## 3 3D Burden Surface Feature Extraction
### Point Cloud Feature Extraction
#### 3.2.2 PointCNN  
The PointCNN is a method for processing point cloud data proposed by [PERSON] in 2018. PointCNN defines a transformation matrix with dimensions \(K\times K\). The function of the transformation matrix is to regularly sort and weight the out-of-order point cloud data to obtain a new set of data. This new set of data has a certain order, and the data elements have adjacent position information, and then ordinary convolution operations can be used to achieve feature extraction. Based on the above ideas, PointCNN proposes the \(\Xi\)-Conv  
Figure 8: The network structure of T-net and PointNet. operation to replace the traditional convolutional operation. The algorithm is expressed as follows:  
\[F_{p}=\text{Conv}(K,\Xi\times p)=\Xi\text{-Conv}(K,p,P,F) \tag{15}\]  
Here, \(K\) is a convolution kernel with dimensions \(K\times K\), \(p\) represents the point cloud data to be convolved or the feature set extracted from the previous layer, \(P\) is the neighbor point set, and \(F\) is the feature set of the neighbor points. For \(\Xi\)-Conv, the operation process is as follows. Algorithm 1 calculates the adjacent point set \(P\) of each point in \(p\) before the \(\Xi\)-Conv operation. In this step, the KNN algorithm or random sampling can be used to extract the neighbor point set \(P\) and then follow the following process:  
1. Subtract the coordinates of point \(p\) from the coordinates of each neighbor point and obtained the relative coordinates of the point. 2. The location information of neighboring points is converted into feature vectors through an \(MLP_{\delta}\) network. 3. Splice the transformed features with their own features to obtain new features. 4. Calculate the \(\Xi\)-matrix through the \(MLP\) network, corresponding to the current input. 5. Use the \(\Xi\)-matrix to process the feature matrix of a specific order. 6. Perform convolution operations to obtain the future of the \(p\) point set. ```
0:\(\mathbf{K},p,\mathbf{P},\mathbf{F}\)
0:\(\mathbf{F}_{p}\)\(\triangleright\) Features \"projected\", or \"aggregated\", into representative point \(p\)
1:\(\mathbf{P}^{\prime}\leftarrow\mathbf{P}-p\)\(\triangleright\) Move \(\mathbf{P}\) to local coordinate system of \(p\)
2:\(\mathbf{F}_{\delta}\leftarrow MLP_{\delta}(\mathbf{P}^{\prime})\)\(\triangleright\) Individually lift each point into \(C_{\delta}\) dimensional space
3:\(\mathbf{F}_{*}\leftarrow[\mathbf{F}_{\delta},\mathbf{F}]\)\(\triangleright\) Concatenate \(\mathbf{F}_{\delta}\), and \(\mathbf{F},\mathbf{F}_{*}\) is a \(K\times(C_{\delta}+C_{1})\) matrix
4:\(\mathcal{X}\leftarrow MLP(\mathbf{P}^{\prime})\)\(\triangleright\) Learn the \(K\times K\mathcal{X}\)-transformation matrix
5:\(\mathbf{F}_{\mathcal{X}}\leftarrow\mathcal{X}\times\mathbf{F}_{*}\)\(\triangleright\) Weight and permute \(\mathbf{F}_{*}\) with the learned \(\mathcal{X}\)
6:\(\mathbf{F}_{p}\leftarrow\text{Conv}(\mathbf{K},\mathbf{F}_{\mathcal{X}})\)\(\triangleright\) Finally, typical convolution between \(\mathbf{K}\) and \(\mathbf{F}_{\mathcal{X}}\)
```  
**Algorithm 1**\(\mathcal{X}\)-Conv operator. The algorithm flow can be expressed as follows:  
\[F_{p}=\Xi\text{-Conv}(K,p,P,F)=\text{Conv}(K,MLP(P-p)\times[MLP_{\delta}(P-p), F]) \tag{16}\]  
According to the core ideas elaborated upon by [PERSON] et al., PointCNN is built based on \(\Xi\)-Conv. Each \(\Xi\)-Conv has four basic properties of \(K\), \(D\), \(P\), and \(C\). \(K\) represents the number of neighboring points, \(D\) represents the expansion coefficient, \(P\) represents the number of points, and \(C\) represents the number of channels (characteristic dimension) of the output. The three inputs are the point set \(p\) that needs to be extracted from the features, the representative point set \(q\), and the feature vector set \(F\). The size of the representative point set \(q\) needs to be consistent with the point set \(p\), and \(q\) is a subset of \(p\). The flow of the \(\Xi\)-Conv operation is as follows:  
1.