# Research on Synthetic Aperture Radar Processing for the Spaceborne Sliding Spotlight Mode
## 5 PFA Algorithm for Sliding Spotlight SAR  
### PFA Algorithm Based on Image Segmentation  
The Polar Format Algorithm is a classical spotlight SAR imaging algorithm [24]. The algorithm uses the polar coordinate scheme to store the data, and effectively solves the problem of the cross-resolution unit moving away from the central scattering point of the imaging area. Since the PFA uses the assumption of the plane wavefront, and the actual wavefront is curved, the error introduced mainly appears as the primary and secondary spatial phase error of the spatial frequency, which corresponds to the geometric distortion and defocusing of image and limits the effective imaging scene size of PFA algorithm [25]. A novel PFA algorithm based on image quadtree partition of image is studied below, which shares the same key idea of the quadtree CBP algorithm and mainly includes three steps:  
Step 1: Sub-image segmentation. The basic LRD algorithm without precise focusing is used to get a blurred SAR image from the whole scene, and then based on the idea of digital spotlight, the echo data can be filtered and segmented into several sub-images through the quadtree method, which is shown in Figure 13.  
Step 2: Motion compensation and PFA sub-imaging. Remove the echo data of each sub-image to the center of each sub-scene, and use PFA algorithm for precise focusing. Taking the sub-scene center as the reference center, the second-order motion compensation for the data of each sub-image is given, and the reference function of compensation is:  
\[\overline{S}_{n}(t,f_{\tau})=\exp\left\{\mathrm{j}\frac{4\pi(f_{\mathrm{c}}+f_ {\tau})}{c}\left[r_{n,0}(t)-r_{0}(t)\right]\right\},\quad n=1,2,\cdots,\ \mathrm{N}. \tag{34}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Conventional CBP** & **Divided to 32 \(\times\) 32 Points** & **Divided to 16 \(\times\) 16 Points** \\ \hline Theoretical Range Resolution & \multicolumn{3}{c}{\(1.00\) m} \\ \hline Simulation Range Resolution & \(1.05\) m & \(1.06\) m & \(1.09\) m \\ \hline Range PSLR & \(-13.71\) dB & \(-13.55\) dB & \(-13.23\) dB \\ \hline Theoretical Azimuth Resolution & \multicolumn{3}{c}{\(1.19\) m} \\ \hline Simulation Azimuth Resolution & \(1.23\) m & \(1.26\) m & \(1.31\) m \\ \hline Azimuth PSLR & \(-13.51\) dB & \(-13.15\) dB & \(-13.03\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation results of point targets simulation. Figure 13: Diagram of pre-filtering process. where \(n\) represents the \(n\)th sub-beam, \(N\) represents the number of sub-images, \(r_{n,\rho}(t)\) represents the instantaneous distance from the satellite to the center of the \(n\)th sub-scene. The data of each sub-image after the motion compensation can be written as:  
\[S_{n}(t,f_{\tau})=\sum_{p\in_{n}}\exp\biggl{\{}\mathrm{i}\frac{4\pi(f_{\mathrm{ c}}+f_{\tau})}{c}\bigl{[}r_{n,\rho}(t)-r_{p}(t)\bigr{]}\biggr{\}}\;. \tag{35}\]  
Through the second-order motion compensation above, the motion error of the center of each sub-beam scene is accurately compensated. Although the compensation is invariant in the each sub-beam, as long as the sub-beam is designed narrow enough to ensure that the sub-scene is within the effective imaging scene radius of the PFA, the residual error of the non-center point can be completely negligible. Step 3: Mosaic of sub-images. The full scene free focus large graph can be obtained by mosaicking each sub-image. In the PFA algorithm based on sub-images, it is not necessary to a divide the full image into very small sub-images like in the CBP algorithm. The main purpose of fast CBP is to reduce the amount of interpolation in the backprojection process to reduce the computation, so the sub-images must be small enough to achieve this purpose effectively. The PFA imaging algorithm itself is very simple and efficient. Therefore, the level of sub-image segmentation can be reduced, as long as each sub-image is within the range of the effective focusing scene.