## 2 Registration Method
### Statistical Randomization-Based PSO (SR-PSO) Algorithm
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**MSE in Micrometer2**} & \\  & \multicolumn{2}{c}{**SR-PSO with ICP**} & \multicolumn{2}{c}{**PSO with ICP**} \\ \hline View pairs & \(\alpha=0.5\) & \(\alpha=1.0\) & \(\alpha=1.5\) & \(\alpha=2.0\) & \\
1 vs. 2 & **5.8628** & 5.8636 & 5.8632 & 5.8809 & **5.8156** \\
2 vs. 3 & 4.8906 & 4.8912 & **4.8860** & 4.8884 & **4.8844** \\
3 vs. 4 & 5.4030 & 5.4024 & **5.4017** & 5.4031 & **5.4055** \\
4 vs. 5 & 5.1326 & **5.1253** & 5.1747 & 5.1261 & 5.1261 \\
5 vs. 6 & 5.6829 & 5.6880 & **5.6828** & 5.6882 & 5.6956 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: MSE from SR\(-\)PSO with iterative closet point (ICP) and PSO with ICP for the regular-tooth model (the best value is in bold). Figure 9: The best SR-PSO registration results for the pairs (**a**) 1 and 2; (**b**) 2 and 3; (**c**) 3 and 4; (**d**) 4, and 5; (**e**) 5 and 6 of the regular-tooth model. For the orthodontic-tooth model, the registration MSEs for the SR-PSO algorithm and the PSO algorithm for each registration pair are shown in Table 8. Figure 12 shows the best registration images of different pairs from the SR-PSO algorithm. Meanwhile, the registration MSEs for the SR-PSO algorithm with the ICP method and the PSO algorithm with the ICP method for each registration pair are shown in Table 9. The best registration images of different pairs from the SR-PSO algorithm with the ICP method are shown in Figure 13. We can see that using only the SR-PSO algorithm provides better results than using only the PSO algorithm. However, the PSO algorithm with the ICP method is better than the SR-PSO algorithm with the ICP method in one of the pair-wise registrations \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**SR-PSO with ICP**} & \multicolumn{2}{c}{**PSO with ICP**} \\ \hline \(\alpha=0.5\) & \(\alpha=1.0\) & \(\alpha=1.5\) & \(\alpha=2.0\) & \\
7.3670 & 7.3668 & **7.3666** & 7.3667 & 17.1150 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: MSE of the final registration of six consecutive views (micrometer\({}^{2}\)) for the regular-tooth model (the best value is in bold). Figure 10: The best SR-PSO with the ICP registration results for the pairs (**a**) 1 and 2; (**b**) 2 and 3; (**c**) 3 and 4; (**d**) 4 and 5; (**e**) 5 and 6 of the regular-tooth model. (4 vs. 5) with a little difference. The final registration MSEs of the six consecutive views are shown in Table 10. The best final 3-D registration image is from the SR-PSO algorithm with the ICP method and \(\alpha=0.5\). It has an MSE of 7.4130 micrometer2 whereas the comparable error of 7.4672 micrometer2 is achieved by the PSO algorithm with the ICP method. We can also see in Figure 14 that the final 3-D registration image of the orthodontic-tooth model also provides a good visualization to human eyes. Figure 12: The best SR-PSO registration results for the pairs (**a**) 1 and 2; (**b**) 2 and 3; (**c**) 3 and 4; (**d**) 4 and 5; (**e**) 5 and 6 of the orthodontic-tooth model. Figure 13: The best SR-PSO with ICP registration results for the pairs (**a**) 1 and 2; (**b**) 2 and 3; (**c**) 3 and 4; (**d**) 4 and 5; (**e**) 5 and 6 of the orthodontic-tooth model.