## 2 Methods
### Identification and Information Extraction of Rock Mass Structural Planes
#### 2.1.3 Application Analysis of Rock Mass Structural Planes Identification and Information Extraction  
4. Visualization Analysis of Normal Vectors  
To further demonstrate the advantage of the Hough space transform in calculating the normal vectors of point clouds near the edges of structural planes, this paper uses both the Hough space transform and the PCA method to calculate the normal vectors of point cloud data for a cube. The normal vectors are then visualized using the PCL point cloud library. If the normal vector of each point is displayed, the result would be too cluttered to discern the distribution of the normal vectors at the edges of structural planes. Therefore, a normal vector is displayed every 10 points, and the length of each normal vector is set to 0.5. The visualization results are shown in Figure 2.  
In the figure, the green parts represent the point cloud, and the white line segments represent the normal vectors. The direction of each white line indicates the normal direction at that point. Each face of the cube represents a structural plane, and each edge represents the boundary of a structural plane. As shown in Figure 2, in flat areas, the normal vectors calculated by both methods perform well. In Figure 2b, the normal vectors near the edges are divergent, and the change in the direction of the normal vectors is relatively smooth. However, in Figure 2d, at the same locations, the normal vectors are perpendicular to the planes on both sides of the edges, and the distribution shows a more distinct variation trend. The Hough space transform method significantly outperforms the PCA method in calculating normal vectors near the edges of structural planes. The main reason is that the PCA method calculates the normal vector of a core point using the spatial relationship of all points in its neighborhood. When the core point is near or on the edge of a structural plane,points from another structural plane can affect the accuracy of the core point's normal vector. In contrast, the Hough space transform method randomly selects two points in the neighborhood to form a plane with the core point and calculates its normal vector. The most voted normal vector is then selected as the core point's normal vector, effectively eliminating the influence of points from other structural planes on the calculation and providing excellent noise resistance. Analysis of Region Growing Results  
The normal vector difference-based region growing algorithm using the Hough space transform involves several key parameters: neighborhood size \(K\), normal vector angle threshold \(\theta_{th}\), surface variation threshold \(SV_{th}\), maximum number of planes \(T_{\max}\), confidence level, and minimum number of cluster points \(C_{\min}\). \(K\) represents the number of points in the neighborhood of point \(P_{i}\); the smaller the \(K\) value, the stronger the seed point's characteristics, and the more fragmented the structural plane's segmentation becomes. The normal vector angle threshold \(\theta_{th}\) and the surface variation threshold \(SV_{th}\) are limiting parameters. The \(\theta_{th}\in[0,\pi]\) is set too high, points with small feature differences will be merged, reducing the number of identified structural planes. The \(SV_{th}\in[0,5]\) is used for seed point determination, and the smaller the value, the longer the algorithm takes, though it has minimal effect on the structural planes recognition result. The maximum number of planes \(T_{\max}\) and the confidence level control the balance between computation time and accuracy. Generally, when \(T_{\max}=800\), the confidence level exceeds 0.95. \(C_{\min}\) represents the minimum number of points within structural planes after region growing, and in practical applications, this value should be determined based on the point cloud density, typically ranging from 50 to 150.