## 3 Main Technical Research
### Research on Point Cloud Registration and Thickening Method Combining ORB-SLAM3 and NDT Technology
#### 3.1.1 Point Cloud Registration  
Point cloud registration refers to the process of transforming the overlapping parts of point clouds into a unified coordinate system by solving the transformation matrix. Registration includes coarse registration and fine registration. Coarse registration is performed when the position difference between two frames of point clouds is large, and the relative pose is completely unknown, providing a better transformation initial value for subsequent fine registration. Fine registration further optimizes the given initial transformation matrix to obtain a more accurate transformation. Currently, widely used point cloud registration methods include iterative closest point (ICP) and normal distribution transformation (NDT) [32].  
For ICP registration, it minimizes the distance between two sets of point clouds by finding the closest point pairs and iteratively calculating and applying translation and rotation transformations. NDT is a probability distribution-based point cloud registration method. It first divides the point cloud data into multiple grids and models the point cloud data in each grid with Gaussian distribution. Then, it achieves registration by finding the transformation that maximizes the overlap between the probability distributions of the two sets of point cloud data. Furthermore, its primary steps are as follows:  
1. Acquisition and Preprocessing of Point Cloud: acquire the target point cloud and the point cloud to be registered and preprocess them.
2. Division of Target Point Cloud: divide the target point cloud into standard grids, with each grid containing at least six points.
3. Modeling: For each point \(x\) in the grid, a Gaussian probability density function \(P(x)\) is modeled based on normal distribution by calculating the mean vector \(q\) and covariance matrix \(C\). This results in a Gaussian probability density function for each grid point: \[P(x)=c\cdot e^{[-\frac{1}{2}(x-q)^{T}C^{-1}(x-q)]}\] (1) Here, c is a normalization constant.
4. Setting the Initial Transformation Matrix: bring the point cloud to be registered close to the target point cloud.
5. Calculation of Score: Insert the points from the point cloud to be registered into \(P(x)\) and calculate the probability values in each grid unit. Calculate these to obtain the total score.
6. Solution for Optimal Parameters: maximize the objective function obtained by multiplying all score values to find the optimal parameters.  
In order to compare the applicability of the two methods in high-speed rail tunnel detection, this paper selects the road point cloud data of the KITTI dataset [33] for comparative testing of the two methods. Considering the characteristics of high-speed rail tunnels with few plane structures, the experimental data is rotated \(10^{\circ}\) along the \(Z\)-axis, and then point cloud registration is performed separately. The experimental results are shown in Figure 3. After registration, the root mean square error (RMSE) of the ICP method is significantly larger than that of the NDT method. In order to further compare the applicability of the two methods to the registration of rotating planes, this paper conducted eight experiments from 0 to 30 degrees, and the RMSE of the two methods is shown in Figure 4. As can be seen from the figure, the registration error of the ICP method increases sharply from 0.4 m at \(8^{\circ}\) to 1.12 m at \(10^{\circ}\), while the registration error of the NDT method remains stable at around 0.45. The standard deviations of RMSE for the two groups of experiments are 0.52 (ICP) and 0.02 (NDT), indicating that the NDT method is more suitable for point cloud registration in the presence of rotation. Therefore, this paper selects the NDT method as the point cloud registration algorithm.