# Point Cloud Deep Learning Network Based on Local Domain Multi-Level Feature
## 3 Proposed Method
### Feature Fusion Based on Dynamic Graph Convolution  
We use the k-nearest neighbor graph structure of the directed graph to represent the local structure of the point cloud G = (V, E), where V = (1,..., n) and E = (V, V) are the vertices and edges, respectively. In the local directed graph, each point xi is the central node of the local neighborhood, the edge feature e\({}_{\bar{y}}\) between the central node and other points in the neighborhood is calculated, and then the feature information of edges is aggregated to represent the feature information of this central node, as shown in Figure 4. Assume  
Figure 3: Structure of edge and relation convolution. (**a**) Edge Convolution Module. (**b**) Relation Convolution Module.  
Figure 2: Proposed network architecture.  
the input point cloud is \(\mathrm{X}=\left\{x_{i}\in\mathrm{R}^{F}\right\}_{i=1}^{n}\). For the object point \(\mathrm{x_{i}}\), the k neighboring points are searched, and the local map is constructed. They can be expressed by the index \((i,j_{1}),\ (i,j_{2}),\ \cdots,(i,j_{k})\). According to the local map of \(x_{i}\), the \(x_{i}\) and the edge feature of neighboring points are obtained, which are expressed as \(\mathrm{e_{ij}}=\mathrm{h}\big{(}x_{i},\ x_{j};\ \theta\big{)}\); \(\mathrm{h}(\cdot)\) is the custom edge feature extraction function; \(\mathrm{e_{ij}}\) is an \(F^{\prime\prime}\) dimensional vector; and \(\theta\) forms the hyperparameter of \(\mathrm{h}(\cdot)\) that needs to be learned.  
We define the edge feature as \(\mathrm{e_{ij}}=\mathrm{h}\Theta(x_{i}x_{j})\), where \(\mathrm{h}\Theta\) is a nonlinear function composed by the learnable parameter \(\Theta\). The complete edge feature of the ith central node is derived by the pooled feature aggregation operation and can be expressed as xi = \(\mathrm{h}\Theta(x_{i}x_{j})\).  
The choice of \(h\Theta\) and feature aggregation operation has a very significant impact on the performance of the network. The first choice is \(h\Theta(x_{i}x_{j})=h\Theta(x_{i})\), which is used in the PointNet, and it only considers global information and ignores local information; the second choice is \(h\Theta(x_{i}x_{j})=h\Theta(x_{j}x_{i})\), which encodes only local information and will lose global information. The DGCNN method chooses \(h\Theta(x_{i}x_{j})=h\Theta(x_{i}x_{j}\)-\(x_{i})\), which combines the advantages of the first and second choice and considers both local and global features. Generally, a Multi-Layer Perceptron is chosen for the learning of the parameter \(\Theta\), and max pooling is used for the aggregation operation of the edge features. Then, the final edge features can be specifically represented as  
\[{x^{\prime}}_{i}=\max\left\{\mathrm{ReLU}(h_{\Theta}(x_{i},x_{j}))\right\} \tag{1}\]  
where \(\mathrm{ReLU}\) is an activation function.  
Compared with the sigmoid and tanh activation function, it is easy to obtain the derivative of \(\mathrm{ReLU}\), and this can improve the training speed of the network, increase the nonlinearity of the network and make the network sparse.  
In this paper, the network structure is optimized so that the input of each layer of edge convolution is a fusion of all previous output features, and the edge features of each layer can be expressed as  
\[e_{ij}^{(l+1)}=h_{\Theta}\Big{(}\Big{[}x_{i}^{l+1},x_{j}^{l+1}\Big{]}\Big{)}, \tag{2}\]  
The final stitching result is  
\[{x^{\prime}}_{i}^{(l+1)}=[x_{0},x_{1},\ldots,x_{l}], \tag{3}\]  
where \([x_{0},x_{1},\ldots,x_{l}]\) denotes the output features of the network from layer 0 to l.