Adaptive Sparse Recovery by Parametric Weighted L Minimization for ISAR Imaging of Uniformly Rotating Targets

Transcription

1 942 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 6, NO. 2, APRIL 2013 Adaptive Sparse Recovery by Parametric Weighted L Minimization for ISAR Imaging of Uniformly Rotating Targets Wei Rao, Gang Li, Member, IEEE, Xiqin Wang, and Xiang-Gen Xia, Fellow, IEEE Abstract It has been shown in the literature that, the inverse synthetic aperture radar (ISAR) echo can be seen as sparse and the ISAR imaging can be implemented by sparse recovery approaches. In this paper, we propose a new parametric weighted L minimization algorithm for ISAR imaging based on the parametric sparse representation of ISAR signals. Since the basis matrix used for sparse representation of ISAR signals is determined by the unknown rotation parameter of a moving target, we have to estimate both the ISAR image and basis matrix jointly. The proposed algorithm can adaptively refine the basis matrix to achieve the best sparse representation for the ISAR signals. Finally the high-resolution ISAR image is obtained by solving a weighted L minimization problem. Both numerical and real experiments are implemented to show the effectiveness of the proposed algorithm. Index Terms Adaptive sparse representation, ISAR imaging, parametric weighted minimization. I. INTRODUCTION T HE inverse synthetic aperture radar (ISAR) is a high resolution radar system that provides 2-D imaging results of non-cooperative moving targets. The motion of a target relative to the radar is usually divided into translational part and rotational part. After the translational motion compensation [1], [2], the target can be modeled as a rotating object. Then the Fourier-based range Doppler (RD) algorithm is usually applied to obtain the ISAR image. However, the RD method suffers from the inherent problems of Fourier transform such as low resolution and high sidelobes. Besides, to achieve considerably high cross-range resolution the observation interval must be long enough, which may make the motion compensation difficult for non-cooperative moving targets. Manuscript received April 23, 2012; revised July 15, 2012; accepted July 18, Date of publication October 15, 2012; date of current version May 13, This work was supported in part by the National Natural Science Foundation of China under Grant , by the National Basic Research Program of China (973 Program) under Grant 2010CB731901, by the Program for New Century Excellent Talents in University under Grant NCET , and by Tsinghua University Initiative Scientific Research Program. The work of X.-G. Xia was supported in part by the National Science Foundation (NSF) under Grant CCF , the Air Force Office of Scientific Research (AFOSR) under Grant FA , and the World Class University (WCU) Program, National Research Foundation, Korea. (Corresponding author: G. Li.) W. Rao, G. Li, and X. Wang are with the Department of Electronic Engineering, Tsinghua University, Beijing , China (corresponding author gangli@tsinghua.edu.cn). X.-G. Xia is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE USA ( xxia@ee.udel.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JSTARS In recent years, compressive sensing (CS) theory [3] [5] presents a novel way to deal with the resolution limitation of the conventional RD method. Some sparsity-driven methods of radar imaging have been proposed to acquire high resolution from very limited data [6] [9]. In [6], [7] the CS technique is used for imaging of static targets, which has shown that compressive sensing radars can achieve better resolution than the conventional Fourier-based methods with less data. In [8], the cross-range signal is parameterized as a summation of many sinusoids reflected from corresponding scatterers in a range cell. The CS technique is used to select the correct sinusoids and, as a result, better cross-range resolution is achieved. In [9], ahybridmatched filter/cs-technique is applied to improve the cross-range resolution. However, in [8] and [9] it is required that the ISAR signal is approximately stationary and the range resolution is the same as the Fourier-based algorithms. The quality of the resulting ISAR image based on the CS technique is greatly related to the sparse recovery algorithm. Since the original sparse recovery by norm minimization is nonconvex and intractable to solve directly, norm minimization is usually used as an approximation of norm minimization [3] [5], and some computational efficient algorithms based on matching pursuit (MP) are also proposed in [10] [12]. In [13], aweighted minimization method is proposed, which outperforms the regular minimization algorithm mainly in the following two aspects: the required number of measurements to recover a sparse signal is much less, and the reconstruction error in noisy environment is significantly reduced. The authors of [14] demonstrated with rigorous analysis that when the support of the signal can be divided into two different subclasses with unequal sparsity fractions, the weighted minimization outperforms the regular minimization substantially. In the practical application of ISAR, the weighted minimization algorithm is applied in [8] to improve the cross-range resolution in low SNR scenarios. Most of the current works on compressive sensing focus on that the basis matrix is fixedandalreadyknowninprior. However, fixed bases as in [8] and [9] are not flexible enough to represent the actual signals, such as sounds, natural images and radar echoes. For instance, the basis matrix in [8] is a fixed Fourier matrix and thus it is not suitable for nonstationary ISAR signals. An iterative algorithm [15] is proposed to retrieve the estimation of the best basis and recover the sparse signal together. However, it requires that the signal is sparse in a tree structured orthogonal basis, such as local cosine basis and wavelet transform basis. In the ISAR imaging problem, /$ IEEE

