# Satellite Attitude Determination and Map Projection Based on Robust Image Matching
## 5 Discussion  
### Inlier Rates and the Number of Repetitions in Robust Estimation  
All four variants from the RANSAC family combined with the geometrical constraint of satellite observation successfully eliminate incorrect matched pairs as shown in Section 4, which results in an accurate satellite attitude at the observation and a map projection. Here, we discuss computational costs for outlier rejection, describe an early stopping mechanism of RANSAC iterations, and apply this mechanism to the baseline RANSAC and PROSAC. In principle, all of the correct feature pairs will conform to the geometric relationship described in Section 3.1. In Section 4.1, we confirmed that their \(\mathbf{V_{\mathrm{C}}}\) and \(\mathbf{\hat{\mathrm{M}}^{\prime}V_{\mathrm{E}}}\) estimates matched within the required 0.02\({}^{\circ}\) accuracy. Therefore, any combination of three correct pairs (the smallest number of pairs to derive the satellite attitude), should provide an accurate satellite attitude and maximize \(L\), the number of inliers. This means it is possible to skip additional RANSAC iterations once we find three correct feature pairs, which helps to reduce computational costs. In addition, because feature extraction from base-map images and pre-processing for the base-map images described in Section 3.2.2 can be performed independently in advance, we can also skip a feature extraction step for base-map images at the rough matching part. Reducing computational costs at each observation would be beneficial, especially for a mission with limited computation resources at ground, and also useful for the immediate derivation of satellite attitude. Based on this idea, we designed an early stopping mechanism to reduce the number of iterations: Once we find \(\mathbf{\hat{\mathrm{M}}^{\prime}}\) whose \(L\) larger than a threshold \(L_{0}\), then we finish the repetitions. We set the threshold to be \(L_{0}=10\) in the experiments in this subsection. The expected number of repetitions for the baseline RANSAC required to find three correct pairs from a population of \(N\) candidates with \(L\) inliers is calculated to be \(\left( \begin{array}{cc}N\\ 3\end{array} \right)/\left( \begin{array}{c}L\\ 3\end{array} \right)\) (see Appendix B). If the proportion of inliers among the extracted feature pairs is sufficiently high, the number of iterations can be reduced to below 10. For example, the expected numbers of repetitions in the Kanto and Yosemite cases were 3.3 and 4.3, respectively, estimated from their \(\left( \begin{array}{cc}N\\ 3\end{array} \right)/\left( \begin{array}{c}L\\ 3\end{array} \right)\) (the values of \(N\) and \(L\) are taken from Table 3). The actual \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathbf{N_{\mathrm{pair}}}\) & \(\mathbf{\widehat{\mathrm{\Delta x}}}\) **(m)** & \(\mathbf{\widehat{\mathrm{\Delta y}}}\) **(m)** & \(RMSE(\mathbf{\mathrm{\Delta x}})\) **(m)** & \(RMSE(\mathbf{\mathrm{\Delta y}})\) **(m)** \\ \hline (a) Kanto, Japan & 92 & 7 & 47 & 123 & 102 \\ (b) Yosemite, US & 107 & \(-25\) & \(-6\) & 121 & 181 \\ (c) Kyushu, Japan & 23 & \(-11\) & \(-9\) & 156 & 170 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean miss-registration errors and RMSEs of matched feature pairs extracted from the projected VIS images and OLI images. Symbols \(\mathrm{\Delta x}\) and \(\mathrm{\Delta y}\) represent the east-west and north–south directions, respectively. repetition numbers, averaged from 1000 trials for each image, were 3.5 for the Kanto case and 4.9 for the Yosemite case. In the Kyushu case (with large cloud coverage), the actual and expected numbers of repetitions were 139 and 145, respectively, much larger than in the other cases. The maximum number of iterations for the baseline RANSAC was also higher in the Kyushu case than in the Kanto and Yosemite cases (1058 versus 25, and 28, respectively, among 1000 trials).