## 3 Measurement System Measures the Core Algorithm
### Measurement Principle  
The detection principles and methods of monocular vision have been extensively studied [11] and are relatively mature. The detection principle coordinate system is shown in Figure 3. \(uO_{0}v\) is the image pixel coordinate system, \(xO_{1}y\) is the image physical coordinate system, \(O_{c}X_{c}Y_{c}Z_{c}\) is the camera coordinate system, \(O_{w}X_{w}Y_{w}Z_{w}\) is the world coordinate system. The length of \(O_{1}O_{c}\) is the focal length \(f\) of the camera.  
Figure 3: Detecting coordinate system relationships in the system.  
Figure 2: Algorithm flowchart for the measurement system.  
Mathematical relationship between the world coordinate system and the camera coordinate system:  
\[\begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\\ 1\end{bmatrix}=\begin{bmatrix}R&T\\ 0&1\end{bmatrix}\begin{bmatrix}X_{w}\\ Y_{w}\\ Z_{w}\\ 1\end{bmatrix} \tag{1}\]  
Camera coordinate system and image mathematical relationship exists between the physical coordinate system:  
\[s\begin{bmatrix}x\\ y\\ 1\end{bmatrix}=\begin{bmatrix}f&0&0&0\\ 0&f&0&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\\ 1\end{bmatrix} \tag{2}\]  
Mathematical relationship between the physical coordinate system of an image and image pixel coordinate system; \(dx,dy\) represents the length of a unit pixel on the \(x\) and \(y\) axes in the coordinate system \(xO_{1}y\):  
\[\begin{bmatrix}u\\ v\\ 1\end{bmatrix}=\begin{bmatrix}1/dx&0&u_{0}\\ 0&1/dy&v_{0}\\ 0&0&1\end{bmatrix}\begin{bmatrix}X\\ Y\\ 1\end{bmatrix} \tag{3}\]  
Mathematical relationship between world coordinate system and image pixel coordinate system:  
\[s\begin{bmatrix}u\\ v\\ 1\end{bmatrix}=\begin{bmatrix}1/dx&0&u_{0}\\ 0&1/dy&v_{0}\\ 0&0&1\end{bmatrix}\begin{bmatrix}f&0&0&0\\ 0&f&0&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}R&T\\ 0&1\end{bmatrix}\begin{bmatrix}X_{w}\\ Y_{w}\\ Z_{w}\\ 1\end{bmatrix} \tag{4}\]  
Due to the impact of the site environment, there is a slight tilt angle between the measurement system and the cross-cutting board in order to correct the skew error of the \(uO_{0}v\) coordinate system's axes. Introducing parameters \(\gamma=f_{x}\bullet\tan\theta\), \(\gamma=f_{x}\bullet\tan\theta\) is a key term in the camera tilt model that describes geometric deformation or correction. Its essence is the product of the focal length and the tangent of the tilt angle, reflecting the coupled influence of tilt installation on the image coordinates. By embedding it into the homography matrix or coordinate mapping formula, high-precision image correction and measurement can be achieved. In practical applications, parameters need to be optimized in combination with calibration data to ensure the robustness of the model. \(\theta\) indicates the degree of inclination of a plane along the \(v\)-axis; its value was obtained through experiments on different angles; \(h\) is the distance from the industrial camera to the roller conveyor; \(l\) represents the maximum field of view of the measurement system. The Equation (4) can be written as follows:\[s\begin{bmatrix}u\\ v\\ 1\end{bmatrix}=\begin{bmatrix}1/dx&\gamma&u_{0}\\ 0&1/dy&v_{0}\\ 0&0&1\end{bmatrix}\begin{bmatrix}f&0&0&0\\ 0&f&0&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}R&T\\ Y_{w}\\ Z_{w}\\ 1\end{bmatrix}\begin{bmatrix}X_{w}\\ Y_{w}\\ Z_{w}\\ 1\end{bmatrix} \tag{5}\] \[=\begin{bmatrix}f_{x}&\gamma&u_{0}&0\\ 0&f_{y}&v_{0}&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}R&T\\ 0&1\end{bmatrix}\begin{bmatrix}X_{w}\\ Y_{w}\\ Z_{w}\\ 1\end{bmatrix}\]