Refocusing of Moving Targets in SAR Images via Parametric Sparse Representation

Abstract

In this paper, a parametric sparse representation (PSR) method is proposed for refocusing of moving targets in synthetic aperture radar (SAR) images. In regular SAR images, moving targets are defocused due to unknown motion parameters. Refocusing of moving targets requires accurate phase compensation of echo data. In the proposed method, the region of interest (ROI) data containing the moving targets are extracted from the complex SAR image and represented in a sparse fashion through a parametric transform, which is related to the phase compensation parameter. By updating the reflectivities of moving target scatterers and the parametric transform in an iterative fashion, the phase compensation parameter can be accurately estimated and the SAR images of moving targets can be refocused well. The proposed method directly operates on small-size defocused ROI data, which helps to reduce the computational burden and suppress the clutter. Compared to other existing ROI-based methods, the proposed method can suppress asymmetric side-lobes and improve the image quality. Both simulated data and real SAR data collected by GF-3 satellite are used to validate the effectiveness of the proposed method.

1. Introduction

Moving target imaging is an important task of synthetic aperture radar (SAR) [1,2,3,4,5,6]. Since SAR imaging algorithms were originally designed for stationary targets, the main challenge for SAR imaging of moving targets is to compensate the phase error caused by non-cooperative motion of targets. It is well known that the Doppler frequency shift caused by target movement results in the imaging position offset along the azimuth direction. Further, the change of the azimuth frequency modulation rate, which is related to the acceleration in range and the velocity in azimuth of the target, introduces defocus of moving targets in regular SAR images [7]. From the viewpoint of process flow, the existing methods of SAR moving target imaging can be generally divided into two types. (1) Methods based on the raw radar data [8,9,10,11,12]. These methods deal with the entire echo data reflected from both stationary background and moving targets. In fact, only a small part of the entire data is related to the moving targets, and therefore, processing on entire data will induce a large amount of clutter. The displaced phase center antenna (DPCA) and along track interferometry (ATI) can effectively suppress the clutter [13,14], which requires multiple channels and therefore is not suitable for single-channel SAR systems. (2) Methods based on the region of interest (ROI) data [15,16,17]. This type of methods can effectively remove clutter and easily detect moving targets. Since only a small image contacting the ROI data is extracted from regular SAR imaging result, the amount of data to be processed is significantly reduced. Zhang et al. [16] proposed a high-resolution SAR imaging method of ground moving targets with defocused ROI data. The authors derive an exact analytic expression of the ROI data without approximation of the slant range. Ref. [16] assumed that the phase error arises from the unknown azimuth and range velocities in a specific model. Then, the azimuth and range equivalent velocities are estimated by a 2-D search, such that the maximum contrast of moving target image is reached. The image of target is obtained by Stolt interpolation with each pair of equivalent velocity parameters. However, the equivalent velocity cannot reflect the actual motion parameters of target in acceleration state, which results in the difficulty of determining the search interval of equivalent velocities. Even if the accurate equivalent velocities are estimated, the model mismatch problem would introduce high asymmetric side-lobes in the imaging result when the target is in an acceleration state.

Since the radar echo reflected from man-made moving targets are usually stronger than the background, in recent years, many sparsity-aware methods have been applied to SAR moving target parameter estimation and imaging [18,19,20,21,22,23,24,25]. Ref. [23,25] summarize the latest application of sparse processing in SAR systems. An approach to motion parameter estimation with low pulse repetition frequency based on compressed sensing (CS) theory is proposed in [19]. Ref. [20] also proposes a method for motion parameter estimation of moving target based on the raw radar data. However, complicated clutter suppression is required before motion parameter estimation. Ref. [21] presents a method for imaging of moving targets with multi-static SAR using an overcomplete dictionary. In Ref. [22], the authors divide a phase error into three subcategories and correct them by using a nonquadratic regularization approach. However, due to the heavy computational burden, this method may not be suitable for these targets occupying a large number of resolution cells. Onhon et al. utilize the sparsity-driven autofocus framework to solve the problem of moving target imaging in [18], where the phase error induced by target movement is corrected by the non-quadratic regularization approach. However, this method ignores the relationship between the phase error and the motion parameters. It needs to independently estimate the phase error of each sample data, thus increasing the unnecessary computational burden.

