1. Introduction
The synthetic aperture radar (SAR) is an important remote sensing technology capable of all-weather and all-time imaging, widely utilized in fields including natural environment monitoring, agricultural mapping, and change detection [
1,
2,
3,
4,
5]. High-resolution wide-swath (HRWS) imaging has emerged as a critical developmental direction for spaceborne SAR systems [
6]. However, traditional SAR systems are hampered by constraints such as the minimum antenna area [
7,
8] and noise-equivalent sigma zero (NESZ), among other parameters, preventing the achievement of HRWS imaging. To overcome these limitations, researchers globally have embarked on extensive research, leading to the proposition of innovative systems. These include the azimuth multichannel SAR system [
9,
10,
11], multiple-input multiple-output (MIMO) SAR technology [
12,
13,
14], and digital beamforming (DBF) technology [
15,
16].
In spaceborne SAR systems, to protect the receiving subsystem from being damaged by high-power transmitted pulses, signals cannot be transmitted and received simultaneously, leading to swath-width blind areas. In traditional SAR systems, these blind areas are fixedly located along the azimuth direction due to the constant pulse-repetition frequency (PRF). Staggered SAR, an innovative imaging system, introduces variable PRF to mitigate the issue of consistently positioned blind areas [
17,
18,
19,
20,
21]. By periodically altering the radar pulse repetition interval (PRI), staggered SAR disperses these blind areas across various slant range gates throughout the swath. Consequently, in staggered SAR, blind areas are not merely parallel strips along the track axis, but are evenly distributed throughout the swath in a pattern dictated by the sequence of PRIs [
22]. This strategic dispersion allows the staggered SAR to significantly widen the swath, achieving HRWS imaging—a capability exemplified by its application in the Tandem-L system [
23,
24,
25].
The unique variable PRF mode inherent in staggered SAR systems introduces two principal challenges: nonuniform sampling and echo data loss, around which existing signal-processing methodologies are centered [
26]. The German Aerospace Center (DLR) has developed an innovative interpolation method utilizing best linear unbiased (BLU) estimation techniques [
22]. This method initially employs BLU interpolation to convert nonuniform echo data into a uniformly sampled dataset. Following this resampling, traditional SAR frequency domain imaging algorithms are applied to derive the final imagery. Nevertheless, it is important to note that the fidelity of BLU interpolation diminishes with a reduction in the oversampling rate, resulting in this algorithm only being applicable to highly oversampled staggered SAR. Although high oversampling rates play a critical role in minimizing azimuth ambiguity within staggered SAR frameworks, it is not desirable for spaceborne SAR systems due to expansive data volumes.
Recent studies have increasingly focused on low-oversampled staggered SAR, which presents a promising approach to substantially diminish the volume of echo data requiring storage [
27,
28]. A notable advancement in this domain is the introduction of a two-step algorithm designed for handling low-oversampled staggered SAR data, as detailed in [
29]. This process begins with the application of the missing-data iterative adaptive algorithm (MIAA) [
30], grounded in spectrum estimation, to infer the complete echo spectrum and recuperate absent echo data. Subsequently, a multichannel reconstruction algorithm is employed to reconstruct the echo signal. Despite its innovative approach, this algorithm encounters two significant obstacles: the high computational demand associated with the iterative recovery of missing data and the diminished accuracy of spectral estimation methods in reconstructing distributed targets. In an effort to refine this approach, [
31] introduced an enhancement to the aforementioned algorithm, integrating it with the BLU interpolation technique. This enhanced method incorporates a preliminary step to determine if the scene of interest encompasses distributed targets. Depending on the outcome, the process either proceeds with BLU interpolation for distributed scenes or applies the original two-step algorithm otherwise. While the strategies proposed in [
29] and [
31] offer viable solutions for low-oversampled staggered SAR, they both hinge on the precise restoration of missing echo data to effectively mitigate azimuth ambiguities.
