# An Automatic Geometric Registration Method for Multi Temporal 3D Models
## 3 Methodology
### Registration Model Solution  
The geometric transformation model from the vertex of the registered 3D model to the vertex of the base 3D model can be defined as 3D similarity transformation, as shown in Equation (7).  
\[\begin{bmatrix}X_{b}\\ Y_{b}\\ Z_{b}\end{bmatrix}=\lambda\cdot\mathbf{R}\cdot\begin{bmatrix}X_{r}\\ Y_{r}\\ Z_{r}\end{bmatrix}+\begin{bmatrix}r_{x}\\ r_{y}\\ r_{z}\end{bmatrix} \tag{7}\]  
where \(\begin{bmatrix}X_{r}&Y_{r}&Z_{r}\end{bmatrix}^{\text{T}}\) is the vertex position of the model to be registered and \(\begin{bmatrix}X_{b}&Y_{b}&Z_{b}\end{bmatrix}^{\text{T}}\) is the corresponding vertex position of the base model. \(\mathbf{R}\) and \(\begin{bmatrix}r_{x}&r_{y}&r_{z}\end{bmatrix}^{\text{T}}\) represents the rotation matrix and the translation from the vertex of the model to be registered to the base model, respectively. \(\lambda\) represents the scale factor from the vertex of the model to be registered to the base model, and when \(\lambda=1\), Equation (7) becomes a 3D rigid transformation model.  
The detailed definition of rotation matrix \(\mathbf{R}\) in Equation (7) is shown in Equation (8).  
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