1. Introduction
Radar has come to play an increasingly important role in several areas, including autonomous navigation, driverless driving, and ground surveillance [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]. Real aperture radar (RAR) can scan an observation area using an antenna to produce two-dimensional range and cross-range imagery. RAR can be used for any imaging geometry, without the need for coherent processing in the cross-range context [
12]. Therefore, RAR is of considerable interest for forward-looking imaging, where synthetic aperture radar (SAR) and Doppler beam sharpening (DBS) techniques fail to work [
13]. As the range resolution is related to the bandwidth of the transmitted signal and the angular resolution is dependent on the aperture of the antenna, the range resolution typically far outweighs the angular resolution in practical applications, and the problem of low angular resolution is a limiting factor of RAR. Fortunately, echo data along the azimuthal direction can be modeled as a convolution relationship between the target scatterings and the antenna pattern. Deconvolution methods provide potential for improvement in the angular resolution.
Sparse recovery approaches comprise a popular class of deconvolution methods. The sparse recovery problem is an underdetermined linear system, where the unknown vector is assumed to be sparse, meaning that most of its elements are zero. Sparse recovery approaches have been widely used in many fields, such as medical imaging, video processing, source localization, and radar imaging [
14,
15,
16,
17]. In radar imaging, SAR, inverse synthetic aperture radar (ISAR), multiple-input multiple-output (MIMO) radar, and ground-penetrating radar have achieved high-resolution imaging with the help of sparse recovery techniques.
For RAR imaging, the target of interest usually only has a few strong scatterers that are sparsely distributed, such as foreign objects on an airport runway. Sparse recovery approaches can be exploited in RAR imaging to enhance the angular resolution [
18]. In [
19], based on Bayesian theory, the authors assumed the target to have sparse characteristics and employed the Laplace distribution to describe the sparse prior distribution, achieving angular super-resolution imaging. In [
20], the authors applied the
norm to express the target sparse prior based on the regularization strategy and converted the angular super-resolution problem into an optimization problem. Theoretically, the two types of methods are essentially equivalent, and they aim to address the
norm minimization problem. Furthermore, the iterative adaptive approach (IAA) from the sparse spectrum estimation field was applied for angular super-resolution imaging in [
21], obtaining similar results to the two above-mentioned sparse approaches.
As mentioned, current sparse recovery algorithms mostly utilize the
norm to represent the sparsity of the target. Indeed, mathematically speaking, the
norm has stronger sparsity compared to the
norm. However, directly solving the
norm minimization is an NP-hard problem. Fortunately, in a recent study [
22], the authors proposed a smoothed
norm (SL0) algorithm, which utilizes a smooth Gaussian function to approximate the
norm and then uses the steepest descent and gradient projection to minimize the approximate
norm. The SL0 algorithm has higher accuracy and faster convergence speed compared to conventional
norm-based sparse recovery algorithms. In radar imaging, we have successfully applied SL0 to SAR [
23], ISAR [
24], and MIMO radar [
25], reducing the computational complexity while improving the imaging quality.
However, little work on the use of the SL0 method for RAR imaging has been reported. After an investigation, we found the existing SL0 algorithm to be unsuitable for RAR imaging. The antenna measurement matrix is seriously ill-posed because of the low-pass characteristic of the antenna in RAR imaging. The SL0 algorithm requires the calculation of the pseudo-inverse of the ill-posed antenna measurement matrix to set the initial value and calculate the gradient projection. For an ill-posed matrix, the error when calculating the pseudo-inverse is too large; as a result, the SL0 algorithm cannot be utilized directly for RAR imaging. Moreover, for the steepest descent, we can easily fall into local minima.
This paper proposes a modified SL0 (MSL0) algorithm. By replacing the ill-posed antenna measurement matrix with the regularization matrix, we avoid the huge error caused by calculating the pseudo-inverse, allowing for the realization of RAR angular super-resolution imaging. In addition, a hard threshold operator is added to prevent local minima and accelerate convergence. Compared with existing angular super-resolution methods, such as the iterative adaptive approach (IAA) [
21] and sparse regularization method [
20], the proposed method has advantages in terms of super-resolution performance and computational efficiency.
The remainder of this paper is organized as follows.
Section 2 introduces the signal model for RAR. In
Section 3, the schemes of the proposed MSL0 method are detailed.
Section 4 presents the simulation results and a performance comparison between existing methods and the proposed MSL0 method, followed by a discussion in
Section 5. In
Section 6, experimental data results are presented to further validate the proposed method. Finally,
Section 7 presents the conclusions.
2. Signal Model for Real Aperture Radar Imaging
As shown in
Figure 1, the RAR acquires echo data through radar beam scanning. The received echo data can achieve high resolution in the range dimension by transmitting a linear frequency modulation (LFM) signal and pulse compression. For the azimuth dimension, according to the sequential process of scanning imaging, the received echo data along the azimuth dimension can be modeled using a one-dimensional convolution model:
where
denotes the azimuth angle,
is the received azimuth echo,
is the target scattering distribution,
is the antenna pattern,
is the noise, and ⊗ is the convolution operation.
