# Adaptive 3D Imaging for Moving Targets Based on a SIMO InISAR Imaging System in 0.2 THz Band
## 4 Three-dimensional Imaging of Moving Targets with InISAR Technology
### Interferometric Imaging  
After image registration, the same scatterer will be projected onto the same pixel in the ISAR images of different channels, then the interferometric operation is performed to get the 3D imaging results. In this paper, channels Rx1, Rx2, and Rx4 are adopted to constitute the typical 'L' antenna array, which form interferometric baselines in the azimuth and vertical directions, respectively. According to (22), the phase responses of a certain scatter in the three images are expressed as:  
\[\phi_{I1}=\frac{\pi}{\lambda R_{p}}(x_{Lci}^{2}-L_{x}x_{p}+z_{Lci}^{2}-L_{z}z_{ p}) \tag{25}\]  
\[\phi_{I2}=\frac{\pi}{\lambda R_{p}}(x_{Lci}^{2}+L_{x}x_{p}+z_{Lci}^{2}-L_{z}z_{ p}) \tag{26}\]  
\[\phi_{I4}=\frac{\pi}{\lambda R_{p}}(x_{Lci}^{2}-L_{x}x_{p}+z_{Lci}^{2}+L_{z}z_{ p}) \tag{27}\]  
Then the interferometric phases in each direction are extracted as:  
\[\Delta\phi_{x}=angle(I_{1}I_{2}^{*})=\frac{2\pi L_{x}}{\lambda R_{p}}x_{p} \tag{28}\]  
\[\Delta\phi_{z}=angle(I_{1}I_{4}^{*})=\frac{2\pi L_{z}}{\lambda R_{p}}z_{p} \tag{29}\]  
This way, the target coordinates in the horizontal and vertical directions can be determined using the following equations:  
\[u=\frac{\lambda R_{p}}{2\pi L_{x}}\Delta\phi_{x} \tag{30}\]\[w=\frac{\lambda R_{p}}{2\pi L_{z}}\Delta\phi_{z} \tag{31}\]  
It should be noted that since the targets are continuously tracked and the target size is far smaller than the target range, there is no need to consider the phase unwrapping under this situation. The target coordinate in the range direction is then calculated with the following formula:  
\[v=\sqrt{R_{p}^{2}-u^{2}-w^{2}} \tag{32}\]  
By now, the coordinates of the target in the range, azimuth and vertical directions are all determined, and the 3D imaging results can be initially obtained. Figure 6 presents the whole flowchart of InSAR imaging procedure with the combined motion compensation method. In order to illustrate the proposed InISAR imaging method based on combined motion compensation in a clear way, practical processing procedures with reference to Figure 6 are presented specifically in Table 2. \begin{table}
\begin{tabular}{p{113.8 pt}|p{113.8 pt}|p{113.8 pt}|p{113.8 pt}} \hline \hline Virtual sum echoes & Rx2 raw echoes & Rx1 raw echoes & Rx4 raw echoes \\ \hline  & & & \\ \cline{2-3}  & & Image registration & Image registration \\ \hline Range compression & Range compression & Range compression & Range compression \\ \hline Envelope alignment & Combined envelope alignment & Combined envelope alignment & Combined envelope alignment \\ \hline Phase correction & Combined phase correction & Combined phase correction & Combined phase correction \\ \hline Cross-range compression & Cross-range compression & Cross-range compression & Cross-range compression & Cross-range compression \\ \hline Rx0 ISAR image & Rx2 ISAR image & Rx1 ISAR image & Rx4 ISAR image \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Practical procedures of the InISAR imaging. Figure 6: The interferometric inverse synthetic aperture radar (InISAR) imaging processing flowchart based on combined motion compensation. However, the section above discusses only the InISAR imaging procedure when the targets move in the front FOV of the radar, which is the same as the typical imaging scenario. But in this paper, since the antenna beams always keep illuminating the targets during tracking, the FOV is varying as the targets move, which is more complex than the typical situation. Thanks to the short wavelength of the THz wave, the Doppler frequency induced by target movement is so obvious that the integration interval can be greatly reduced to achieve a better resolution in azimuth direction. Therefore, the whole observation can be separated into several imaging windows, during which the instantaneous 3D images are obtained, hence making it possible to achieve a dynamic imaging to acquire information about the target status during movement.