Frequency-Scaling-Based Spaceborne Squint SAR Sparse Imaging

In synthetic aperture radar (SAR), due to the characteristics of squint data in spaceborne imaging geometry, a more accurate range cell migration correction operation is necessary. Furthermore, imaging degradation due to ignoring range-variant filtering needed for secondary range compression limits SAR imaging. In addition, owing to enormous computational complexity required for squint SAR data processing, the existing imaging methods, for instance, chirp scaling algorithm and range Doppler algorithm, are no longer sufficient, especially for large-scale scenes. In order to solve above problems, this article presents a novel spaceborne squint SAR sparse imaging method, which could not only eliminate the effects of squint to a certain extent, but also improve the performance of focused SAR image. Compared with the matched-filtering-based squint imaging algorithm, the proposed method can obtain the SAR images with higher quality from fully sampled or downsampled echo data.


I. INTRODUCTION
S YNTHETIC aperture radar (SAR) is an active earth observation system that is installed on various flight platforms and has all-weather and all-day surveillance capabilities [1]. Nowadays, it has been widely used in disaster monitoring, resource exploration, military reconnaissance, etc. [2]. Conventional matched filtering (MF)-based SAR imaging algorithms, e.g., range Doppler algorithm with range cell migration correction (RCMC) in the range Doppler domain [3] and the chirp scaling algorithm (CSA) without interpolation operation [4], could obtain a focused image accurately at low squint angle. However, at higher squint, the presence of squint angle will cause the shift of Doppler center, which requires accurate correction The authors are with the Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China, and also with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing  in the imaging process [5]. Simultaneously, the widening of the main lobe and the horizontal increase of sidelobes to the edge of range swath will reduce the image quality of spaceborne SAR under the squint case [6]. In order to solve the problems caused by the increase in the squint angle, several methods were proposed. In 1996, a nonlinear frequency modulation (FM) chirp scaling algorithm (NCSA), an extension of CSA, was proposed by Davidson et al. [7], which provided more accurate secondary range compression (SRC) to accommodate the moderate squint angle. As an accurate and efficient MF-based method to achieve range-variant SRC, NCSA makes high-squint SAR imaging possible. In the following decade years, research based on chirp scaling at high squint angle has been developed. In 2001, Yeo et al. [8] proposed a high-squint subaperture algorithm, which focuses SAR image at high squint angle in the stripmap mode. In 2007, Zhang and Zhai [9] introduced a new method for an accurate estimation of Doppler center based on CSA. Simultaneously, Wang et al. [10] proposed an algorithm based on the quartic phase model, which was applied to describe the rangedependent characteristics of SAR signals. In 2009, a new method was proposed by Zhong et al. [11], which was extended from NCSA for focusing the bistatic SAR data under a wide range swath and a large baseline cases. In 2015, to process the highly squinted missile-borne SAR data, Chen et al. [12] presented a novel fractional CSA. Furthermore, in 2014, Tang et al. [13] introduced an ω − K algorithm to diving missiles carrying SAR, which can handle missiles with wide swaths and high squint angles. Then, Sun et al. [14] constructed an improved ω − K algorithm for spatial high-magnification L-band SAR imaging. Meanwhile, Fan et al. [15] presented a modified range Doppler algorithm for handling high-squint SAR, which modifies the migration of ranging units in the ranging frequency domain and avoids interpolation operation. However, the above studies discussed the problem of squint imaging with fully sampled echo; how to achieve high-quality imaging from fully sampled and downsampled squint echo data still requires future research. Sparse signal processing is a theory developed since the 1990s for processing the large amounts of data. In 2001, Cetin and Karl [16] combined a sparse signal processing technique with SAR imaging for the first time. In 2006, Candes et al. [17], [18] used compressed sensing to process data that realized the reconstruction of targets from incomplete samples, thus breaking the requirements of Shannon-Nyquist sampling theory. In 2010, Patel et al. [19] proposed an imaging model based on an accurate observation matrix to achieve sparse image This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ reconstruction by solving an L 1 -norm minimization problem. It can use the sparse prior information of the observed scene to reduce the amount of data through sparse sampling, so as to improve the efficiency of data processing. Nevertheless, sparse SAR imaging methods based on the conventional observation matrix can hardly be used in practical applications because of the huge computation in the matrix construction as well as iterative recovery. In order to solve above problems, an approximated observation-based sparse SAR imaging method was introduced and achieved range and azimuth decoupled by constructing an echo simulation operator [20], [21]. It is found that this method could reduce the computational cost of sparse imaging to the same order as MF and make large-scale sparse data processing possible.
