1. Introduction
Airborne radar plays an important role in many fields for its all-day and all-weather imaging ability [
1,
2]. In general, airborne radar collects the echo through antenna scanning along with the platform movement. In range direction, the antenna continuously transmits a large bandwidth signal. After pulse compression, the range resolution is negatively correlated with the bandwidth, i.e.,
where
is range resolution,
c is light speed and
B is the bandwidth of transmitted signal. Therefore, high range resolution can be achieved with large bandwidth signals. In azimuth, the resolution is limited to antenna size. According to Rayleigh criterion, the adjacent targets with an interval less than Rayleigh distance (RD) cannot be distinguished, where RD is the space between the peak of the antenna pattern and the first zero-crossing [
3,
4]. In order to distinguish adjacent targets with small spacing, radar needs to emit a narrow beam. However, narrower beams require a larger antenna aperture. Due to platform limitations, the antenna aperture of airborne radar is usually limited, resulting in lower azimuth resolution.
The application of super-resolution technology can make the resolution break Rayleigh limit [
5,
6], which makes it possible to improve the azimuth resolution without increasing the aperture of airborne radar. In fact, many super-resolution methods have been proposed in recent years. In [
7], the Tikhonov regularization (TREGU) method was proposed to improve the resolution. However, this method encounters over smoothing, which makes the improvement of resolution limited. In [
8], truncated singular value decomposition (TSVD) was utilized. This method suppresses the noise amplification by truncating small singular values, but its performance is poor in the condition of low signal-to-noise ratio (SNR). Iterative adaptive approach (IAA) can further suppresses noise, but it suffers from high computational complexity [
9]. The sparse regularization method introduces the prior information of the target, and has a good effect of improving the resolution of the sparse target [
10,
11]. However, all of the above methods only consider the improvement of resolving ability, and do not consider the preservation of target contour information. Total variation (TV) method, which introduces the gradient constraint of targets, can effectively preserve the contour of targets. Recently, TV method has been widely used in imaging restor and radar imaging [
12,
13,
14]. In [
15], we proposed a one-dimensional TV method to improve the azimuth resolution of airborne radar. Unlike optical image restoration, in airborne radar imaging, range resolution has been improved by pulse compression, so the proposed TV method only introduces azimuth TV norm. The experiments show that the TV method can preserve the target contour information of airborne radar. However, the computational complexity is very high due to matrix inversion, and the computational complexity is
. For airborne radar imaging, the azimuth samples
N is determined by scanning range
, scanning speed
and pulse repetition frequency (PRF), i.e.,
Usually, N is large, which leads to the inefficiency of the algorithm. Therefore, it is necessary to study how to realize fast inversion to reduce the computational complexity.
In recent years, many researches have devoted to solve the problem of high computational complexity caused by matrix inversion. These methods utilized the special structure of coefficient matrix to achieve fast inversion, the computational complexity then can be decreased [
16,
17,
18]. In previous research, we have found that the coefficient matrix of TV method has an approximate Toeplitz structure, which makes it possible to achieve fast inversion using the Toeplitz structure. In fact, literature [
19] indicated the concepts of displacement structure and displacement rank, as well as revealing that the operation can be compressed by using a Toeplitz matrix. Subsequent research has proven that the displacement rank of a Toeplitz matrix is very small and, so, its inverse matrix also has a displacement structure, showing that the inversion of Toeplitz matrix can be fast realized [
20]. Utilizing the low displacement rank features of Toeplitz matrices, the fast inversion of Toeplitz matrix has been achieved using Gohberg–Semencul (GS) representation [
21,
22].
In this paper, a GS representation based fast TV (GSFTV) method is proposed realize fast super-resolution imaging as well as preserve the contour information in airborne radar imaging. Firstly, the received signal of airborne radar is analyzed. It can be found that the azimuth echo can be modeled as a convolution of target scattering and antenna pattern. Secondly, the azimuth gradient constraint of the target is introduced in the regularization framework to transform the super-resolution problem into a TV regularization problem, and the TV regularization problem is solved by split Bregman algorithm (SBA). Thirdly, to solve the problem of high computational complexity caused by matrix inversion, we approximate the coefficient matrix to Toeplitz matrix, and use GS representation to realize fast inversion. The computational complexity will be decreased from to . Then we will prove that the error caused by the approximation is quite small and can be ignored through numerical analysis. Finally, the performance of the proposed GSFTV is demonstrated by experiments.
The reminder of the paper is organized as follows.
Section 2 analyzes the received signal and models the echo model of airborne radar imaging. In
Section 3, the traditional TV method is reviewed and the computational complexity is analyzed. In
Section 4, the proposed GSFTV is deduced in detail. In
Section 5, some experiments are conducted to verify the superior performance of the proposed GSFTV method. The conclusion is discussed in
Section 6.
2. Signal Model of Airborne Radar Imaging
Airborne radar scans the imaging region along with the movement of the aircraft. The schematic diagram of airborne radar imaging is shown in
Figure 1. The aircraft flies at altitude
H and speed
v.
is the scanning speed of the antenna and
is pitching angle. When the antenna is scanning the target
P, we define the azimuthal angle is
, and the distance between the target and the radar is
. After time
t, the radar moving distance is
. At this time, the distance between radar and target
P is
, and the azimuth angle of the radar beam is
with
.
According to the trigonometric relation, the range history at time
t can be obtained as
where
.
It can be approximated as
In practical applications, since the time for the antenna beam to sweep across the target is very short and the radar has a large working distance, the quadratic term in (
4) is very small and can be ignored. Thus the range history can be finally approximated as
Considering both the range resolution and working distance, the radar transmits linear frequency modulated (LFM) signal, i.e.,
where rect
is a rectangle window,
is the fast time,
is carry frequency and
is chirp rate. After antenna scanning, the received signal is
where
is the target scattering distribution,
represents the modulation effect of antenna pattern and
is time delay.
Matched filtering is a widely utilized technology to obtain high range resolution. After matched filtering, the received signal becomes
For airborne radar, antenna scanning is accompanied by platform movement, which results in the echo of the same range unit being dispersed in different units. Therefore, range walk correction is needed to eliminate the influence of platform motion. After that, the received echo can be modeled as a convolution of antenna pattern and target scattering distribution [
3,
23], i.e.,
where
is the noise and
is the convolution matrix structured by antenna pattern, i.e.,
Based on the convolution model of airborne radar imaging, the target distribution can be recovered by deconvolution, but this process is extremely ill-posed.
6. Conclusions
In this paper, a GSFTV method was proposed to solve the problem of low azimuth resolution in airborne radar imaging. The proposed GSFTV method can efficiently improve the azimuth resolution and break the Rayleigh limit. In this process, the contour information of target is preserved. So its processed result is more clear than other traditional methods. Besides, utilizing the GS representation, the computational complexity of each iteration is decreased from to , which greatly increases the computing efficiency in practice. Although we make some approximations in order to realize the acceleration, we also proved that these approximations can be ignored.
Through simulation and real data processing, we demonstrated that the proposed GSFTV method almost no performance degradation compared with traditional TV method. Hardware test results show that the efficiency of the proposed GSFTV is much higher than that of the traditional TV method, and the more the number of azimuth points, the greater the computing advantage of the proposed GSFTV method.