1� �

ð17Þwhere am =A(:,m) corresponds to the mth column of theRD image and Wm ¼ diag γ0m γ1m ⋯ γ �N ‐1ð Þm

� �N� �N

is the diagonal matrix corresponding to the Laplace distri-butions. Introducing nonidentical scale parameters in (20)

provides immediate benefits. We want the probabilisticmodel to favor certain configurations for the magnitudesand locations of the significant scattering centers that pro-mote the performance of the BSR imaging. For this pur-pose, a long tailed distribution with small scale parameterfor each prominent scattering center imposes high prob-ability of a large-valued pixel, and large scale parametersfor other signal components lead to small value predic-tion. The MAP estimator is then given by

A Sð Þ ¼ arg maxA∈CN

��M

pðS A; σ2�� �

⋅ p A γ00; γ01⋯; γ N�−1ð Þ M−1ð Þ

��� � ihð18Þ

Which is also equivalent to

A Sð Þ ¼ arg maxA∈CN

��M

nloghpðS A; σ2�� �i

þ log p A γ00; γ01⋯; γ N�−1ð Þ M−1ð Þ

��� � ih oð19Þ

Substituting Eqs. (4) and (17) into (19), the MAP esti-mator becomes

A Sð Þ ¼ arg maxA∈C �N�M

−1σ2

S‐FAk k2F−XM−1

m¼0

Wmamk k1( )

¼ arg minA∈C �N�M

XM−1

m¼0

sm‐Famk k2Fþσ2 Wmamk k1� �( )

ð20Þ

where sm = S(:,m) denotes the mth column of S corre-sponding to am. In contrast to BSR in (15), the image ismodeled by the nonidentical Laplace distribution in (20).We designate it improved BSR (IBSR). IBSR imaging candiscriminate pixels containing the strong scattering cen-ters from the weak ones, which emphasizes the spatialconfiguration and strong components in image forma-tion. To this end, each element of the image is treatedindividually to favor the recovery of the structuralorganization of the target scattered field. To encouragethe probability of scattering center reconstruction, andenforce the noise be near zero, small scale parametersare preferable for scattering centers, while large scale pa-rameters for noise components. Furthermore, similar tothe optimization in weighted compressive sensing [50],the weights in the l1-norm penalty term in (20) will in-fluence the contributions of different components to thepenalty function improving reconstruction accuracy andefficiency. Therefore, if proper scale parameters are uti-lized in (20), different distributions of signal and noisecan be obtained, leading to more precise signal recoveryand more effective noise suppression.

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 5 of 19

3 Statistics estimation and imaging solver

A. Statistics estimation and imaging procedure

It is widely accepted that in the Bayesian optimizationin (15), if μ is set too large, weak scatterers together withnoise will be rejected in the reconstruction of imageleaving only the dominant scattering centers. If μ is toosmall, then a large amount of noise may be tolerated inthe image. Therefore, for successful SR imaging via theMAP estimator in (15), we must choose σ2 and γ opti-mally. In ISAR imaging, strong scattering centers of thetarget are usually clustered densely in only a small frac-tion of the whole RD plane. Therefore, some pure noisepixels are evidently distinguishable from the target sup-port in some range bins, with which the noise variancecan be estimated straightforwardly. For the other factorγ, supplementary approaches can be employed. As isknown, conventional SR methods are usually precise androbust enough to extrapolate the short aperture to a lim-ited length, such as twice the original aperture length. Inreal applications, we find that in the BSR optimization,using γ estimated from the SR image generated by Burg’sBWE performs well when its SR factor is small. Thismotivates us to combine BWE techniques as an auxiliarytool to provide a coarse SR image to determine the stat-istical parameters for constructing the Bayesian SR im-ages. In this section, we design a scheme for parameterestimation by combining the BWE and CFAR detectiontechniques together with a stage-by-stage procedure toachieve a desired SR robustly. The main characteristic ofthis process is extrapolation of aperture of the signal bya factor of two and updating the estimate of the Laplacescale parameter each stage. In this procedure, the scaleparameter estimation depends on the SR image withdouble BWE from last stage, yielding better BSR andIBSR images. Therefore, the resolution is enhanced stepby step until a desired SR is achieved. The procedureruns in the following three steps:

1) We apply Fourier transform (FT) to the range-compressed data to generate the coarse RD imageand estimate the noise variance by using the purenoise samples. Those noise samples are also used todevelop a CFAR threshold to remove noise in thebackground of the coarse image. It should beemphasized that the noise within the target regioncannot be removed; however, due to the high SNRgain from two-dimensional coherent integration, thisnoise does not affect much. The de-noised image isthen transformed back into the time domain by aninverse Fourier transform (IFT). This is followed byBurg’s BWE to extrapolate the aperture twice, andthen the scale parameter is estimated from the BWE

image based on the ML rule. This step functions asthe initialization.

2) The Bayesian optimization for an image with a SRfactor of two is developed with the statistics fromthe last step and then solved by a Quasi-Newtonsolver (this solver will be introduced in the followingsection). The reconstructed SR image is transformedback into the time domain by IFT, after which,Burg’s BWE with double extrapolation is employedto obtain an image with SR at a factor of four and toestimate the scale parameter for the next stage.

3) We repeat the second step until a desired SR isreached.

Step 1 plays a significant role in determining the noisevariance and coarse target support in the RD plane.With suitable motion compensation, a RD image of sizeN ×M is obtained by a FT in the azimuth direction. Thestatistical parameters can be estimated by using the RDimage. Since motion compensation usually removes alltranslational motion, the target is centralized in the RDplane, and thus, the pixels corresponding to high Dop-pler frequency contain noise only. In practice, we usuallyuse the range bins excluding the target support as thepure noise samples. These pure noise samples can beused to form the ML estimate of σ2. Suppose that wechoose Nε pure noise samples and place them in a vectorwith real and imaginary parts given by εr nð Þ½ �N ε�1 and

εi nð Þ½ �N ε�1 , respectively, the ML estimate σ 2 of σ2 isgiven by

σ 2 ¼ 1N ε

XN ε

n¼0

ε2r nð Þ þXN ε

n¼0

ε2i nð Þ" #

ð21Þ

This estimate is usually accurate due to the abundanceof noise samples. The CFAR detector for strong scatter-ing centers in the RD image can be developed subse-quently. The RD image has high SNR gain from thetwo-dimension coherent integration, and so strong scat-tering centers are discernible from noise in amplitude.In general, discriminating strong scatterers from noise inthe RD domain corresponds to distributed target detec-tion under homogeneous Gaussian noise. Noting σ 2 alsocorresponds to the power of noise, a CFAR threshold isdeveloped as [51].