In this article, we proposed a novel frequency-scaling-based spaceborne squint SAR sparse imaging method. The proposed method first incorporates a small nonlinear FM component in the received range signals before regular chirp scaling operation. Then, it constructs an approximated observation-based imaging model via NCSA, and finally, considered scene reconstruction is achieved by solving an L 1 -norm regularization problem. Compared with MF-based algorithms, i.e., NCSA, the proposed method could reduce sidelobes and suppress clutter, show better SAR imaging performance, and even allow the large-scale sparse imaging from downsampled squint echo.
The rest of this article is organized as follows. In Section II, a brief introduction of NCSA for squint imaging is provided. In Section III, we introduce the proposed spaceborne squint SAR sparse imaging method from model construction to regularization recovery. Experimental results along with the performance analysis based on simulated and real data are shown in Section IV. Finally, Section V concludes this article.
II. SQUINT SAR IMAGING VIA NCSA CSA realizes the scale transformation of the chirp signal by FM operation, but it only considers the distance-related RCMC. Ignoring the distance dependence of SRC, CSA could be effectively performed by just changing the frequency rate of range matching filters. Nevertheless, even at moderate squint angle, this approximation will lead to a significant degradation of the image. Therefore, with the increase in the squint angle, the change of SRC is no longer negligible. While the NCSA adds a small nonlinear FM component before regular chirp scaling operation, which resolves the dependence of SRC on distance. In the following, we will introduce the implementation process of NCSA so as to further apply it to the subsequent sparse SAR imaging model. The transmitted chirp signal with carrier frequency f c and modulation rate K r can be expressed as where T r is the pulse duration and τ is the time of chirp signal, which is also the range time in the SAR system. After the echo signal received by the radar is quadrature demodulated, the baseband signal of a point target can be written as where A 0 is the complex constant coefficient, c is the speed of light, η is the azimuth time, R(η; r) is the instantaneous range, η c is the beam center offset time, and w a and w r are the azimuth and range weights of the antenna pattern, respectively. The flow diagram of NCSA is shown in Fig. 1. Before regular CSA processing, the following nonlinear FM filter is added in the 2-D frequency domain, i.e., where f η and f τ are the azimuth range frequencies, respectively, v is the equivalent velocity, R r is the reference slant range, and A m is the correction coefficient. The signal trajectory in the range Doppler domain is where τ d (f η ; r) is the scatterer trajectory of signal, τ r (f η ) is the reference trajectory, and Δτ (f η ; r) is the difference between τ d (f η ; r) and τ r (f η ).
After adding the nonlinear FM filter, the signal in the range Doppler domain can be written as (ignoring constant coefficient) where K mr is the changed range modulation rate at reference slant range. Then, a chirp scaling operation is performed in the range Doppler domain. In order to ensure that all the signals have same range migration after chirp scaling, the desired trajectory τ s (f η ; r) can be expressed as where Δτ (f ηr ; r) is the difference between τ s (f η ; r) and τ r (f η ) at azimuth reference frequency. To obtain the desired trajectory, we need to scale Δτ (f η ; r) to Δτ (f ηr ; r). Therefore, the chirp scaling operation is completed by multiplying where τ r is the reference trajectory, α is a scaling factor that is linearly related to Δτ (f η ; r) and Δτ (f ηr ; r), β represents a nonlinear scale to take into account the nonlinear relationship between Δτ (f η ; r) and Δτ (f ηr ; r) in an orbital geometry, and K s is the slope of the variation. After above phase multiplication, the differential range cell migration will be corrected. In order to remove the range dependence of SRC, Δτ (f η ; r) needs to be set to zero. Under this constraint, A m can be expressed as Then, after transforming signal to the 2-D frequency domain, the signal is , and Δτ is Δτ (f η ; r). Range compression and consistent RCMC will be implemented by multiplying .
(11) It should be noted that the reference azimuth frequency f ηr has to be out of the azimuth frequency band of the signal. This is due to the fact that the desired trajectory and the scatterer trajectory intersect at f ηr . Therefore, there is no chirp scaling effect on this range line, and hence, the desired effect of eliminating range dependence of SRC could not be obtained.
Finally, azimuth compression and residual phase compensation will be applied in the range Doppler domain by multiplying Then, the final signal of focused target in the 2-D time domain becomes where θ(η, τ ) is the phase of the target.