In this paper, we propose a parametric sparse representation method for SAR imaging of moving targets with ROI data. The parametric sparse representation technique has been utilized to ISAR imaging of rotating targets [26,27,28], SAR motion compensation [29], and moving target motion parameters estimation [12]. Differing from the previous work where the process flow starts from entire raw data, the proposed method in this paper is based on ROI data. Firstly, the complex image in ROI that contains the defocused moving target is extracted from the regular SAR image. Then, the principal components of the ROI data, which correspond to the dominant scatterers of a moving target, are represented by a parametric transform that is uniquely determined by a phase compensation parameter. By updating the reflectivities of the target and the parametric transform in an iterative fashion, the phase compensation parameter can be accurately estimated and the focused image of moving target can be obtained when the iterative process converges. Differing from the method in [16], which realizes the phase compensation by searching 2-D equivalent velocities of moving targets, in this paper, the phase compensation is achieved by estimating the phase compensation parameter in an iterative fashion. Experimental results based on both simulated and space-borne SAR data demonstrate that the proposed method outperforms the method in [16] in terms of the imaging quality of moving targets. Simulations also show that, compared to the method in [16], the proposed algorithm in this paper has a higher tolerance for model mismatch problem and is capable of providing satisfactory refocused image even when the target is moving in acceleration state.

The rest of this paper is organized as follows. The SAR signal model is reviewed in Section 2. The proposed method for moving target imaging is formulated in Section 3. The performance of the proposed method is evaluated, with both simulated and measured data, in Section 4. Conclusions are presented in Section 5.

2. Signal Model

The geometry relationship between the platform and a moving target for side-looking SAR is shown in Figure 1. The horizontal axis denotes the azimuth direction, and the vertical axis denotes the slant-range direction. Assume that the SAR platform flies straight at a speed of $V$ along the x-axis, and the velocities of a moving target in azimuth and range are $v_x$ and $v_r$, respectively.

The slow time is denoted as $t_s=nT$ with an integer $n$, and $T$ is the pulse repetition time. Thus, the radar antenna phase center (APC) position at $t_s$ is $\left(V{t}_s,0\right)$, and the position of the moving target is $\left(x_0+v_x{t}_s,r_0+v_r{t}_s\right)$, where $\left(x_0,r_0\right)$ is the initial position when $t_s=0$. The instantaneous distance between the moving target and the radar can be expressed as

$$R\left(t_s\right)=\sqrt{{\left(V{t}_s-x_0-v_x{t}_s\right)}^2+{\left(r_0+v_r{t}_s\right)}^2}$$

(1)

Suppose that the radar transmits a linear frequency-modulated (LFM) signal as

$$s\left(t,t_s\right)=\mathrm{rect}\left(t/T_p\right)\cdot \exp \left(j2\pi f_{c}t+j\pi \gamma t^2\right),$$

(2)

where $t$ is the fast time, $T_p$ is the pulse width, $\mathrm{rect}(\cdot )$ denotes the rectangular function, $f_c$ is the carrier frequency, and $\gamma$ is the chirp rate. The baseband echo of the moving target can be expressed as

$$s_r\left(t,t_s\right)=\sigma \cdot \mathrm{rect}\left(\frac{t-2R\left(t_s\right)/c}{T_p}\right)\cdot \mathrm{rect}\left(\frac{t_s}{T_a}\right)\cdot \exp \left(-j4\pi f_{c}R\left(t_s\right)/c+j\pi \gamma {\left(t-2R\left(t_s\right)/c\right)}^2\right),$$

