## 3 The Proposed Method
### Rcmc Based on Hough-SOKT  
When the squint angle or the radial velocity of the target is large, the Doppler center shift is large, and the range walk is usually larger than the range curvature. In order to reduce the coupling effect of range walk and range curvature, the first-order term in Equation (3) must be corrected to remove the main component in the range migration. In the two-dimensional time domain, the range walk appears as an oblique line in the image, and its slope is proportional to the radial velocity of the target [23]. Therefore, Hough transform can be used to transform the estimation of the range walk into the estimation of walk inclination to realize range walk correction. In the two-dimensionaltime domain, Hough transform uses the trajectory information of the moving target to estimate parameters, so there is no need to worry about the influence of Doppler spectrum ambiguity on the estimation result of the Doppler centroid. The schematic diagram of the two-dimensional time domain Hough transform is shown in Figure 2, where \(\theta\) is the angle between the target's moving direction and the azimuth direction, and the Hough transforms a straight line in the coordinate space into a peak point in the parameter space \((\rho,\theta)\). The inclination of the line can be estimated by detecting the position of the peak in the transform domain. The relationship between the estimated value of the radial velocity \(v_{r}\) and \(\theta\) can be written as [23]  
\[\widehat{v_{r}}=\frac{c\cdot PRF}{2f_{s}}\tan(\theta) \tag{9}\]  
where \(PRF\) and \(f_{s}\) denote the azimuth and range sampling frequencies, respectively. The estimated Doppler center is \(f_{ref}=2\widehat{v_{r}}/\lambda\). In the azimuth-time domain and range-frequency domain, the required phase compensation for the range walk in Equation (6) can be written as [10; 24]  
\[H_{fdc}(f_{r},t_{a})=\exp\left\{-j2\pi f_{ref}\frac{f_{r}+f_{c}}{f_{c}}t_{a}\right\} \tag{10}\]  
For Equation (6), error in \(v_{r}\) leads to the residual of the range walk after correction. Therefore, the signal of the moving target after range walk correction can be expressed as  
\[S_{2}(f_{r},t_{a})=Aw_{a}(t_{a})rect\left(\frac{f_{r}}{B}\right)\exp\left\{-j \frac{4\pi(f_{r}+f_{c})}{c}\left(R_{0}+a^{\prime}1_{a}+a_{2}t_{a}^{2}+a_{3}t_{ a}^{3}\right)\right\} \tag{11}\]  
where \(a^{\prime}_{1}=a_{1}+\widehat{v_{r}}\) is the residual first-order term after range walk correction. After the range walk correction, the residual amount of the first-order term is small, and the main component of the residual range migration is the range curvature. The SOKT is used to correct the range curvature [10; 19; 25]. The new azimuth-time variable is denoted by \(t_{m}\), and the azimuth-time scale transformation relation is as follows [8]  
\[t_{a}=\sqrt{\frac{f_{c}}{f_{r}+f_{c}}}\cdot t_{m} \tag{12}\]  
Substituting Equation (12) into Equation (11), the azimuth-time domain and range-frequency domain signal after the SOKT is  
\[S_{3}(f_{r},t_{m})=Aw_{a}(t_{m})rect\left(\frac{f_{r}}{B}\right)\exp\left\{-j \Phi_{1}(f_{r},t_{m})\right\} \tag{13}\]  
Figure 2: Hough transform schematic (the red line is the estimated range walk, and the black curve is the real range migration). The expression of \(\Phi_{1}(f_{r},t_{m})\) in the above equation is written as  
\[ \begin{split}\Phi_{1}(f_{r},t_{m})&=\frac{4\pi(f_{r}+f _{c})}{c}R_{0}+\frac{4\pi}{c}\sqrt{f_{c}(f_{r}+f_{c})}a^{\prime}{}_{1}t_{m}\\ &\quad+\frac{4\pi f_{c}}{c}a_{2}t_{m}^{2}+\frac{4\pi f_{c}}{c} \sqrt{\frac{f_{c}}{f_{c}+f_{r}}}a_{3}t_{m}^{3}\end{split}  \tag{14}\]  
In the range-frequency domain and azimuth-time domain, the coupling between the quadratic term \(t_{m}^{2}\) and \(f_{r}\) has been removed after the SOKT, indicating that the range curvature has been corrected. Generally, the SAR systems satisfies \(f_{r}\ll f_{c}\), and there is an approximate relationship [19]  
\[