## 2 Detection Principles and Self-Deflection Analysis
### System Composition and Detection Principles
\begin{bmatrix}r_{\text{r}1}&r_{\text{r}2}&t_{\text{r}1}\\ r_{\text{r}3}&r_{\text{r}4}&t_{\text{r}2}\\ 0&0&1\end{bmatrix}\begin{bmatrix}x_{\text{p}}\\ y_{\text{p}}\\ 1\end{bmatrix}  \tag{3}\]  
where (\(x_{\text{r}}\),\(y_{\text{r}}\)) represents the registered laser point cloud data, and \(\mathbf{R_{\text{r}}}\) and \(\mathbf{T_{\text{r}}}\) are the rotation matrix and translation vector corresponding to the registration transformation, respectively, containing four rotation parameters from \(r_{\text{r}1}\) to \(r_{\text{r}4}\) and two translation parameters from \(t_{\text{r}1}\) to \(t_{\text{r}2}\). To obtain the surface deformation, it is also necessary to acquire the point cloud's axial coordinates perpendicular to the bore profile plane, which can be achieved using a dot laser based on the time-of-flight method of ranging. By measuring the distance \(z_{\text{p}}\) from the dot laser to the reflector on the borescope trolley, the axial coordinates \(z_{n}\) of the \(n\)-th (where \(n=1,\ldots,N\)) profile point cloud can be obtained:  
\[z_{n}=z_{\text{p}}+\Delta z_{\text{p}}+l \tag{4}\]  
with \(\Delta z_{\text{p}}\) representing the axial offset of the dot laser installation within the bore, and \(l\) representing the distance from the reflector to the ring laser plane. Then, the left and right surface deformations \(\delta_{\chi,n}\) and the upper and lower surface deformations \(\delta_{y,n}\) of the bore satisfy  
\[\left\{ \begin{array}{l}\delta_{x,n}=x_{\text{r},n}-x_{\text{s}}\\ \delta_{y,n}=y_{\text{r},n}-y_{\text{s}}\end{array} \right. \tag{5}\]  
Figure 2: Schematic of laser triangulation for acquiring bore profile point clouds. The bore straightness is defined as the deviation of the profile center point at each position from the line connecting the center points of the first and last profiles, as shown in Figure 3. Assuming the coordinates of the \(n\)-th profile center point in the world coordinate system \((x_{\text{w}},y_{\text{w}},z)\) are \((x_{\text{w},\text{o},n},y_{\text{w},\text{o},n},z_{n})\), then the bore straightness can be expressed as  
\[\left\{ \begin{array}{l}u_{n}=(x_{\text{w},\text{o},n}-x_{\text{w},\text{o},1} )-(x_{\text{w},\text{o},\text{N}}-x_{\text{w},\text{o},1})\frac{z_{n}-z_{1}}{ z_{\text{N}}-z_{1}}\\ v_{n}=(y_{\text{w},\text{o},n}-y_{\text{w},\text{o},1})-(y_{\text{w},\text{o}, \text{N}}-y_{\text{w},\text{o},1})\frac{z_{n}-z_{1}}{z_{\text{N}}-z_{1}}\\ \end{array} \right. \tag{6}\]  
where \(u_{n}\) is the side straightness, and \(v_{n}\) is the arch straightness. During the capture of the inner bore profile by the camera mounted on the bore borescope trolley, the dot laser spot image on the reflector is also recorded within the same frame. By translating the registered profile center point (\(0,0,z_{n}\)) using the registered dot laser spot center (\(x_{\text{r},\text{p},n},y_{\text{r},\text{p},n},z_{n}\)) as the origin, the coordinates of the \(n\)-th profile center point in the world coordinate system can be obtained as follows:  
\[\left\{ \begin{array}{l}x_{\text{w},\text{o},n}=-x_{\text{r},\text{p},n}\\ y_{\text{w},\text{o},n}=-y_{\text{r},\text{p},n}\\ \end{array} \right. \tag{7}\]  
According to (6) and (7), the bore straightness (\(u_{n}\), \(v_{n}\)) described by the registered point cloud data can be obtained as follows:  
\[\left\{ \begin{array}{l}u_{n}=(x_{\text{r},\text{p},\text{N}}-x_{\text{r}, \text{p},1})\frac{z_{n}-z_{1}}{z_{\text{N}}-z_{1}}-(x_{\text{r},\text{p},n}-x_ {\text{r},\text{p},1})\\ v_{n}=(y_{\text{r},\text{p},\text{N}}-y_{\text{r},\text{p},1})\frac{z_{n}-z_{ 1}}{z_{\text{N}}-z_{1}}-(y_{\text{r},\text{p},n}-y_{\text{r},\text{p},1})\\ \end{array} \right. \tag{8}\]