J Indian Soc Remote Sens DOI 10.1007/s12524-017-0709-3 RESEARCH ARTICLE 2D Superresolution ISAR Imaging via Temporally Correlated Multiple Sparse Bayesian Learning Xiaowei Hu1 • Ningning Tong1 • Xingyu He1 • Yuchen Wang1 Received: 7 April 2016 / Accepted: 1 September 2017 Ó Indian Society of Remote Sensing 2017 Abstract In inverse synthetic aperture radar (ISAR) imaging, the image resolution is always limited by the bandwidth and the observation time. Sparse recovery (SR) is recently proposed to improve the range resolution or cross-range resolution effectively. However, for the two dimensional superresolution case, a SR-induced range cell migration (RCM) occurs among the High-Resolution Range Profiles (HRRPs) and definitely degrades the ISAR image. After that translational motion compensation is completed, the common sparsity of HRRPs is exploited to suppress the RCM in this paper. Furthermore, by taking the temporal correlation of HRRPs into account, an ISAR imaging method based on temporally correlated Multiple Sparse Bayesian Learning is proposed to improve the imaging quality. Simulated data and real data results demonstrate the effectiveness of the proposed method. Keywords Inverse synthetic aperture radar 2D superresolution Sparse recovery Temporally correlated Multiple Sparse Bayesian Learning Introduction In inverse synthetic aperture radar (ISAR) imaging, the range-Doppler image is usually obtained by using two-dimensional (2D) fast Fourier transform (FFT). However, the 2D resolution is always limited by the bandwidth and the rotation angle during the observation time. Considering the sparsity of target image, sparse recovery (SR) method is & Xiaowei Hu huxiaowei1987625@163.com 1 Air Force Engineering University, Xi’an 710051, China recently proposed to improve the range resolution (Wang et al. 2011; Zhu et al. 2011; Zhang et al. 2011) or crossrange resolution (Xu et al. 2011; Liu et al. 2014). And some fine SR methods, such as the Sparse Bayesian Learning (SBL) (Wipf and Rao 2004), prove to be capable of obtaining satisfactory resolution with limited bandwidth and short observing time. However, a problem will emerge when SR is used for the 2D superresolution. In the highresolution range profile (HRRP) synthesis using SR, a problem called range cell migration (RCM) (Gao et al. 2015) will arise in the HRRPs, especially for the case of complex targets imaging. This irregular RCM will seriously influence cross-range profile synthesis using SR and obviously deteriorate the ISAR image finally. The RCM, mainly caused by the problem of basis mismatch (Chi et al. 2011) in SR, cannot be addressed by the conventional compensation methods which are aiming at the migration through resolution cells (MTRC). Gao et al. (2015) propose a skillful SR-based imaging framework to avoid the RCM effectively, however, it may produce several false scatters and will degrade the final image in some degree. Assuming that the translational motion has been already compensated, the signal of targets should retain the common profile in the HRRPs domain. It means that the position of nonzero components will not change, i.e., HRRPs hold the common sparsity (Cotter et al. 2005). In this paper, a refined SBL method considering the common sparsity, is used to restrain the RCM. It is called the temporally correlated Multiple Sparse Bayesian Learning (TMSBL) (Zhang et al. 2011). By exploiting the common sparsity, the irregular RCM is expected to be aligned in the HRRP synthesis. Furthermore, components in each nonzero row of HRRPs are usually temporally correlated. However, most of existing SR algorithms (Cotter et al. 2005; Zhang et al. 2011) all ignore the temporal correlation 123 J Indian Soc Remote Sens and assume that components are independent and identically distributed (i.i.d.). All of them result in the degraded HHRPs. TMSBL models the temporal correlation using a block sparse Bayesian learning framework, and can obtain a better recovery performance than other algorithms in the case of temporal correlation. Therefore, a improved ISAR image can