BSBL-based multiband fusion ISAR imaging

: Multiband fusion imaging can effectively improve the range resolution of inverse synthetic aperture radar (ISAR) imaging. In this study, the block sparse Bayesian learning (BSBL) method is applied to multiband fusion imaging to achieve high-resolution ISAR imaging of a block-structured target. The BSBL method is suitable for the ISAR imaging of numerous and continuous scatterers because it considers the block structure characteristics of the signal. The validity of the proposed method is verified by the simulation and real-data experimental results.


Introduction
A high-resolution inverse synthetic aperture radar (ISAR) image can provide detailed structure information that can improve the efficiency of target identification and classification in space surveillance [1]. The high cross-range resolution can be achieved by increasing the accumulation angle of ISAR imaging, while increasing range resolution should increase the transmit signal bandwidth. Multiband fusion imaging technology can effectively improve the radar range resolution by fusing the target frequency responses, which are measured by multiple radars with different frequency bands at the same view angle, to a higher bandwidth frequency response in the signal level without changing the existing radar configuration, and thus attracts many researchers to study this technology [1,2].
The traditional multiband radar signal fusion technology is mainly based on the modern spectral estimation methods [3,4]. Even though these methods have high estimation accuracy, they need to know the number of target scatterers, which is difficult to accurately estimate in a practical scenario. In recent years, sparse reconstruction theory has developed rapidly, and it has been successfully applied in ISAR imaging [5]. Compared with traditional ISAR imaging methods, the sparse representation-based methods can effectively reconstruct the target image. The existing sparse representation methods for ISAR imaging are based on the reality that the target scatterers are discretely distributed in the imaging region. Moreover, the target of ISAR imaging usually has some block structural characteristics, i.e. nonzero scattering coefficients occur in blocks in the imaging scene, such as aircrafts and satellites with complex structures. Thus, the block sparse representation method is more suitable for ISAR imaging of these numerous and continuous scatterers because it considers the block structure characteristics of the signal [6].
Compared with the traditional sparse representation algorithm, the block sparse signal recovery algorithm can remove the irrelevant components and obtain higher recovery performance [6]. The existing block sparse reconstruction methods are mainly based on l p (0 < p ≤ 1) regularisation [7] and sparse Bayesian learning (SBL) [8]. The SBL-based methods automatically estimate the signal parameters by Bayesian inference, which have much less local minimum than the l p regularisation method. Zhang Zhilin studied the characteristics of the block sparse Bayesian learning (BSBL) recovery algorithm in detail; the results showed that the BSBL algorithm had a better effect than the conventional SBL algorithm in recovering the block structure target [9].
In this paper, the BSBL algorithm is applied to multiband signal fusion ISAR imaging to get a high-resolution image of the target. Both the range dimension and cross-range dimension of the target image have the block structural characteristic, so the multiband fusion ISAR imaging based on the BSBL method can be achieved through a cross-range decoupled sparse reconstruction. Simulation and real-data experiments will be used to verify the effectiveness of the method.