2 RAO et al.: ADAPTIVE SPARSE RECOVERY BY PARAMETRIC WEIGHTED MINIMIZATION FOR ISAR IMAGING OF UNIFORMLY ROTATING TARGETS 943 the basis matrix is related to the target motion parameter the rotation rate. Since in most ISAR applications the moving target is non-cooperative, the basis matrix is usually unknown and should be estimated too, which is different from the typical compressive sensing model in [3] [5]. A parametric sparse representation model of the ISAR signal for uniformly rotating targets has been proposed in [16] where the basis matrix depends on the unknown rotation rate of the target (Detailed discussion on the sparse representation of the ISAR signal can also be seen in Section II). In order to find the best sparse representation of the received ISAR signal, we need to know the rotation rate. In [16], the rotation rate is estimated by searching a candidate rotation rate which maximizes the resulting ISAR image contrast. Nevertheless, such a technique must repeat the image formation procedure via sparse recovery for a number of candidate rotation rates, and it is obviously time consuming. In [17], we proposed a computationally efficient parametric MP algorithm to simultaneously retrieve the rotation rate and the high-resolution ISAR image by using the orthogonal matching pursuit (OMP) [11] for sparse recovery. Since the sparse signal recovered by the weighted minimization is much more precise than OMP [12], in this paper we extend [17] and propose a parametric weighted minimization method to retrieve both the high-resolution ISAR image and the rotation rate. The joint estimation of the ISAR image and unknown rotation rate is solved by alternately estimating the ISAR image and rotation rate in an iterative manner. It will be shown that the parametric weighted minimization can adaptively modify the rotation rate from an initial value to the true value by some iterations. Compared to [17], the proposed method in this paper offers the advantage of the weighted minimization approach that a more accurate focused ISAR image can be obtained. The remainder of the paper is as follows. In Section II, the parametric sparse representation of ISAR echoes isbriefly introduced and then the sparsity-driven imaging problem is formulated. The proposed weighted minimization algorithm for iteratively estimating the rotation rate and ISAR image is given in Section III. Some experimental results are given in Section IV to verify the algorithm. Finally, some conclusions are drawn in Section V. II. PROCEDURE FOR PAPER SUBMISSION Sparse representation is an important issue in compressive sensing problems. In the ISAR applications, the basis matrix is related to the target motion parameter the rotation rate [16], [17].Inthissectionwewillfirstgive a brief review of the sparse representation of ISAR echoes. A. Parametric Sparse Representation of ISAR Signals Suppose the translational motion compensation of a maneuvering target has been done and thus only the rotational motion is considered. The target rotation rate can be considered as constant in a limited observation interval. If we uniformly divide the spatial domain into discrete spatial positions, where is the number of discretized range bins and is the number of discretized cross-range bins. Then Fig. 1. Rotation model of a uniformly rotating scatterer. according to [16], [18], the range variation of a dominant scatterer with spatial coordinate is determined at slow time as follows (as shown in Fig. 1): where refers to the constant rotation rate, refers to the initial position of the dominant scatterer, refers to the range resolution and refers to the cross-range resolution of the discrete spatial position. Suppose that there are samplesinfasttime(rangedimension) and samples in slow time (cross-range dimension), respectively. If the reflectivity of the scatterer is,since the echoes generated by the dominant scatterer are only some delayed versions of the transmitted signal and the delay times can be easily calculated according to (1), the ISAR data matrix denoted by can also be determined. For a target containing many dominant scatterers, the received ISAR signal is the summation of all signals reflected from all possible dominant scatterers in the spatial domain: where denotes the reflectivity of the scatterer with spatial coordinate. If there is no scatterer in the discrete spatial position,then. According to [16], (2) can be rewritten as: (1) (2) (3) is an matrix composed of all the vectors : where is the vectorization of the matrix : (4) (5)

