## 3 Method
### DGCNN-Based Encoder  
To capture the context information from discrete point sets, it is essential to extract the correlation of points inside the local neighborhood. Both encoders of FoldingNet and DGCNN use graph-based layers to extract the local geometric information in each point's neighborhood and a _max pooling_ layer to aggregate information. The local feature of FoldingNet at \(i\)-th vertex is computed as follows:  
\[x^{\prime}_{i}=\underset{j\in(i,j)\in E}{\max}h_{\text{G}}\big{(}x_{j}-x_{i} \big{)} \tag{1}\]  
Figure 1: The architecture of our 3D Autoencoder (AE) branch, consisting of a DGCNN (Dynamic Graph Convolutional Neural Network)-based encoder module (**top**) and a folding-based decoder module (**bottom**).  
Figure 2: The architecture of semantic segmentation step of our label-efficient Deep Learning (DL) method.  
In the edge function \(h_{\Theta}\big{(}x_{j}-x_{i}\big{)}\), \(x_{i}\) is the central point belonging to point set \(X=\{x_{1},\ x_{2},\cdots,\ x_{n}\}\ \subseteq\ \mathcal{R}\), \(x_{j}\) is the local neighbor around the central point \(x_{i}\), and \(h_{\Theta}\) is implemented by a fully connected multi-perceptron layer, which includes learnable parameters. FoldingNet obtains the local information by encoding \(x_{j}-x_{i}\) edge features. Then the learned local information is aggregated by a local max pooling operation on the constructed graphs \(G=(V,\ E)\), where \(V=\{1,\ 2,\ \cdots,\ N\}\) and \(E\ \subseteq|V|\times|V|\) are the vertices and the edges, respectively, and \(N\) is the number of vertices.  
However, FoldingNet ignores the capture of the global-shape information. On the other hand, the operation on the constructed graph \(G\) of DGCNN is the EdgeConv operation, which may extract both local geometric and global-shape information from the constructed graph. Firstly, the EdgeConv layer computes an edge feature set of size \(k\) for each input point cloud through an asymmetric edge function (Equation (2)):  
\[\mathrm{e}_{ij}=h_{\Theta}\big{(}x_{i},\ x_{j}-x_{i}\big{)} \tag{2}\]  
In this edge function, EdgeConv captures the global shape by encoding the coordinates of \(x_{i}\) and then obtains the local information by encoding \(x_{j}-x_{i}\). The output feature is aggregated by edge features from each connected vertex and itself in the constructed graph:  
\[{x^{\prime}}_{i}=\max_{j:(i,j)\in E}h_{\Theta}\big{(}x_{i},\ x_{j}-x_{i}\big{)} \tag{3}\]  
Thus, EdgeConv can explicitly combine the global-shape structure information with local neighborhoods' information.  
Furthermore, in FoldingNet, the graph is constructed by computing pairwise distances using initial input point coordinates. Hence their graph \(G\) is fixed. In contrast, we calculate the pairwise distance in the feature space at each layer and choose the nearest \(k\) points per each central point, and then we dynamically construct \(G^{l}=\left(V^{l},\ E^{l}\right)\) at layer \(l\). Meanwhile, the receptive field becomes larger while such dynamic graph updates in each EdgeConv layer. The local information around central points and global information in different receptive fields are aggregated and stacked in the last layer before the max pooling layer.  
The procedure for producing feature representations in DGCNN-based encoder is displayed in Figure 1. We remove the encoder of FoldingNet and replace it with three EdgeConv layers in the architecture of DGCNN. The outputs of the three EdgeConv layers are concatenated and then passed to a feature-wise max pooling layer to produce a \(\mathrm{C_{out}}\)-dimensional \"codeword\" \(\theta\). The outcomes of three EdgeConv layers and the \"codeword\" \(\theta\) are stored in the pre-trained AE model, which will be the basis for our segmentation network.