T ¼ffiffiffiffiffiffiffiffiffiσ 2⋅λ

pð22Þ

where λ is a scale coefficient used to achieve a desiredconstant false alarm probability. For an extensive math-ematical study of the CFAR detector, see [34] and [51].To calculate λ, the software provided by Glen Davidson[52] and Matlab function npwgnthresh.m can be useddirectly. Pixels with amplitude larger than the CFAR

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 6 of 19

threshold are determined as target components, whilethose below the threshold are classified as noise and setto zeros to achieve the de-noised image. Only the noisecomponents in the background region are detected andremoved, while noise mixed with the target region re-mains. However, due to the high SNR gain from thedouble coherent integration, the remaining noise has lit-tle affect. The de-noised image is then transformed backinto the time domain where Burg’s BWE is used to ex-trapolate the data doubly. Therefore, a SR image can beachieved by a FT after Burg’s BWE processing. For clar-ity, we define �A1 ¼ �A1 n;mð Þ½ �N1�M as the BWE imagewith size N1 ×M (N1 = 2N). Then, the parameter scaleγ1 of the image with SR at a factor of two is ready to beestimated by the ML rule. The ML estimator of γ1 isformed by finding γ1 that maximizes the log-likelihoodfunction

log½P �A1 γ1�� Þ� ¼ log

γ212π

� �N1⋅M

⋅ exp −γ1 �A1

�� ��1

�" #

ð23ÞDifferentiating with respect to γ1 yields

∂ log½P �A1 γ1�� Þ�

∂γ1¼ 2N1⋅M

γ1−XN1−1

n¼0

XM−1

m¼0

�A1 n;mð Þ�� �� ð24Þ

Setting (24) equal to zero so that γ1 is a critical pointleads to the ML estimator

γ 1 ¼2N1⋅MXN1−1

n¼0

XM−1

m¼0

�A1 n;mð Þ�� �� ð25Þ

This shows that the ML estimator of γ1 is the recipro-cal of the amplitude mean of Ā1. Without loss of gener-ality, in the initialized step, all variables are indicated bythe subscript g = 1, and the initialization step is embed-ded in the first stage of the whole procedure.Steps 2 and 3 implement the BSR imaging in an itera-

tive manner. By using σ 2 and γ 1 , the optimization for aSR image with double aperture extrapolation which cor-responds to the first stage of SR imaging can be devel-oped. For clarity, we use subscript g as the stage number,and in the first stage, we have g = 1. BSR optimization inthe gth stage is given by

Ag ¼ arg minA∈C �N g�M

S‐FgAg

�� ��2Fþσ 2γ g Ag

�� ��1

n oð26Þ

where Ag denotes the �Ng �M �Ng ¼ 2gN� �

SR image,and the N � �Ng Fg can be calculated by using Eq.(3). Similarly, Burg’s BWE is applied to Âg in orderto estimate the Laplace scale parameter for the next

stage. We transform Âg back into the azimuth timedomain and use Burg’s BWE to generate the �Ngþ1

�M BWE image �Agþ1 ¼ �Agþ1 n;mð Þ� N1�M which is

then used in the ML estimation of the next Laplace

scale parameter γ gþ1 ¼ 2Ngþ1⋅MXNgþ1−1

n¼0

XM−1

m¼0

�Agþ1 n;mð Þ�� �� . The BSR

optimization for the next stage can be performed usingγ gþ1 , σ

2 , and Fg + 1. We increment g and repeat this pro-

cessing until a desired SR is obtained. For example, if wewant to achieve a SR factor of eight, we need three stagesfor BSR imaging together with the initialization step, andin each stage, we need to perform the Burg’s BWE andBSR imaging once. For a clear description of the proced-ure, we list the corresponding variables in Table 1 and givea detailed flowchart of the BSR with G stages in Fig. 1.It should be noted that the Burg’s BWE plays an essen-

tial role in the estimation of the Laplace scale parameter.Implemented in a recursive way, Burg’s BWE is stable inaperture extrapolation. In real data processing, the orderof the linear prediction model in the BWE is typicallyset high, such as one third of the data length, to ensureestimation precision of the largest coefficients withoutbringing spurious scattering centers [10]. Additionally,in the Burg’s BWE, the retention of primary data andphase coherency of the extrapolated signal are meaning-ful for the precise estimation of the scale parameter.However, in [53], detailed theoretic analysis reveals thatBurg’s BWE has limited SR ability at a factor of 2.6 fortwo identical point-scatterers, and beyond that, thenegative effect becomes evident, which is the mainreason why the extrapolation factor of two in eachstage is set.Similarly, IBSR imaging with the optimization in (20)

can be also performed by using the parameter estimationand stage-by-stage procedure presented above. UnlikeBSR imaging, IBSR utilizes the nonidentical Laplace distri-bution, and the estimation of the scale parameters shouldbe reconsidered. As mentioned before, the ISAR imagerepresents the local spatial structure and energy-assembling characteristic of the target scattering field,which means that neighboring components tend to belarge or small simultaneously. Therefore, we estimate thescale parameter corresponding to a certain component by

Table 1 Variable description in the stage-by-stage procedure

g Subscript of the stage number

�Ng ¼ 2g⋅N Azimuth dimension of SR image in the gth stage

Ag SR image �Ng �M� �

Âg Estimation of Ag (�Ng �M)