III. HIGH-SQUINT SAR SPARSE IMAGING
On the basis of investigating the signal with a high squint angle, we introduce a sparse imaging technique for improving imaging performance. In this section, we first present 1-D sparse imaging model. In the 2-D sparse imaging model, it is still necessary to vectorize echoes and images to be reconstructed during the computation, which leads to huge computational cost. Therefore, we use the approximated observation matrix to construct a high-squint SAR sparse imaging model and reconstruct the sparse image by solving an L 1 -norm regularization problem.

A. Observation-Matrix-Based Sparse Imaging
The sparse reconstruction algorithm could be used in 1-D SAR echo data processing. By vectorizing the 2-D backscattering coefficient X of considered scene and echo data Y, we could obtain the 1-D sparse SAR imaging model where x = vec(X), y = vec(Y) (vec(·) represents the superposition of columns), Φ is the observation matrix, which is determined by the SAR parameters and imaging geometry relationship, and n 0 is the noise vector. If the considered scene is a sparse scene and, meanwhile, Φ satisfies the restricted isometry property condition, x can be reconstructed by solvinĝ where λ is the regularization parameter.

B. Approximated Observation-Based Sparse Imaging
Similar to (14), the 2-D sparse SAR imaging model can be expressed as [22] Y = ΦX + N (16) where Φ is the observation matrix of the considered scene, and N is the noise matrix. Let L(·) denote the MF-based SAR imaging operation, such as NCSA in this article. Thus, we have [21], [22] with where • is the Hadamard product, F a and F r are the azimuth and range Fourier transforms, respectively, and F −1 a and F −1 r are the azimuth and range inverse Fourier transforms, respectively. Let the binary matrices Φ r and Φ a denote the range and azimuth downsampling matrices, respectively. Thus, the 2-D sparse SAR imaging model in (16) can be rewritten as [22] with G(·) being the procedure of SAR echo data collection (the inverse process of L(·)), i.e., where (·) * is the conjugate transpose operator.

C. Iterative Recovery
For the model in (16), the scene can be reconstructed bŷ whereX is the reconstructed 2-D backscattering coefficient. Several algorithms, such as complex approximated message passing and iterative soft thresholding (IST), are available for solving the optimization problem in (21). Without loss of generality, the IST algorithm is selected for the iterative recovery in this article, whose flow diagram is depicted in Fig. 2 [21] and the detailed procedure is listed in Table I, where K represents the sparsity of observed scene.

D. Computational Cost Analysis
Assume that the size of 2-D echo data Y is N η (Azimuth) × N τ (Range) and the size of discrete surveillance scene X is N P (Azimuth) × N Q (Range). Let M = N η × N τ and N = N P × N Q ; then, the computational complexity of the CSA and NCSA imaging methods can be expressed as O(M log(M )) [21]. The computation of the proposed method includes two main parts, i.e., the calculations of an inverse and an MF procedure, whose complexity is O(M log(M )), and a thresholding operation with the complexity O(N ) for each iteration. Let I denote the number of required iterative steps for the accurate reconstruction of the proposed method; thus, its total computational complexity is in the order of O(I(M log(M ) + N )) [21]. Since the proposed algorithm usually converges within ten steps, it means that it has similar computational complexity to traditional MF, which makes the sparse reconstruction of the large-scale scene possible.

