## 3 Methodology
### Feature-Enhanced Point Transformer
#### 3.1.1 Feature Position Enhancement  
During the process of point cloud information encoding in the point transformer network, each data point and its neighboring points are considered as a local space, which contains the coordinates and additional information of the point, as well as the coordinates and additional information of neighboring points related to the point. The point transformer layer performs feature aggregation and self-attention calculations by inputting these local spaces and utilizing the relative positions between each point in these spaces and its neighboring points. The network encoding structure is shown in Figure 5. In this way, the point transformer layer can effectively process point cloud data and improve its feature representation ability.  
In order to utilize the geometric and structural relationships of point clouds and fully utilize the input point cloud information more effectively, this study designed a new point transformer layer structure based on the basic structure of the point transformer layer, which enhanced the geometric information of point clouds through local spatial information encoding. The network design not only preserved the original coordinates of each point, but also added information on the three-dimensional coordinates of the central sampling point, the coordinates of the point relative to the central sampling point, and the Euclidean distance. By adding this information, the geometric and structural information of the local space of the point cloud could be encoded more completely. This improved point transformer layer could more concretely display the local spatial information of point clouds. By adding information, such as the original coordinates, relative positions, and relative distances of the points, the geometric structure and spatial information of point clouds could be learned more accurately and effectively.  
Figure 4: The network structure of the point transformer.  
For a single point, the corresponding MLP processing content changed in the point transformer layer after a local spatial information encoding enhancement. The single-point feature acquisition operation in the original local space can be represented by Formula (2), while the improved single-point information encoding can be represented by Formula (2):  
\[\delta=\theta(p_{i}\oplus p_{j}\oplus(p_{i}-p_{j})\oplus\|p_{i}-p_{i}\|) \tag{2}\]  
In this formula, \(p_{i}\) is the coordinate of the sampling point in the local space; \(p_{j}\) is the coordinate of the sampling point in the local space; \((p_{i}-p_{j})\) is the coordinate difference between the sampling point and a certain domain point; \(\|p_{i}-p_{i}\|\) is the Euclidean distance between the sampling point and a certain domain point; and encoding function \(\theta\) is an MLP with two linear layers and one nonlinear ReLU layer.  
The improved network structure adopted a design based on a local spatial encoding information enhancement, which fully encoded the local spatial information of each point, including the coordinates, relative coordinates, and Euclidean distance of the central sampling point. During the network processing stage, MLP neural networks were used to aggregate features for each point and connect the encoded information with the original point information so that the feature vectors of each point contained rich spatial information. After the ReLu activation function operation, the information of the point cloud was further compressed and connected to the point cloud coordinates to obtain the feature vectors of the entire local space, as shown in Figure 6.  
Compared to the original point transformer network, the enhanced point transformer network based on local spatial geometric information encoding could better learn the spatial geometry and structural information of point clouds and present a better performance. The experimental results further validate this hypothesis.