Āg BWE SR image �Ng �M� �

in the gth stage

Fg Partial Fourier matrix �Ng � N� �

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 7 of 19

using its neighbors, such as the 5 × 5 neighborhood sur-rounding it. It should be noted that in reality, the num-ber of the neighboring components should be selectedoptimally since if we use only a few neighboring pixels,severe deviation would be introduced in the result,while if we use a large number of neighbors, they wouldtend to not follow similar distributions leading to poorIBSR performance as well. A useful scheme is to useonly those with similar magnitudes to the central pixelin a small neighboring window 11 (such as 3 × 3) in thefirst stages, and in the last stages select those with simi-lar magnitudes in a large window (such as 5 × 5) to esti-mate the Laplace scale for the central pixel. The scaleparameter is estimated as the reciprocal of the mean ofthese pixel values similar to the ML estimation in (25).With the help of CFAR detection and Burg’s BWE, theresolution is enhanced stage-by-stage until it reaches

the predetermined factor. Similarly, for the gth stage,we have the IBSR optimization as

Ag ¼ arg minA∈C �N g�M

XM−1

m¼0

sm‐Fgam gð Þ�� ��2

Fþσ 2 Wm gð Þam gð Þ�� ��

1

�( )

ð27ÞFor clarity, a flowchart for IBSR imaging with G stages

is provided in Fig. 2.

B. An effective solver to the IBSR optimization

As mentioned in the above section, many algo-rithms and software [54–57] are ready to be appliedin order to solve the optimizations of (15) and (20).Herein, we apply a Quasi-Newton solver to them.This takes advantage of the fast Fourier transform

Fig. 1 Flowchart of BSR imaging

Fig. 2 Flowchart of IBSR

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 8 of 19

(FFT), promising high efficiency and the possibility ofreal-time implementation of BSR and IBSR imaging.In the following content, we only deduce the solverto the optimization (20), but it can be extended to(15) straightforwardly. Noting that the optimizationsin (15) and (20) are non-differentiable, we need tomodify them to be consistent with the gradient-basedoptimization algorithms. In order to overcome thenon-differentiability of the l1-norm around the originin (20), a useful approximation [27, 58] is employedby

anmk k1≈ anmj j2 þ τ� �1=2 ð28Þ

where | ⋅ | stands for the modulus operator, and τ is asmall nonnegative parameter. To ensure the approxima-tion as rigid as possible, τ should be set small. The IBSRoptimization in (18) can be reformatted as

A Sð Þ ¼ arg minA∈CN

��M

(XM−1

m¼0

sm‐Famk k2F

þσ2XN�−1n¼0

γnm⋅ anmj j2 þ τ� �1=2!)

ð29Þ

The approximation in (28) keeps the optimizationsin (20) and (29) consistent, and the solution to (29)tends to that of the optimization (20) as τ approachesto zero. Therefore, a precise solution to optimization(20) can be achieved by solving (29) if and only if avery small τ is used in (28). In the following experi-ments, τ = 10−6. The selection of τ is still an openproblem, and it is empirically small to get rid of largeapproximation errors. Due to the independencebetween range cells, solving the two-dimensionaloptimization (29) is equivalent to solving the one-dimension optimization.

am Sð Þ ¼ arg minam∈C �N�1

sm‐Famk k2Fþσ2X�N−1

n¼0

γnm⋅ anmj j2 þ τ� �1=2

ð30Þ

for each range cell separately. That is, in order to re-construct the mth column of A, we must solve (30).

The conjugate gradient of f amð Þ ¼ sm‐Famk k2 þ 2σ2

X�N−1

n¼0

Wm n; nð Þ⋅ anmj j2 þ τ� �1=2

with respect to am is

given by

∇am� f amð Þ ¼ 2FHFam þ σ2⋅U amð Þ⋅Wmam−FHsm¼ H amð Þ⋅am−FHsm

ð31Þwhere the Hessian matrix H(am) is approximately givenby

H amð Þ≈2FHFþ σ2⋅U amð Þ⋅Wm ð32Þand

U amð Þ ¼ diag

"1= a0mj j2 þ τ� �1=2

; 1= a1mj j2 þ τ� �1=2

; ⋅⋅⋅;

1= a �N−1ð Þm��� ���2 þ τ

� �1=2#

ð33ÞBecause the Hessian approximation relies on the ob-

jective am, an iterative solver to (30) is presentedthrough:

a hþ1ð Þm ¼ H a hð Þ

m

�h i‐1FHsm ð34Þ

where a hð Þm is the estimator of A(:,m) in the hth iteration.

To accelerate the iteration, the conjugate gradient algo-rithm (CGA) [59] can be applied to avoid the Hessianmatrix inversion in (34). A fixed threshold δCG is usedfor a complete run of the CGA. Without prior informa-tion about A, it is initialized as Â0 = FHS. We iterate h

until a hþ1ð Þm −a hð Þ

m

��� ���2= a hð Þ

m

��� ���2≤ρ , where ρ is a small param-

eter for the predetermined threshold, or h reaches a pre-determined maximum iteration number.In the following computational cost analysis of the

Quasi-Newton solver, we keep track of the multiplica-tions. The main computational cost of the solver forIBSR optimization is in using CGA to iterate (34). Onemay note that the term FHF corresponds to the partialFourier matrix F, allowing us to use a FFT to efficientlycalculate FHFam in (31). We perform an inverse fastFourier transform (IFFT) to am, set the components cor-responding to the vacant aperture to zero, and thenapply a FFT dramatically reducing the computationalload. Let the number of the CGA iterations to solve (34)be NCGA. The computational cost of the FFT and IFFTin solving (34) is 2NCGA �N log2 �N flops. If there are NQN

iterations in the Quasi-Newton solver, the total compu-tation cost is approximately 2NQNNCGA �N log2 �N flops,while the computational load of calculating matrix inver-sion through the Cholesky factorization [60] is up to

NQN �N 3=3 flops. Since NCGA is usually on the order of

several tens, the efficiency improvement via FFT andCGA is obvious. In practice, we can start with a relaxedCGA tolerance δCG and reduce it as we iterate the

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 9 of 19

Quasi-Newton solver. For further improvements, thereare optimization algorithms such as the preconditionedconjugate gradient algorithms [60], which can reducethe number of iterations of the standard CGA.

