1. Introduction
Synthetic aperture radar (SAR) is an important imaging technology that has been applied in environmental protection and marine observation [
1,
2]. In recent years, the sparse signal processing method based on CS [
3] has been implemented in microwave imaging [
4,
5]. It can recover the scene by solving
regularization.
In [
6], Çetin et al. proposed a sparsity-driven SAR imaging model for achieving autofocusing and moving targets imaging. Zhang et al. [
7] explored the principles and applications in sparse microwave imaging. Patel et al. [
8] analyzed different azimuth sampling methods based on the CS model. Luo et al. [
9] developed a multiple scatterers detection method for SAR tomography with CS approach. Hossein et al. [
10] proposed a polarimetric SAR estimator under the frame of CS. In [
11], Zhu reviewed the CS-based super-resolving algorithm. Zhang et al. [
12] proposed a novel 3D SAR imaging algorithm based on 2D compressive sensing. It not only provides super-resolution performance, but also reduces the storage of data acquisition and processing. Compared with matched filtering (MF), microwave imaging based on sparse signal processing can improve the image quality by suppressing noise and sidelobes as well as azimuth ambiguities with the downsampled data [
7,
13].
In compressive sensing (CS), the sparsity is usually considered as a given parameter. However, it is unknown practically. For many functions of CS, we need to know this parameter. Therefore, the estimation of sparsity is crucial for sparse SAR imaging. In some cases, the sparsity can be estimated directly based on prior information, which is obtained from the historical data. In other cases, we can only get the range of the sparsity based on the prior information, rather than an accurate value. In the process of accurately reconstructing a large number of these scenarios, it is more advantageous to estimate the sparsity automatically than to select the sparsity manually. In [
14], several methods, such as Stein’s unbiased risk estimator, L-curve, and generalized cross-validation, have been presented for automatically estimating the regularization parameter. Adaptive parameter estimation for sparse SAR imaging can be achieved by these methods. However, these methods are deduced based on an observation matrix. The observation matrix-based sparse SAR imaging achieves decouplingby vectorizing the raw data matrix, which will entail huge computational and memory costs. Therefore, it is challenging to adopt these adaptive parameter estimation methods based on an observation matrix into a large-scale scene reconstruction.
An azimuth-range decouple-based sparse SAR imaging method has been proposed [
7,
15]. The coupling of the 2D data can be removed by constructing an echo simulation operator to replace the observation matrix, which can effectively relieve the computational complexity [
16]. This method has been widely used in TOPS SAR [
17], Sliding Spotlight SAR [
18], displaced phase center antenna (DPCA) imaging [
19], wide-angle SAR (WASAR) [
20] and ground moving target indication (GMTI) [
21]. We can combine it with automatic parameter estimating methods to achieve an adaptive parameter estimation of the large-scale sparse SAR imaging. However, considering that finding the optimal regularization parameter requires iterative processing, the total computational cost of the adaptive parameter estimation method based on azimuth-range decouple is still large.
A complex image-based sparse SAR imaging method is proposed in [
22,
23]. Combining this method with the automatic parameter estimating methods, we get the adaptive parameter estimation method based on a complex image. The complex image-based sparse SAR imaging method only considers the threshold operation of the complex image, which can further reduce the computational and memory costs. In this paper, for the case of the downsampled raw data, we propose an efficient adaptive parameter estimation method. The complex image-based sparse SAR imaging method is adopted first to pre-estimate the parameter. Then, the parameter iteration range is updated according to the pre-estimated parameter. Finally, we introduce the azimuth-range decouple operators into the parameter estimation and deduct the efficient adaptive parameter estimation method for sparse SAR imaging.
The rest of this paper is organized as follows. In
Section 2, we introduce the sparse SAR signal models and the automatic regularization parameter estimation method. In
Section 3, we give details of the proposed method. The computational complexity of different methods is also analyzed in this section.
Section 4 presents the simulated and real data results to analyze the performance of the proposed method. The conclusions are presented in
Section 5.
3. Efficient Adaptive Parameter Estimation for Sparse SAR Imaging
In this section, the parameter estimation method based on azimuth-range decouple and the parameter estimation method based on complex image are introduced. Next, we introduce the proposed method in detail. Finally, the computational complexity of these methods is analyzed.
3.1. The Adaptive Parameter Estimation Method Based on Azimuth-Range Decouple
Combining the azimuth-range decouple operators with GCV, we can get the adaptive parameter estimation method for sparse SAR imaging. Compared with the adaptive parameter estimation method based on observation matrix, this method can reduce the computational complexity. Considering that
is a large diagonal matrix of
, the computational cost of the trace of it is also large, we replace the trace operator
with the sum operator. Equation (7) can be rewritten as follows:
which is the cost function of the adaptive parameter estimation method based on azimuth-range decouple, where
is the echo simulation operator and
is shown in Equation (4).
There are several algorithms to achieve the sparse reconstruction, such as iterative soft thresholding (IST) [
31] and complex approximated message passing (CAMP) [
32,
33]. In this paper, we choose CAMP as sparse reconstruction algorithm, which has been applied to constant false-alarm rate (CFAR) detection in sparse SAR imaging [
34].
