# Research on Synthetic Aperture Radar Processing for the Spaceborne Sliding Spotlight Mode
## 4 Convolution Backprojection Algorithm for Sliding Spotlight SAR
### Fast CBP Algorithm Based on Image Segmentation
In order to avoid energy leakage caused by defocusing, motion re-compensation are needed, and then linear RD algorithm and low-pass filtering can be applied. In summary, the fast filtering process includes:  
(a) Motion re-compensation to the center of the each sub-image. Figure 6: Schematic diagram of fast CBP algorithm based on image segmentation. For each sub-image, the motion compensation function is constructed based on the the central point in the sub-image, and the echo data is compensated by pulse by pulse. Taking sub-image A as an example, the phase compensation factor is:  
\[\phi_{sA}(t,f_{\tau})=\exp\left[\mathrm{j}\frac{4\pi}{c}f_{\tau}(R_{sA}-R_{a}) \right]\cdot\exp\left[\mathrm{j}\frac{4\pi}{c}f_{c}(R_{sA}-R_{a})\right]. \tag{27}\]  
where \(R_{sA}=R_{sA}(t)\) is the instantaneous distance between the phase center of the antenna and the center of sub-image A, and it can be described in coordinates:  
\[R_{sA}=\sqrt{(x_{a}-x_{sA})^{2}+(y_{a}-y_{sA})^{2}+(z_{a}-z_{sA})^{2}}. \tag{28}\]  
The new phase history is obtained after phase compensation:  
\[S_{WCA}(t,f_{\tau})=\mathrm{rect}\bigg{(}\frac{t}{T_{a}}\bigg{)}\cdot\iint \limits_{(x,y)\in W}g(x,y)\cdot\mathrm{rect}\bigg{(}\frac{f_{\tau}}{B_{r}} \bigg{)}\cdot\exp\bigg{[}\mathrm{j}\frac{4\pi}{c}(f_{c}+f_{\tau})(R_{sA}-R_{t })\bigg{]}\mathrm{dx}\mathrm{dy}. \tag{29}\]  
(b) Two dimensional imaging and window interception. At this time an image of the full scene can still be obtained by two-dimensional FFT, but now the center of the scene has been transferred to the center of each sub-image. Then it is convenient to extracting the sub-image data from the central part of the large two-dimensional data array of the full image according to the subscripts. The array size after interception is \(N/2\times N/2\). (c) Returning to the phase history domain. The scene of this sub-image is reduced by half in both range and azimuth, so the sampling rate can be also reduced by half in both range and azimuth domain. After returning the image to the space-frequency domain through IFFT, the amount of data is reduced to the \(1/4\) of original, only containing the information of sub-image A. The signal returning to data domain is:  
\[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]0  
Here the sampling points is half of the original, and the sampling interval is doubled. So the sampling intervals in \((t,f_{\tau})\) domain are \(2T_{0}\) and \(2F_{s}/N\). The discrete values of \(t\) and \(f_{\tau}\) are:  
\[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]1  
\[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]2  
The flow diagram of fast filtering is shown in \[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]3  
\[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]4 Flow diagram of fast filtering. Still taking sub-image A as an example, \(S_{sA}(t,f_{\tau})\) is the down-sample signal \(P_{\theta\_A}(U)\) in frequency domain containing only information of sub-image A, so the reconstruction formula is changed into the following:  
\[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]5  
The backprojection process is still realized by interpolation and summation. The pixel numbers of each sub-image is \(N/2\times N/2\), and the pulse numbers for backprojection is also reduced to \(N/2\), so the number of interpolation for each sub-image is \(N^{3}/8\) and total is \(N^{3}/2\) for all of the four sub-images, only half comparing to the normal process. Step 4: Sub-images mosaic. The full image is obtained by sub-images mosaic according to the original segmentation rules.