4 Performance analysis

A. Simulation for SR analysis

In this subsection, we analyze the SR power of BSR andIBSR by comparing them with two compressive sensing-based SR imaging approaches. Namely, we compare BSRand IBSR to the CS [29] and improved CS (ICS) [61] im-aging methods under different signal-to-noise ratio (SNR)conditions. All of the methods are sparsity-driven. Boththe CS and ICS imaging methods are implemented by thecvx software [56], which usually finds the globally optimalsolution to the l1-norm constraint problem. Meanwhile,the BSR and IBSR optimizations are solved by the Quasi-Newton solver by using the CGA and FFT. It should beemphasized that, as pointed out in [32], the needle-like re-sponses of SR methods tell neither the location accuracynor capability of resolving two close-located scattererswith limited noisy pulses. In general, the performance ofall SR approaches degrades in some degree with the in-crease in noise. This is due to the strong coherence be-tween Fourier basis vectors, amplitude fluctuation of thescatterers, and the increasing presence of false points atadjacent bins that are usually involved. Since it is difficultto find a theory supporting the SR power comparison, weapproach the problem experimentally by analyzing the re-construction of a signal containing two closely spacedscatterers. Taking both the reconstruction accuracy andthe presence of false points into account, we use the MeanSquare Error (MSE) and a specified probability of detec-tion PD to evaluate the SR power of different methods.The MSE is defined as

MSE ¼ a‐að ÞH ⋅ a‐að Þ�N

ð35Þ

where a and â is the ideal and the reconstructed signal,respectively, and �N denotes the length of the reconstruc-tion. We repeat the SR reconstruction 1000 times withdifferent noise levels and use the average MSE as anevaluating metric.Assuming that there are only two closely located scat-

terers, probability of detection corresponds to the hy-pothesis test:

H0: There is at most one scatterer present in thereconstruction signal;H1: There are exactly two scatterers inside the givenDoppler bins of the reconstruction signal.

Note that we take it as a failure when artifacts occurin the reconstruction, which means that the definition ofPD is slightly different from that in [32]. In general, thenumber of scatterers in the reconstructed signal can beconverted into the problem of model selection by someinformation theory rules before the SR reconstruction[32]. In the manner of signal detection, we utilize theCFAR technique by a predefined threshold to determinethe number of signal components. The threshold is cal-culated according to the added noise variance before theSR reconstruction. The false alarm rate is adjusted ex-perientially according to the given noise variance tomaintain the threshold value constant at 0.3, which is se-lected optimally based on the performance of IBSR. Inthe following experiments, bins of the reconstructed sig-nal exceeding the threshold are determined to be signalcomponents and the total number of components is out-putted for the hypothesis test.First, we apply the methods to one-dimensional syn-

thetic signals covering the Doppler range −255 to256 Hz. All signals contain N = 32 regularly sampled ac-quisitions with sample interval 1/512 s, providing a Ray-leigh resolution of 16 Hz. Each signal contains twoscatterers corresponding to two closely located Dopplerpoints whose amplitudes are equally set to unity andhave zero phases. We consider four cases of differentdistances between the two scatterers to analyze the SRpower of the methods. In the four cases, the two scat-terers are located at Doppler frequencies of 0/8 Hz, 0/4 Hz, 0/2 Hz, and 0/1 Hz, respectively. To separate thetwo scatterers in all cases with the given central 32 sam-ples within the dwell time [−31/512, 32/512] (s), we per-form the SR methods with SR factors 2, 4, 8, and 16,corresponding to the reconstructed signal lengths 64,128, 256, and 512, respectively. Complex-valued Gauss-ian noise is added in to the synthetic signals to generateSNRs from 0 to 20 dB, wherein SNR is defined as theenergy ratio of a single point and the added noise.For comparison, typical results from the sparsity-

driven methods with SNR = 5 dB are presented in Fig. 3.The first and second rows correspond to the resultsfrom CS- and ICS-based imaging, while the last tworows show the BSR and IBSR images, respectively. FromFig. 3, we find that reconstructing the ideal signal withonly 32 noisy samples is achievable by all sparsity-drivenapproaches. These approaches work well in cases of rela-tively high SNRs. We can find that under SNR = 5 dB,the two scatterers are discriminated optimally and thenoise is also suppressed effectively in all cases. Neverthe-less, as we increase noise, the performance of each ap-proach is expected to degrade producing fluctuatingamplitudes and artifacts. In general, the imaging per-formance of ICS and IBSR degrades more gracefullythan that of CS and BSR. To see this, the MSEs of the

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 10 of 19

reconstructions are plotted in Fig. 4. From Fig. 4, wenote that when the SNR is larger than 10 dB, eachmethod can provide optimal SR reconstruction sincethey can provide very small MSEs and high probabilityof detection. Performance degrades with the increaseof noise. From Fig. 4, we can note that BSR is compar-able to CS-based imaging only in the high SNR situa-tions. This indicates that BSR should be more sensitiveto strong noise. In the low SNR cases, ICS and IBSRusually provide better performance than the others,due to the weighting processing in the l1-norm optimi-zations. In particular, IBSR provides the lowest MSEand the highest probability of detection in case 1 and2, which demonstrates the improvement achieved byusing the nonidentical Laplace probability assumption.However, this performance is not maintained with theincrease of SR factor and decrease of SNR. In case 3and 4, we find that IBSR provides the lowest MSE andthe highest probability of detection only in the highSNR situations, with its performance degrading dra-matically with the decrease of SNR. Especially in case3 and 4 when SNR decreases under 5 dB, IBSR is worsethan ICS and close to BSR and CS. This is probablydue to Laplace scale parameters not being estimated

accurately by the proposed iterative procedure due tothe strong noise interference and capability limitationsof Burg’s BWE. It is easy to understand that when MSEincreases to an unacceptably high level or the probabil-ity of detection decreases under a low threshold, theSR reconstruction can be regarded as a failure, bywhich one can have a concrete definition of SR. In thissense, ICS and IBSR should be the better options inreal applications, and IBSR should be the best from theview of MSE in first and second cases. From the twopoint-scatterers simulation, both MSE and probabilityof detection curves provide two perspectives of BSRand IBSR:

1) The performance of BSR can approach cvx-basedCS imaging, especially in the cases of relative highresolution. Therefore, although BSR never performsbetter than CS-based imaging, BSR is preferable insome ideal applications due to its high efficiency.