(3)

where $T_a$ is the synthetic aperture time, and the target scattering coefficient $\sigma$ is assumed to be constant during the observation time interval. By taking the two-dimensional Fourier transform, the signal of the moving target in 2-D frequency domain can be expressed as

$$\begin{array}{l}{S}_r\left(f_r,f_a\right)\\ =\sigma \cdot W_r\left(f_r\right)\cdot W_a\left(f_a\right)\cdot \exp \left\{j\cdot \left[-2\pi f_a\frac{\delta }{v_e^2}-\pi \frac{f_r^2}{\gamma }-\frac{4\pi \left[x_0{v}_r+r_0\left(V-v_x\right)\right]}{c{v}_e}\sqrt{{\left(f_c+f_r\right)}^2-\frac{c^2{f}_a^2}{4v_e^2}}\right]\right\},\end{array}$$

(4)

where $f_r$ and $f_a$ are the range and azimuth frequencies, respectively, $W_r\left(f_r\right)$ and $W_a\left(f_r\right)$ are the range and azimuth envelope function, respectively, $\delta =x_0\left(V-v_x\right)-r_0{v}_r$ and $v_e=\sqrt{{\left(V-v_x\right)}^2+v_r^2}$. After conventional matched filtering and Stolt interpolation, (4) becomes [16]

$$\begin{array}{l}{S}_{r\_Stolt}\left(f_r,f_a\right)\\ =\sigma \cdot W_r\left(f_r\right)\cdot W_a\left(f_a\right)\cdot \exp \left\{j\cdot \left[-2\pi f_a\frac{\delta }{v_e^2}-\frac{4\pi \left[x_0{v}_r+r_0\left(V-v_x\right)\right]}{c{v}_e}\sqrt{{\left(f_c+f_r\right)}^2+\frac{c^2{f}_a^2}{4}\left(\frac{1}{V^2}-\frac{1}{v_e^2}\right)}\right]\right\}.\end{array}$$

(5)

It can be seen from (5) that the high-order residual items still exist after the conventional SAR imaging processing, which means that the image of the moving target is defocused. To achieve a refocused image of moving target, it is necessary to compensate the residual phase items. The filter for residual phase compensation can be constructed as [16]

$$H_1\left(f_r,f_a\right)=\exp \left\{j\cdot \left[\frac{4\pi R_{ref}}{c}\sqrt{{\left(f_c+f_r\right)}^2+\frac{c^2{f}_a^2}{4}\left(\frac{1}{V^2}-\alpha \right)}-\frac{4\pi R_{ref}}{c}\left(f_c+f_r\right)\right]\right\},$$

(6)

where $R_{ref}$ is the reference distance, and $\alpha$ is the phase compensation parameter, which is the parameter to be estimated in this paper. The phase compensation parameter is related to the motion parameters of target, and it can be expressed as $\alpha =1/\left[{\left(V-v_x\right)}^2+v_r^2\right]$ when the target is in uniform motion state. The phase compensation parameter becomes $\alpha =1/V^2$ when the target is stationary. This results in $H_1\left(f_r,f_a\right)=1$ in (6), which means that no phase compensation is required for the data reflected from stationary targets. In the proposed method, we can initialize the phase compensation parameter as $1/V^2$. By taking the two-dimensional inverse Fourier transform in (5), we can obtain the regular SAR complex image of observed scene, in which the sub-image that containing moving target is the so-called ROI data. The extracted ROI sub-image from the focused stationary background can be expressed as

$$s_{r\_ROI}\left(\overline{t},{\overline{t}}_s\right)=\sigma \cdot W_r\left(\overline{t}-\frac{2R_{ROI}\left({\overline{t}}_s\right)}{c}\right)\cdot W_a\left({\overline{t}}_s-\frac{\delta }{v_e^2}\right)\cdot \exp \left(-j\frac{4\pi }{\lambda }{R}_{ROI}\left({\overline{t}}_s\right)\right),$$

(7)

where $\overline{t}$ and ${\overline{t}}_s$ denote the sampling index of ROI date in range and azimuth time domain, respectively, and

$$R_{ROI}\left({\overline{t}}_s\right)=\sqrt{{\left(\frac{x_0{v}_r+r_0\left(V-v_x\right)}{v_e}\right)}^2-{\left({\overline{t}}_s-\frac{\delta }{v_e^2}\right)}^2\cdot \left(\frac{1}{V^2}-\frac{1}{v_e^2}\right)},$$

(8)

In the next section, we will describe how to estimate the phase compensation parameter and refocus target image with the ROI data.