In the past decade, sparse signal processing algorithms based on compressive sensing (CS) theory [
32,
33] have been successfully introduced to SAR signal processing and markedly enhanced SAR imaging quality [
34,
35]. Studies indicate that, under the condition where the observation matrix adheres to the restricted isometric property (RIP) [
36], it is feasible to recover the original sparse signal from a smaller set of samples than what is mandated by the traditional Shannon–Nyquist sampling theorem [
37,
38]. The pioneering work of Çetin et al. in 2001 [
39] introduced the concept of applying regularization theory to SAR imaging, aiming at the enhancement of image features. Following this, Patel et al. in 2010 [
40] formulated a more comprehensive CS-based SAR model, facilitating scene reconstruction by tackling the
L1 regularization problem. This model has since been widely adopted in CS-based SAR imaging [
41]. Subsequently, Zhang et al. [
42] and Çetin et al. [
43] furthered this line of inquiry by integrating sparse signal processing with SAR imaging, coining the concept of “sparse SAR imaging”. This approach leverages solving the
(
1) regularization problem to recover scenes of interest. Compared with matched filtering (MF), sparse SAR imaging exhibits superior capabilities in noise and sidelobe suppression, as well as in mitigating azimuth ambiguities attributed to under-sampling. The introduction of the approximate observation concept based on azimuth-range decoupling [
44] has enabled the application of sparse SAR imaging to large-scale scene reconstructions. This technique has found applications across various constant PRF modes, including strip-map [
45], sliding spotlight [
46], terrain observation by progressive scans (TOPS) [
47], and variable PRF modes [
48]. Nevertheless, few studies have applied sparse signal processing to the staggered SAR system. In addition, SAR imaging methods that utilize the internal information of images have been focused on research. A novel SAR imaging strategy was proposed in [
49]. This method segregates the SAR image into sparse and low-rank matrices, reflecting the image’s redundancy characteristics, thereby framing the SAR imaging process as a joint sparse and low-rank matrix recovery problem. In [
50], a new structural sparse representation-based SAR imaging approach is proposed to effectively depict SAR image structures. This approach establishes an adaptive sparse space to accurately represent the varying local structures of images.
The
L1 regularization problem, when translated into an equivalent convex quadratic optimization issue, can be resolved with notable efficiency. However,
L1 regularization often introduces additional bias into estimations and affects the reconstruction accuracy [
51]. Recent studies in CS have highlighted the unique advantages of
L1/2 regularization [
52]. As a nonconvex penalty,
L1/2 regularization is lauded for its unbiasedness, capacity to enforce sparsity, and oracle properties, delivering solutions that are notably sparser than those yielded by
L1 regularization [
53]. Moreover, region-based features play a crucial role in applications such as target classification and image segmentation. To this end, the total variation (TV) norm of image magnitude has been integrated as a constraint within the SAR imaging model [
54], facilitating the maintenance of continuity in the backscattering coefficient across distributed targets within specified areas [
55,
56]. Obviously, combining
L1/2 regularization with TV regularization can significantly enhance the quality of reconstructed SAR images. In [
57], the authors proposed an SAR imaging method based on
and TV composite norm regularization. However, this method will inevitably result in huge computational and memory costs. In recent years, deep learning methodologies, particularly those involving deep convolutional neural networks (CNNs), have demonstrated exceptional prowess across various domains, including image restoration [
58] and speech signal processing [
59]. The fusion of deep learning techniques with sparse SAR imaging will emerge as a promising avenue for future research.
In this paper, we propose a sparse SAR imaging method for low-oversampled staggered mode via compound regularization. This method integrates L1/2-regularization-based sparse signal processing to mitigate azimuth ambiguities, while the incorporation of the TV regularization term boosts the reconstruction accuracy of distributed targets. Our proposed method uniquely accounts for the positioning of blind areas within the sparse SAR imaging model, constructing a blind-area index matrix that signifies echo data loss. This allows for the suppression of azimuth ambiguities without necessitating the recovery of missing data. Furthermore, acknowledging the nonuniform sampling inherent in the staggered mode, our approach employs nonuniform Fourier-transform techniques in the formation of imaging and echo simulation operators, diverging from the traditional azimuth-range decouple operators used in conventional SAR systems. Additionally, we incorporate the TV regularization term into our sparse reconstruction model, facilitating the precise reconstruction of distributed targets. The effectiveness of our method is demonstrated through both simulated data and real spaceborne SAR data experiments.
The remainder of this paper is structured as follows.
Section 2 delves into the sparse imaging and reconstruction models specific to staggered SAR.
Section 3 outlines the reconstruction process employing
L1/2&TV regularization, detailing the construction of imaging and echo simulation operators for staggered SAR and introducing our proposed method.
Section 4 is dedicated to numerical simulations and experiments with real data.
Section 5 thoroughly analyzes the experimental outcomes. Finally,
Section 6 concludes the paper with a succinct summary.
5. Discussion
In this section, we will discuss the experimental results of the previous section in detail. Firstly, the point target simulation experimental results will be discussed. As shown in
Figure 2a, the 1-D imaging result of the MF method for the point target located inside the blind areas contains multiple pairs of weak azimuth ambiguities and one pair of strong azimuth ambiguities.
