# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA
### Algorithm Flow
When the value of grids interval exceeds \(K\), the target scene will be out of the range of the focused images and the phenomenon of image overlap appears. To show the imaging effect of the algorithm clearly, the amplitude images and interferometric images shown below are the part of the target scene in the focus images when the grid interval is less than \(K\). Figure 11 is the results of the simulation data and shows the amplitude and interferometric images of the target scene. As can be seen from it that with the increase of grid interval, the target scene size increases, the details of the target scene get richer and the interference fringes in the interferograms become clearer. The main characters of the cone in Figure 11a,b, and Figure 11c are well restored and the interferometric fringes are distinct. The reduction of the proportion of the target scene in the focusing image causes the reduction of the resolution of the target scene, which also leads to the loss of the detail information of the restored images in Figure 11a,b. As shown in Figure 11d, when the grid interval exceeds \(K\), the target scene exceeds the scope of the focused image, resulting in image aliasing and image focusing failure. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Grid Number** & **ENT** & **SHA** & **RES** & **COH** & **SNR** & **SPD** \\ \hline \(1502\times 2000\) & 1.324 & 466.82 & \(5.0\times 10^{6}\) & 0.811 & 0.448 & \(2.1\times 10^{7}\) \\ \(2000\times 2000\) & 1.326 & 466.41 & \(6.9\times 10^{6}\) & 0.807 & 0.447 & \(2.7\times 10^{7}\) \\ \(4000\times 4000\) & 1.332 & 452.90 & \(9.4\times 10^{6}\) & 0.787 & 0.440 & \(3.7\times 10^{7}\) \\ \(5000\times 5000\) & 1.341 & 451.52 & \(1.1\times 10^{7}\) & 0.786 & 0.440 & \(4.3\times 10^{7}\) \\ \(7000\times 7000\) & 1.372 & 450.94 & \(1.6\times 10^{7}\) & 0.772 & 0.436 & \(6.3\times 10^{7}\) \\ \hline \hline \end{tabular} \(\mathrm{SHA}=\mathrm{sharpness.}\)

\end{table} 
Table 7: Images’ quality indicators under different grid numbers. Figure 10: Quality indicators of real-data. Similarly, Figure 12 shows only the scope of the target scene of real-data. It is clear that with the increase of grid interval within the scope of \(K\), the information of restored images increase. Figure 12d shows the restored results when the grid interval is \(K\). Patently, spectrum aliasing results in a sharp decrease in image focusing. What differs from the simulation results above is the loss of the focused images information in azimuth with the reduction of grids interval. This is because the image information on both sides of azimuth is obtained from radar half aperture data. The frequency range of the back-projection decreases with the reduction of the grid spacing, resulting in the loss of the azimuthal data of the image and the degradation of the image quality. With the shrink of grids interval, the target scene information in the azimuthal edges will get smaller until it completely disappears. Figure 11: The results of the simulated cone: (**a**) 0.5 K, (**b**) 0.75 K, (**c**) K, and (**d**) 1.5 K. (Top: amplitude. Bottom: interferogram.). Figure 12: The results of experiments: (**a**) 0.25 K, (**b**) 0.5 K, (**c**) 0.75 K, and (**d**) 1.5 K. (Top: amplitude. Bottom: interferogram.). The experimental results show that the quality of amplitude images and the interference images get better with the increase of the grid interval within the range of \(K\). When the value of the grid interval exceeds \(K\), the focus image quality will decrease dramatically due to the problem of spectrum aliasing. Therefore, the optimal frequency grid interval of the proposed algorithm is \(K\). ### Scene Area

The grid division formulas demonstrate that the interval of grids and the number of grids are calculated from image scene size \(W\) and vary with the size of the imaging scene. Figure 13 is the target scene of imaging results with the imaging scene size is the 1, 1.5, 2, and 2.5 times of the original scene, respectively.