3. Parametric Sparse Representation Method for Moving Target Imaging

In this section, we formulate the parametric sparse representation method for imaging of moving targets. We denote the original image size as $N_r\times N_a$, and the ROI data size as $n_r\times n_a$. Note that the data size is significantly reduced, but the signal bandwidth of moving target remains unchanged. We define the range and azimuth frequency of ROI data as ${\overline{f}}_r$ and ${\overline{f}}_a$, respectively. By taking two-dimensional Fourier transform in Equation (7), the ROI data in 2-D frequency domain can be expressed as

$$\begin{array}{l}{S}_{r\_ROI}\left({\overline{f}}_r,{\overline{f}}_a\right)\\ =\sigma \cdot W_r\left({\overline{f}}_r\right)\cdot W_a\left({\overline{f}}_a\right)\cdot \exp \left\{j\cdot \left[-2\pi {\overline{f}}_a\frac{\delta }{v_e^2}-\frac{4\pi \left[x_0{v}_r+r_0\left(V-v_x\right)\right]}{c{v}_e}\sqrt{{\left(f_c+{\overline{f}}_r\right)}^2+\frac{c^2{\overline{f}}_a^2}{4}\left(\frac{1}{V^2}-\frac{1}{v_e^2}\right)}\right]\right\}.\end{array}$$

(9)

The phase compensation filter of ROI data can be rewritten as

$$H_{2\left(\alpha \right)}\left({\overline{f}}_r,{\overline{f}}_a\right)=\exp \left\{j\cdot \left[\frac{4\pi R_{ref}}{c}\sqrt{{\left(f_c+{\overline{f}}_r\right)}^2+\frac{c^2{\overline{f}}_a^2}{4}\left(\frac{1}{V^2}-\alpha \right)}-\frac{4\pi R_{ref}}{c}\left(f_c+{\overline{f}}_r\right)\right]\right\}.$$

(10)

The refocused image of the moving target can be obtained through a refocusing transform $\mathsf{\Gamma }(\cdot )$:

$$\mathsf{\Theta }=\mathsf{\Gamma }\left(s_{r\_ROI}\right)={\mathsf{\Psi }}_r^{-1}\cdot \left[\left({\mathsf{\Psi }}_r\cdot s_{r\_ROI}\cdot {\mathsf{\Psi }}_a\right)\circ H_{2\left(\alpha \right)}\right]\cdot {\mathsf{\Psi }}_a^{-1},$$

(11)

where $\mathsf{\Theta }$ is the refocusing result of moving target, $\mathsf{\Psi }$ and ${\mathsf{\Psi }}^{-1}$ are the Fourier matrix and the inverse Fourier matrix, respectively. The subscripts $r$ and $a$ denote the range direction and azimuth direction, respectively, and $\circ$ denotes Hadamard product. As shown in Equation (11), through a series of matrix operations, the refocusing transform $\mathsf{\Gamma }(\cdot )$ can achieve the conversion from the defocused ROI data to the refocusing result of moving target. It is obvious that the image quality of $\mathsf{\Theta }$ depends on the phase compensation parameter $\alpha$, i.e., different phase compensation parameter produces different filter function $H_{2\left(\alpha \right)}$ and, therefore, different refocusing result $\mathsf{\Theta }$. Thus, $\mathsf{\Theta }$ can be regarded as a function of $\alpha$ and denoted by ${\mathsf{\Theta }}_{\left(\alpha \right)}$. With a wrong value of $\alpha$, the refocusing transform $\mathsf{\Gamma }\left(s_{r\_ROI}\right)$ will induce the high-order phase error in azimuth and range, and thus a blurred image is most likely obtained. With accurate estimate of $\alpha$, the phase error of the ROI data can be well compensated, and accordingly, a well-focused moving target image will be obtained. From the above consideration, refocusing of the moving target can be carried out in an iterative fashion, i.e., the sparse imaging result $\mathsf{\Theta }$ and the phase compensation parameter $\alpha$ are iteratively updated.

