Wideband interference mitigation for synthetic aperture radar based on the variational Bayesian method
Introduction
Synthetic aperture radar (SAR) has full-time, all-weather imaging characteristics with long-range, high resolution capabilities that make it suitable for a wide range of applications in the fields of resource exploration, marine observation, geological mapping, and environmental perception [1], [2], [3], [4], [5]. However, the large-bandwidth transmission signal employed by SAR to achieve high-resolution imaging makes the resulting images susceptible to radio frequency interference (RFI), which degrades the imaging quality and hinders the subsequent interpretation of SAR images by generating inaccuracies in the estimation of critical Doppler features, such as the centroid and modulation rate. Strong RFI significantly reduces the signal-to-noise ratio of the echo signal and can even induce saturation of the SAR receiver [1,6]. Therefore, substantial efforts have been devoted toward mitigating the effects of RFI on SAR imaging.
Generally, RFI can be divided into narrowband interference (NBI) and wideband interference (WBI) according to the ratio of the bandwidth of the interference to that of the target echo signal. Here, WBI represents a bandwidth ratio that is greater than 1%, while NBI is the opposite, and typically concentrates within a limited number of frequency bins [7]. The difference between these two types of interference is illustrated in Fig. 1(a) and (b), where NBI appears as a horizontal bright line in the range spectrum image, while WBI appears as some bright stripes. Meanwhile, NBI mitigation methods are almost inappropriate for WBI because WBI has a wider bandwidth and more randomly varied center frequencies than NBI [8], [9], [10]. This has led to the intensive development of mitigation algorithms suitable for WBI over the past decades. Specifically, these mitigation algorithms can be classified into two categories: data-driven algorithms and model-driven algorithms.
This category of WBI mitigation algorithms typically seeks to design a reasonable filter based on the different characteristics of the WBI and the target echo signal in time, frequency, or time–frequency (TF) domains. The precise methodology mainly includes notch filtering [11], [12], [13], [14], [15], subspace projection [15], [16], spatial adaptive filtering [17], [18], [19], or deep learning [20]. Here, notch filtering is simple and easily implemented. However, the direct filtering out of identified bandwidths by a notch filter often results in the partial loss of the target echo signal, and this can interfere with the estimation of accurate imaging parameters, resulting in defocused images. This signal loss can be addressed by introducing subspace projection to construct appropriate WBI and target echo signal subspaces, and then applying filtering only to the WBI subspace components. However, the performance of subspace projection algorithms is significantly compromised for WBI with statistical characteristics that vary greatly over time. A series of spatial adaptive filtering algorithms have been developed to take advantage of the spatial information in multi-channel SAR echo signals cooperated with space–time adaptive processing [17] and adaptive beamforming [18]. Nevertheless, these WBI mitigation algorithms are limited by the spatial freedom of the actual antenna equipment, and inapplicable to single-channel SAR. In contrast with these conventional WBI mitigation algorithms, artificial learning techniques have been demonstrated to be very successful in interference identification and mitigation algorithms based on the excellent feature extraction and image generation capabilities of deep learning networks. For example, Fan et al. utilized deep convolutional neural networks to detect and reconstruct WBI in the TF domain [20]. However, the performance of WBI mitigation algorithms based on artificial learning is restricted by the quantity and quality of the available training samples, which are often limited, particularly for strongly time-varying WBI.
This category of WBI mitigation algorithms generally seeks to develop mathematical representations of WBI, and applies model analysis for detecting and eliminating WBI. For example, multi-parameter mathematical models are applied to characterize WBI, and the model parameters are estimated to reconstruct WBI [21], [22], [23]. However, the performance of these methods depends heavily on the completeness and accuracy of the mathematical model. Moreover, the process required for solving a multi-parameter model requires extensive computation time and hardware resources. This issue has been improved by utilizing matrix factorization and regularization constraint theories [24], [25], [26], [27], [28]. However, model-driven WBI mitigation algorithms typically introduce constraints in the model based on the characteristics of the signal data to increase the precision of WBI reconstruction, which tends to decrease the robustness and generalization ability of the model for a wider range of WBI characteristics.
As discussed above, existing WBI mitigation methods are subject to various advantages and disadvantages. Nonetheless, other more typical limitations must also be discussed as well. For example, some WBI mitigation methods convert the characteristics of WBI into NBI, thereby achieving WBI mitigation using NBI-based methods [14,15]. However, the intrinsic characteristics of WBI are not utilized completely, and this leads to low robustness for SAR data captured under different conditions. Moreover, existing WBI mitigation methods always require the fine-tuning of several hyperparameters, which greatly restricts the adaptability of the methods to variations in SAR signals [22], [23], [24], [25]. Conventional interference mitigation algorithms also typically assume that a SAR echo signal conforms to a complex Gaussian distribution. Therefore, the Frobenius norm is utilized during optimization iterations. However, it is difficult to obtain a good performance for complex SAR data because the methods are more sensitive to non-Gaussian noise and outliers [24,25,29]. Accordingly, the development of robust and self-adaptive WBI mitigation algorithms utilizing the intrinsic characteristics of WBI is essential for applying SAR in complex electromagnetic environments.
The present work addresses these issues by proposing a WBI mitigation scheme based on variational Bayesian theory, which is denoted herein as the VB-WBIM algorithm. Firstly, we utilize the short-time Fourier transform (STFT) to characterize echoes as spectrograms in the TF domain. Then, the existence of WBI is identified adaptively according to statistical analyses of TF spectrograms based on kurtosis. Moreover, a low-rank matrix factorization (LRMF) model with the norm is constructed according to the low-rank characteristics of WBI and the Laplace distribution assumption of the target echo signal in the TF domain to represent the WBI mitigation problem as an optimization process. Furthermore, the WBI is reconstructed accurately by formulating a hierarchical Bayesian complete posterior model with model parameters estimated via the variational Bayesian inference (VBI). Finally, an interference-free echo signal is obtained by applying the interference cancelation and the inverse short-time Fourier transform (ISTFT). In the proposed scheme, the entire WBI mitigation algorithm is realized pulse by pulse, and parallel computation can be introduced to improve the efficiency.