2) Based on the nonidentical statistical assumptionand accommodation of statistical parametersestimation implemented by CFAR and Burg’sBWE, IBSR is more robust to noise and limitationof measurements.

Fig. 3 Typical SR results of different methods under SNR = 5 dB

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 11 of 19

However, when the SR factor increases up to 8 and16, performance degrades dramatically. The reason forthis phenomenon lies in that, in these cases, Burg’sBWE cannot provide accurate estimation of the non-identical Laplace scale parameters. In other words,when the two point-scatterers are spaced too closely,the assistance from Burg’s BWE in the IBSR iterationis still achievable, but not when strong noise is in-volved. In this sense, the SR power of IBSR is limitedby Laplace parameter estimation. On the other hand,this phenomenon also indicates that if some operationsare capable of improving the accuracy of statistical es-timation, the performance of IBSR can be further en-hanced in some degree. This should be accounted forin further study.

B. Real data set description and evaluation metrics

In the following, real ground-based data is used for aperformance analysis of BSR and IBSR for SR. Real dataexperimental comparisons of the Burg’s BWE, CS, ICS,BSR, and IBSR are given. The performance analysis inthe following is carried out by considering mainly twofactors: the pulse amount and the noise interference.The data set of a Yak-42 airplane is recorded by a C-

band (5.52 GHz) ISAR experimental system. The systemtransmits a 400-MHz linear modulated chirp signal with25.6-μs pulse duration, resulting in a range resolution of0.375 m. The received signal is de-chirped and I/Q sam-pled. In this data set, the pulse repetition frequency is100 Hz, i.e., 256 pulses within dwell time [−1.28, 1.28](s) are utilized. Motion compensation, including rangealignment and autofocus processing, is performed to re-move the translational motion in advance. The alignedprofiles are shown in Fig. 5a. The RD image generated

Fig. 4 Comparison of SR performance. a MSEs with different sparsity-driven methods. b PD with different sparsity-driven methods

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 12 of 19

by all 256 pulses is shown in Fig. 5b for evaluating theexperimental results. In the following experiments, SNRis defined as the energy ratio between the original dataset and the added noise.To provide a quantitative evaluation for the following

SR imaging experiments, we consider some metrics thatare directly relevant to the performance of automatictarget recognition (ATR). The first metric is the coherencebetween the reconstructed SR image and the referenceimage in Fig. 5b, which represents the comparability ofthese two. In general, high coherence ensures high recog-nition probability in template-based classifiers [61]. Thecoherence is defined as follows.

Coherence ¼ A⊙A� �Ak kF ⋅ A

�� ��F

ð36Þ

where A and A ¼ anm½ � �N�M denote the original and re-constructed SR image, respectively, ⊙ denotes the

Hadamard product, and < · >, the operator for summingup all components of a matrix. Notably, the coherence isdefined with respect to the magnitude. Since the phasesof the image are random, they usually provide few con-tributions to feature extraction for ATR [62]. Therefore,we calculate the coherence only with the magnitude ofimage in (36).Another evaluation metric is the target-to-background

ratio (TBR). By applying the adaptive watershed segmen-tation method [63] on the RD image to separate the tar-get and the background regions and then counting thetarget energy (within the target region) and noise energy,and the TBR definition is given by

TBR ¼ 10⋅ log10

Xn;mð Þ∈Τ

anmj j2

Xn;mð Þ∈Β

anmj j2

0BB@

1CCA ð37Þ

where T and B are the target and background region, re-spectively, shown in Fig. 6. In our special case of imagingwith limited noisy measurements, the main disadvantageof approaches based on parametric modeling is the intro-duction of false scattering centers. This could have a nega-tive effect on the description of the configuration andfeatures of the target. Clearly, TBR effectively considersthe false points and energy leakage of target support. It isalso relevant to the SNR of the reconstruction. Moreover,it can measure target preservation and noise suppressionwith the help of the predetermined target region. We de-fine the noise energy as

Noise ¼Xn;mð Þ∈Β

anmj j2 ð38Þ

The false points and noise in the background of thereconstructed image could be treated as scatters in ATR,

Fig. 5 Data set and RD image with 256 pulses. a Aligned rangeprofiles. b RD image

Fig. 6 Target region

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 13 of 19

potentially leading to performance degradation. Both theTBR and noise energy metrics should be significant forfeature-based ATR processing. We repeat the SR recon-struction 100 times with different noise in the followingexperiments to analyze coherence, TBR, and noise en-ergy. The mean values of coherence, TBR, and noise en-ergy from different SR approaches have been exploitedto evaluate the experimental results in the following.

C. Performance versus pulses amount

To study how the sample amount affects the recoveryperformance, we compare the reconstructed results ofBSR, IBSR, and current SR algorithms by varying thesample number. With different amount of pulses andconstant SNR (10 dB), the SR images are generated with256 Doppler bins. The pulse amount decreases from 128to 16. In Fig. 7, we give typical images with four differentpulse numbers (128, 64, 32, and 16), corresponding to 2,4, 8, and 16 times SR. By this, we investigate the per-formance of the five approaches with specific pulsenumbers. In all processing of BSR and IBSR, CFAR is set

to 10 − 4 in the initialization step, the threshold for theQuasi-Newton solver is ρ = 10− 3, and the terminatingtolerance for CGA is δCG = 10− 5. Typical experimentalresults are illuminated in Fig. 7. In Fig. 7, the first rowpresents the aperture patterns with different pulse num-bers, the second row gives RD images after Burg’s BWE,the third and fourth rows contain CS and ICS images,respectively, and the fifth and the bottom row show BSRand IBSR images, respectively.From the comparisons, it is clear that ICS and IBSR

perform better in scattering center extraction and noisesuppression. In addition, IBSR imaging is more stablethan BSR with a small number of pulses, as it generatesdense and clear images with significant information andenergy preserved to some extent. In order tocharacterize the performance quantitatively, the curvesfor the mentioned metrics with respective to the pulseamount are given in Fig. 8. Figure 8a provides the coher-ence between the recovered SR images and the referenceimage. Figure 8b, c shows the TBR and noise energy, re-spectively. It is worth noting that the performance of theIBSR and ICS performs well even under low SNR