3.1. Update the Sparse Solution

At the p-th iteration, denotes the estimate of the phase compensation parameter ${\alpha }^{\left(p\right)}$. Accordingly, the filter function $H_{2\left({\alpha }^{\left(p\right)}\right)}$ and refocusing transform $\mathsf{\Gamma }\left(s_{r\_ROI}\right)$ can be constructed according to Equations (10) and (11). The refocusing process described in (11) is reversible, and the inverse transform can be written as

$$s_{r\_ROI}={\mathsf{\Gamma }}^{-1}\left(\mathsf{\Theta }\right)={\mathsf{\Psi }}_r^{-1}\cdot \left[\left({\mathsf{\Psi }}_r\cdot \mathsf{\Theta }\cdot {\mathsf{\Psi }}_a\right)\circ H_{2\left(\alpha \right)}^{*}\right]\cdot {\mathsf{\Psi }}_a^{-1},$$

(12)

where ${(\cdot )}^{*}$ denotes the conjugate operation. Since the moving target is usually sparse in 2-D space domain, we can obtain the moving target imaging results by solving the following unconstrained problem [30,31]

$$\underset{\mathsf{\Theta }}{\min }{\Vert s_{r\_ROI}-{\mathsf{\Gamma }}^{-1}\left(\mathsf{\Theta }\right)\Vert }_2^2+\lambda {\Vert \mathsf{\Theta }\Vert }_1,$$

(13)

where ${\Vert \cdot \Vert }_1$ and ${\Vert \cdot \Vert }_2$ denote $l_1$ and $l_2$ norms, respectively, and $\lambda >0$ is the regularization parameter that balances the recovery error and the sparsity of the solution. Given the value of ${\alpha }^{\left(p\right)}$, Equation (13) is a standard problem of sparse signal recovery. In this paper, we utilize the soft iterative thresholding algorithm [32] to solve Equation (13). The main steps are summarized below.

Soft Iterative Thresholding Algorithm

Input: $s_{r\_ROI}$, $\lambda$, ${\alpha }^{\left(p\right)}$, $\mathsf{\Gamma }(\cdot )$, and ${\mathsf{\Gamma }}^{-1}(\cdot )$

Initialization: Let the iterative counter $k=1$, residual matrix $R_0=s_{r\_ROI}$, and ${\widehat{\mathsf{\Theta }}}_0={\mathbf{0}}^{n_r\times n_a}$.

Iteration: at the k-th iteration $\left(k>1\right)$

(1) Update the sparse result by ${\widehat{\mathsf{\Theta }}}_k=\mathrm{soft}\left({\widehat{\mathsf{\Theta }}}_{k-1}+\mathsf{\Gamma }\left(R_{k-1}\right),\lambda \right)$, where $\mathrm{soft}\left(x,\lambda \right)=\mathrm{sign}\left(x\right)\cdot \max \left(\left|x\right|-\lambda ,0\right)$.

(2) Update the residual matrix by $R_k=s_{r\_ROI}-{\mathsf{\Gamma }}^{-1}\left({\widehat{\mathsf{\Theta }}}_k\right)$.

(3) Increment k, and return to Step (1) until the stopping criterion is met. Here the stopping criterion is ${\Vert {\widehat{\mathsf{\Theta }}}_k-{\widehat{\mathsf{\Theta }}}_{k-1}\Vert }_2/{\Vert {\widehat{\mathsf{\Theta }}}_{k-1}\Vert }_2\le \epsilon$. The selection of the threshold value $\epsilon$ is related to the precision requirement.

Output: ${\widehat{\mathsf{\Theta }}}^{\left(p\right)}={\widehat{\mathsf{\Theta }}}_k$.

