# A Novel Mosaic Method for Spaceborne ScanSAR Images Based on Homography Matrix Compensation
## 5 Experimental Results and Discussion
### Efficiency Analysis  
The time needed for keypoint detection and matching will be greatly reduced due to the improved methods we proposed. However, it is difficult to estimate the degree of efficiency optimization because the distribution of keypoints in spaceborne ScanSAR images is not uniform. In this paper, assuming that keypoints follow a uniform distribution, we roughly estimated the improvement in the efficiency with our proposed methods. The total time needed for keypoint detection by the SIFT algorithm and matching with the traditional method is:  
\[t_{0}=t_{S}+t_{m} \tag{18}\]  
where \(t_{S}\) is the time needed for keypoint detection by the SIFT algorithm and \(t_{m}\) represents the time needed for matching. Assuming that \(u\) is the computational load, the computational efficiency of the SIFT algorithm is \(O(u)\) and the computational efficiency of matching is \(O(u^{2})\). The time after overlapping area extraction is:  
\[t_{1}=r_{o}t_{S}+(r_{0})^{2}t_{m} \tag{19}\]  
where \(r_{o}\) is the overlapping rate of the reference image and the image to be stitched. The time after overlapping area extraction and downsampling is:  
\[t_{2}=\frac{r_{o}t_{S}}{n}+\Big{(}\frac{r_{o}}{n}\Big{)}^{2}t_{m} \tag{20}\]  
where \(1/n\) is considered the scaling coefficient of the image. The range of keypoint detection and matching is reduced to equal parts of overlapping areas when performed in parallel, so the time after overlapping area extraction, downsampling and division is:  
\[t_{3}=\frac{r_{o}t_{S}}{Mn}+\Big{(}\frac{r_{a}}{Mn}\Big{)}^{2}t_{m} \tag{21}\]  
where \(M\) is the number of parts after division. Thus, compared with the traditional method, the efficiency of keypoint detection and matching can be considerably improved. To evaluate the influence of the improved methods presented in Section 4 on the mosaic efficiency of multiple large-area images, several mosaic experiments with different situations of overlapping area extraction, different scaling coefficients and different numbers of divided parts were performed on six GF-3 ScanSAR images (two images in the azimuth and three subswath). The basic parameters of the six GF-3 ScanSAR images are presented in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Range of Keypoint** \\ **Detection** \\ \end{tabular} } & \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{\begin{tabular}{c} **Number of** \\ **Keypoints** \\ **in Azimuth1\_** \\ **Subswath2** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Number of** \\ **Matched** \\ **Keypoints** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **\(E_{m}\)/\%** \\ **\end{tabular} } \\  & 843,159 & 931,704 & 53,269 & 94.92 \\ Overlapping area & 93,866 & 93,507 & 51,812 & 98.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Keypoint detection and matching performances of the whole images and overlapping areas. From Figure 9a, we know that the time can be reduced to approximately \(1/10\) when the overlapping area extraction is applied to the mosaic of large-area images. The efficiency of stitching was improved significantly. As shown in Figure 9b, mosaic time can also be reduced effectively by downsampling overlapping areas, especially when the scaling coefficient is from 0.5 to 0.05. The smaller the scaling coefficient is, the faster the stitching speed. However, the degree of improvement in efficiency is not significant when the scaling coefficient changes from 0.1 to 0.05, which means that too much scaling is unnecessary. In addition, Figure 9c shows that parallel operation greatly contributed to the efficiency of mosaics, especially for large-area images. As the number of divided parts increased, the stitching speed became faster. Moreover, all situations of the image mosaic contrast experiments we implemented are presented in Figure 9d.