be achieved by using TMSBL. the noise vector and the expected HRRP in the nth observing angle, respectively. Then Eq. (1) can be rewritten in a matrix form S ¼ s0 ; . . .; sn ; . . .; sðN1Þ ¼ U h0 ; . . .; hn ; . . .; hðN1Þ þ e0 ; . . .; en ; . . .; eðN1Þ ¼ UH þ E ð2Þ ISAR Signal Model In ISAR echo model, the radar target is usually considered in the far-field and is composed of finite scattering centers with different scattering coefficients. Thus, the echo signal from a radar target can be seen as the sum of complex scattered signals from each scattering center. In this paper, we consider that the echo is collected in a short interval and the small angle approximation holds. Then the echo signal of a target with Q scatterers can be expressed as (Wang et al. 2015) sðm; nÞ ¼ Q X rq exp j4p=c fmx xq þ fny yq þ eðm; nÞ: q¼1 ð1Þ where m ¼ 0; 1; . . .; M 1, n ¼ 0; 1; . . .; N 1. rq denotes the scattering coefficient of the qth scatterer with a coordinate of ðxq ; yq Þ. c is the propagation speed of the wave. fmx ﬃ fm ¼ m df , fny ﬃ fc hn ¼ fc ndh, are the sampling results in the frequency domain and the angle domain. df , dh are the frequency step and angle step, respectively. fc denotes the center frequency. eðm; nÞ denotes the additional white Gaussian noise with zero mean. By applying the 2D FFT to the above signal, an ISAR image of the target can be obtained. However, due to the narrow bandwidth and the short observing interval, the resolution of the image is usually not satisfactory in both the range direction and cross-range direction. where S 2 CMN , E 2 CMN , H 2 CMN denote the signal matrix, noise matrix, and the HRRPs matrix, respectively. U 2 CMM is a partial Fourier matrix, which is defined as 2 3 1 1 1 Þ 11 1ðM1 61 7 X X 7 .. .. .. U¼6 4 ... 5 . . . XðM1Þ1 XðM1ÞðM1Þ where X ¼ exp j 2p . For the superresolution case, M M should be bigger than M. Now the common sparsity of HRRPs is modelled by row of H a Gaussian prior assigning to the mth 1 ¼ 0; 1; . . .; M pðHm m ð3Þ ; cm ; RÞ N ð0; cm RÞ; 1 where cm is an unknown variance parameter controlling the row sparsity in H, and R is a positive definite matrix that captures the temporal correlation of H. Then Eq. (2) can be re-expressed in vector form h þ e s ¼ U ð4Þ where s ¼ vecðST Þ, h ¼ vecðHT Þ, e ¼ vecðET Þ, ¼ U I N , vecðÞ denotes the vectoring operation, and U denotes the Kronecker product. Suppose that elements in the noise vector e are independent and each is Gaussian with noise variance k. Then the Gaussian likelihood of (4) is given p sh; k N ðUh; kI Þ ð5Þ The prior of common sparsity can be vectorized as cm ; R; 8m N ð0; C RÞ p h; ð6Þ TMSBL-Based ISAR Imaging In this paper, a new ISAR imaging method based on TMSBL is proposed to improve the 2D ISAR resolution. This method consists of two steps: the HHRP synthesis via TMSBL and ISAR imaging via SBL, which will be discussed in the following. HHRP Synthesis Let sn ¼ ½sð0; nÞ; . . .; sðM 1; nÞTM1 , en ¼ ½eð0; nÞ; . . .; eðM 1; nÞTM1 , and hn 2 CM1 denote the signal vector, 123 where C ¼ diagðc0 ; c1 ; . . .; cM1 Þ. By combining the likelihood and the prior above, we can get the posterior density of h which is also Gaussian N lh; Rh p hjs; k; cm ; R; 8m ð7Þ where T s k lh ¼ RhU h i1 TU k Rh ¼ ðC RÞ1 þU ð8Þ ð9Þ J Indian Soc Remote Sens With a reasonable approximation 1 1 1 T T ðC RÞU kI þ UCU R (Zhang kI þ U and Rao 2011), Eqs. (8) and (9) can be simplified as 1 ð10Þ lH ¼ CUT kI þ UCUT S 1 ð11Þ RH ¼ C1 þ UT U k Hyperparameters k; C; R can be estimated with the Type-II maximum likelihood which marginalizes over the weights and then performs the maximum-likelihood estimation. The estimated results are shown below 1 T 1 cm ¼ H R Hm ð12Þ þ ðRH Þmm ; 8m N m h i T T 1 kS UHk2F kTr UCU kI þ UCU ð13Þ þ k¼ M MN ~ R R¼ ; R ~ where ~¼ R F M1 X HTm cm Hm ð14Þ Experimental Section Simulated Data Experiments m¼0 where TrðÞ denotes the trace of a matrix. An iterative procedure is produced by the learning rules Eqs. (10), (11), (12), (13) and (14), with which all hyperparameters can be estimated, and the maximum a posterior (MAP) estimate