3.2. Update the Estimate of Phase Compensation Parameter

Given the sparse solution ${\widehat{\mathsf{\Theta }}}^{\left(p\right)}$, the phase compensation parameter estimate can be updated by

$${\alpha }^{\left(p+1\right)}={\alpha }^{\left(p\right)}+\zeta \cdot \rho ,$$

(14)

where $\zeta$ and $\rho$ are iterative direction and iterative step-length, respectively. Further, the parameters $\left\{\zeta ,\rho \right\}$ can be estimated by minimizing the recovery error, that is

$$\min {\Vert s_{r\_ROI}-{\mathsf{\Gamma }}^{-1}\left({\widehat{\mathsf{\Theta }}}^{\left(p\right)}\right)\Vert }_{\mathrm{F}},$$

(15)

where ${\Vert \cdot \Vert }_{\mathrm{F}}$ denotes F-norm of matrix. Substituting Equations (12) into (15), we have

$$\min {\Vert \mathbf{Y}-\mathbf{U}\circ H_{2\left(\alpha \right)}^{*}{\mathsf{\Psi }}_a^{-1}\Vert }_{\mathrm{F}},$$

(16)

where $\mathbf{Y}={\mathsf{\Psi }}_r\cdot s_{r\_ROI}$, and $\mathbf{U}={\mathsf{\Psi }}_r\cdot {\widehat{\mathsf{\Theta }}}^{\left(p\right)}\cdot {\mathsf{\Psi }}_a$. Define $B_{\left(\alpha \right)}=\mathbf{U}\circ H_{2\left(\alpha \right)}^{*}$. According to (9) and (10), we have the expression of each element of $B_{\left(\alpha \right)}$ as

$$B_{\left(\alpha \right)}\left({\overline{f}}_r,{\overline{f}}_a\right)=U\left({\overline{f}}_r,{\overline{f}}_a\right)\cdot \exp \left\{-j\left[\frac{4\pi R_{ref}}{c}\sqrt{{\left(f_c+{\overline{f}}_r\right)}^2+\frac{c^2{\overline{f}}_a^2}{4}\left(\frac{1}{V^2}-\alpha \right)}-\frac{4\pi R_{ref}}{c}\left(f_c+{\overline{f}}_r\right)\right]\right\}$$

(17)

By taking the first-order Taylor expansion of $B_{\left(\alpha \right)}$, we have

$$B_{\left(\alpha \right)}=B_{\left({\alpha }^{\left(p\right)}\right)}+{\frac{d{B}_{\left(\alpha \right)}}{d\alpha }|}_{\alpha ={\alpha }^{\left(p\right)}}\cdot \mathrm{\Delta }\alpha$$

(18)

where $\mathrm{\Delta }\alpha$ is the first-order increment. According to Equation (17), the derivative of $B_{\left(\alpha \right)}$ can be expressed as

$$\frac{d{B}_{\left(\alpha \right)}\left({\overline{f}}_r,{\overline{f}}_a\right)}{d\alpha }=B_{\left(\alpha \right)}\left({\overline{f}}_r,{\overline{f}}_a\right)\cdot \left(j\frac{\pi R_{ref}c{\overline{f}}_a^2}{2\sqrt{{\left(f_c+{\overline{f}}_r\right)}^2+\frac{c^2{\overline{f}}_a^2}{4}\left(\frac{1}{V^2}-\alpha \right)}}\right)$$

(19)

Then, Equation (16) can be rewritten as

$$\min {\Vert \mathbf{Y}-B_{\left({\alpha }^{\left(p\right)}\right)}\cdot {\mathsf{\Psi }}_a^{-1}-{\frac{d{B}_{\left(\alpha \right)}}{d\alpha }|}_{\alpha ={\alpha }^{\left(p\right)}}\cdot {\mathsf{\Psi }}_a^{-1}\cdot \mathrm{\Delta }\alpha \Vert }_{\mathrm{F}}$$

(20)

