# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 6 Discussion
### Moving Target Refocusing  
Although moving targets suffer azimuth defocusing in the SAR image obtained by the proposed imaging method, we can refocus them after target detection and data extraction. Figure 9 presents a flowchart of moving target detection and refocusing. First, the DPCA processing is applied to the registered SAR images to cancel the clutter. Then, the detection and data extraction are implemented. Finally, the operation of refocusing is applied to the extracted target data.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Range Displacement (m)**} & \multicolumn{2}{c}{**Azimuth Displacement (m)**} & \multicolumn{2}{c}{**ATI Phase (rad)**} \\ \cline{2-7}  & \begin{tabular}{c} **Theoretical** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Measured** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Theoretical** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Measured** \\ **Value** \\ \end{tabular} & \
\begin{tabular}{c} **Theoretical** \\ **Value** \\ \end{tabular} &
\begin{tabular}{c} **Measured** \\ **Value** \\ \end{tabular} \\ \hline Target 1 & 3.99 & 3.75 & 5904.7 & 5914 & 0.5555 & 0.5643 \\ \hline Target 2 & 2.19 & 3.75 & 4329.1 & 4325 & \(-\)0.4375 & \(-\)0.4676 \\ \hline Target 3 & 4.18 & 3.75 & 6592.2 & 6478 & 0.5833 & 0.5544 \\ \hline Target 5 & 6.61 & 7.5 & 7821.3 & 7808 & 0.7291 & 0.7229 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Characteristic Measurement of Moving Targets.  
A refocusing method exploiting the maximum contrast criterion is proposed below. Assuming that the inverse operations of azimuth compression and RCMC have been applied to the extracted data, then we can utilize the following method to refocus the target  
\[s_{refocused}\Big{(}t_{r},t_{a};\hat{l}_{1},\hat{l}_{2},\hat{l}_{3}\Big{)}=IDFT_{ 2}\Big{\{}S(f_{r},f_{a})\cdot H_{refocus}\Big{(}f_{r},f_{a};\hat{l}_{1},\hat{l}_ {2},\hat{l}_{3}\Big{)}\Big{\}}, \tag{49}\]  
with  
\[H_{refocus}\Big{(}f_{r},f_{a};\hat{l}_{1},\hat{l}_{2},\hat{l}_{3}\Big{)}=\exp \bigg{\{}2\pi f_{r}\bigg{(}\frac{3c\hat{l}_{3}\hat{l}_{3}}{16f_{c}^{2}\hat{l}_{ 3}^{2}f_{a}^{2}}+\frac{c\hat{r}_{a}^{2}}{8f_{c}^{2}\hat{l}_{2}}+\frac{c\hat{l} _{3}}{16f_{c}^{2}\hat{l}_{3}^{2}f_{a}^{3}}\bigg{)}\bigg{\}}\exp\bigg{\{}-\pi \lambda\bigg{(}\frac{1}{4\hat{l}_{2}}f_{a}^{2}+\frac{3\hat{l}_{1}\hat{l}_{3}}{8 \hat{l}_{2}^{3}}f_{a}^{2}+\frac{\hat{l}_{3}\lambda}{16\hat{l}_{2}^{3}}f_{a}^{ 3}\bigg{)}\bigg{\}}, \tag{50}\]  
where \(S(f_{r},f_{a})\) is the signal after range compression and SRC, \(IDFT_{2}\) (\(\cdot\)) indicates 2-D IFT. In addition, \(\hat{l}_{1}\), \(\hat{l}_{2}\), and \(\hat{l}_{3}\) are the estimates of the coefficients of the range equation. They are estimated via the following maximum-contrast based way  
\[(\hat{l}_{1},\hat{l}_{2},\hat{l}_{3})=\arg\max_{l_{1},l_{2},l_{3},\lambda_{2}} \Big{\{}Contrast\Big{[}s_{refocused}(t_{r},t_{a};\hat{l}_{1},\hat{l}_{2},\hat{ l}_{3})\Big{]}\Big{\}}, \tag{51}\]  
with  
\[Contrast\big{[}s_{refocused}(t_{r},t_{a};\hat{l}_{1},\hat{l}_{2},\hat{l}_{3} )\big{]}=\frac{\sqrt{E\Big{\{}\Big{[}\big{[}\big{|}s_{refocused}(t_{r},t_{a}; \hat{l}_{1},\hat{l}_{2},\hat{l}_{3})\big{|}^{2}-E\Big{(}\big{|}s_{refocused}(t _{r},t_{a};\hat{l}_{1},\hat{l}_{2},\hat{l}_{3})\big{|}^{2}\big{)}\big{]}^{2} \Big{\}}}}{E\Big{\{}\big{|}s_{refocused}(t_{r},t_{a};\hat{l}_{1},\hat{l}_{2}, \hat{l}_{3})\big{|}^{2}\Big{\}}}, \tag{52}\]  
where \(E(\cdot)\) defines the spatial average, \(Contrast(\cdot)\) indicates the image contrast.  
Figures 10 and 11 present the refocusing results of target 1 and target 2. One can see that these two targets are all well focused.