## 2 Registration Method
### Statistical Randomization-Based PSO (SR-PSO) Algorithm
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Number of particles & \(K\) & 100 \\ Number of iterations & \(T\) & 2000 \\ Constriction coefficient & \(\chi\) & 0.7298 \\ & \(\alpha\) & \(0.5,1.0,1.5,2.0\) \\  & \(\varphi\) & 4.1 \\ Personal learning coefficient & \(c_{1}\) & 1.4962 \\ Global learning coefficient & \(c_{2}\) & 1.4962 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical randomization-based particle swarm optimization (SR-PSO) algorithm’s parameters settings in the experiment. Equation (20) is \(\mathbf{H}_{b\_select}^{-1}=\left[ \begin{array}{rrrr}0.995691&0.122637&-0.006306&- 0.116511\\ -0.012102&1.016839&0.012930&0.045549\\ 0.030081&0.212238&1.000898&0.065096\\ 0&0&0&1\end{array} \right]\). After comparing \(\mathbf{H}_{g}^{-1}\) with \(\mathbf{H}_{b\_select}^{-1}\), \(\mathbf{H}_{b\_select}^{-1}\) is better, hence \(\mathbf{H}_{g}^{-1}=\mathbf{H}_{b\_select}^{-1}\). Figure 3 depicts the registration mean-squared error (MSE) of the global best of the SR-PSO algorithm. We can see that, as the iteration proceeds, the MSE decreases and moves towards 3.22 \(\times\)\(10^{-31}\) at the 2000 th iteration. Figure 4 shows the registration progress image of the global best particle in the 1 st, 50 th, 500 th, and 2000 th iterations. It can be seen that the registration result in the last iteration is an almost perfect match. Figure 4: Registration results from the global best particle at the (**a**) 1 st, (**b**) 50 th, (**c**) 500 th, and (**d**) 2000 th iterations. Figure 3: Mean–squared error (MSE) of SR–PSO algorithm. The registration mean-squared errors (MSEs) using the SR-PSO algorithm with and without refining with the ICP method are shown in Table 3. The best result is when \(\alpha=1.5\) with an extremely tiny error of 3.22 \(\times\) 10\({}^{-31}\). The best final registration result is also shown in Figure 5.  
The best final transformation matrix is  
\[\mathbf{H}^{-1}=\left[ \begin{array}{cccc}0.901895&0.421703&0.439339&0.1\\ -0.143742&0.826257&-0.237726&-0.1\\ -0.294063&0.244980&0.819157&0.1\\ 0&0&0&1\end{array} \right] \tag{24}\]  
Although the MSEs of the final registration with the other values of \(\alpha\) are not as good, the final registration images are almost aligned as shown in Figure 6. The reason for this mistake might be because the SR-PSO algorithm does not find the correct global minimum. Figure 5: The best final registration. Figure 6: (a) The superimposed image before registration, the final registration with \(\alpha\) is equal to (b) 1.0, and (c) 2.0. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{\(\alpha\)} \\ \cline{2-5}  & **0.5** & **1.0** & **1.5** & **2.0** \\ \hline SR-PSO without ICP & 7.68 \(\times\) 10\({}^{-2}\) & 9.23 \(\times\) 10\({}^{-2}\) & **3.22 \(\times\) 10\({}^{-31}\)** & 9.68 \(\times\) 10\({}^{-2}\) \\ SR-PSO with ICP & 7.68 \(\times\) 10\({}^{-2}\) & 9.23 \(\times\) 10\({}^{-2}\) & **3.22 \(\times\) 10\({}^{-31}\)** & 9.68 \(\times\) 10\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The registration mean-squared error (MSE) in pixels\({}^{2}\) (the best value is in bold). Now, we are ready to implement this algorithm on tooth reconstruction. We used the EinScan series, a commercial 3-D scanner, to collect a point-cloud data set in several consecutive views from two commercial tooth models. Examples of data acquisition setup and the point-cloud data are shown in Figure 7.  
We implemented this system on two different tooth models, i.e., a regular tooth model and an orthodontic tooth model. We used only six consecutive point-cloud coordinate \((x,y,z)\) views with an interval of 30 degrees in each model. The information of the tooth point-cloud data is shown in Table 4.  
In this dataset, we randomly sampled each view into 60% of the original view. We also assumed that there was overlapping between each consecutive view.