Figure 2b presents the imaging result of the BLU interpolation method under the same conditions. Comparing
Figure 2a with
Figure 2b, it can be found that, under low-oversampling conditions, the results of the BLU interpolation method and the MF method were almost the same, and the azimuth ambiguities could not be suppressed. It can be seen from
Figure 3 that, when the azimuth oversampling rate increases, the ISLR values of the three imaging methods all decreased. Moreover, when the oversampling rate is greater than 1.5, that is, in the case of high oversampling, the BLU interpolation method performs better in suppressing azimuth ambiguities caused by nonuniform sampling, but it is not suitable for lower oversampling rates (from 1.1 to 1.5), as shown in
Figure 3b. It should be explained that the recovery accuracy of the BLU interpolation algorithm proposed in [
22] depends largely on the PRF of the radar system [
26]. The accuracy of the interpolation decreases when the PRF is close to the Nyquist frequency and improves as the oversampling rate increases [
27,
29]. Therefore, the imaging results in
Figure 2 and the quantitative calculation results in
Table 3 and
Figure 3 are consistent with the theoretical analysis. For the point target, the
L1/2&TV regularization actually degenerated into the
L1/2 regularization.
Figure 2c presents the imaging result of the
L1/2-regularization-based method when the point target is located inside the blind areas. Combined with the quantitative results in
Table 3, it indicates that the proposed method can effectively suppress the azimuth ambiguities caused by nonuniform sampling and echo data loss in the low-oversampled staggered mode, reducing the ISLR value by about 10 dB. In addition, the AASR values in
Table 3 also show that the proposed method had a certain suppression effect on the strong azimuth ambiguities caused by the nonideal AAP. By comparing the results of
Figure 2 vertically and combining them with
Figure 3, we can find that the values of ISLR were different when the point target was located inside the blind areas, at the boundary of the blind areas, and outside the blind areas, respectively. This is because they lose different samples within one PRI variation period. It is obvious that the ambiguity of the point target was the lowest when it was located outside the blind areas. Comparing
Figure 3c with
Figure 3a, it can be found that the oversampling rate–ISLR relationship curve of the
L1/2-regularization-based method was much lower than that of the MF method under various conditions. Comparing
Figure 3c with
Figure 3b, although the results of the
L1/2-regularization-based method and the BLU interpolation method were similar under high-oversampling conditions, the former performed significantly better than the latter under low-oversampling conditions. Therefore, the proposed sparse SAR imaging method can effectively solve the problem of azimuth ambiguities caused by nonuniform sampling and echo data loss in the low-oversampled staggered mode.
Secondly, the results of the distributed-targets simulation experiments will be discussed.
Figure 4b,c show the imaging results of the MF method and BLU interpolation method for the low-oversampled staggered mode, respectively. It can be seen from
Figure 4b that, compared with the point target, for the distributed target, the azimuth ambiguities caused by nonuniform sampling were more dispersed. Although the BLU interpolation method could maintain the scattering characteristics of the distributed target, it could not effectively suppress the azimuth ambiguities under low-oversampling conditions, resulting in almost the same imaging results as the MF method, as shown in
Figure 4c.
Figure 4d shows the imaging result of the
L1/2&TV regularization-based method proposed in this paper. It can be seen that the proposed method can effectively suppress the azimuth ambiguities caused by nonuniform sampling without recovering the missing echo data and reduce the reconstruction error of the distributed target. Furthermore, the quantitative calculation results in
Table 4 indicate that, under the low-oversampling condition (oversampling rate 1.1), the proposed method can significantly reduce the NRMSE value, while the NRMSE value of the BLU interpolation method is not significantly different from that of the MF method. Combined with the imaging results in
Figure 4, it can be concluded that the proposed method can reconstruct distributed targets more accurately. This can be explained by the TV norm characteristic [
55,
56], which can maintain the continuity of the backscattering coefficient of distributed targets. From
Figure 5, it can be seen that, in the presence of noise, the proposed
L1/2&TV-regularization-based method can obtain lower NRMSE values than the BLU interpolation method, verifying the accuracy of this method for distributed target reconstruction under low-oversampling conditions. In addition, by comparing
Figure 5a,b, it can be seen that, when the distributed target is located at the boundary of the blind areas, the NRMSE values of both methods are lower than those when the target is located inside the blind areas. This indicates that the reconstruction accuracy is related to the number of samples lost within one PRI variation period, and the higher the data loss rate, the lower the reconstruction accuracy. From
Figure 6, it can be seen that both the BLU interpolation method and the
L1/2&TV-regularization-based method had reduced NRMSE values as the oversampling rate increased, regardless of whether the distributed target was located inside or at the boundary of the blind areas. This illustrates that the reconstruction accuracy was improved, and the larger the oversampling rate, the better the effect. When the oversampling rate was greater than 1.5, that is, in the case of high oversampling, the NRMSE values of the BLU interpolation method significantly decreased, being even lower than those of the
L1/2&TV-regularization-based method. However, when the oversampling rate was less than 1.2, the reconstruction accuracy of the
L1/2&TV-regularization-based method was superior to that of the BLU interpolation method, further verifying that the proposed method could improve the reconstruction accuracy of distributed targets under low-oversampling conditions.
