## 3 3D Burden Surface Feature Extraction
### Point Cloud Feature Extraction
#### 3.2.1 PointNet  
PointNet is a network model specifically for training point cloud data proposed by [PERSON] in 2017 [21]. The network structure is shown in Figure 7.  
The PointNet network takes \(n\) points in a set of point clouds as input, each containing three dimensions \((x,y,z)\). Then, the input data are spatially transformed. The feature transformation is realized through the convolutional layer. Then, the feature space is spatially transformed, and the feature is reduced in the maximum pooling layer to obtain the compressed global feature. Finally, the high-dimensional features are fused by the fully connected layer, and the score of the point in each class of \(K\) is the output. It is an extended network of classification networks that fuses the local features of each point and outputs the score of each point belonging to each part. The PointNet network has three key modules:  
1. It uses the maximum pooling layer as a symmetric function that aggregates information from all points. The high-dimensional features extracted from the point cloud converge through a symmetry function and finally approach the function defined on the point set: \[f(\{x_{1},\ldots,x_{n}\})\approx g(h(x_{1}),\ldots,h(x_{1}))\] \[f:2^{\mathbb{R}^{N}}\rightarrow\mathbb{R},h:\mathbb{R}^{N} \rightarrow\mathbb{R}^{K}\text{ and }g:\underbrace{\mathbb{R}^{K}\times \cdots\times\mathbb{R}^{K}}_{n}\rightarrow\mathbb{R}\] (13) Here, \(h\) is simulated by a multilayer perceptron (MLP) and combined with the maximum pooling function to simulate \(g\). With the set of \(h\), the network can learn \(f\) to capture different properties of the point cloud data. 2. The structure combines the local and global features. After calculating the global features of the point cloud data, the global features and the local features of each point are connected as a new feature matrix, and then the new features of each point  
Figure 7: The network structure of PointNet. are extracted from the new feature matrix so that the local and global information is considered for each point. 3. Two federated alignment networks rotate and align each point and feature in the point cloud data. PointNet uses T-net to learn the structure of point cloud data and obtain a spatial transformation matrix. This method can be extended to feature space alignment. We add another T-net network after the convolutional layer to learn the feature structure information of this layer and perform spatial transformation processing on the current high-dimensional features. However, the dimensionality of the features is much higher than that of the point cloud data, which leads to increased computational complexity, making it hard to converge. The orthogonal transformation matrix does not lose information, so PointNet adds the orthogonalization function of the transformation matrix to the final loss function: \[L_{reg}=\left\|I-AA^{T}\right\|_{F}^{2}\] (14) Here, \(A\) is the spatial transformation matrix learned by T-net, which has the effect of aligning features. We build PointNet and adjust the network hyperparameter separately to realize the feature extraction of the burden surface. The network structure is shown in Figure 8.  
In the first T-net, \(k=3\), corresponding to the \((x,y,z)\) coordinate values, realizes the rotationally symmetric transformation of the point cloud data. The second T-net (\(k=64\)) outputs a 64-dimension spatial transformation matrix. This T-net is the point cloud data after a convolution transformation, and the characteristics of the obtained points are data-aligned. The final output is a 1 \(\times\) 4 matrix representing four types of the burden surface. This method implements classification through the last fully connected layer (FC-4). The output results are also used for the calculation of loss values and the adjustment of network parameters. A total of 256 features were extracted from the fully connected layer (FC-256), and a point cloud feature matrix was obtained, which was used for subsequent research.