^{②} (Key Laboratory of Electromagnetic Space Information, Chinese Academy of Sciences, Hefei 230027, China);
^{③} (Academy of Photoelectric Technology, Hefei University of Technology, Hefei 230009, China)
^{②} (中国科学院电磁空间信息重点实验室 合肥 230027)
^{③} (合肥工业大学光电技术研究院 合肥 230009)
High resolution radar imaging has been widely used in target scattering diagnostics and recognition. As we all know, high resolution in range dimension is derived from the bandwidth of the transmitting signal and in the cross range dimension from synthetic aperture of multiple spatial positions. Under the fixed bandwidth and the synthetic aperture, traditional Matched Filter (MF) based methods for radar imaging suffer from low resolution and high sidelobes limited by the synthetic aperture^{[1]}.
In order to improve the resolution and suppress the sidelobes, many high resolution methods have been applied to radar imaging. For example, the recently introduced theory of Compressed Sensing (CS) provides an idea to improve the resolution and reduce the amounts of measurement data under the constraint of sparsely distributed target prior, which has been widely explored for applications of radar imaging^{[2–4]}. However, conventional CS methods are confronted with a range of problems in practical scenarios, such as complexity in calculation, high Signal-to-Noise Ratio (SNR) requirement, model mismatch caused by off grid problem^{[5]}, phase mismatch^{[6]}, frequency error^{[7]} and position error^{[8]}. To avoid the off grid problem of CS, modern spectral estimation methods like MUltiple SIgnal Classification (MUSIC), matrix pencil and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) have been used in radar imaging for resolution improvement^{[9]}. However, most those methods suffer from performance degradation when there is little prior knowledge of the exact numbers of the scatters or under low SNR condition. Recently, the atomic norm minimization algorithm^{[10]} based on continuous compressed sensing is introduced to enhance the SNR of the received echo and using Vandermonde decomposition to eliminate the grid mismatch. Nevertheless, this method can only be tailored to a specific model and brings huge computational cost.
Consideration the aforementioned fact while combining the sparsity low rank matrix recovery technology and deconvolution algorithm, we introduce a high resolution radar imaging method based on the MF result. Firstly we establish the convolution model of target’s backscatter coefficients and the Point Spread Function (PSF), and then we want to use the deconvolution method like Wiener filter to improve the radar imaging resolution. However, the performance improvements of those methods depend on high SNR, and their super resolution performance is visibly affected by the low pass character of the PSF^{[11]}. Although the MF result has enhanced the SNR, we can further improve the echo SNR by the sparsity and low rank matrix recovery. Low rank matrix recovery has been applied in many signal processing applications to estimate a low rank matrix from its noisy observation^{[12, 13]}. Combinng the sparsity of the echo matrix, we modify the low rank matrix recovery and introduce it to radar echo denoising, which can improve the performance of the two-Dimensional (2D) deconvolution. Finally, some experimental results are conducted to verify the effectiveness of the proposed method.
Notation: (·)^{T}, (·)^{H} and (·)^{*} denote the transpose, the conjugate transpose and the conjugate operation, respectively.
Considering a typical arrangement for radar imaging in which an object with scattering reflectivity
Transmitting a stepped-frequency signal with frequency
In this equation:
In far-field and small rotation angle case,
where
Then the received echo can be written as follow under some approximated conditions:
where
After discrete imaging region with P×Q grids, the received echo in Eq. (4) can be described as the following 2D linear signal model:
where
Considering the targets present sparse point scattering characteristic under high frequency scattering in most practical application scenarios, we present our method to improve the resolution of radar imaging under sparse target constraint using 2D deconvolution algorithm with low rank sparsity echo matrix denoising.
3 2D Deconvolution with Echo DenoisingAs we all know, the MF algorithm which is based on the maximum signal to noise ratio is the most stable and commonly used radar imaging method. However, due to limitation of the synthetic aperture and bandwidth, the standard MF method suffers from relatively low resolution and high sidelobes, especially under the requirements of high resolution. The received echo after MF from Eq. (5) can be obtained by:
From the result of Eq. (6), the echo of the surface target after MF can be described as the sum of all the wave scattered at the points on the surface grid, i.e.,
where we define the PSF as:
here,
We can find that Eq. (6) can be seen as the 2D convolution of the PSF and target backscatter coefficients:
Inspired by this, we can recovery the backscatter coefficients using deconvolution algorithm to improve the imaging quality. Firstly, we should analyze the characteristic of the PSF and its influence on the deconvolution result.