Figure 6 also illustrates the impact of the data loss rate on the accuracy of distributed target reconstruction, which was more evident in low-oversampling cases. The above discussion of the distributed target simulation experimental results prove that the proposed sparse SAR imaging method can improve the reconstruction accuracy of distributed targets and enhance the region-based features of SAR images in the low-oversampled staggered mode.
Thirdly, the results of real data experiments will be discussed. The effectiveness of the proposed method in suppressing azimuth ambiguities is discussed first.
Figure 7c presents the imaging result of the BLU interpolation method for the first sea–land boundary scenario. It can be seen that azimuth ambiguities caused by nonuniform sampling still existed in the image, proving its limitations under low-oversampling conditions.
Figure 7d is the imaging result of the
L1/2&TV regularization-based method. It can be seen that, due to the azimuth–range decouple operators, the azimuth ambiguities and false targets in the image were effectively suppressed.
Figure 8 further illustrates the conclusion. From the comparison results in
Figure 8, it can be seen that the BLU interpolation method could not effectively suppress azimuth ambiguities, so it was not suitable for the low-oversampled staggered mode. Compared with the BLU interpolation method, the
L1/2&TV regularization-based method could significantly improve the suppression effect of azimuth ambiguities caused by nonuniform sampling and echo data loss. Next, the effectiveness of the proposed method in improving the accuracy of the distributed target reconstruction is discussed. It can be seen from
Figure 9 that there were lots of coherent speckles in the image reconstructed by MF, and the continuity of the island areas could not be satisfied, as shown in
Figure 9a. The reconstructed image result using the proposed method was more uniform and continuous, and the coherent speckles and noise in the image were effectively suppressed, as shown in
Figure 9b. This proves that the proposed method could maintain the continuity of the backscattering coefficient of distributed targets and improve the reconstruction accuracy of distributed targets for the low-oversampled staggered mode. The numerical calculation results in
Table 6 are discussed at the end. As multi-look processing is not performed on real data and the BLU interpolation method actually utilizes matched filtering for imaging processing, the ENL values of the reconstruction result are approximately equal to 1, as shown in the first row of
Table 6. In contrast, the
L1/2&TV-regularization-based method can effectively improve the ENL values of the reconstruction result, as shown in the second row of
Table 6, further demonstrating that the proposed method can enhance the region-based features of SAR images.
Finally, the comparison of simulation and real data experimental results will be discussed. For the point target simulation, we designed two experiments, as stated above. Both point target simulation experiments verified the effectiveness of the proposed method for suppressing azimuth ambiguities in low-oversampled staggered SAR. However, it should be noted that the point target simulation experiments only considered the
L1/2 regularization term in compound regularization. Similarly, the distributed targets simulation verified the effectiveness of the proposed method for improving the reconstruction accuracy of distributed targets in low-oversampled staggered SAR. But these experiments mainly considered the TV regularization term in compound regularization. Therefore, it is necessary to select appropriate scenarios in real data experiments and verify the effects of
L1/2 regularization and TV regularization simultaneously, as shown in
Figure 7 and
Figure 10. Since the selected scenarios included both strong point targets and distributed targets, such as island areas, compound regularization can be effectively verified. Comparing
Figure 8 with
Figure 2, it can be concluded that
L1/2&TV regularization can indeed suppress the azimuth ambiguities caused by nonuniform sampling and echo data loss in the low-oversampled staggered mode. In addition, the ISLR indicators of the azimuth profiles of different imaging methods shown in
Figure 8 were quantitatively calculated. The result was that the ISLR value of
L1/2&TV regularization was 12 dB lower than that of BLU interpolation, which was consistent with the calculation results in
Table 3. Comparing
Table 6 with
Table 4, it can be found that two different evaluation indicators (NRMSE and ENL) both demonstrated that
L1/2&TV regularization had more significant advantages in distributed target reconstruction. In summary, the comparison between simulation and real data experiments fully demonstrates the effectiveness of compound regularization in both azimuth ambiguity suppression and accurate reconstruction of distributed targets.
It should be pointed out that, if we want to better suppress the azimuth ambiguities caused by nonideal AAP, the group sparsity property [
69] should be considered when constructing the imaging model, and the nonuniform sampling problem should be solved during the group sparse reconstruction process [
70]. This is also work we will carry out in the future.