3 944 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 6, NO. 2, APRIL 2013 where denotes the -th column vector of and denotes the transpose of the matrix. (with dimension ) is the vectorization of ISAR data matrix,and is a vector: The parametric sparse representation of the ISAR signal is shown as (3), where the basis matrix depends on the rotation rate that is usually unknown for non-cooperative moving targets.onecanseethat represents the reflectivities of all possible scatterers at all positions in the spatial domain, i.e., is the vectorization of the ISAR image. Since the dominant scatterers are usually sparsely distributed in the spatial domain in ISAR scenario, the number of dominant scatterers is always much smaller than the number of discrete spatial positions, i.e., only elements are dominant in (6) and the rest are close to zero. B. Problem Formulation Since ISAR image resolution is determined by the size of discrete spatial position, to achieve better ISAR image resolution, large values of and are necessary. However, when is larger than, (3) becomes underdetermined and has infinite solutions. Considering is sparse, we could find out the sparsest one among infinite solutions of (3) as the true solution. If the rotation rate of the target is already known, the ISAR imaging problem becomes a sparse recovery problem as follows [16]: where denotes norm, i.e. the number of nonzero elements in, denotes the basis matrix constructed by the rotation rate and denotes norm. is the error threshold determined by the noise energy. Detailed discussion about the parameters will be introduced in Section III-B. As shown in [13], the weighted minimization can provide an accurate approximation of norm minimization. Therefore, (7) can be approximated by: where denotes the norm and the weighted matrix is a diagonal matrix with on the diagonals as designed in [13]. In many ISAR applications the target is non-cooperative and the rotation rate of the target cannot be known in prior. In this case, the problem becomes much more complicated because joint estimation of the rotation rate and the sparse solution is required: (9). Most of the current works on compressive sensing focus on that the basis matrix is fixed and already known in prior, as shown in (7) and (8), while the basis matrix in (9) depends on an unknown parameter that should be estimated too. This is the (6) (7) (8) difference between the parametricsparserepresentationandthe typical sparse representation. The joint estimation of the unknown basis matrix and sparse signal greatly increases the difficulty of sparse recovery. A possible method is searching the correct rotation rate which maximizes the corresponding ISAR image contrast in a wide feasible region as in [16]. Obviously, such a technique is time-consuming because we have to repeat the sparse recovery procedure for all the rotation rate candidates. Another idea is that we can start from some basis matrix, and adaptively refine the basis matrix in an iterative manner. When the iteration goes to the convergence, the correct basis matrix which can best represent the received ISAR echo is achieved. This is the basic idea of the proposed algorithm in this paper. III. PROPOSED ALGORITHM In this section, a parametric weighted minimization algorithm is proposed. The proposed algorithm can reduce the complexity of solving (9) in an iterative manner. In each iteration, the method updates the ISAR imageandtherotationrateestimate alternately by sequentially minimizing the weighted norm and the recovery error. A. Algorithm Description Suppose the estimation of the rotation rate is after the th iteration, the steps of the th iteration are detailed in the rest of this section. For the current rotation rate estimation, one can construct a basis matrix according to (4). Then the weighted minimization [13] can be applied to obtain the ISAR image. (10) Several weighted matrices have been proposed in the previous works. In [8], the weight for the -th element is defined as,where is the conjugate transpose of the -th column of and is a tunable positive number. More discussions about the parameters will be introduced in the next subsection. Candes et al. proposed an iterative weighted minimizationmethodin[13],whichalternates between estimating the sparse signal and redefining the weights. The weighted matrix is initialized as an identity matrix, or equivalently, the method starts with a non-weighted minimization. The steps of the algorithm are briefly introduced here. Suppose the weight matrix is after iterations, then the sparse signal estimation is obtained by solving: (11), the weights are up- After we get the sparse signal dated as: where denotes the matrix. (12) denotes the -th element of the sparse signal and -th weight on the diagonals of the weight