3.1 Analysis of PSFThe PSF can be evaluated as:
We can calculate the 2D mainlobe width which represents the radar imaging resolution as follows:
Eq. (9) indicates that the MF result can be seen as the convolution result of backscatter coefficients and
As we have get 2D convolution form as Eq. (9), here we consider to use the direct deconvolution algorithm to recovery target backscatter coefficients. Firstly, we transform Eq. (9) into the spatial frequency domain using 2D Fourier transform as:
where,
Theoretically, the target scattering information could be restored by deconvolution as:
However,
In order to alleviate the ill-posed problem, we use Winner filtering algorithm and sparse low rank matrix recovery to improve the quality of imaging result.
The result after Winner filter algorithm can be written as^{[14]}:
where
We can prove that the echo matrix after MF is sparse and low rank in Appendix A and by using this characteristic, the echo SNR can be improved. Consider the problem of estimating a sparse low rank matrix X from its noisy observation Y:
Define the sparse low rank matrix recovery problem as:
where
By applying Augmented Lagrangian Method (ALM), we can get the optimization problem:
And the update rules for solving this problem are as follows:
where,
where,
See Appendix B for the detailed derivation of Eq. (18) and Eq. (19).
The flowchart of the proposed method is shown in Fig. 2 by combining the sparse low rank matrix recovery with the 2D deconvolution.
The parameters in the simulation are given in Tab. 1. In this experiment, we set four-point targets, the imaging results are shown in Fig. 3.
As shown in Fig. 3(a), due to the limitation of synthetic aperture and bandwidth, the MF method suffers from relatively low resolution and high sidelobes which make it difficult to distinguish between four-point targets even there is no noise. Fig. 3(b)–Fig. 3(d) show the imaging results reconstructed by MF and proposed method including the intermediate denoising results when SNR = –15 dB. It can be clearly seen that the effect of denoising compared Fig. 3(c) with Fig. 3(a) and Fig. 3(b), the echo SNR is further improved by the sparsity and low rank matrix recovery during the proposed intermediate denoising procedure. The final imaging result is shown in Fig. 3(d), from which we can see that the proposed method has a better reconstruction precision with higher resolution imaging of four distinguishable point targets.
5 Experiment ResultsThe experimental scene is shown in Fig. 4(a), which is the same with the model in Fig. 1. The radar system consists of a pair of horn antennas, a turntable whose rotation angle can be precisely controlled by the computer, and an Agilent VNA N5224A which is used for transmitting and receiving the stepped-frequency signal with bandwidth of 10 GHz from 28 GHz to 38 GHz and number of frequencies N equals to 256 (Frequency interval
As we know, image entropy can be considered as a metric for measuring the smoothness of the probability density function of image intensities^{[15]}. The imaging entropy is defined as:
where
In this experiment, we set
Fig. 5 shows the results of the MF and our proposed method for the mental spheres. The one-dimensional
The parameters for this experiment are set as follows,
Fig. 7 shows the imaging results of the scissors reconstructed by MF and proposed method. It can be seen from the results that the proposed method has a high reconstruction precision with a shaper shape of scissors.
The entropies of the imaging results by MF and our proposed method are given in Tab. 2 to quantitatively assess the performance. The proposed method has a low entropy which means the proposed method can improve the resolution and verifies its superiority.
We introduce a robust deconvolution method with enhancing SNR technology to realize high resolution radar imaging. Compared to other high resolution methods, our proposed method is simple and robust. Although the signal model and experiments are performed for turntable radar situation with SF waveform, the method can be directly generalized to other practical radar systems based on other types of signals.
Appendix A Proof of the sparsity and low rank characteristic
To prove the echo matrix after MF is sparse and low rank, the following lemma is needed.
Lemma 1^{[16]}: For matrix A and B, the ranks of the product of A and B satisfy the inequality below:
From Eq. (5) and Eq. (6), we can see that the echo matrix after MF can be written as:
We have supposed that the target has sparse distribution, so the target backscatter coefficients matrix
Appendix B Derivation of Eq. (18) and Eq. (19)
For Eq. (18), the optimization problem can be described as Eq. (B-1), and it has a closed-form solution just as Eq. (18) according to Ref. [13].
For Eq. (19), it is the same with Eq. (18), which can written as
It also has a closed-form solution as Eq. (19) according to Ref. [17].
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