# 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
In Equation (1), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(b_{1}\), \(b_{2}\), \(b_{3}\), \(c_{1}\), \(c_{2}\), \(c_{3}\) are the elements of the rotation matrix of rigid body transformation between the world coordinate system \(\text{O}_{\text{w}}\text{-}\text{X}_{\text{w}}\text{Y}_{\text{w}}\text{Z}_{ \text{w}}\) and the line-scan camera coordinate system \(\text{O}_{\text{c}}\text{-}\text{X}_{\text{c}}\text{Y}_{\text{c}}\text{Z}_{ \text{c}}\). The rotation matrix is an orthogonal matrix related to the camera installation posture (\(\alpha\), \(\beta\), \(\gamma\)), as shown in Equation (3). \[\left[ \begin{array}{cc}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{array} \right]=\left[ \begin{array}{cc}\cos\beta\cos \gamma&\cos\beta\sin\gamma&-\sin\beta\\ \sin\alpha\sin\beta\cos\gamma-\cos\alpha\sin\gamma&\sin\alpha\sin\beta\sin \gamma+\cos\alpha\cos\gamma&\sin\alpha\cos\beta\\ \cos\alpha\sin\beta\cos\gamma+\sin\alpha\sin\gamma&\cos\alpha\sin\beta\sin \gamma-\sin\alpha\cos\gamma&\cos\alpha\cos\beta\end{array} \right] \tag{3}\]  
In Equation (1), the constraint condition \(Z_{\text{P}}=g(X_{\text{P}})\) is the equation of conveyor belt arc cross-section in the world coordinate system by polynomial curve fitting. Let the range of the conveyor belt content corresponding to the overlapping region of 2 adjacent images in the \(X_{\text{w}}\) axis direction of the world coordinate system be \([X_{\text{P1}}\), \(X_{\text{P2}}]\). The overlapping region of 2 adjacent images only considers the \(v\) axis coordinates of image pixels. Let the pixel range of the overlapping region of the left image be \([v_{1}\), n]. Let the pixel  
Figure 4: Imaging schematic diagram of the line-scan camera. range of the overlapping region of the right image be [1, \(v_{2}\)]. According to Equation (1), the overlapping region estimation model can be further established, which satisfies the relationship of Equation (4):  
\[\left\{ \begin{array}{l}v_{1}=v_{0}+\Delta v_{v=v_{1}}-f^{\prime}\frac{c_{3}(X_ {P1}-X_{01})-a_{3}(Z_{P1}-Z_{01})}{a_{1}(Z_{P1}-Z_{01})-c_{1}(X_{P1}-X_{01})}\\ n=v_{0}+\Delta v_{v=n}-f^{\prime}\frac{c_{3}(X_{P2}-X_{01})-a_{3}(Z_{P2}-Z_{01 })}{a_{1}(Z_{P2}-Z_{01})-c_{1}(X_{P2}-X_{01})}\\ 1=v_{0}+\Delta v_{v=1}-f^{\prime}\frac{c_{3}(X_{P1}-X_{02})-a_{3}(Z_{P1}-Z_{02 })}{a_{1}(Z_{P1}-Z_{02})-c_{1}(X_{P1}-X_{02})}\\ v_{2}=v_{0}+\Delta v_{v=2}-f^{\prime}\frac{c_{3}(X_{P2}-X_{02})-a_{3}(Z_{P2}-Z_{ 02})}{a_{1}(Z_{P2}-Z_{02})-c_{1}(X_{P2}-X_{02})}\\ s.t.\hskip 14.226378 ptZ_{P1}=g(X_{P1})\hskip 14.226378 ptZ_{P2}=g(X_{P2})\end{array} \right. \tag{4}\]  
where, \(X_{01}\), \(Y_{01}\), \(Z_{01}\) and \(X_{02}\), \(Y_{02}\), \(Z_{02}\) are the installation position coordinates of the cameras corresponding to the 2 adjacent images, and \(a_{1}\), \(a_{3}\), \(c_{1}\), \(c_{3}\) and \(a_{1}\)', \(a_{3}\)', \(c_{1}\)', \(c_{3}\)' are the elements of the rotation matrix of the cameras corresponding to the 2 adjacent images. When \(v\) is \(v_{1}\), \(u\), \(1\), and \(v_{2}\), the values of \(\Delta v_{v_{v}=v1}\), \(\Delta v_{v_{v}=n}\), \(\Delta v_{v_{v}=1}\), and \(\Delta v_{v_{v}=v2}\) are the values of \(\Delta v\), respectively. The overlapping region of 2 adjacent images can be estimated by solving the values of unknown variables \(v_{1}\), \(v_{2}\), \(X_{\text{w}1}\), and \(X_{\text{w}2}\).