The optimal regularization parameter is estimated by minimizing Equation (9). However, considering that finding the optimal regularization parameter requires the iterative processing, the total computational cost of the adaptive parameter estimation method based on azimuth-range decouple is still large.
3.2. The Adaptive Parameter Estimation Method Based on Complex Image
Compared with the azimuth-range decouple-based sparse SAR imaging method, the complex image-based sparse SAR imaging method only considers the threshold operation, which can further reduce the computational and memory costs. Combining it with GCV, we can get the adaptive parameter estimation method for sparse SAR imaging based on complex image. Equation (7) can be rewritten as follows:
which is the cost function of the adaptive parameter estimation method based on complex image, where
is shown in Equation (6).
3.3. The Proposed Method
The proposed method is mainly for the case of the downsampled data. On the one hand, although the adaptive parameter estimation method based on azimuth-range decouple can estimate the sparsity accurately, as mentioned above, the total computational cost of this method is large. On the other hand, due to the energy dispersion and ambiguities, the estimated sparsity of the parameter estimation method based on complex image will be greater than the true value, and we cannot simply use the parameter estimation method based on complex image to replace the parameter estimation method based on azimuth-range decouple. Therefore, we need to find a method to adaptively estimate the sparsity accurately while having the lower computational complexity. A good solution is to combine these two adaptive methods together, utilizing the complex image to pre-estimate the parameter and reduce the iteration range, then estimating the accurate parameter with raw data.
The proposed method has three steps. First, set the iteration range of sparsity to and adaptively estimate the sparsity based on the complex SAR image which is reconstructed by the downsampled raw data. The pre-estimated sparsity is set to , which is greater than the true value due to ambiguities and energy dispersion caused by downsampling. Second, update the iteration range from to . Third, get the adaptive reconstructed image and the optimal adaptive result of sparsity on the new range based on raw data.
The details of adaptive parameter estimation based on azimuth-range decouple are shown in Algorithm 1, where is the range of the sparsity; is the threshold function of CAMP.
Algorithm 1: The adaptive parameter estimation method based on azimuth-range decouple |
1: Input: downsampled SAR raw data , parameter , |
2: Initialization: , , |
3: while and |
1) ; |
2) ; |
|
|
3) ; |
4) Calculate and according to (9) |
5) if else |
6) |
4: end while |
5: Output: the reconstructed image and the adaptive parameter |
3.4. Analysis of Computational Complexity
The computational complexity of different adaptive parameter estimation methods is analyzed in this section. A common characteristic of the adaptive parameter estimation methods mentioned above is that regularization parameter iterations are required. The difference lies in the different sparse reconstruction algorithms.
The measure of the computational complexity is the floating point operation (FLOP). Each FLOP represents a real addition operation or a real multiplication operation. In the observation matrix-based sparse SAR imaging method and azimuth-range decouple-based sparse SAR imaging method, the main calculation includes the imaging process, the echo simulation process, and the threshold process. The computational complexity of the threshold process is FLOPs, where is the scene size. In the observation matrix-based sparse SAR imaging method, the imaging process and echo simulation process are two matrix multiplications. The main computational complexity of a single-step iteration of the observation-matrix-based sparse SAR imaging method is FLOPs, where is the sampling number. This computational complexity is approximately proportional to the quadratic square of the scene size.
In this paper, the chirp scaling [
35] operator is chosen as the imaging operator. Therefore,
and
can be expressed as follows:
where
and
are the azimuth Fourier transform (FFT) operators and azimuth inverse Fourier transform (IFFT) operators,
and
are the range FFT operators and range IFFT operators,
,
and
are three complex phase matrix. Chirp scaling and inverse chirp scaling both contain two FFTs, two IFFTs, and three time complex phase multiplications. According to [
2], the computational complexity of FFT and IFFT with length
is
FLOPs, and the computational complexity of a complex multiplication operation is six FLOPs. Assuming that the data are sampled in the manner of uniform/nonuniform downsampling, the main computational complexity of a single-step iteration of the azimuth-range decouple-based sparse SAR imaging method is
FLOPs, which is approximately proportional to the product of the linear logarithm of the scene size.
The complex image-based sparse SAR imaging method includes only threshold process. The computational complexity of a single-step iteration of this method is FLOPs, which is much lower than the azimuth-range decouple-based sparse SAR imaging method.
Let
represent the iteration steps of the recovery for sparse reconstruction algorithms. Let
and
denote the number of iteration steps required for regularization parameter convergence when the iteration ranges of sparsity are
and
, respectively. Assuming that
,
,
, the scene size
, and the downsampling rate
, the computational complexity of different adaptive parameter estimation methods is shown in
Table 1.
Since the proposed method utilizes the complex image as the prior information to pre-estimate the parameter, the iteration range of the sparsity is reduced when the adaptive parameter estimation is processed in the raw data domain. Therefore, the proposed method has the lower computational complexity compared with the parameter estimation method based on azimuth-range decouple. For example, if the scene size is and the downsampling rate is 80%, the proposed method can increase the computational efficiency about 3‒4-fold.