Define

$${\mathrm{\Xi }}^{\left(p\right)}={\left[\mathbf{Y}-B_{\left({\alpha }^{\left(p\right)}\right)}\cdot {\mathsf{\Psi }}_a^{-1}\right]}_{Vector},$$

(21a)

$${\mathrm{{\rm Z}}}^{\left(p\right)}={\left[{\frac{d{B}_{\left(\alpha \right)}}{d\alpha }|}_{\alpha ={\alpha }^{\left(p\right)}}\cdot {\mathsf{\Psi }}_a^{-1}\right]}_{Vector},$$

(21b)

where ${[\cdot ]}_{Vector}$ denotes the operation of stacking vectors one underneath the other sequentially, i.e., the size of ${\mathrm{\Xi }}^{\left(p\right)}$ and ${\mathrm{{\rm Z}}}^{\left(p\right)}$ are $n_r{n}_a\times 1$. Define

$$C^{\left(p\right)}=\left[\begin{array}{c}\mathrm{real}\left({\mathrm{{\rm Z}}}^{\left(p\right)}\right)\\ \mathrm{imag}\left({\mathrm{{\rm Z}}}^{\left(p\right)}\right)\end{array}\right],$$

(22a)

$$D^{\left(p\right)}=\left[\begin{array}{c}\mathrm{real}\left({\mathrm{\Xi }}^{\left(p\right)}\right)\\ \mathrm{imag}\left({\mathrm{\Xi }}^{\left(p\right)}\right)\end{array}\right],$$

(22b)

where $\mathrm{real}(\cdot )$ and $\mathrm{imag}(\cdot )$ denote the real and imaginary parts of a complex number, respectively. Then, the real-valued solution of the first-order increment can be directly obtained as

$$\mathrm{\Delta }{\alpha }^{\left(p\right)}={\left({\left[C^{\left(p\right)}\right]}^{\mathrm{T}}{C}^{\left(p\right)}\right)}^{-1}{\left[C^{\left(p\right)}\right]}^{\mathrm{T}}{D}^{\left(p\right)},$$

(23)

where ${[\cdot ]}^{\mathrm{T}}$ denotes the transpose operation. Then the iterative direction and iterative step-length can be calculated as

$$\{\begin{array}{l}\zeta =\mathrm{\Delta }\alpha /\left|\mathrm{\Delta }\alpha \right|\\ \rho =\kappa \cdot \left|\mathrm{\Delta }\alpha \right|\end{array},$$

(24)

where $\kappa$ is the convergence parameter associated with the convergence speed, and it can be determined empirically. In the experiments of this paper, we set $\kappa =10$. Finally, the phase compensation parameter can be updated by Equation (14).

By iteratively updating $\mathsf{\Theta }$ and $\alpha$ as described above, the complete procedure of the parametric sparse representation method for moving target imaging is summarized in Figure 2. The iterative process is terminated until $\left|{\alpha }^{\left(p+1\right)}-{\alpha }^{\left(p\right)}\right|<\eta$, where $\eta$ is the convergence threshold.

4. Experimental Results

The experimental results based on simulated data and real SAR data collected by GF-3 satellite are presented in this section to demonstrate the effectiveness of the proposed method.

4.1. Simulated Data

First of all, the proposed method is verified by using simulated data. The main system parameters are as follows. The carrier frequency is 10 GHz, the scene center range is 10 km, the platform velocity is 150 m/s, the transmitted bandwidth is 300 MHz, and the pulse duration is $2.2\mathsf{\mu }\mathrm{s}$. The simulated scene contains 2 stationary reference scatterers (S1–S2) and a rigid-body moving target consisting of 4 scatterers (M1–M4), as shown in Figure 3. We first assume that the target only has constant-speed components in azimuth and range without acceleration. The actual velocity components of the target are set to be $v_x=10\mathrm{m}/\mathrm{s}$ and $v_r=5\mathrm{m}/\mathrm{s}$. Figure 4 shows the imaging result obtained by the range migration algorithm (RMA) with the entire data, where the defocused sub-image containing the moving target, i.e., ROI, is indicated by the red dashed box.