of H can be obtained, too. Thus, the HRRPs without RCM can be synthesized in this step. ISAR Imaging The next step aims to achieve the superresolution in the cross-range direction and get the high resolution ISAR image finally. This process can be formulated as an optimization problem x^m ¼ arg min khm Wxm k2 þskxm k1 ð15Þ x^m CN where kk1 denotes the l1 norm of a vector. hm ¼ HTm th range cell of the denotes the signal vector in the m HRRPs. X 2 CMN denotes the high resolution ISAR th range cell. image, and xm ¼ X Tm is the vector in the m NN s is is the partial Fourier matrix, where N\N. W2C the regularized factor. Equation (15) can be solved by Fig. 1 Flow charts of different methods. a SBL-based method; b TMSBL-based method using the SBL algorithm. One can find the details of SBL’s application in superresolution in the paper of Liu et al. (2014). After applying SBL in each range cell, the high resolution ISAR image can be achieved finally. The main difference between the proposed method and the conventional SR method is focused on the range compression, in which TMSBL is adopted for the proposed method, while SBL is adopted for the conventional one. In conclusion, the process flow of the proposed method is depicted in Fig. 1b. And the conventional SR method (using SBL algorithm for both range and cross-range compression) is given in Fig. 1a for comparison. In the following, the above two methods are called the SBL and TMSBL-based method, respectively. A simulated data of Boeing 727, which is downloaded from an online source (http://airborne.nrl.navy.mil/vchen/tftsa. html), is used in this simulation. Radar parameters of this data are shown in Table 1. Motion compensation is completed originally. In the experiment, a partial successive data with the size of 64 64 is utilized to test the 2D superresolution performance of the proposed method. The expected 2D superresolution image is with the size of 128 128. Some existing methods of 2D IFFT, SBL and FSI in the paper of Gao et al. (2015) are used for comparison. The results (HRRPs and ISAR image) under the signal-to-noise rate (SNR) 30 dB are shown in Fig. 2. We can see from Fig. 2a, b that, the conventional IFFT method results in blurred dominant scatters, duo to the limited bandwidth and coherent time. Figure 2c, d show the results Table 1 Parameters of online data Carrier frequency (GHz) Waveform bandwidth (MHz) Radar PRF (KHz) No. of bursts Data length 9 150 20 64 64 9 256 HRRPs Radar echo Motion compensation SBL SBL ISAR image SBL ISAR image a HRRPs Radar echo Motion compensation TMSBL b 123 J Indian Soc Remote Sens 120 20 100 Range cell 40 Range cell Fig. 2 Simulated data results. a HRRPs via IFFT; b ISAR image via IFFT; c HRRPs via SBL; d ISAR image via SBL; e HRRPs via FSI; f ISAR image via FSI; g HRRPs via TMSBL; h ISAR image via TMSBL 60 80 80 60 40 100 20 120 10 20 30 40 50 60 20 40 Pulse-train index 60 80 100 120 100 120 100 120 100 120 Cross-range cell a b 120 20 100 Range cell Range cell 40 60 80 80 60 40 100 20 120 10 20 30 40 50 60 20 Pulse-train index 40 60 80 Cross-range cell c d 120 20 Range cell Range cell 100 40 60 80 80 60 40 100 20 120 20 40 60 80 100 20 120 40 60 80 Cross-range cell Pulse-train index e f 120 20 Range cell Range cell 100 40 60 80 80 60 40 100 20 120 10 20 30 40 Pulse-train index g 123 50 60 20 40 60 80 Cross-range cell h J Indian Soc Remote Sens ðl;nÞ2RT 35 IFFT 30 SBL 25 TMSBL FSI TBR using SBL method. Obvious RCM arises in the HRRPs and many false scatters exist in the final image. Results using FSI and the proposed method are shown in Fig. 2e–h, respectively. We can see in Fig. 2e, g that, RCMs can be effectively restrained using both FSI and TMSBL. What is more, the proposed method produces less sidelobes than FSI, which is shown in Fig. 2f, h. It preliminarily demonstrates the superiority of the proposed method. Furthermore, the above methods are tested under different noise level. To measure the performance quantificationally, the target-to-background ratio (TBR) is defined by 0 1 , X X 2 2 TBR ¼ 10 log 10@ jXðl; nÞj jXðl; nÞj A 20 15 10 5 0 0 where X denotes the recovered ISAR image, RT and RB are the predetermined target and background regions, respectively. TBR measures the target energy preservation and can reflect the quality of the recovered ISAR image. The higher quality the image is, the higher TBR is. In order to get the target and background regions, we apply 128 128 points 2D IFFT to the full data instead of the partial data, and a high-quality ISAR image can be obtained. 