## 2 Registration Method
### Statistical Randomization-Based PSO (SR-PSO) Algorithm
We also selected representative points inside the overlapping area using the voxel hull method [33; 34; 35] before we implemented the registration process with the parameter setting shown in Table 2. Since there were six consecutive views, we utilized the hierarchical registration to increase the registration performance as shown in Figure 8 with \(F=6\). At each level, we selected the best final registration result (SR-PSO algorithm with the ICP method) to survive to the next level of the hierarchical registration. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Object View** & **Object Name** & **Points Number** \\ \hline \multirow{6}{*}{Regular tooth model} & 1 & Scan\_0.asc & 28,807 \\  & 2 & Scan\_1.asc & 28,970 \\  & 3 & Scan\_2.asc & 28,983 \\  & 4 & Scan\_3.asc & 25,809 \\  & 5 & Scan\_4.asc & 17,303 \\  & 6 & Scan\_5.asc & 21,739 \\ \cline{2-4}  & Total & Six views & 151,592 \\ \hline \multirow{6}{*}{Orthodontic tooth model} & 1 & Scan\_0.asc & 25,301 \\  & 2 & Scan\_1.asc & 25,772 \\ \cline{1-1}  & 3 & Scan\_2.asc & 22,432 \\ \cline{1-1}  & 4 & Scan\_3.asc & 17,167 \\ \cline{1-1}  & 5 & Scan\_4.asc & 22,537 \\ \cline{1-1}  & 6 & Scan\_5.asc & 24,148 \\ \cline{1-1} \cline{2-4}  & Total & Six views & 137,357 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Tooth dataset information. Figure 7: (**a**) Hardware setup and (**b**) point-cloud view. We also compared the results with those from the regular particle swarm optimization (PSO) algorithm with the ICP method. The registration process from the PSO algorithm with the ICP method is the same as the SR-PSO algorithm with the ICP method. However, the inertia weight update equation in PSO [29] is  
\[w(t+1)=w_{\text{max}}-t\frac{(w_{\text{max}}-w_{\text{min}})}{(T)} \tag{25}\]  
The parameter setting was similar to those used in the SR-PSO algorithm except that \(w_{min}\), \(w_{max}\), \(c_{1}\), and \(c_{2}\) were set to 0.4, 0.9, 2, and 2, respectively. For the regular-tooth model, the registration MSEs for the SR-PSO algorithm and the PSO algorithm for each registration pair are shown in Table 5. Meanwhile, the registration MSEs for each of the two algorithms with the ICP method for each registration pair are shown in Table 6. Figures 9 and 10 show the best registration image of each consecutive pair from the SR-PSO algorithm without and with the ICP method, respectively. Again, the local optima are the cause of high MSE in pair-wise registration. Although Table 6 shows that the PSO algorithm with the ICP method is a little bit better than the SR-PSO algorithm with the ICP method in some pair-wise registrations, i.e., 1 vs. 2, 2 vs. 3, and 3 vs. 4, the difference is only in the order of 0.002 to 0.05 micrometer2. When considering the final registration results of six consecutive views shown in Table 7, we find that the best result from the SR-PSO algorithm with the ICP method and \(\alpha=1.5\) is 7.3666 micrometer2 whereas that from the PSO algorithm with the ICP method is 17.1150 micrometer2. The result from the SR-PSO algorithm with the ICP method is better than that from the PSO algorithm with the ICP method. The final registration result of the regular-tooth model is shown in Figure 11. We can see that the final registration image provides a good visualization to human eyes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**MSE in Micrometer2**} & \\  & & \multicolumn{3}{c}{**SR-PSO**} & **PSO** \\ \hline View pairs & \(\alpha=0.5\) & \(\alpha=1.0\) & \(\alpha=1.5\) & \(\alpha=2.0\) & \\
1 vs. 2 & **5.9300** & 6.1910 & 6.3274 & 6.3470 & 7.0328 \\
2 vs. 3 & 5.0026 & 4.9070 & **4.8937** & 4.9507 & 5.3218 \\
3 vs. 4 & 5.8824 & **5.4310** & 6.0370 & 5.5086 & 8.1904 \\
4 vs. 5 & **5.2666** & 5.3131 & 5.2807 & 32.312 & 6.7879 \\
5 vs. 6 & 6.0431 & 5.8163 & 5.9118 & **5.8166** & 6.5468 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: MSE from SRPSO and PSO for the regular-tooth model (the best value is in bold and the worst value is underlined). Figure 8: Hierarchical multiple-views registration.