IV. EXPERIMENTS AND PERFORMANCE ANALYSIS
In this section, several experiments based on echo data of point targets and real scenes are used to validate the proposed method. Without loss of generality, we set targets in three different positions (see in Fig. 3), and the squint angle θ is set as 5 • and 10 • , respectively. T2 is at the scene center; T3 and T1 are 5 km away from T2 in the range and azimuth directions, respectively. The rest of the experimental parameters are listed in Table II. Since the imaging results of three point targets are similar, we only list the recovered image of them from fully sampled echo data with θ = 10 • . The rest of the cases only take the imaging results corresponding to T2 as an example. Fig. 4 shows the imaging results of point targets from fully sampled data by traditional CSA, NCSA, and the proposed sparse imaging method with θ = 5 • and θ = 10 • , respectively. It can be seen that due to the existence of squint angle, CSA can no longer recover the point target with significant energy dispersion. However, both the NCSA and the proposed method can obtain a well-focused image of the target even at the scene edges. In the simulation based on full-sampled echo with θ = 10 • , we use the peak-to-sidelobe ratio (PSLR) and the integrated sidelobe ratio (ISLR) to compare the sidelobe suppression ability of different    Tables III and IV, respectively. Since the results of the CSA have been severely defocused, we only calculate the PLSR and the ISLR of the NCSA and the proposed method for comparison. It shows that compared to NCSA-based results, the images reconstructed by the proposed method show better quality with less sidelobes. To further support our viewpoint, we perform 25% random sampling for the collected fully sampled echo of a point target with θ = 5 • and θ = 10 • . Fig. 5 shows the imaging results of different methods. From Fig. 5, it is found that due to the lack of data and existence  of high squint, the CSA has been unable to recover the target. The other MF-based algorithm, i.e., NCSA, could also not recover the target anymore with severe energy dispersion in the azimuth direction. However, the proposed method reconstructs the point target with better performance even under the high-squint case.
In order to quantitatively evaluate the noise and clutter suppression capability of the proposed method, we introduce the  target-to-background ratio (TBR) to validate the performance of different methods [23], which is defined as where I represents the target area, B represents the background region near I, and N B is the number of pixels in B. Table V shows the TBR values of the images recovered by different methods with θ = 5 • and θ = 10 • , respectively. It can be seen that whether using fully sampled or downsampled data, TBR values in the recovered images of the proposed method averaged over 65 dB, significantly outperforming CSA and NCSA. Furthermore, we introduce the proposed method to process the echo of real large-scale scenes. The sizes of the scenes in  Table VI. Fig. 6 shows the reconstructed image of the sparse salt pan area by different methods with θ = 5 • . Fig. 7 depicts the imaging results of the coastal scene by different methods at θ = 10 • . From Figs. 6 and 7, it is clearly seen that the reconstructed images by the NCSA with 25% samples have obvious azimuth energy dispersion, thus causing failed recovery. However, similar to above simulated results, the proposed method could also achieve the high-squint sparse imaging of complicated scenes even from 25% downsampled data. In addition, it also has better image performance with less noise and clutter.

V. CONCLUSION
In this article, a novel spaceborne squint SAR sparse imaging method was proposed and applied to real data processing. With the help of approximated observation, the proposed method made the sparse reconstruction of large-scale scenes possible. Furthermore, experimental results showed that the proposed method can eliminate the degradation of image quality caused by the squint angle and effectively reduce the sidelobes, noise, and clutter compared with MF-based algorithms. In addition, it also shows the high-quality imaging ability of the sparse surveillance area from downsampled data.