# A CSAR 3D Imaging Method Suitable for Edge Computation
## 3 A CSAR 3D Imaging Method Suitable for Edge Computation
### Principle of the Method
Ignoring the changes to \(\sigma_{\mathrm{p}}\), the scattering coefficient of the target and the sine change track of the ideal point target echo after distance compression is as follows:  
\[s_{\mathrm{r0}}(\tau,\varphi)=\delta\bigg{[}\frac{2 Bz_{\mathrm{p}}\cos(\psi)}{ c}+\frac{2 Br_{\mathrm{p}}\sin(\psi)}{c}\cos(\varphi-\theta_{\mathrm{p}})\bigg{]} \tag{9}\]  
The IRT of \(s_{\mathrm{r0}}(\tau,\varphi)\) is as follows:  
\[g(x,y)=\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}s_{ \mathrm{r0}}(\tau,\varphi)\exp\bigg{\{}-j2\pi\frac{2 Bz_{\mathrm{i}}\cos(\psi) }{c}\bigg{\}}\exp\big{\{}-j2\pi\big{[}xk_{\mathrm{x}}+yk_{\mathrm{y}}\big{]} \big{\}}dk_{\mathrm{x}}dk_{\mathrm{y}} \tag{10}\]  
where \(k_{\mathrm{x}}=\frac{2B\sin(\psi)}{c}\cos(\varphi)\) and \(k_{\mathrm{y}}=\frac{2B\sin(\psi)}{c}\sin(\varphi)\). When the height of the imaging plane is consistent with the actual size of the target, that is \(z_{\mathrm{i}}=z_{\mathrm{p}}\), a sine curve in the range-compressed echo signal is transformed into a point \((x_{\mathrm{i}},y_{\mathrm{i}})\) in the imaging plane \(x-y\) after IRT, and the relationship between the amplitude \(A_{\mathrm{i}}\) and initial phase \(\theta_{\mathrm{i}}\) of the point corresponding to the original sine curve and the point coordinates is as follows:  
\[\hat{A_{\mathrm{i}}}=\sqrt{{x_{\mathrm{i}}}^{2}+{y_{\mathrm{i}}}^{2}},\ \hat{ \theta_{\mathrm{i}}}=\arctan\frac{y_{\mathrm{i}}}{x_{\mathrm{i}}} \tag{11}\]  
The IRT of the compressed echo of the N ideal point target distance is:  
\[I(x,y)=\sum\limits_{i=1}^{N}\delta(x-A_{\mathrm{i}}\sin\theta_{\mathrm{i}}) \delta(y-A_{\mathrm{i}}\cos\theta_{\mathrm{i}}) \tag{12}\]  
The height of the imaging plane is inconsistent with the actual height of the target; that is, \(z_{\mathrm{i}}\
eq z_{\mathrm{p}}\). Based on CSAR confocal 3D imaging theory [13], only when the imaging height is consistent with the target height can the target on the 2D image be accurately focused; On the contrary, at other imaging heights, the target on the 2D image will be defocused into a circle, and the radius of the circle has a linear relationship with the deviation between the imaging height and the target height. Then, the sine curve is transformed into a circle in the imaging plane \(x-y\) by IRT, and the relationship between the radius of the circle \(\Delta r\) and the height deviation \(\Delta h=z_{\mathrm{p}}-z_{\mathrm{i}}\) can be defined as follows:  
\[\Delta r=\Delta h\cdot\tan(\psi) \tag{13}\]  
In CSAR imaging mode, the incident angle \(\psi\) is constant, and the relationship between the radius of the circle \(\Delta r\) and the height deviation \(\Delta h\) is linear. As shown in Figure 1, the target focus point \(P\), which is the vertex of the cone formed by circles with different heights, is the intersection point of the cone generatrix passing through the vertex. Hough transform is an effective method to detect the intersection of many straight lines [14]. In the Hough-transformed space \((\rho,\theta)\) of the binary image, the meeting of two line segments on the image is the sine curve that passes through two peak points simultaneously. According to the coordinates \((\rho_{1},\theta_{1})\) and \((\rho_{2},\theta_{2})\), which are the two peak points, the intersection coordinates of two line segments in the image can be calculated via the Hough transformation formula as follows:  
\[\left\{ \begin{array}{l}\rho_{1}=x_{0}\cos\theta_{1}+y_{0}\sin\theta_{1}\\ \rho_{2}=x_{0}\cos\theta_{2}+y_{0}\sin\theta_{2}\end{array} \right.\Rightarrow \left\{ \begin{array}{l}x_{0}=\frac{\rho_{2}\sin\theta_{1}-\rho_{1}\sin\theta_ {2}}{\sin(\theta_{1}-\theta_{2})},x_{0}\in[0,x_{\max}]\\ y_{0}=\frac{\rho_{1}\cos\theta_{2}-\rho_{2}\cos\theta_{1}}{\sin(\theta_{1}- \theta_{2})},y_{0}\in[0,y_{\max}]\end{array} \right. \tag{14}\]