# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA
### Algorithm Flow
To quantitatively evaluate the focusing effect and phase-preserving performance of FDBPA, amplitude images are evaluated by image entropy (ENT) and image sharpness (SHA) in this article, while residue, coherence (COH), signal-to-noise ratio (SNR), and sum absolute phase difference (SPD) are used to evaluate the quality of InSAR interferometric phase. Table 2 shows the detailed measurement values of the indicators above. ENT represents the image entropy, and the formula of ENT is as follows:

\[ENT=-\sum_{j=1}^{M}\sum_{i=1}^{N}p(c(i,j))\ln(p(c(i,j))) \tag{16}\]

where \(p(\cdot)\) is the probability of the occurrence of each gray level. Images entropy shows how chaotic the image is. Generally, the smaller the ENT, the better the quality of the amplitude image. However, this rule is not applicable in all cases. Especially for the experiment data, owing to the complexity of the real scene, the evaluation criteria of obtained results are not absolute. Therefore, we pay more attention to the comparison of the indicators of multiple algorithms in this paper. The formula of image sharpness can be expressed as follows:

\[SHA=10*\log\left(\sum_{i}\sum_{i}\left|f(i,j)\right|^{4}\right) \tag{17}\]

where \(f(i,j)\) is the pixel value of the focused image. In contrast to ENT, the larger the value of SHA, the better quality of the image. The residue (RES) is one of the most important indicators to represent the interference phase quality. It seriously affects the phase unwrapping in InSAR imaging processing. The definition of RES is as follows:

\[RES=\sum_{i=1}^{4}\Delta_{i}=\left\{ \begin{array}{ll}0&\textit{ notresidue}\\ 2\pi&\textit{positiveresidue}\\ -2\pi&\textit{negitiveresidue}\end{array} \right. \tag{18}\]

Figure 2: Simulation scene and antennas. (**a**) Original cone scene. (**b**) The trajectories of dynamic antennas and ideal antennas. where \(\Delta_{i}\) denotes the wrapping phase gradient of the pixel. SPD denotes absolute phase difference (SPD). For noiseless phases, the phase gradient of adjacent pixel points should be a relatively small value since the terrain is slowly changing. When there is noise in the phase, the phase gradient of adjacent pixels will increase sharply. Based on this principle, [PERSON] et al. proposed the sum of phase gradients (SPD). Compared with the residue based on the wrapping phase gradient, the absolute phase gradient and the absolute phase gradient are calculated based on the absolute phase, so they can more accurately reflect the denoising ability of the filtering algorithm. A detailed explain about this can be seen in Reference [46]. The equation of SPD can be written as follows:

\[SPD=\sum_{m}\sum_{n}\biggl{(}\frac{1}{8}{\sum_{i=1}^{-1}{\sum_{j=1}^{-1}}}| \phi(m,n)-\phi(m+i,n+j)|\biggr{)} \tag{19}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Indicator** & **RD** & **FFBP** & **TDBP** & **FDBP** \\ \hline Entropy & 1.762 & 1.742 & 1.930 & 1.911 \\ Sharpness & 0.00069 & 0.0011 & 0.0031 & 0.0036 \\ Residue & \(9.1\times 10^{5}\) & \(7.8\times 10^{5}\) & \(3.8\times 10^{5}\) & \(3.6\times 10^{5}\) \\ coherence & 0.820 & 0.821 & 0.869 & 0.884 \\ SNR & 0.4506 & 0.4509 & 0.4648 & 0.4692 \\ SPD & \(4.49\times 10^{6}\) & \(3.63\times 10^{6}\) & \(1.89\times 10^{6}\) & \(1.80\times 10^{6}\) \\ time(s) & 5.3 & 63.7 & 144.3 & 68.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quality indicators of imaging results. Figure 3: The imaging results of the simulated cone: (**a**) range Doppler (RD), (**b**) fast factorized back-projection algorithm (FFBP), (**c**) time-domain back-projection algorithm (TDBPA), and (**d**) frequency-domain back-projection algorithm (FDBP) (top: amplitude, middle: interferograms, and bottom: coherence-map). SPD is calculated based on the absolute phase, so the quality of interferograms can be accurately reflected. In general, the small residue and SPD means the higher the quality of the interferograms, while the coherence coefficient and SNR are the opposite.