2D median filtering is applied to the ISAR image and four times the mean value of all pixels is set as the threshold to separate the target region and the background region. Points located within the target region RT are regarded as true signal, and the rest are noise. After counting the target energy (within the target region) and noise energy (within the background region) of the reconstructed image, the target-to-background ratio is obtained. TBRs of different methods with SNR = 0:5:40 dB are shown in Fig. 3. We can see that the proposed method holds the highest TBR under various SNRs among all the methods. 20 30 40 SNR(dB) ðl;nÞ2RB ð16Þ 10 Fig. 3 Experimental results with different SNRs and 128 complex samples are used. 2D IFFT, SBL, FSI and TMSBL methods are applied to obtain a superresolution image with the size of 256 128. The experimental results are shown in Fig. 4. Because of the shortage of data length, only a low resolution ISAR image is got via the IFFT method. The SBL method can improve the resolution but with a RCM problem. The FSI method is able to alleviate the RCM in HRRPs, however, it produces some false scatters in the ISAR image. The proposed method results in a well-focused ISAR image while alleviating the RCM in HRRPs. All these prove the better performance of our method. In Table 2, TBR of ISAR images via different methods are given to compare the image quality quantitatively. The full-data ISAR image is obtained by applying 256 128 points 2D IFFT to the full real data of YaK-42 plane Obviously, the image via TMSBL holds the highest TBR value, which suggests the superiority of our proposed method over other existing methods in real applications. Real Data Experiments In this part, a set of real data of the Yak-42 plane is used to test the performance of the proposed method. The signal bandwidth is 400 MHz with carrier frequency 10 GHz, which is corresponding to a range resolution of 0.375 m. The pulse repetition frequency is 100 Hz. 128 successive pulses are saved and 256 complex samples are collected for each pulse. The translational motion has been compensated with existing method for this data set. In the superresolution experiment, a limited data with 64 successive pulses Conclusion In this paper, a TMSBL-based method is proposed to enhance the 2D resolution of ISAR image. After motion compensation, accurate HRRPs can be obtained by exploiting the common sparsity and the temporal correlation. An improved ISAR image is generated by aligning the SR-induced RCM successfully. Experiment results verify the effectiveness of the proposed method. 123 J Indian Soc Remote Sens 250 50 200 Range cell Range cell Fig. 4 Real data results. a HRRPs via IFFT; b ISAR image via IFFT; c HRRPs via SBL; d ISAR image via SBL; e HRRPs via FSI; f ISAR image via FSI; g HRRPs via TMSBL; h ISAR image via TMSBL 100 150 200 150 100 50 250 10 20 30 40 50 60 20 Pulse-train index 40 60 80 100 120 100 120 100 120 100 120 Cross-range cell a b 250 200 Range cell Range cell 50 100 150 200 150 100 50 250 10 20 30 40 50 60 20 Pulse-train index 40 60 80 Cross-range cell c d 250 200 Range cell Range cell 50 100 150 200 150 100 50 250 20 40 60 80 100 120 20 Pulse-train index 40 60 80 Cross-range cell e f 250 200 Range cell Range cell 50 100 150 200 150 100 50 250 10 20 30 40 Pulse-train index g 123 50 60 20 40 60 80 Cross-range cell h J Indian Soc Remote Sens Table 2 ISAR image quality Method IFFT SBL FSI TMSBL TBR 11.6555 10.9478 13.3470 20.4262 Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 61701526, 61372166, 61571459). 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