Fig. 7 SR imaging with 128, 64, 32, and 16 pulses

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 14 of 19

Fig. 8 Performance with different pulse numbers. a Coherence. b TBR. c Noise energy

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 15 of 19

conditions. They provide higher coherence, TBR, andlower noise energy than the others. Figure 8a shows thatthe coherence corresponding to all algorithms increaseswith the increase of pulse amount. However, thesparsity-driven approaches can provide higher coherencethan Burg’s BWE even in the case of very limited pulses.Burg’s BWE also presents good coherence performance,which lies in the fact that in Burg’s BWE, the retentionof primary data and phase coherency of the extrapolatedsignal are useful. In Fig. 8b, we note that Burg’s BWEprovides the lowest TBR because of its poor de-noising.CS, ICS, BSR, and IBSR give very high TBRs in a stableway. In particular, ICS and IBSR present TBR up to25 dB, indicating that they suppress the majority of noisewhile preserving target energy optimally, which is alsoclear in Fig. 8c. The metrics show that we achieve betterperformance by imposing proper statistics on the targetand the nonidentical Laplace distribution is suitable torepresent the sparsity of the image.

D. Performance versus SNR

The aim of this experiment is to analyze the influenceof SNR on the performance of the approaches. To com-pare the noise tolerance of the four methods, we use 64pulses corresponding to the dwell time [−0.32, 0.32] (s).As the original data set has high SNR after range com-pression, we generate experimental data with differentSNRs (15, 10, 5, and 0 dB) by adding complex-valuedwhite Gaussian noise into the data. Then, four timesaperture extrapolation is carried out with Burg’s BWE,and a hamming windowed Fourier transform is then ap-plied to yield a SR image with 256 Doppler bins. Corres-pondingly, SR images with the same Doppler binsobtained by CS and ICS are constructed with precisenoise level. For BSR and IBSR implemented by thestage-by-stage flows, we need three iterations to yield256-Doppler-bin images with 64 pulses. In Fig. 9, wegive the resulting images with different SNRs. The re-covered images via Burg’s BWE, CS, ICS, BSR, and IBSRare shown from the first row to the fifth in Fig. 9, re-spectively. By comparing the results, we note that all theapproaches provide good performance in the case of

Fig. 9 SR imaging with SNR 15, 10, 5, and 0 dB

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 16 of 19

high SNR (15 and 10 dB) and the images are formed ina clean and dense way. Nevertheless, artifacts increaseapparently, and some scattering centers are missingwhen we decrease SNR in Burg’s BWE images. Thesparsity-driven methods are capable of providing goodperformance under strong noise. Distinctively, ICS andIBSR usually generate images with much fewer artificialbackground points and denser target images due to theweighting processing. Putting it quantitatively, themetric curves with respective to the SNR are given inFig. 10. Figure 10a provides the coherence between therecovered SR images and the reference image. Figure 10b,c shows the TBR and noise energy, respectively. InFig. 10a, we note that all algorithms provide stable co-herence with varied SNRs. However, BSR provides com-parable coherence to CS. ICS and IBSR give the highestcoherence. In Fig. 10b, we note that TBR of all ap-proaches decreases proportionally to the increase ofnoise energy. However, Burg’s BWE generates the lowestTBR due to the lack of noise suppression. All sparsity-driven approaches perform much more stably, yieldinghigh TBR up to 20 dB regardless of the decreasing SNR.In the aspect of TBR, IBSR is the best. In Fig. 10c, thenoise energy of SR images obtained by the sparsity-driven approaches is distinctive from Burg’s BWE. It isvery low and almost constant with the decrease of SNR.Both the last and current experimental results show thatsparsity-driven methods are optimal in SR imagingunder noise conditions. In particular, ICS and IBSR canovercome strong noise and generate high-quality SR im-ages effectively.

5 ConclusionsIn this paper, we present SR imaging algorithms basedon BCS. By combining the statistics estimation andCS, they are implemented by solving optimizationswith l1-norm optimization, derived from the MAP es-timations. The sparsity level of the reconstructedimage is determined by extracting the statistics fromdata stage by stage. Both BSR and IBSR are robust tostrong noise and can reconstruct an image while sup-pressing strong noise. Based on the fact that the scat-tered field usually has distinctive spatial configuration,the nonidentical Laplace distribution is introduced inIBSR to discriminate prominent scattering centersfrom weak ones and noise in the l1 penalty term. Thenonidentical Laplace distribution promotes perform-ance in terms of noise tolerance and limited pulses.The contribution of this paper illuminates the possibil-ity of substantially improving the performance of SRimaging by leveraging statistical models coincidingwith the image optimally. Surely, there are other novelstatistics models that can also be used to represent thesparsity and local feature of radar image ideally.

Recent studies show that Gaussian mixture models[64] are optimal candidates to describe the structuraldependence and sparsity of a target. This will be sur-veyed in future work. The improved statistical com-pressive sensing would be also beneficial for multi-antenna setting for SAR [65] and ISAR imaging formaneuvering targets [66, 67].

Fig. 10 Reconstruction metrics with different SNRs. a Coherence.b Target-to-background ratio. c Noise energy

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 17 of 19

AcknowledgementsThe authors thank the anonymous reviewers for their valuable comments.This work was supported by the National Natural Science Foundation ofChina (No. 61301280 and 61301293).

Competing interestsThe authors declare that they have no competing interests.

Author details1National Lab of Radar Signal Processing and the Collaborative InnovationCenter of Information Sensing and Understanding, Xidian University, Xi’an710071, People’s Republic of China. 2School of Mathematical and StatisticalSciences, University of Texas-Rio Grande Valley, Edinburg TX 78539, USA.