Multiband fusion ISAR imaging echo model
The radars for multiband fusion ISAR imaging discussed in this paper are adjacently configured, and the target response of each sub-band is reflected by ideal scatterers. We should note here that when the radars with multiple different frequency bands observe the target simultaneously, the echoes measured by radars are incoherent due to the difference of the radar positions, the signal transmission time, and the initial phases of the radar systems. The incoherence will affect the performance of multiband fusion, so the echo data are pre-processed according to the approach proposed in [10]. In a later discussion, we focus on the fusion process of the pre-processed data.
The fusion of radar echo signals in two different sub-bands is considered in this paper. Suppose the radars transmit chirp signals. The carrier frequency and the bandwidth of sub-band 1 are f c1 and B 1 , respectively. The carrier frequency and the bandwidth of sub-band 2 are f c2 and B 2 , respectively. After translational motion compensation, the echo signals can be equivalent to the turntable model, as shown in Fig. 1. The pulse-compressed echo spectra of the two sub-bands are given by where K is the number of scatterers, σ k is the scattering coefficient of the arbitrary scatterer k on the target, R k is the range between the scatterer k and the radar. For sub-band 1, i = 1, f (1) is the fast time frequency of the baseband signal; is the starting frequency of the full band; Δ f is the frequency sampling interval; n 1 = 0, 1, …, N 1 − 1; N 1 = B 1 /Δ f is the number of samples in subband 1. For sub-band 2, i = 2, f (2) is the fast time frequency; is the number of samples in sub-band 2; N is the number of samples in the full band, N 1 + N 2 ≤ N. Let R 0 be the range between the target centroid and the radar; then where (x k , y k ) is the coordinate of the scatterer k in the target coordinate system, x k = r k cos θ 0 , y k = r k sin θ 0 , and r k is the distance between the scatterer k and the target centroid. Δθ m is the accumulative rotation angle of the target relative to the radar within m pulses; m = 0, 1, …, M − 1; M is the total number of echo pulses.
Substituting (2) into (1) and ignoring the constant phase, (1) can be written as Since Δθ m is small during the imaging process, cos Δθ m ≃ 1, sin Δθ m ≃ Δθ m . It is considered that the target is constantly rotating at the rotation velocity of ω c during the imaging time. Δθ m = ω c t m , t m = mT r is the slow time, and T r is the pulse repetition time (PRT). Then (3) can be further written as Considering the high range resolution after multiband fusion processing, the migration through resolution cell (MTRC) effect of scatterers is not negligible even with a small cumulative angle during the imaging time. Therefore, in this paper, the keystone transform of each sub-band is applied to correct the MTRC before the fusion processing. The keystone transformation of the two subbands based on the same frequency f 0 through variable substitution and sinc interpolation can eliminate the time-frequency coupling of the last term on the right side of (4): After the keystone transformation, the signals can be expressed as

Multiband signal sparse reconstruction in range dimension
3.1.1 Sparse representation of a multiband signal: Firstly, we analyse the sparse representation of multiband signals in the range dimension. For simplicity, we let which is constant for the mth pulse. Then (6) can be written as Let ω y = 2Δ f y k /c, ω y ∈ 0, 1 . Denote ω y as ω y = p/P, p = 0, 1, …, P − 1, P ≥ N. The sparse representation of the fusion signal of the mth pulse can be written as where where α P × M is the range profile matrix, and α m is the sum of σ km at the same range of the mth pulse; H and Ψ are the sampling matrix and the Fourier dictionary matrix, respectively. where In (10), I is the identity matrix and 0 is the zero matrix. The basis matrix of the multiband fusion Ψ f is obtained by

Reconstruction based on BSBL:
Considering the block structure of the target in range dimension, the BSBL method is used to reconstruct the range profile α m in (9). The signal model with noise can be rewritten as The most common block structure is that the vector α m contains g blocks, and the size of each block is d: Therefore, P = dg. Only g 0 (g 0 ≪ g) signal blocks are nonzero in signal α m . The BSBL algorithm [9] assumes that the ith (i = 1, 2, …, g) signal block follows the Gaussian distribution, and the blocks are mutually uncorrelated. Then the signal model can be given as where Γ = diag −1 γ 1 B 1 , …, γ g B g , γ i denotes the correlation of the ith block signal, and γ i B i is the covariance matrix. The matrix B i ∈ ℂ d × d represents the correlation structure information within the ith signal block. N denotes the complex Gaussian distribution.
For the measurement vector s m , it satisfies the following probability density distribution: where the noise v is independent and identically distributed, and it satisfies p v = N 0, β −1 I , β −1 is the noise power. According to (15) and (16), we can obtain the posterior probability density function and likelihood function where In order to estimate the parameters γ i , B i and β, the cost function ℒ can be obtained by the Type II likelihood maximisation method [9]: After obtaining the estimated values of the parameters γ i , B i and β, the maximum posteriori estimation of α m equals the mean of the posterior probability: So we can obtain the estimated range profile matrix α P × M from multiband signals via the BSBL sparse reconstruction pulse by pulse.

