## 2 Registration Method
### Statistical Randomization-Based PSO (SR-PSO) Algorithm
\right.\] (18) If the \(l^{\text{th}}\) intermediate particle (\(x^{tml}\)) is the best particle among the intermediate particles and it is better than the global particle, then the global best particle will be replaced by \(x^{tml}\). To increase the chance of exploration and exploitation of \(x^{g}\), we randomly select the individual best \((x_{k}^{b})\) from all particles (instead of randomly selecting a particle except the best one as in [23]) using  
\[x_{select}^{b}=\text{rand}(\mathbf{X}^{b})\text{ where }\mathbf{X}^{b}=\left\{x_{1}^ {b},\ldots,x_{K}^{b}\right\} \tag{19}\]  
To test the algorithm, we ran the experiment of the generated original shape and its 3-D transformation as shown in Figure 2.  
The transformation matrix used to transform the original shape described by the target point-cloud (Figure 2a) to the transformed shape described by the source point-cloud (Figure 2b) is  
\[\mathbf{H}=\left[ \begin{array}{cccc}0.882050&-0.285362&-0.555884&-0.061153\\ 0.225174&1.041540&0.181496&0.063487\\ 0.249299&-0.413927&0.966936&-0.163016\\ 0&0&0&1\end{array} \right] \tag{22}\]  
Hence, the transform matrix from the source point-cloud (Figure 2b) to the target point-cloud (Figure 2a) is  
\[\mathbf{H}^{-1}=\left[ \begin{array}{cccc}0.901895&0.421703&0.439339&0.1\\ -0.143742&0.826257&-0.237726&-0.1\\ -0.294063&0.244980&0.819157&0.1\\ 0&0&0&1\end{array} \right] \tag{23}\]  
In the experiment, the parameters for the SR-PSO algorithm were set as shown in Table 2. To demonstrate how the SR-PSO algorithm works in each step, we provide an example as follows:  
Figure 1: Diagram of the proposed 3-D tooth model reconstruction system. Figure 2: **(a)** The original shape (Target point-cloud) and (**b**) the 3-D transformation (Source point-cloud). Suppose an initialized particle is  
\[\mathbf{H}^{-1}=\left[ \begin{array}{cccc}0.731187&-0.409449&0.035758&0.590480 \\ 0.142237&0.573472&1.013846&1.008356\\ -0.099053&-0.640562&1.135987&1.336424\\ 0&0&0&1\end{array} \right],\]  
then after the velocity and position are updated, the transformation is  
\[\mathbf{H}^{-1}=\left[ \begin{array}{cccc}0.791731&-0.221107&0.249697&0.47944 7\\ -0.022301&0.854962&0.732671&0.571793\\ -0.010562&-0.459633&1.176127&0.893047\\ 0&0&0&1\end{array} \right].\]  
The individual best of this particle is \(\mathbf{H}_{g}^{-1}=\left[ \begin{array}{cccc}0.731187&-0.409449&0.035758&0.5 90480\\ 0.142237&0.573472&1.013846&1.008356\\ -0.099053&-0.640562&1.135987&1.336424\\ 0&0&0&1\end{array} \right]\) and the global best in this iteration is \(\mathbf{H}_{g}^{-1}=\left[ \begin{array}{cccc}1.004627&0.362394&-0.003666 &0.024357\\ -0.764060&0.540426&-0.902233&1.839488\\ 0.567330&0.659548&1.130398&-0.219057\\ 0&0&0&1\end{array} \right]\). Then, all five intermediate particles are \(\mathbf{H}_{tml}^{-1}=\left[ \begin{array}{cccc}0.995780&0.115222&-0.006401&-0. 116843\\ -0.012096&1.016352&0.012924&0.045527\\ 0.032553&0.004589&0.998258&0.055794\\ 0&0&0&1\end{array} \right],\)  
\[\mathbf{H}_{tml}^{-1}=\left[ \begin{array}{cccc}1.012918&0.166930&-0.011245 &-0.209060\\ 0.071820&1.028852&0.106682&0.048913\\ 0.065354&-0.028643&0.978727&0.171113\\ 0&0&0&1\end{array} \right],\]  
\[\mathbf{H}_{tml}^{-1}=\left[ \begin{array}{cccc}1.064280&0.408684&-0.434687& 0.419931\\ 0.324798&1.345943&0.210470&0.898905\\ 0.221952&0.184071&1.084612&-0.470323\\ 0&0&0&1\end{array} \right],\]  
\[\mathbf{H}_{tml}^{-1}=\left[ \begin{array}{cccc}0.112204&-0.176563&0.677722& -1.436055\\ -0.037890&0.631210&0.318046&0.523522\\ -0.659836&-0.011208&0.253057&0.588023\\ 0&0&0&1\end{array} \right],\text{and}\]  
\[\mathbf{H}_{tml}^{-1}=\left[ \begin{array}{cccc}0.599898&-0.757315&-0.454490&1. 325230\\ 0.730332&0.576859&-0.516980&-0.534417\\ 0.809368&-0.039595&0.673142&2.244765\\ 0&0&0&1\end{array} \right].\]  
The best of intermediate particles is \(\mathbf{H}_{tml}^{-1}\) and it is better than \(\mathbf{H}_{g}^{-1}\), hence \(\mathbf{H}_{g}^{-1}=\mathbf{H}_{tml}^{-1}\). Then \(\mathbf{x}_{select}^{b}\) from