4 RAO et al.: ADAPTIVE SPARSE RECOVERY BY PARAMETRIC WEIGHTED MINIMIZATION FOR ISAR IMAGING OF UNIFORMLY ROTATING TARGETS 945 The iteration terminates when attains a specific maximum number. Then the corresponding ISAR image is assigned to,whichisthefinal solution of (10). Moreover, the iteration number can be small because most of the benefit comesfromthefirst few reweighting iterations. Therefore few reweighting iterations are typically needed in practice [13]. Actually we can choose in this paper to reduce the computational cost. The iterative weighted algorithm tends to obtain better estimation of the nonzero coefficient locations. Even though some inaccurate signal estimates are generated at the beginning, the estimations (including the amplitudes and positions of the nonzero elements in the sparse signal) become more accurate after several reweighting iterations [13]. After the new ISAR image is obtained, the support set of the nonzero values in is updated as: (13) where is a zeroing threshold and is the -th element of. Obviously denotes the locations of the dominant scatterers. Denote as the elements of indexedinset and as the columns of indexedinset.since the length of observation is generally much greater than the sparsity of, we can see that the number of columns of is much less than the number of rows. All the above discussions are aimed to solve the weighted minimization in (10) to get the new ISAR image and update the corresponding support set. The next step is to refine the basis matrix. For an inaccurate estimation,theconstructed basis matrix cannot best represent the received signal. The discrepancy between and can further be reduced as follows. For more accurate rotation rate estimation, the basis signals in can represent better. Thus we can refine the rotation rate estimation by minimizing the residual error: (14) We need to solve both unknown parameters and in (14). According to the work of Golub and Pereyra [20], (14) can be solved by optimizing the parameters and separately. Thus we can solve (14) by two steps. Firstly, for a fixed,sincethe length of is less than, the optimal solution for is the least square (LS) solution calculated by: (15) where denotes conjugate transpose matrix and denotes the matrix pseudo-inverse. Secondly, substituting in (14) as (15), and the new estimation is thus updated as: (16) Due to the complicated expression of, it is hard to calculate the explicit expression of the gradient of the cost function. Here we use one dimensional linear search to find the estimation for in (16). The most complex computation in (16) is to solve an LS problem of dimension, whose computationalcostis [24]. Considering the optimization of the minimization problem in (10) is of computational complexity, where the exponent is generally no less than 2 [4], [26]. The one dimensional search problem in (16) needs much less computational load compared to the weighted minimization procedure. After the new estimation is obtained, the basis matrix is refined and a new ISAR image is obtained accordingly by solving (10). This procedure is repeated until the estimation of rotation rate converges. The practical convergence in the iteration can be determined by checking the relative change of and between the th and the th iterations. In our numerical experiments, the iteration is terminated when the relative change is less than or equal to a threshold (for example, ). Since the proposed parametric weighted minimization method can adaptively refine the rotation rate estimation, compared to the method in [16] which must traverse all the possible rotation rate candidates, the number of sparse recovery procedure is notably reduced. According to the above discussion one can see that the rotation rate searching procedure needs much less computation compared to the sparse recovery procedure. From an overall perspective, the efficiency of the proposed method is determined by the number of the sparse recovery procedures and the calculation of the rotation rate search only takes a small part. Generally the proposed method is computationally costly, especially for large probed imaging scenarios. Fortunately, the computation time of the weighted minimization (10) can be reduced by other fast algorithms such as TNIPM algorithm in [25] and GPSR algorithm in [26], with empirical complexity and respectively, where. Besides, we would like to refer the reader to [27] for a fast GPU parallel implementation to further reduce the computation time. In summary, the detailed steps of the parametric weighted minimization algorithm are given in Table I. B. Some Discussions First, the estimation of the error threshold in (10) is still an open problem in sparse recovery area [13], [21]. The performance of the weighted minimization is greatly related to. If is too high, a part of the signal component is not recovered, therefore some real scatterers are missing. Otherwise, if is too small, some noise components are treated as signal and artifacts are generated in ISAR image. In this paper, we consider that the discrepancy between and in (14) is mainly caused by the noise. Thus the error threshold is determined by the noise level. The noise energy can be determined by the method in [22], which is also applied in [8] for the same reason. Real ground-based data are used in [8] to verify the method and it is proved to be reasonable. After the noise energy is obtained, the error threshold in (10) is chosen via the method provided in [21], where the inverse of noise cumulative distribution function is used to decide how much the error threshold should be larger than the noise energy. For more complex scenarios when the discrepancy between and contains other factors, we refer the reader to [23] for error threshold.