Received: 29 April 2016 Accepted: 7 July 2016

References1. Q Zhang, Y-Q Jin, Aspects of radar imaging using frequency-stepped chirp

signals. EURASIP J Appl Signal Process 2006, 43–43 (2006). Available: http://dx.doi.org/10.1155/ASP/2006/85823

2. SL Borison, SB Bowling, KM Cuomo, Super-resolution methods for widebandradar. Lincoln Lab J 5, 441–61 (1992)

3. JL Walker, Range-doppler imaging of rotating objects. Aerospace andElectronic Systems, IEEE Transactions on AES-16(1), 23–52 (1980)

4. SH, Image recovery: theory and practice (Academic, Orlando, 1987)5. J Gudnason, J Cui, M Brookes, Hrr automatic target recognition from

superresolution scattering center features. Aerospace and ElectronicSystems, IEEE Transactions on 45(4), 1512–24 (2009)

6. LM Novak, G Owirka, AL Weaver, Automatic target recognition usingenhanced resolution SAR data. Aerospace and Electronic Systems, IEEETransactions on 35(1), 157–75 (1999)

7. H-J Li, N Farhat, Y Shen, A new iterative algorithm for extrapolation of dataavailable in multiple restricted regions with application to radar imaging.Antennas and Propagation, IEEE Transactions on 35(5), 581–8 (1987)

8. K Suwa, M Iwamoto, A two-dimensional bandwidth extrapolation techniquefor polarimetric synthetic aperture radar images. Geoscience and RemoteSensing, IEEE Transactions on 45(1), 45–54 (2007)

9. I Gupta, High-resolution radar imaging using 2-D linear prediction.Antennas and Propagation, IEEE Transactions on 42(1), 31–7 (1994)

10. TG Moore, BW Zuerndorfer, EC Burt, Enhanced imagery using spectral-estimation-based techniques. Lincoln Lab J 10(no. 2), (1997)

11. RO Lane, KD Copsey, AR Webb, A Bayesian approach to simultaneousautofocus and superresolution ( 2004), pp. 133–42. Available: +http://dx.doi.org/10.1117/12.541504

12. A Mohammad-Djafari, G Demoment, Maximum entropy Fourier synthesiswith application to diffraction tomography. Appl Optics 26(9), 1745–54(1987)

13. J Li, P Stoica, Efficient mixed-spectrum estimation with applications to targetfeature extraction. Signal Processing, IEEE Transactions on 44(2), 281–95 (1996)

14. Z Bi, J Li, Z-S Liu, Super resolution SAR imaging via parametric spectralestimation methods. Aerospace and Electronic Systems, IEEE Transactionson 35(1), 267–81 (1999)

15. A Lazarov, Iterative MMSE method and recurrent Kalman procedure for ISARimage reconstruction. Aerospace and Electronic Systems, IEEE Transactionson 37(4), 1432–41 (2001)

16. Z-S Liu, R Wu, J Li, Complex ISAR imaging of maneuvering targets via thecapon estimator. Signal Processing, IEEE Transactions on 47(5), 1262–71 (1999)

17. M Palsetia, J Li, Using apes for interferometric SAR imaging. ImageProcessing, IEEE Transactions on 7(9), 1340–53 (1998)

18. J Li, P Stoica, An adaptive filtering approach to spectral estimation and SARimaging. Signal Processing, IEEE Transactions on 44(6), 1469–84 (1996)

19. H Stankwitz, RJ Dallaire, J Fienup, Nonlinear apodization for sidelobe controlin SAR imagery. Aerospace and Electronic Systems, IEEE Transactions on31(1), 267–79 (1995)

20. X Xu, R Narayanan, Enhanced resolution in SAR/ISAR imaging using iterativesidelobe apodization. Image Processing, IEEE Transactions on 14(4), 537–47(2005)

21. J Tsao, B Steinberg, Reduction of sidelobe and speckle artifacts inmicrowave imaging: the clean technique. Antennas and Propagation, IEEETransactions on 36(4), 543–56 (1988)

22. R Bose, A Freedman, B Steinberg, Sequence clean: a modifieddeconvolution technique for microwave images of contiguous targets.Aerospace and Electronic Systems, IEEE Transactions on 38(1), 89–97 (2002)

23. D Donoho, Compressed sensing. Information Theory, IEEE Transactions on52(4), 1289–306 (2006)

24. E Candes, T Tao, Near-optimal signal recovery from random projections:universal encoding strategies? Information Theory, IEEE Transactions on52(12), 5406–25 (2006)

25. E Candes, M Wakin, An introduction to compressive sampling. SignalProcessing Magazine, IEEE 25(2), 21–30 (2008)

26. K Cetin, WC Karl, AS Willsky, Feature-preserving regularization method forcomplex-valued inverse problems with application to coherent imaging.Opt Eng 45(no. 1), 017 003–017 003–11 (2006). Available: +http://dx.doi.org/10.1117/1.2150368

27. M Cetin, W Karl, Feature-enhanced synthetic aperture radar imageformation based on nonquadratic regularization. Image Processing, IEEETransactions on 10(4), 623–31 (2001)

28. G Zweig, Super-resolution Fourier transforms by optimisation, and ISARimaging. Radar, Sonar and Navigation, IEE Proceedings 150(no. 4), 247–52 (2003)

29. L Zhang, M Xing, C-W Qiu, J Li, Z Bao, Achieving higher resolution ISARimaging with limited pulses via compressed sampling. Geoscience andRemote Sensing Letters, IEEE 6(3), 567–71 (2009)

30. XX Zhu, R Bamler, Tomographic SAR inversion by l1 normregularization—the compressive sensing approach. IEEE Trans GeosciRemote Sens 48(10), 3839–46 (2010)

31. A Budillon, A Evangelista, G Schirinzi, Three-dimensional SAR focusing frommultipass signals using compressive sampling. Geoscience and RemoteSensing, IEEE Transactions on 49(1), 488–99 (2011)

32. XX Zhu, R Bamler, Super-resolution power and robustness of compressivesensing for spectral estimation with application to spaceborne tomographicSAR. Geoscience and Remote Sensing, IEEE Transactions on 50(1), 247–58(2012)

33. S Ji, Y Xue, L Carin, Bayesian compressive sensing. Signal Processing, IEEETransactions on 56(6), 2346–56 (2008)

34. H Rohling, Radar CFAR thresholding in clutter and multiple target situations.Aerospace and Electronic Systems, IEEE Transactions on AES-19(4), 608–21(1983)

35. Y Wang, H Ling, V Chen, ISAR motion compensation via adaptive joint time-frequency technique. Aerospace and Electronic Systems, IEEE Transactionson 34(2), 670–7 (1998)