By extracting the ROI sub-image from the regular SAR image, the data size is reduced to $n_r\times n_a$, i.e., $30\times 1051$. In the proposed method, the initialized value of phase compensation parameter is set as ${\alpha }^{\left(0\right)}=1/{150}^2$. The convergence threshold in Figure 2 is set as $\eta ={\alpha }^{\left(0\right)}/{10}^4$. The convergence process of the phase compensation parameter estimation is shown in Figure 5. For comparison, the algorithm proposed in [16] is also employed in this experiment. The defocused ROI data and refocused results are shown in Figure 6. By 2-D searching, the method of [16] can accurately estimate the target equivalent velocity. From Figure 6b,c, we can see that the image produces by the proposed method has a stronger contrast. The reason is that the sparse based method can effectively suppress the side-lobes.

In the second simulation, the target in acceleration state is considered. The actual motion parameters of the point target are set to be $v_x=10\mathrm{m}/\mathrm{s}$, $v_r=5\mathrm{m}/\mathrm{s}$, azimuth acceleration $a_x=1\mathrm{m}/{\mathrm{s}}^2$, and range acceleration $a_r=1\mathrm{m}/{\mathrm{s}}^2$. By extracting the ROI sub-image from the regular SAR image, the data size is $41\times 2201$. The convergence process of the phase compensation parameter estimation is shown in Figure 7. The refocusing results obtained by the method in [16], and the proposed method are shown in Figure 8a,b, respectively. It is clear that the focusing performance of the method in [16] is seriously deteriorated. The reason is that this method cannot eliminate the high-order phase error caused by accelerative motion of the target. Moreover, we take the scatterer M3 on the target as example to analyze the refocusing quality. The range and azimuth profiles obtained by different methods are shown in Figure 9 and Figure 10, respectively. One can find that the sub-image produced by the proposed method can effectively suppress the asymmetric side-lobe and improve the image quality, which benefits from the superiority of the sparse constraint.

4.2. Space-Borne Measured Data

We now present the experimental results by using the GF-3 space-borne SAR data containing moving ships. The regular SAR image of the sea surface is shown in Figure 11. We can see that the whole image is focused well, but some ships are defocused due to their movements. Three ROIs named T1, T2, and T3 containing moving ships are cropped from the original complex image, respectively. These three ships are processed sequentially using the method in [16] and the proposed method, respectively. Figure 12 shows the convergence processes of the phase compensation parameter of these moving ships. The refocused results obtained by the method in [16] and the proposed method are compared in Figure 13, Figure 14 and Figure 15. It can be observed that the proposed method can successfully reconstruct images of moving ships and significantly suppress the side-lobes. To quantitatively compare different algorithms in terms of the image quality, the image entropy values of the refocused ROI images are listed in

Table 1. The image entropy of the refocused ROI images.

Targets and Methods	Ship T1		Ship T2		Ship T3
	Ref. [16]	PSR	Ref. [16]	PSR	Ref. [16]	PSR
Image Entropy	6.4149	3.5389	6.3011	3.9863	5.077	4.0953

. The smaller value of image entropy means better focusing effect. As shown in Table 1, the proposed method can provide better image quality than the method in [16].

5. Conclusions

In this paper, we have presented a parametric sparse representation method for moving target imaging in SAR with ROI data. The ROI data extracted from regular SAR images is represented in a sparse fashion through a parametric refocusing transform. Then, the spare image of the target and the phase compensation parameter are estimated by solving a joint the optimization problem through an iterative process. The proposed method works on the complex ROI data, rather than the raw entire data, which is helpful in reducing the amount of data and alleviating the clutter. Particularly, the proposed method can suppress asymmetric side-lobes and improve the image quality of moving targets, compared to the method in [16]. Experimental results based on both simulated and real space-borne SAR data validate the effectiveness of this method on refocusing the image of moving target using ROI data. Through our experiments, with the initial values as set in Section IV, we have never encountered the case where the proposed algorithm does not converge. A theoretical convergence analysis of the proposed algorithm will be studied in our future work.