## 3 Crack Identification and Measurement System
### Measurement System Based on Camera and Laser Rangefinder
#### 3.1.2 Geometric Transformation Formula of Pixel Length  
It can be seen from Equation (4) that the length of a crack can be automatically counted by obtaining the distance to the measured target. However, the distance \(U\) obtained by the laser rangefinder is not very accurate, because the camera's photosensitive sensor and the  
Figure 3: Crack measurement equipment composed of camera, laser rangefinder, laptop and tripod. origin of the laser rangefinder are not certainly in the same plane. This study provides a formula to correct the position of the camera and laser rangefinder as shown in Equation (9):  
\[U^{\prime}=X+U, \tag{9}\]  
where X is a constant and represents the distance between the camera imaging plane and the origin of the laser rangefinder. Moreover, because of the difference between the half and full frame types, the focal length \(f\) involved in the calculation is different from that in the nominal case, and the correction formula is expressed as  
\[f^{\prime}=Y\times f, \tag{10}\]  
where Y is another constant that represents the focal length correction factor. Inputting Equations (9) and (10) into Equation (4), we obtain Equation (11):  
\[L=\frac{pix\times\varepsilon(U^{\prime}-f^{\prime})}{f^{\prime}}=\frac{pix \times\varepsilon(U+X-Yf)}{Yf}, \tag{11}\]  
This formula is a linear transformation in which a relative geometry relationship of equipment is maintained. In practice, if the positions of the camera and laser rangefinder are fixed, the constants X and Y can be uniquely determined. To apply Formula (11) to the measurement of cracks, if the object plane is not parallel to the images, the main objective is expected to transform the plane to a parallel position on the image plane. The rotation can be decomposed into horizontal and vertical orientation by \(\theta_{h}\) and \(\theta_{v}\) angles, respectively, as shown in Figure 4, and the random measuring point P\({}_{1}\) of the laser rangefinder is defined as the center of rotation. Because the rotation is axisymmetric, the side plane generated by horizontal clockwise rotation is taken as an example, and the geometric relation is proposed in Figure 5a. The distance between point P\({}_{1}\) and the center point of the laser rangefinder R is the measuring distance \(U\). P\({}_{2^{\prime}}\) is another random point measured by the laser rangefinder by rotating \(\alpha\) degrees horizontally on the side plane, and P\({}_{2}\) is the same point in the ideal parallel plane. To calculate the horizontal angle \(\theta_{h}\), the measured distance of the line between points R and P\({}_{2^{\prime}}\) (\(L_{\text{RP}_{2^{\prime}}}\)) can be decomposed into the sum of \(L_{\text{RE}}\) and \(L_{\text{EP}_{2^{\prime}}}\) and derived as follows:  
Figure 4: Decomposition of unparallel plane into horizontal and vertical orientation. \[L_{\text{RE}}=U\times\cos\alpha, \tag{12}\]  
\[L_{\text{EP}_{2}{}^{\prime}}=L_{\text{RP}_{2}{}^{\prime}}-L_{\text{RE}}, \tag{13}\]  
\[L_{\text{EP}_{1}}=U\times\sin\alpha, \tag{14}\]  
\[\beta=\arctan(\frac{L_{\text{EP}_{1}}}{L_{\text{EP}_{2}{}^{\prime}}}), \tag{15}\]  
\[\theta_{h}=\pi/2-\beta-\alpha, \tag{16}\]  
where the distance \(U\), \(L_{\text{RP}_{2}{}^{\prime}}\) and rotation angle \(\alpha\) can be obtained by the laser rangefinder. If the horizontal imaging pixel length \(\it{lpix}\) is used to calculate the target length by Equation (11), the result will represent the length \(L_{\text{P}_{1}\text{P}_{3}}\) instead of the target \(L_{\text{P}_{1}\text{P}_{2}{}^{\prime}}\). The pixel length used to calculate the distance can be expressed as  
\[U^{\prime}=X+U, \tag{9}\]0  
\[U^{\prime}=X+U, \tag{9}\]1  
However, since the position of point P\({}_{2}{}^{\prime}\) is random in the horizontal direction, the proportional relationship between \(\it{lpix}\) and \(\it{lpix}^{\prime}\) should represent the proportion of the arbitrary horizontal pixel distance on the realistic side plane and the corresponding horizontal pixel distance on the ideal parallel plane.