5 946 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 6, NO. 2, APRIL 2013 TABLE I ISAR IMAGING VIA PARAMETRIC WEIGHTED MINIMIZATION recovery by solving (10). The next section will show the proposed algorithm works well with and. This implies that the resolution of the proposed algorithm is at least two times better than that of the conventional Fourier based method. IV. EXPERIMENT RESULTS Fig. 2. Rotation rate estimation versus iteration times (the Mig-25 experiment). The second problem is related to the tunable positive number as mentioned above. As indicated by Candes et al. in [13], the recovery process tends to be reasonably robust to the choice of. should be slightly smaller than the expected nonzero magnitudes of the sparse signal. An emphatically value for is introduced in [13]. Let denote a reordering of in decreasing order of magnitude. Set, where. The third problem is related to the resolution enhancement of the proposed algorithm. High range resolution and cross-range resolution can be obtained when the discretized number and are large, respectively. However, and should satisfy that [13], [28] to guarantee the stable sparse A. Verification of the Proposed Algorithm In this example, noise corrupted data with is generated by adding white Gaussian noise into Mig-25 data provided by Dr. V. C. Chen 1 to verify the proposed algorithm. The main parameters of the stepped frequency radar are as follows. The central frequency is 9 GHz. The transmitted bandwidth is 512 MHz, including stepped frequencies in a burst and totally bursts are used in this experiment. As mentioned in [19], the correct rotation rate of the target is about 8.25 degrees/second, which may be used to judge the correctness of rotation rate estimation. Suppose the spatial domain we are interested in contains 40 range cells and is discretized into. And the cross-range domain (or the Doppler domain) is discretized into. The proposed algorithm is utilized to recover the ISAR signal and estimate the rotation rate with an initial value. In this experiment, the error threshold is chosen as 1.02 times the noise energy. As shown in Fig. 2, after 5 iterations the proposed method converges to, which agrees with the estimation in [19]. Fig. 3 shows how the normalized residual error changes with 1

6 RAO et al.: ADAPTIVE SPARSE RECOVERY BY PARAMETRIC WEIGHTED MINIMIZATION FOR ISAR IMAGING OF UNIFORMLY ROTATING TARGETS 947 Fig. 3. Normalized residual error energy under different basis produced by different rotation rate. the change of rotation rate estimate in the second iteration. The support is calculated according to. If we fix the support, the next estimate which minimizes the residual error can be calculated from (16). Also we can see the next estimation is from Fig. 3. Compared to [16] which must traverse all the possible rotation rate candidates, the proposed method can adaptively find the best basis matrix by only 5 iterations. Thus the number of sparse recovery procedure is significantly reduced. The final ISAR image reconstructed by the proposed method is shown in Fig. 4(c), where a well focused image with high resolution and low sidelobes is obtained. Due to the low SNR, some artifacts appear in the final ISAR image. However, the effect of these artifacts on the resulting ISAR image can almost be ignored. For comparison, the conventional Fourier-based ISAR image is shown in Fig. 4(a), where the range resolution is m and cross-range resolution is m for each pixel. One can see that the dominant scatterers are almost covered by the noise because of the low SNR and short coherent time. Fig. 4(b) shows the image obtained by the two-dimensional Capon method [29]. One can see Fig. 4(b) is still blurred since the method is based on a two-dimensional sinusoidal signal model but the ISAR signal is not stationary in this experiment. Better resolution image is obtained by the proposed algorithm, as shown in Fig. 4(c), where the size of each pixel is improved to m and m respectively. B. Applied to Real Data The proposed algorithm has also been validated with real data of the Yak-42 airplane. This system works at C-band (5.52 GHz) and transmits chirp signals