After discretization, Equation (
1) can be transformed into the following matrix form:
where
is the received azimuth echo vector;
is the target scattering distribution vector;
is the noise vector;
N represents the number of azimuthal sampling points, which is determined based on the pulse repetition frequency (PRF), scanning scope in the azimuth, and antenna scanning velocity; and
H denotes the
antenna measurement matrix composed of the antenna pattern samples shown in Equation (
3):
where
are the samples of the antenna pattern, and the sampling number is determined according to the PRF, antenna beamwidth, and antenna scanning velocity. As the antenna pattern shown in
Figure 2 has low-pass characteristics in the frequency domain, the antenna measurement matrix
H is a seriously ill-posed matrix.
Figure 1.
Echo acquisition procedure of a real aperture scanning radar.
Figure 1.
Echo acquisition procedure of a real aperture scanning radar.
4. Simulation Results for Super-Resolution Algorithms
In this section, simulation results are given to verify the performance of the proposed method. The proposed MSL0 method was compared with the Tikhonov regularization method [
26], truncated singular value decomposition (TSVD) method [
27], Wiener filter method [
28], sparse regularization method [
20], and IAA [
21]. We compared the super-resolution imaging performance and computational efficiency of the six methods through point simulation.
To quantify and compare the super-resolution performance of these six methods, we applied the structure similarity index measure (SSIM) [
29] and the mean square error (MSE) [
30].
The SSIM is defined as follows:
where
and
are the mean and standard deviation of the vectors
and
, respectively, and
represents the correlation coefficients corresponding to the vectors
and
, respectively. The SSIM is a quantitative measure between the super-resolution imaging result and the target true scene. The value of the SSIM ranges between −1 and 1, where 1 indicates that the super-resolution imaging result is identical to the target true scene. A larger SSIM value indicates better super-resolution performance.
The MSE is used to measure the relative error between the super-resolution imaging result and the target true distribution, which is defined as
where
M and
N denote the range cell and azimuth cell, respectively.
For the simulations, we adopted the ideal
antenna pattern and considered additive white Gaussian noise. The main simulation parameters are listed in
Table 1. Our point targets, as shown in
Figure 4a, were located at
and
with normalized amplitudes of 1. As shown in
Figure 4b, as the interval between the two targets was narrower than the antenna beamwidth, the echoes of the targets were mixed in the azimuth dimension. The results of the Tikhonov regularization, TSVD, and Wiener filtering methods are presented in
Figure 4c–e, which show limited resolution improvements. The sparse regularization method and the IAA, as shown in
Figure 4f,g, respectively, resolved the two targets to a certain extent, but their super-resolution performance was weaker than that of the MSL0 method, as shown in
Figure 4h.
Table 2 provides a comparison of the various imaging indices, where the computation times are reported from tests using Matlab 2020a on an AMD Ryzen5 4500U with 16 GB of RAM. From the simulation parameters in
Table 1, the dimensions of the matrices involved can be calculated: the echo matrix
y has a size of
, whereas the antenna measurement matrix
H has a size of
. As shown in
Table 2, the MSL0 method obtained the largest SSIM and the smallest MSE, indicating that its super-resolution performance was superior to that of the other methods. Furthermore, compared with the sparse regularization method and the IAA, the proposed method exhibited a certain advantage in terms of computation speed.
6. Experimental Data Results
Next, experimental data were applied to validate the super-resolution performance of the six considered algorithms. The experiment was carried out at the University of Electronic Science and Technology of China, Chengdu, China. In this experiment, the radar was fixed on top of a building, as shown in
Figure 8a, and the targets were the two corner reflectors on the lawn in the radar forward-looking area shown in
Figure 8b. The main parameters of the radar system are listed in
Table 3.
After collecting the echoes, we used different super-resolution methods to process the echo data using the Matlab 2020a software. The echoes of the two corner reflectors and the corresponding super-resolution processing results are marked with red ovals in
Figure 9.
Figure 9a depicts the real beam echo after pulse compression, which exhibited coarse resolution. The results for the Tikhonov regularization, TSVD, and Wiener filter methods are shown in
Figure 9b–d. Visually, their resolution improvement effects were restricted. The results of the sparse regularization method and the IAA are presented in
Figure 9e,f, where the two corner reflectors can be distinguished; however, some shadows between the two targets can be observed. Because of its superior super-resolution performance, the proposed MSL0 method achieved better shade-free processing results, as shown in
Figure 9g.
The profile results in
Figure 10 show that the Tikhonov regularization, TSVD, and Wiener filtering methods exhibited limited resolution improvement effects. The two targets were not completely separated by the sparse regularization method and the IAA, whereas the proposed MSL0 method achieved complete separation of the two targets.
To further evaluate the experimental data processing results of the above methods, we introduced image entropy, which is commonly utilized to evaluate the overall quality of the radar imagery since entropy increases with an increase in the image blurring level [
33]. From the perspective of image processing, angular super-resolution is the removal of antenna pattern blurring, which is essentially an entropy reduction process. Therefore, lower entropy indicates better super-resolution performance [
19]. Image entropy is defined as
where the probability distribution function is
where
is an element of the related image.
The size of the echo matrix in this experiment was
.
Table 4 compares the results of the image entropy and computational times for the various methods. The proposed MSL0 method yielded the smallest entropy, indicating that it had the best super-resolution performance in the considered practical application scenario. In terms of execution time, the proposed MSL0 method exhibited certain advantages when compared with the sparse regularization method and the IAA.