36. W Ye, TS Yeo, Z Bao, Weighted least-squares estimation of phase errors forSAR/ISAR autofocus. Geoscience and Remote Sensing, IEEE Transactions on37(5), 2487–94 (1999)

37. J Wang, D Kasilingam, Global range alignment for ISAR. Aerospace andElectronic Systems, IEEE Transactions on 39(1), 351–7 (2003)

38. D Zhu, L Wang, Y Yu, Q Tao, Z Zhu, Robust ISAR range alignment viaminimizing the entropy of the average range profile. Geoscience andRemote Sensing Letters, IEEE 6(2), 204–8 (2009)

39. TJ Kragh, Monotonic iterative algorithm for minimum-entropy autofocus, inAdaptive Sensor Array Processing (ASAP) Workshop],(June 2006) (2006)

40. Q Zhang, T-S Yeo, G Du, S Zhang, Estimation of three-dimensional motionparameters in interferometric ISAR imaging. Geoscience and RemoteSensing, IEEE Transactions on 42(2), 292–300 (2004)

41. F Berizzi, G Corsini, Autofocusing of inverse synthetic aperture radar imagesusing contrast optimization. Aerospace and Electronic Systems, IEEETransactions on 32(3), 1185–91 (1996)

42. MAT Figueiredo, Adaptive sparseness for supervised learning. PatternAnalysis and Machine Intelligence, IEEE Transactions on 25(9), 1150–9 (2003)

43. E Vera, L Mancera, S Babacan, R Molina, A Katsaggelos, Bayesiancompressive sensing of wavelet coefficients using multiscale laplacianpriors, in Statistical Signal Processing, 2009. SSP’09 (IEEE/SP 15th Workshopon, 2009), pp. 229–32

44. NL Pedersen, D Shutin, CN Manchón, BH Fleury, Sparse estimation usingBayesian hierarchical prior modeling for real and complex models ( , 2011).arXiv preprint arXiv:1108.4324

45. DF Andrews, CL Mallows, Scale mixtures of normal distributions. J R Stat SocSer B Methodol 36(no. 1), 99–102 (1974). Available: http://www.jstor.org/stable/2984774

46. S Yu, AS Khwaja, J Ma, Compressed sensing of complex-valued data. SignalProcess 92(no. 2), 357–62 (2012). http://www.sciencedirect.com/science/article/pii/S0165168411002568

Zhang et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:80 Page 18 of 19

47. S Chen, D Donoho, M Saunders, Atomic decomposition by basis pursuit.SIAM J Sci Comput 20(no. 1), 33–61 (1998). http://epubs.siam.org/doi/abs/10.1137/S1064827596304010

48. C Zhu, Stable recovery of sparse signals via regularized minimization.Information Theory, IEEE Transactions on 54(7), 3364–7 (2008)

49. E van den Berg, MP Friedlander, In pursuit of a root (2007). preprint50. EJ Candes, MB Wakin, SP Boyd, Enhancing sparsity by reweighted l 1

minimization. J Fourier Anal Appl 14(5–6), 877–905 (2008)51. M Richards, Fundamentals of radar signal processing, ser. Professional

Engineering (Mcgraw-hill, 2005). Available: http://books.google.com/books?id=VOvsJ7G1oDEC

52. G Davidson, Radar toolbox (2003). cited August 2013. [Online]. Available:http://www.radarworks.com/software.htm

53. P-R Wu, A criterion for radar resolution enhancement with burg algorithm.Aerospace and Electronic Systems, IEEE Transactions on 31(3), 897–915(1995)

54. JF Sturm, Using SeDuMi 1.02, a Matlab toolbox for optimization oversymmetric cones. Optimization methods and software 11(1–4), 625–53(1999)

55. VCSDL Donoho, I Driori, Y Tsaig, Sparselab (2007). Available: http://sparselab.stanford.edu/

56. SBM Grant, Y Ye, cvx: Matlab software for disciplined convex programming ( ,,2013). Available: http://www.stanford.edu/~boyd/cvx/

57. Y Zhang, Yall1 ( 2011). Available: http://www.caam.rice.edu/~optimization/L1/YALL1/

58. CR Vogel, ME Oman, Fast, robust total variation-based reconstruction ofnoisy, blurred images. Image Processing, IEEE Transactions on 7(6), 813–24(1998)

59. O Axelsson, Iterative solution methods (Cambridge University Press, 1996)60. SP Boyd, L Vandenberghe, Convex optimization (Cambridge Univ Press,

2004)61. L Zhang, M Xing, C-W Qiu, J Li, J Sheng, Y Li, Z Bao, Resolution

enhancement for inversed synthetic aperture radar imaging under low SNRvia improved compressive sensing. Geoscience and Remote Sensing, IEEETransactions on 48(10), 3824–38 (2010)

62. M Cetin, WC Karl, D Castanon, Feature enhancement and ATR performanceusing nonquadratic optimization-based SAR imaging. Aerospace andElectronic Systems, IEEE Transactions on 39(4), 1375–95 (2003)

63. MN Saidi, MN Saidi, A Toumi, B Hoeltzener, A Khenchaf, D Aboutajdine, ISARdata dynamics: target shapes features extraction for the design of ISARretrieval system, in Synthetic Aperture Radar (EUSAR) (European Conferenceon,, 2008), pp. 1–4

64. G Yu, G Sapiro, Statistical compressed sensing of gaussian mixture models.Signal Processing, IEEE Transactions on 59(12), 5842–58 (2011)

65. J Lopez, Z Qiao, Array geometries, signal type, and sampling conditions forthe application of compressed sensing in MIMO radar, in SPIE Defense,Security, and Sensing (International Society for Optics and Photonics,, 2013),p. 871. 702–871702

66. W Rao, G Li, X Wang, X-G Xia, Adaptive sparse recovery by parametricweighted l1 minimization for ISAR imaging of uniformly rotating targets.IEEE J Selected Topics in Applied Earth Observations and Remote Sensing6(2), 942–52 (2012)

67. L Zhang, J Duan, Z Qiao, M Xing, Z Bao, Phase adjustment and isar imagingof maneuvering targets with sparse apertures. Aerospace and ElectronicSystems, IEEE Transactions on 50(3), 1955–73 (2014)

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