with. The bandwidth of the transmitted signal is 400 MHz with 25.6 pulse-width. In order to reduce the computation cost, down sampling is implemented by factor of 3 on the range dimension and by a factor of 4 on the cross-range dimension. The number of pulses is, and the number of range cells is.wediscretize the spatial domain by range bins and Doppler bins. The proposed algorithm is applied to retrieve both the ISAR image and the rotation rate estimate, with an initial value Fig. 4. Imaging results of Mig-25: (a) the Fourier based method, (b) the Capon method, and (c) the proposed parametric weighted minimization. and the error threshold in (10) is chosen as. After 5 iterations the rotation rate estimate converges at,asshowninfig.5.the ISAR image obtained via the weighted minimization is given in Fig. 6(c). Fig. 6(a) shows the conventional Fourier-based ISAR image, which suffers from low resolution and high sidelobes due to the short coherent time. The range resolution is m and cross-range resolution is 2.1 m respectively. The

7 948 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 6, NO. 2, APRIL 2013 Fig. 5. Rotation rate estimation versus iteration times (the Yak-42 experiment). image result of the Capon method [29] is shown in Fig. 6(b), which makes larger sidelobes and more artifacts compared to Fig. 6(c). One can see the magnitudes of all the spatial positions with strong scatterers in Fig. 6(a) and Fig. 6(b) are also large in Fig. 6(c). This shows the effectiveness of the weighted minimization method. Similarly, better resolution compared to the Fourier based method is obtained by the proposed algorithm since the size of discrete spatial position is improved to 0.45 m 1.05 m. C. Improvement Compared to Existing Methods We have mentioned that the weighted minimization can further improve the recovery of signals from noisy data. In what follows, we aim to demonstrate how the proposed parametric weighted minimization improves the performance of ISAR image compared to the method in [16] and [17]. Considering that the groundtruth on the amplitude and position of the scatterers of the Mig25 artifact and Yak-42 airplane is unknown, a simulated radar echo is applied in this experiment to demonstrate the improvement of the proposed algorithm. The radar transmits chirp signals with 300 MHz bandwidth and. Totally only 25 chirp signals are received. The carrier frequency is 5.52 GHz and the echoes are sampled with 400 MHz for pulse compression. The rotation rate of the target is set to be 10 degrees/second. The dominant scatterers of the target are shown in Fig. 7 and all the reflectivities of the dominant scatterers are 1. The real number of the dominant scatterers is 50. Noise corrupted data with are generated by adding Gaussian noise into the simulated data. In this experiment, we compared the accuracy of the proposed method in estimating the rotation rate with the image contrast maximization method in [16] with the simulated ISAR echoes. For each method, we run 60 trials to obtain the histograms of the rotation rate estimation. The error threshold in (10) is chosen as. The results of [16] and the proposed method are shown in Fig. 8(a) and Fig. 8(b) respectively. One can see most of the estimates in Fig. 8(b) are approximately 10 degrees/second. Apparently, the proposed algorithm can obtain Fig. 6. ISAR imaging results of a Yak-42 aircraft: (a) by using the Fourierbased method, (b) by the Capon method, and (c) by the proposed algorithm. more accurate rotation rate estimation than that of [16], which has larger variance. In the following part of this section, we aim to demonstrate the goodness of the method compared to the parametric MP [17]. Both the parametric MP and parametric weighted minimization method are applied to recover the final ISAR image with different data size when, 20 and 15. The results of the first trial are shown in Fig. 9. The ISAR images obtained by

