## 2 Methods
### Data Set
#### 2.1.2 Point Cloud Data Preprocessing
The method fits a plane of \(ax+by+cz=d\) by solving the normal vector \(\left\{a,b,c\right\}\) and the parameter d, so that the distance from the 3D point cloud \(\left(x_{j},y_{j},z_{j}\right)\) (\(j=1,2,3\ldots n-1\)) to the plane is minimized. Since the normal vector \(\left\{a,b,c\right\}\) needs to meet the condition \(a^{2}+b^{2}+c^{2}=1\), the problem can be transformed into minimizing the function \(f\) using the Lagrange multiplier method [19], as shown in Equation (3). \[f=\sum_{j=1}^{n-1}\left[ax_{j}+by_{j}+cz_{j}-d\right]^{2}-\lambda\left(a^{2}+b ^{2}+c^{2}-1\right) \tag{3}\]  
where \(\lambda\) is the Lagrange multiplier; \(j\) is the number of 3D point cloud data points; and \(n\) is the total number of three-dimensional point cloud data. It can be observed from the above discussion that the plane fitted by the eigenvalue method along the three directions of \(x\), \(y\) and \(z\) can maintain good stability in different  
Figure 3: Depth map before noise reduction vs. depth map after noise reduction: **(a)** Outliers in point cloud images; **(b)** Point cloud depth map after noise reduction. coordinate systems. Therefore, this paper employs the eigenvalue method to address the tilt issue in the 3D point cloud data. The steps are as follows:  
First of all, the partial derivative of Formula (3) to \(d\) is obtained, and its derivative is 0. Then  
\[d=a\big{(}Ex_{j}\big{)}+b\big{(}Ey_{j}\big{)}+c\big{(}Ez_{j}\big{)} \tag{4}\]  
where \(E_{xj}\), \(E_{yj}\), \(E_{zj}\) is the average value of each 3D point \(x_{j}\), \(y_{j}\), \(z_{j}\); \((E_{xj}\), \(E_{yj}\), \(E_{zj})\) is the mean matrix of the point \((x_{j},y_{j},z_{j})\). The coordinates of each 3D data point were subtracted from the mean vector to obtain the centered data matrix. This matrix was then transposed and multiplied, with the results of all the sample data summed to calculate the covariance matrix (Cov), as shown in Equation (5):  
\[\textbf{Cov}\big{(}x_{j},\,y_{j},\,z_{j}\big{)}=\sum_{j=1}^{n} \begin{bmatrix} \big{(}x_{j}-Ex_{j}\big{)}\big{(}x_{j}-Ex_{j}\big{)}&\big{(}x_{j}-Ex_{j}\big{)} \big{(}y-Ey_{j}\big{)}&\big{(}x_{j}-Ex_{j}\big{)}\big{(}z_{j}-Ez_{j}\big{)}\\ \big{(}x_{j}-Ex_{j}\big{)}\big{(}y-Ey_{j}\big{)}&\big{(}y-Ey_{j}\big{)}&\big{(} y-Ey_{j}\big{)}&\big{(}y-Ey_{j}\big{)}&\big{(}y-Ey_{j}\big{)}\big{(}z_{j}-Ez_{j} \big{)}\\ \big{(}x_{j}-Ex_{j}\big{)}\big{(}z_{j}-Ez_{j}\big{)}&\big{(}y-Ey_{j}\big{)}& \big{(}z_{j}-Ez_{j}\big{)}&\big{(}z_{j}-Ez_{j}\big{)}\end{bmatrix}  \tag{5}\]  
The minimum eigenvalue \(\lambda_{min}\) of the covariance matrix \(\text{Cov}\left(x_{j},y_{j},\,z_{j}\right)\) and its corresponding eigenvectors \((a,b,c)\) are solved. The eigenvector is substituted into Formula (5) to obtain the parameter d and determine the fitted plane equation \(ax+by+cz=d\). Finally, the 3D point cloud coordinates are corrected:  
\[Z_{j}=z_{j}+\frac{a}{c}x_{j}+\frac{b}{c}y_{j}+\frac{d}{c} \tag{6}\]  
where \(Z_{j}\) is the \(j\)th three-dimensional point cloud coordinates after correction, and \(z_{j}\) is the \(j\)th three-dimensional point cloud coordinates before correction. The macroscopic texture depth maps of the asphalt before and after correction are shown in Figure 4:  
Because the collected depth map point cloud data is too fine to accurately reconstruct the asphalt texture, the macroscopic texture depth map of the asphalt is sampled to a resolution of 208 \(\times\) 144 to reduce the burden of subsequent computer processing and modeling and improve processing efficiency.