Sparse reconstruction in cross-range dimension
Considering that the scatterers of target are also distributed in blocks in the cross-range direction, a high-quality ISAR image can be obtained by using the BSBL reconstruction method in the crossrange dimension of the target's range profiles. According to (7), let ω d = 2 f 0 x k ω c t′ r /c, t r ′ = t′ m /m, ω d ∈ 0, 1 , and it can be denoted as ω d = q/Q, q = 0, 1, …, Q − 1, Q ≥ M. The range profile matrix α P × M in the pth range cell can be written as the following sparse representation: where σ P × Q is the ISAR image matrix and A M × Q is the Fourier basis matrix: where Finally, the ISAR image of the target can be obtained by reconstructing σ in (23) at each range cell via the BSBL method introduced in Section 3.1.2.

Simulation and real-data experimental results
The effectiveness of BSBL reconstruction algorithm for multiband fusion ISAR imaging is verified by simulation and measured data experiments.

Simulation of the scatterer model
The simulation parameters are as follows: the frequency of subband 1 is 10-10.5 GHz, the frequency of sub-band 2 is 11.5-12 GHz, and the frequency sampling interval Δ f is 10 MHz. The target scatterer model is shown in Fig. 2. The target includes 4 discrete scatterers and 32 continuous block scatterers, the target centroid is located at (0, 0). The interval of continuous block scatterers is 0.15 m in range dimension, which is half of the theoretical resolution of the sub-band, the interval of discrete scatterers is 0.3 m in range dimension, and the amplitudes of the scatterers are all 1. The target rotates constantly at a distance of 3,000 m away from radars during the imaging time, and the accumulated rotation angle is 1.72°. In the simulation, Gaussian white noise is added and the SNR is 20 dB after the pulse compression. Fig. 3 shows the two-sub-band fusion ISAR imaging results of the scatterer target. Figs. 3a and 3b are the ISAR images reconstructed by the traditional SBL method and the proposed BSBL method, respectively. In Fig. 3a, it can be seen that the image of the block structure target is defocused and blurred. The positions of the continuous block scatterers are incorrectly reconstructed by the SBL method, although the discrete scatterers are correctly reconstructed. While Fig. 3b shows the high-quality ISAR image of the target, both the discrete scatterers and the continuous block scatterers are correctly reconstructed by the  proposed BSBL method. The position and structure of the target in the image are consistent with the model. Therefore, the BSBL method outperforms the SBL method in multiband fusion ISAR imaging of block structure targets.

Experiment of Yak-42 real data
The measured Yak-42 aircraft data are also used to verify that a better quality ISAR image can be reconstructed using the BSBL method than using the SBL method. The radar system parameters are consistent with those in [11]. A total 128 echo pulses are used for imaging. The first 64 and the last 64 sampling points of full band are selected as the data of sub-band 1 and sub-band 2 for decoupled fusion imaging, respectively. Fig. 4a shows the result of range doppler (RD) imaging of the full-band data. It can be seen that defocus exists in the cross-range direction. Fig. 4b shows the result of the BSBL reconstruction using the data of sub-band 1. Due to the narrow bandwidth, the range resolution is low and the outline of the aircraft are very blurry. Fig. 4c shows the result of multiband fusion imaging using the SBL method. After the fusion imaging, the resolution of the image is obviously improved, but the image is still defocused, as shown by the circle mark. Fig. 4d shows the result of multiband fusion imaging using the proposed BSBL method. Compared with the first three images, Fig. 4d provides more complete outline of the aircraft, especially the nose and wings, and the image quality is better. Fig. 4 indicates that the fusion ISAR image reconstructed with the BSBL method is better than the SBL method.

Conclusions
This paper proposes to apply the BSBL method to multiband fusion ISAR imaging. With the exploitation of the block structure characteristic of targets in multiband fusion ISAR imaging, the BSBL-based method can obtain a better fusion ISAR image than the traditional SBL-based method. Simulation and real-data experiment results show that the proposed method can reconstruct high-quality images in multiband fusion ISAR imaging.