8 RAO et al.: ADAPTIVE SPARSE RECOVERY BY PARAMETRIC WEIGHTED MINIMIZATION FOR ISAR IMAGING OF UNIFORMLY ROTATING TARGETS 949 Fig. 7. The positions of the dominant scatterers on the simulated airplane. Fig. 9 shows the images obtained by parametric weighted minimization when, 20 and 15, respectively. Totally 50 dominant scatterers are finally found in the former two examples and the positions of the scatterers are exactly the same as in Fig. 7. Only 2 dominant scatterers are missing when. The reconstruction errors and are utilized to evaluate the recovered ISAR image quality in Fig. 9, where presents the ISAR image obtained by the weighted parametric minimization method and presents the image obtained by the method in [17]. Apparently the reconstruction error by the proposed algorithm is much smaller. From the above discussion one can see that the proposed algorithm can provide better ISAR image compared to the parametric MP, both in the positions of the dominant scatterers and the recovery error. We repeat the above experiment across 10 trials for different generated-noise-corrupted data. Table II shows the rotation rate estimation of the two methods with different data size. The last column in Table II calculates the variance of the rotation rate estimation defined as,where is the rotation rate estimation in the i-th trial. Table III shows the number of accurate recovered dominant scatteres positions obtained by the two methods. And the last column in Table III shows the mean number of accurate positions recovered in each occasion. Since the rotation rate is updated by minimizing a residual error as in (16), one can see a more accurate support set estimation can improve the estimation accuracy of the rotation rate. Considering the weighted minimization can provide more precise sparse solution compared to the MP algorithms [13], which is also verified by the results in Table III, the proposed algorithm can provide more accurate rotation rate estimation compared to [17], especially in small amount data occasions. From another point of view, the proposed algorithm requires less data for ISAR formation compared to [17]. One can see that the proposed algorithm outperforms the parametric MP in the following aspects: more accurate rotation rate estimation, more accurate ISAR image both in the positions of the dominant scatterers and the recovery error, and less data for ISAR image formation. V. CONCLUSION Fig. 8. Histograms of the rotation rate estimation: (a) result from image contrast maximization method in [16]; (b) result from the proposed method in this paper. the parametric MP when, 20 and 15 are shown in the left part of the figure. The numbers of accurate recovered dominant scatteres positions are 44, 38 and 15, respectively. One can see some artifacts are generated in the final ISAR images, especially when the data size is small. The right column part of This paper has presented an adaptive sparse recovery algorithm named parametric weighted minimization to solve the sparse recovery problem when the basis matrix depends on an unknown model parameter. The proposed algorithm can adaptively update the model parameter estimation and find the best sparse representation for the measurement vector in the ISAR application. Both simulated data and real data are provided to illustrate the effectiveness of the algorithm and it is shown that the proposed algorithm can retrieve high-resolution ISAR image and the rotation rate simultaneously, even in low SNR scenario. Compared to some homologous algorithms [16], [17], both the rotation rate estimation accuracy and the resulting ISAR image quality are improved. The proposed algorithm tends to work well when the size of radar echoes is large and an appropriate initial value of rotation rate is given. In this paper, the values of the initial rotation rate in the experiments are generally close to 0, which is equivalent

9 950 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 6, NO. 2, APRIL 2013 Fig. 9. Imaging results of simulated data obtained via the parametric MP [17] and the parametric weighted minimization in this paper when,20and 15, respectively. TABLE II ROTATION RATE ESTIMATION RESULTS TABLE III NUMBER OF ACCURATE RECOVERED DOMINANT SCATTERERS POSITIONS The data represent the number of accurate recovered dominant scatteres positions obtained by the parametric MP and parametric weighted when, 20 and 15, respectively.