According to the above discussion, the main contributions of this paper can be summarized as follows.
- (1)
The TF representation can analyze and describe the local changes of signal energy intensity at different times and frequencies by transforming the signal sequence into a two-dimensional TF plane. Therefore, it conveniently combines temporal and frequency information in a SAR echo signal comprehensively, and enables the detection of WBI based on a simple analysis of the extent to which the statistical characteristics of a TF spectrogram deviate from a Gaussian distribution. As a result, the proposed scheme addresses the limitations associated with most existing interference mitigation methods, which are only appropriate for NBI detection [10,30].
- (2)
The proposed scheme addresses the conventional assumption of a complex Gaussian distribution for SAR echo signals by introducing the Laplace distribution to describe the SAR echo signal without interference, and thereby provide a –LRMF model, which can bring about a more satisfactory mitigation performance.
- (3)
The hierarchical Bayesian posterior model promotes the robustness and generalization ability of the WBI mitigation algorithm with respect to unobserved SAR data. The norm LRMF method aims to describe the observed SAR data specifically, but it has the potential overfitting problem due to the deterministic model. The proposed algorithm utilizes the probability to introduce the uncertainty of interference mitigation model, which promotes the robustness on the unobserved SAR data.
- (4)
The adoption of VBI for estimating these parameters in the present scheme addresses the lack of adaptability to changes in signal characteristics commonly encountered in existing interference mitigation methods. Moreover, estimating parameters using VBI also addresses the problems associated with parameter estimation based on the usual expectation-maximization algorithm, which involves a multiple-integration process that makes the estimation process extremely complicated and intractable for complex models.
The remainder of this paper is organized as follows. Section 2 presents the accurate identification of WBI in TF spectrograms via the kurtosis and establishes the LRMF-based WBI reconstruction model. Section 3 extends the WBI reconstruction model to the Bayesian framework, and presents the parameter estimation process via VBI. We also present a theoretical evaluation of the variational lower bound for establishing the degree of convergence of hierarchical Bayesian posterior models. Section 4 integrates the proposed VB-WBIM algorithm within the SAR imaging process and presents the quantitative evaluation metrics employed in the following section. Section 5 presents the results of WBI mitigation experiments conducted with simulated and measured SAR data. Finally, Section 6 concludes this paper.
Section snippets
Flowchart of the proposed method
The framework of the proposed VB-WBIM algorithm is illustrated in Fig. 2. The proposed algorithm is applied in parallel for all SAR data based on the independence of single pulses. Firstly, the original SAR data is converted into the TF domain by the STFT, and the existence of WBI is identified adaptively based on the statistical differences measured according to kurtosis in the TF domain. Then, the WBI component is accurately reconstructed. Finally, the target echo signal is recovered by
Proposed mitigation methodology based on VBI
In this section, we extend the LRMF reconstruction model of WBI to the Bayesian framework and estimate the model parameter utilizing VBI. Meanwhile, we give a theoretical evaluation of variational lower bounds for the hierarchical Bayesian posterior models to assess the convergence degree.
SAR imaging and evaluation
The above-discussed VBI process yields the parameters of the posterior model for WBI reconstruction. Then, the target echo signal is recovered through WBI cancelation in the TF domain and by applying the ISTFT. This process is expressed as follows:where , and represent the converged estimation results of WBI factors.
The performance of the VB-WBIM algorithm can be evaluated both qualitatively and quantitatively. Qualitative evaluation is conducted by
Experimental results
In this section, in order to verify the effectiveness and generalization of VB-WBIM, the WBI mitigation experiments with simulated and measured SAR data is performed. Here, we compare the performance of the proposed VB-WBIM algorithm with those obtained for range-spectrum notch filtering (RSNF) [12], instantaneous-spectrum notch filtering (ISNF) [15], instantaneous-spectrum Eigen-subspace filtering (ISEF) [15], and go decomposition (GoDec) [28].
The WBI-corrupted SAR data is simulated by
Conclusion
The present work addressed the challenges associated with the large bandwidth and complex modulation of the WBI within SAR echo signals, which necessitates the development of WBI mitigation algorithms with strong adaptability and robustness that are lacking in existing algorithms based on filtering and model analysis. To this end, we proposed a new WBI mitigation algorithm based on VBI for SAR data. The algorithm first identifies the existence of WBI by taking advantage of the fact that WBI
CRediT authorship contribution statement
Yi Ding: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. Weiwei Fan: Conceptualization, Methodology, Validation, Formal analysis, Writing – original draft, Writing – review & editing. Mingliang Tao: Writing – review & editing. Zijing Zhang: Writing – review & editing. Li Wang: Writing – review & editing. Feng Zhou: Formal analysis, Writing – review & editing. Bingbing Lu: Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
Thanks to all the anonymous reviewers and editors for their comments and suggestions to this paper, so that the content of this paper can be more rigorous and plentiful. Especially, thanks the professor Deyu Meng in Xi'an Jiaotong University for his research on LRMF, which inspires us to explore the WBI mitigation cooperated with LRMF and VBI. Meanwhile, the authors are grateful for the open-source sentinel-1 satellite SAR data of ESA, which effectively supports the verification experiments of
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