# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA
### Algorithm Flow
Their computational complexity can be described as \((N_{a}\times N_{r}\log N_{r})\) and \(\left(N_{a}\times\left(2N_{r}\log N_{r}+2N_{x}\times N_{y}\right)\right)\), respectively; \(N_{a}\) and \(N_{r}\) are the number of imaging cells in the azimuth direction and range direction, respectively; and \(N_{x}\times N_{y}\) is the imaging scene size. According to the detailed algorithm flow above, the computational complexity of the proposed method consists of four parts. The computational complexity of range compression is \((N_{a}\times N_{r}\log N_{r})\) which is consistent with the InSAR imaging algorithm based on TDBPA. The complexity of the division of spectrum grid, back projection, and 2D inverse FFT are \(\left(N_{x}\times N_{y}\right)\), \(\left(N_{a}\times\left(2N_{r}\log N_{r}\right)+\log N_{a}\right)\), and \(\left(N_{x}\times N_{y}\log\left(\max(N_{x},N_{y})\right)\right)\), respectively. The main complexity of time-domain-based InSAR imaging comes from TDBPA, and the proposed algorithm suffers from the huge computational complexity mainly from back-projection. Hence, the decisive complexity of TDBPA and FDBPA are \(\left(2N_{a}\times\left(N_{r}\log N_{r}+N_{x}\times N_{y}\right)\right)\) and \(\left(2N_{a}\times N_{r}\log N_{r}\right)\), respectively. Obviously, FDBPA can significantly improve the computational efficiency of InSAR imaging, and the gap between the imaging efficiency of TDBPA and FDBPA becomes more obvious with the increase of scene area. It demonstrates the advantages of FDBPA in InSAR real-time imaging with large scenes. ## 4 Results

To verify the feasibility and effectiveness of the proposed algorithm, the simulated and actual echo data are processed by TDBP, FDBP, RD, and FFBP algorithm, respectively. In all the experiments, a computer with Intel Core i7-8700K 3.7 GHz CPU, 32 GB RAM, and NVIDIA GeForce RTX 2060 6 GB hardware capabilities, \"MATLAB\" programming language, CUDA 10.1, CUDNN software has been used. ### Simulation Data

To compare the performance of the proposed algorithm with several classical imaging algorithms, the simulation data are processed by imaging algorithms based on TDBPA, FDBPA, RDA, and FFBPA, respectively. The main simulation parameters of InSAR are shown in Table 1. The original simulation scene is a cone with the size of \(250m\ \times\ 250m\ \times\ 65m\). The reflection coefficient of each target point in the simulation scene is different from another according to its distance from the antenna and the unique incident angle. Figure 2a shows the simulation cone scene model. Antenna jitter is added in the trajectory and the actual trajectory is shown in Figure 2b. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Meaning** & **Value** \\ \hline \(f_{c}\) & Carrier frequency & 10 GHZ \\ \(f_{s}\) & Sampling frequency & 390 MHZ \\ \(B_{r}\) & Bandwidth & 300 MHZ \\ \(D\) & Antenna aperture length & 0.9993 m \\ _PRF_ & Pulse repetition frequency & 500 \\ \(H\) & Platform height & 4000 m \\ \(\theta\) & Incident angle & 45\({}^{\circ}\) \\ \(B\) & Baseline length & 5 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. The amplitude images and interferograms are shown in Figure 3. For comparing the results under the same situation, TDBPA and FFBPA both select the slant plane as the imaging plane. The top of Figure 3 shows the amplitude images of imaging results. Patently, all these algorithms can produce focused images well while the focused scene produced by TDBPA and FDBPA are clearer. The bottom of Figure 3 shows the interferometric images; they have the consistent distribution of interferometric fringes which demonstrates that they reflect the consistent terrain information. However, it is obvious that the interferogram obtained by the RD algorithm appears more noise points.