## 2 Registration Method  
In a 3-D registration problem, the geometry transform can be calculated from the relation between two point-cloud images. The more they are related, the higher quality of the transformation. The highest quality transformation corresponds to the minimization of the following statement:  
\[\mathbf{H}^{*}=\text{argmin}O(\mathbf{H}(\mathbf{Q}),\mathbf{P}) \tag{1}\]  
where \(\mathbf{P}\) is the target point-cloud matrix (\([\boldsymbol{p}_{i}]_{M\times 4}\), \(M\) is the number of target point-cloud points), \(\mathbf{Q}\) is the source point-cloud matrix (\([\boldsymbol{q}_{j}]_{N\times 4}\), \(N\) is the number of source point-cloud points), and \(\mathbf{H}\) is the geometry transform. Finally, \(O(\cdot)\) is an objective function. Since the transformation \(\mathbf{H}\) is estimated by finding the nearest neighbor [26] between a set of point-pairs (\(\boldsymbol{p}_{j}\), \(\boldsymbol{q}_{j}\)), the minimum error of the distance between two corresponding points can be considered [27]. Using the mean squared error (MSE), hence, the minimization problem, in this case, is calculated as  
\[\mathbf{H}^{*}=\text{argmin}_{\mathbf{H}}\frac{1}{N}\underset{j=1}{\overset{N}{ \sum}}\Big{(}\boldsymbol{q}_{j}\cdot\mathbf{H}^{T}-\boldsymbol{p}_{j}\Big{)} ^{2} \tag{2}\]  
\[\boldsymbol{p}_{j}=\underset{\boldsymbol{p}_{i}\in\mathbf{P}}{\text{argmin}} \boldsymbol{q}_{j}\mathbf{H}^{T}-\boldsymbol{p}_{i} \tag{3}\]  
The transformation \(\mathbf{H}\) is the 3-D transformation with 15 unknown parameters, i.e., 3 parameters from scaling (\(\mathbf{S}\)), 3 parameters from translation (\(\mathbf{T}\)), 3 parameters from rotation (\(\mathbf{R}\)), and 6 parameters from shearing (\(\mathbf{SH}\)) [28]. The matrix \(\mathbf{H}\) is computed as  
\[\mathbf{H} =\mathbf{T}\times\mathbf{S}\times\mathbf{R}\times\mathbf{SH}, \tag{4}\]  
\[\text{where }\mathbf{T}=\left[ \begin{array}{cccc}1&0&0&t_{x}\\ 0&1&0&t_{y}\\ 0&0&1&t_{z}\\ 0&0&0&1\end{array} \right],\mathbf{S}=\left[ \begin{array}{cccc}s_{x}&0&0&0\\ 0&s_{y}&0&0\\ 0&0&s_{z}&0\\ 0&0&0&1\end{array} \right],\text{and }\mathbf{SH}=\left[ \begin{array}{cccc}1&sh_{1}&sh_{2}&0\\ sh_{3}&1&sh_{4}&0\\ sh_{5}&sh_{6}&1&0\\ 0&0&0&1\end{array} \right],\]  
\[\mathbf{R}=\left[ \begin{array}{cccc}\cos(\theta_{y})\cos(\theta_{z})&-\cos (\theta_{y})\sin(\theta_{z})&\sin(\theta_{y})&0\\ \sin(\theta_{x})\sin(\theta_{y})\cos(\theta_{z})+\cos(\theta_{x})\sin(\theta_{ z})&-\sin(\theta_{x})\sin(\theta_{y})\sin(\theta_{z})+\cos(\theta_{x})\cos(\theta_{z})&- \sin(\theta_{x})\cos(\theta_{y})&0\\ -\cos(\theta_{x})\sin(\theta_{y})\cos(\theta_{z})+\sin(\theta_{x})\sin(\theta_{ z})&\cos(\theta_{x})\sin(\theta_{y})\sin(\theta_{z})+\sin(\theta_{x})\cos(\theta_{z})&\cos( \theta_{x})\cos(\theta_{y})&0\\ 0&0&1\end{array} \right] \tag{5}\]Hence,  
\[\mathbf{H}=\left[ \begin{array}{cccc}a&d&g&l\\ b&e&i&m\\ c&f&j&n\\ p&q&r&s\end{array} \right] \tag{6}\]  
where  
\[ \begin{array}{c}a=s_{x}(cycz)+sh_{3}s_{x}(-cysz)+sh_{5}s_{x}(sy),\\ b=s_{y}(sxysycz+cxsz)+sh_{3}s_{y}(-sxysyz+cxcz)+sh_{5}y(-sxcy),\\ c=s_{z}(-cxsycz+sxsz)+sh_{3}s_{z}(cxsysz+sxcz)+sh_{5}s_{z}(cxcy),\\ d=sh_{1}s_{x}(cycz)+s_{x}(-cysz)+sh_{6}s_{x}(sy),\\ e=sh_{1}s_{y}(sxysycz+cxsz)+s_{y}(-sxysysz+cxcz)+sh_{6}s_{y}(-sxcy),\\ f=sh_{1}s_{z}(-cxsycz+sxsz)+s_{z}(cxsysz+sxcz)+sh_{6}s_{z}(cxcy),\\ g=sh_{2}s_{x}(cycz)+sh_{4}s_{x}(-cysz)+s_{x}(y),\\ i=sh_{2}s_{y}(sxysycz+cxsz)+sh_{4}s_{y}(-sxysysz+cxcz)+s_{y}(-sxcy),\\ j=sh_{2}s_{z}(-cxsycz+sxsz)+sh_{4}s_{z}(cxsysz+sxcz)+s_{z}(cxcy),\\ cx=\cos(\theta_{x}),cy=\cos(\theta_{y}),cz=\cos(\theta_{z}),\\ sx=\sin(\theta_{x}),sy=\sin(\theta_{y}),sz=\sin(\theta_{z}).\end{array}  \tag{7}\]  
It is worthwhile noting that \(a\) through \(j\) are non-rigid transformations resulting from the combination of scaling, shearing, and rotation properties. Meanwhile, \(l\), \(m\), and \(n\) are simply \(t_{x}\), \(t_{y}\), and \(t_{z}\), respectively. Moreover, \(p\), \(q\), and \(r\) are set to 0 since they are perspective property values. Finally, \(s\) is always set to 1 because of the scaling factor.