## 2 Method
### Dynamic Correction Model  
In a smooth running condition, the deep learning model LatDisLanes has high recognition accuracy and a satisfactory recognition result. However, if the inclination angle changes drastically due to the fluctuation of road surface and centripetal force under high speed, larger recognition error will result. In this study, a dynamic correction model is developed by analyzing the influence of inclination angle on lateral distance recognition and the error is reduced in this way. The camera position in the world coordinate system is depicted in Figure 7, wherein \(\theta\), \(\alpha\), and \(\beta\) are inclination, yaw, and roll angle of the vehicle, respectively. Since this study focuses on the lateral distance between the lane markings and vehicle, the effect that is produced by \(\alpha\) and \(\beta\) is so little that it can be neglected. The \(\theta\) is closely related to the lateral distance on \(y\)-axis. By studying the relationship among vehicle coordinates, camera coordinates, image coordinates, and pixel coordinates, this study ultimately obtains the variation of \(\Delta y_{w}(\theta)\) in the world coordinate system induced by \(\Delta\theta\). The process is as follows:  
1. Transformation from world coordinate to camera coordinates  
Let \(P(x_{w}\), \(y_{w}\), \(z_{w})\) be a point in the world coordinate system, then the corresponding point in camera coordinate system is \(P(x_{c}\), \(y_{c}\), \(z_{c})\), and camera position in world coordinate is \(x_{0}\) = [0,0,\(\mu\)]. The transformation from world coordinates to camera coordinates is obtained by:  
\[x_{c}=R(x_{w}-x_{0}) \tag{18}\]  
where \(R=\left[ \begin{array}{ccc}\cos\alpha&0&-\sin\alpha\\ -\sin\theta\sin\alpha&\cos\theta&-\cos\alpha\sin\theta\\ \cos\theta\sin\alpha&\sin\theta&\cos\theta\cos\alpha\end{array} \right]\) is a rotation matrix. Figure 6: The LatDisLanes network architecture. Figure 7: Three-dimensional (3D) world and camera coordinate systems, where \(\theta\), \(\alpha\), and \(\beta\) are inclination, yaw and roll angle of the vehicle, respectively. Suppose that point \(P^{\prime}(x_{c},y_{c},z_{c})\) in the camera coordinate system corresponds to the point \(Q(x,y)\) in image coordinate system, then if we let\(f\) be a focal length of camera, according to the relationship of perspective projection, the transformation from camera coordinates to image coordinates is obtained by:  
\[\left[ \begin{array}{c}x\\ y\end{array} \right]=\left[ \begin{array}{c}f\frac{x_{c}}{z_{c}}\\ f\frac{y_{c}}{z_{c}}\end{array} \right] \tag{19}\]  
3. Transformation from image coordinates to pixel coordinates  
If the point \(Q^{\prime}(x,y)\) in pixel coordinate system corresponds to the point \(Q(x,y)\) in the image, then the mapping between \(Q\) and \(Q^{\prime}\) can be obtained by:  
\[\left[ \begin{array}{c}u\\ v\end{array} \right]=\left[ \begin{array}{ccc}s_{u}&s_{1}&u_{0}\\ 0&s_{v}&v_{0}\end{array} \right]\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right]=\left[ \begin{array}{c}f\frac{s_{u}x_{c}}{z_{c}}+u_{0}\\ f\frac{y_{c}}{z_{c}}+v_{0}\end{array} \right] \tag{20}\]  
where (\(u_{0}\), \(v_{0}\)) is the cardinal point, \(s_{u}\) and \(s_{v}\) are the pixel distances in the horizontal and vertical directions in images, and \(s_{1}\) is the skew factor. The radial and tangential distortions are not taken into account in this study, thus \(s_{1}\) is set to 1 [42].