# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA
### Algorithm Flow
The coherence (COH) between complex images is often used to measure the similarity between images and the quality of the interferograms. The coherence is defined as follows:

\[COH=\frac{\left|E[S_{1}S_{2}{}^{*}]\right|}{\sqrt{E\left|\left[S_{1}\right]^{2} \right|E\left|\left[S_{2}\right]^{2}\right|}} \tag{20}\]

where \(*\) is the conjugate sign, and \(S_{1}\) and \(S_{2}\) denote the complex image information. Moreover, the signal-to-noise ratio can be obtained by the formula \(SNR=COH/(1+COH)\). As we can see from Table 2, all these image quality indexes of FDBPA are close to TDBPA. ENT, RES, and the value of SPD of FDBPA are slightly lower than that of TDBPA, while SHA, COH, and SNR of FDBPA are the opposite. These data demonstrate that FDBPA achieves similar imaging accuracy and phase-preserving property with TDBPA in processing the simulation data. Patently, the performances of RDA and FFBPA are closer to each other and worse than TDBPA and FDBPA. Although the ENT of RDA and FFBPA is lower than that of TDBPA and FDBPA, it is obvious that FDBPA and TDBPA have a better focusing effect according to the amplitude images in Figure 3.

To verify the imaging efficiency of FDBPA, imaging simulations under different scene sizes were implemented with imaging algorithms based on TDBPA, FDBPA, RDA, and FFBPA, respectively. The records of running time are shown in Table 3. It demonstrates that (1) FDBPA is more efficient than TDBPA when the imaging scene size is the same. It can enhance efficiency by three times at least compared with TDBPA. (2) As the increase of the target area, the advantages of FDBPA become more apparent (3) The imaging efficiency of FFBPA is similar to that of FDBPA, but with the increase of scene size, its performance is lower than that of FDBPA finally. (4) RDA has the fastest imaging speed. Simulations results in this chapter verify the feasibility of FDBPA in InSAR imaging. It can improve computational efficiency while keeping similar imaging accuracy and property of phase-preserving with TDBPA. It means that the InSAR imaging algorithm based on FDBPA has great application advantages and prospects. ### Experiment Data

The measured data of this experiment are obtained from an airborne InSAR system based on the operation mode of single send and double receive. The airborne platform flew 5230 m above the mountainside and collects echo data of the scene. The main system parameters of InSAR are as follows: the carrier frequency, \(f_{c}=16\,\mathrm{GHz}\); the signal bandwidth, \(B_{r}=800\,\mathrm{MHz}\); the sampling frequency, \(f_{s}=1.2\,\mathrm{GHz}\); the antenna aperture length, \(D=0.4\,\mathrm{m}\); the pulse repetition frequency, \(PRF=1352\); the platform height, \(H=5230\,\mathrm{m}\); the radar incident angle, \(\theta{=0}^{\circ}\); the baseline length, \(B=3\,\mathrm{m}\); and the reference distance, \(R0=4987\,\mathrm{m}\). This experiment selects part of raw scene data that can reflect the imaging effect to be processed by TDBPA, FDBPA, RDA, and FFBPA, respectively. The imaging scene is a rolling hillside. As shown in the amplitude images of Figure 4, all of these algorithms can well restore the scene, while TDBPA and FDBPA have better focus effects. Except for the slope depression, the traces of artificial mining, the trees also can be clearly distinguished \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Scene Size** & **RD(s)** & **FFBP(s)** & **TDBP (s)** & **FDBP (s)** \\ \hline \(1024\times 1024\) & 0.41 & 218.82 & 877.53 & 344.89 \\ \(2048\times 2048\) & 1.41 & 746.74 & 3523.32 & 1135.66 \\ \(4096\times 4096\) & 6.40 & 3531.60 & 14,443.20 & 3991.83 \\ \(8192\times 8192\) & 22.25 & 19365.08 & 60,541.36 & 14920.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of imaging time. in the amplitude images of Figure 4c,d. Interferograms contain terrain elevation information of the target scene. Interferograms with consistent interferometric fringes in Figure 4 show that the elevation information obtained by these used algorithms is uniform.