10 RAO et al.: ADAPTIVE SPARSE RECOVERY BY PARAMETRIC WEIGHTED MINIMIZATION FOR ISAR IMAGING OF UNIFORMLY ROTATING TARGETS 951 to say that we think the target almost doesn t rotate in the beginning. Even so, the rotation rate estimation can get closer to the practical value and finally converges after several iterations. Although a local convergence may occur when the radar data size is small, we have never encountered an experiment where the proposed algorithm did not converge during the experiment. However, a theoretical convergence analysis of the technique could be a useful direction of future work. REFERENCES [1] J. Wang and D. Kasilingam, Global range alignment for ISAR, IEEE Trans. Aerosp. Electron. Syst, vol. 39, no. 1, pp , Jan [2] M.Martorella,F.Berizzi,andB.Haywood, Contrastmaximization based technique for 2-D ISAR autofocusing, IEE P Radar, Son. Nav., vol. 152, no. 4, pp , Aug [3] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, vol. 52, no. 4, pp , Apr [4] E. 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Candés and Y. Plan, A probabilistic and RIPless theory of compressed sensing, IEEE Trans. Inf. Theory, vol.57,no.11,pp , [29] Z. S. Lieu, R. Wu, and J. Li, Complex ISAR imaging of maneuvering targets via the Capon estimator, IEEE Trans. Signal Process., vol. 47, no. 5, pp , May Wei Rao received the B.S. and M.S. degrees from Huazhong University of Science and Technology, Wuhan, China, in 2007 and 2009, respectively. He is now working toward the Ph. D. degree in the Department of Electronic Engineering, Tsinghua University, Beijing, China. His current research interests are in the areas of compressed sensing and ISAR imaging. Gang Li (M 08) received the B.S. and Ph.D. degrees from Tsinghua University, Beijing, China, in 2002 and 2007, respectively. In July 2007, he joined the Department of Electronic Engineering, Tsinghua University, Beijing, China, where he is currently an Associate Professor. His current interests include radar imaging and sparse signal processing. He has authored and coauthored more than 50 journal and conference papers. Xiqin Wang received the Ph.D. degree in electronic engineering from Tsinghua University, Beijing, China, in From May 2000 to August 2003, he was a Visiting Scholar and an Assistant Researcher with Partners for Advanced Transit and Highways, University of California at Berkeley. He is currently a Professor with the Department of Electronic Engineering, Tsinghua University. His research interests include radar signal processing, image processing, and electronic system design and applications. Xiang-Gen Xia (M 97 S 00 F 09) received the B.Sc. degree in mathematics from Nanjing Normal University, Nanjing, China, the M.Sc. degree in mathematics from Nankai University, Tianjin, China, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983, 1986, and 1992, respectively. He was a Senior Research Staff Member at Hughes Research Laboratories, Malibu, CA, during In September 1996, he joined the Department of Electrical and Computer Engineering, University of Delaware, Newark, where he is the Charles Black Evans Professor. His current research interests include space-time coding, MIMO and OFDM systems, digital signal processing, and SAR and ISAR imaging. He has over 230 refereed journal articles published and accepted, and seven U.S. patents awarded. He is the author of the book Modulated Coding for Intersymbol Interference Channels (Marcel Dekker, 2000). Dr. Xia received the National Science Foundation (NSF) Faculty Early Career Development (CAREER) Program Award in 1997, the Office of Naval

11 952 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 6, NO. 2, APRIL 2013 Research (ONR) Young Investigator Award in 1998, and the Outstanding Overseas Young Investigator Award from the National Nature Science Foundation of China in He also received the Outstanding Junior Faculty Award of the Engineering School of the University of Delaware in He is currently an Associate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON SIGNAL PROCESSING, Signal Processing (EURASIP), and the Journal of Communications and Networks (JCN). He was a guest editor of Space-Time Coding and Its Applications in the EURASIP Journal of Applied Signal Processing in He served as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 1996 to 2003, the IEEE TRANSACTIONS ON MOBILE COMPUTING during 2001 to 2004, IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY during2005to 2008, the IEEE SIGNAL PROCESSING LETTERS during 2003 to 2007, and the EURASIPJournalofAppliedSignalProcessingduring2001to2004.Dr.Xia served as a Member of the Signal Processing for Communications Committee from 2000 to 2005 and a Member of the Sensor Array and Multichannel (SAM) Technical Committee from 2004 to 2009 in the IEEE Signal Processing Society. Dr. Xia is Technical Program Chair of the Signal Processing Symposium, Globecom 2007 in Washington D.C. and the General Co-Chair of ICASSP 2005 in Philadelphia, PA.