## 3 Crack Identification and Measurement System
### Measurement System Based on Camera and Laser Rangefinder
#### 3.1.2 Geometric Transformation Formula of Pixel Length
When the camera and laser rangefinder are fixed, the horizontal baseline distance \(BL\) can be determined. Since the optical axes are parallel, it can be seen that  
\[U^{\prime}=X+U, \tag{9}\]2  
\[U^{\prime}=X+U, \tag{9}\]3  
\[U^{\prime}=X+U, \tag{9}\]4  
\[U^{\prime}=X+U, \tag{9}\]5 The geometric relationship between the combination of the camera and laser rangefinder and the target plane in clockwise direction: (**a**) horizontal length transformation; (**b**) vertical proportional relationship. \[U^{\prime}=X+U, \tag{9}\]6  
\[U^{\prime}=X+U, \tag{9}\]7  
\[U^{\prime}=X+U, \tag{9}\]8  
\[U^{\prime}=X+U, \tag{9}\]9  
where the relationship between \(hpix\) and \(hpixx^{\prime}\) is determined by the rotation angle \(\theta_{h}\) and shooting distance \(U^{\prime}\). Equation (24) demonstrates that the distance from the center of rotation P\({}_{1}\) to a random point in the horizontal direction can be represented by the imaging pixel \(hpix\), and when introducing Formula (24) into Formula (17), the distance \(L_{\text{P}_{1}\text{P}_{2}^{\prime}}\) can be obtained as follows:  
\[f^{\prime}=Y\times f, \tag{10}\]0  
Meanwhile, the horizontal rotation of the target plane will not only create distortion in the horizontal direction but will also cause the displacement and deformation in the vertical direction as presented in Figure 5b and expressed as follows:  
\[f^{\prime}=Y\times f, \tag{10}\]1  
where \(VL\) and \(VL^{\prime}\) are the vertical length on the ideal parallel plane and realistic side plane, respectively. The vertical length can be calculated by Formulas (11) and (26) with the vertical pixel length \(vpix\), and \(VL^{\prime}\) is obtained as follows:  
\[f^{\prime}=Y\times f, \tag{10}\]2  
When the realistic plane rotates in the counterclockwise direction, as shown in Figure 6a, the relationship between \(hpix\) and \(hpixx^{\prime}\) can be derived similarly. Formula (24) can be converted as follows:  
\[ \begin{split}& L_{\text{P}_{1}\text{P}_{3}}-L_{\text{FP}_{3}}=L_{ \text{P}_{1}\text{P}_{2}^{\prime}}\times\cos\theta_{h}\\ &\Rightarrow lpix^{\prime}=\frac{hpix}{\cos\theta_{h}+\sin \theta_{h}\times\left(\frac{pix\times(U^{\prime}-f^{\prime})}{U^{\prime}f^{ \prime}}-\frac{Bi}{U^{\prime}}\right)}\end{split}  \tag{28}\]  
Figure 6: The geometric relationship between the combination of the camera and laser rangefinder and the target plane in counter-clockwise direction: (**a**) horizontal length transformation; (**b**) vertical proportional relationship. The length between point P\({}_{1}\) and point P\({}_{2^{\prime}}\) in the side plane can be measured by Formulas (17) and (28) as follows:  
\[L_{\text{P}_{1}\text{P}_{2}^{\prime}}=\frac{hpix\times\varepsilon(U^{\prime}-f^{ \prime})\times U^{\prime}f^{\prime 2}}{U^{\prime}f^{\prime}\cos\theta_{h}+\sin \theta_{h}\times(hpix\times\varepsilon(U^{\prime}-f^{\prime})-BL\times f^{ \prime})}, \tag{29}\]  
Similarly, when the plane is rotated counterclockwise horizontally as shown in Figure 6b, the pixel length in the vertical direction also changes proportionally as presented in Equation (30) as follows:  
\[VL^{\prime}=\frac{VL}{U^{\prime}}\times(U^{\prime}-L_{\text{P}_{2}^{\prime} \text{F}}), \tag{30}\]  
It is worth noting that the proposed system is based on the camera located on the left side of the laser rangefinder, and the base line distance \(BL\) does not only exist in the horizontal but also in the vertical direction. However, it can be seen from Formulas (24) and (28) that when the camera is close to the laser rangefinder or measured from long-distance, the shooting distance \(U^{\prime}\) is much larger than the base line distance \(BL\), implying that \(BL\) can be ignored to simplify the model. The position of a laser point is a key parameter in the proposed system that represents the center of target plane rotation and the origin of calculating the distance between pixels on both sides of the crack.