# Adaptive Multi-View Image Mosaic Method for Conveyor Belt Surface Fault Online Detection
## 3 Materials and Methods
### Grayscale-Based Overlapping Region Registration  
#### 3.1.1 Overlapping Region Estimation  
The overlapping region estimation model of 2 adjacent images is established according to the geometry of the conveyor belt and the parameters, installation position, and posture of multi-view cameras. For a conveyor belt, the horizontal distance between its two ends is set as \(B\), the speed is \(V\). Suppose it runs smoothly and at a uniform speed. The sensor pixel size of the line-scan camera is w, the number of pixels is \(n\), the line sampling frequency is \(f_{h}\), and the lens focal length is \(f\). For the conveyor belt, the world coordinate system \(\rm O_{w}\)-\(\rm X_{w}\)\(\rm Y_{w}\)\(\rm Z_{w}\) is established. The \(\rm X_{w}\) axis passes through the bottom end of the arc cross-section of the conveyor belt and is parallel to the ground, and the direction is from left to right. The \(\rm Z_{w}\) axis passes through the leftmost endpoint of the arc cross-section of the conveyor belt and is perpendicular to the ground, and the direction is from bottom to top. The intersection point \(\rm O_{w}\) of the \(\rm X_{w}\) axis and the \(\rm Z_{w}\) axis is the origin, and the origin coordinate is (0, 0, 0). The direction of \(V\) is the same as the \(\rm Y_{w}\) axis. Then, in the world coordinate system \(\rm O_{w}\)-\(\rm X_{w}\)\(\rm Y_{w}\)\(\rm Z_{w}\), the coordinate of the camera installation position \(\rm O_{c}\) is (\(\rm X_{0}\), \(\rm Y_{0}\), \(\rm Z_{0}\)), the unit is millimeter; the camera installation posture is (\(\alpha\), \(\beta\), \(\gamma\)), the unit is degrees (\({}^{\circ}\)). In the line-scan camera imaging process, the space points are projected onto the image sensing pixels to form an image, and the image pixel coordinate system is \(\rm O_{f}\)-uv. The imaging schematic diagram of the line-scan camera is shown in Figure 4.  
The static imaging of a line-scan camera can only obtain the one-dimensional image of the conveyor belt. Because of the relative motion between the conveyor belt and line-scan camera, the line-scan camera can obtain a two-dimensional image of the conveyor belt by dynamic imaging. Suppose that the coordinate of any point \(P\) on the conveyor belt in the world coordinate system \(\rm O_{w}\)-\(\rm X_{w}\)\(\rm Y_{w}\)\(\rm Z_{w}\) is (\(\rm X_{P}\), \(\rm Y_{P}\), \(\rm Z_{P}\)), and the coordinate of the point \(P\) imaged in the image pixel coordinate system \(\rm O_{f}\)-uv is (\(u\), \(v\)). According to the  
Figure 3: Flow chart of the adaptive multi-view image mosaic (AMIM) method. fundamental relationship of the perspective projection geometric imaging model of the imaging characteristics of the area-scan camera and the line-scan camera [23; 24], one can obtain the dynamic imaging model of the line-scan camera, is seen in Equation (1):  
\[\left\{ \begin{array}{l}u=\frac{a_{2}(X_{p}-X_{0})+b_{2}(Y_{p}-Y_{0})+c_{2}(Z_{p }-Z_{0})}{b_{2}V}\\ v=v_{0}+\Delta v-f^{\prime}\frac{c_{1}(X_{p}-X_{0})-a_{1}(Z_{p}-Z_{0})}{a_{1}(Z_ {p}-Z_{0})-c_{1}(X_{p}-X_{0})}\\ s.t.~{}Z_{p}=g(X_{p})\end{array} \right. \tag{1}\]  
where \(v_{0}\) is the pixel coordinate value of the main imaging point O of sensor pixel center of the line-scan camera, which can be generally set as \(v_{0}=n/2\). \(f^{\prime}=f/\text{w}\). \(V^{\prime}\) represents the relative motion between the conveyor belt and the line-scan camera in each image acquisition cycle, satisfying \(V^{\prime}=V/f_{h}\). Compared with \(V^{\prime}\), the relative motion of the conveyor belt caused by vibration and offset can be ignored. The results show that \(\triangle v\) is the imaging distortion, and \(\triangle v\) increases gradually from \(v_{0}\) to both sides. The Equation is as follows:  
\[\Delta v=k_{1}(v-v_{0})^{5}+k_{2}(v-v_{0})^{3}+k_{3}(v-v_{0})^{2} \tag{2}\]  
where \(k_{1}\), \(k_{2}\), and \(k_{3}\) are distortion parameters, which can be obtained by camera calibration.