# Boundary-Inner Disentanglement Enhanced Learning for Point Cloud Semantic Segmentation
## 3 Methods
### Boundary-Inner Entanglement Measurement  
Consider a point cloud with n points, denoted by \(X=\{\chi_{1},\cdots,\chi_{i},\cdots,\chi_{n}|\chi_{i}=(p_{i},f_{i}),p_{i}\in R^{ 3},f_{i}\in R^{d}\}\), where \(p_{i}=(x_{i},y_{i},z_{i})\) represents Euclidian 3D coordinates, \(f_{i}\) represents additional feature attributes such as color, surface normal, and intensity, and \(d\) represents feature dimensions. With point \(\chi_{i}\) as a centroid, its neighboring points \(N_{i}\) are identified using the simple K-nearest neighbors (KNN) algorithm. A point \(\chi_{i}\) is annotated as a boundary point if there exists a point in a different category in the neighborhood; otherwise, it is annotated as an inner point. Accordingly, a boundary point set \(B_{l}\) can be generated from the ground truth:  
\[B_{l}=\{\chi_{i}\in X|\exists\chi_{k}\in N_{i},l_{k}\
eq l_{i}\}, \tag{1}\]  
where \(l_{i}\) represents the ground truth of the center point \(\chi_{i}\). Some basic variables involved in this paper are summarized as follows:  
* \(X\) denotes the input point cloud;
* \(n\) denotes the number of points in \(X\);
* \(\chi_{i}\) denotes point \(i\) in \(X\);
* \(l_{i}\) denotes the ground truth of \(\chi_{i}\);
* \(p_{i}=(x_{i},y_{i},z_{i})\) denotes the Euclidian 3D coordinates of \(\chi_{i}\);
* \(f_{i}\) denotes the feature of \(\chi_{i}\);
* \(B_{l}\) denotes the boundary point set in \(X\). For training the point cloud with the ground truth, its boundary information @boundary is generated according to Equation (1). As shown in Figure 1, the generated boundaries (red regions) are located at the joints between objects belonging to different categories, delineating a clear semantic contour of the 3D objects. Specifically, @boundary\({}_{i}\) is equal to 1 if point \(\chi_{i}\) belongs to \(B_{l}\); otherwise, it is equal to 0. Table 1 compares the segmentation results of the control group and experimental group, where the control group takes the initial point cloud as input, whereas the experimental group takes the point cloud endowed with boundary information as input. The mean intersection over union (mIoU), overall accuracy (OA), and mean class accuracy (mACC) are used as evaluation metrics to quantitatively compare the results of the different methods, which are respectively computed as  
\[\text{mIoU}=\frac{1}{S}\sum_{s=1}^{S}\frac{\sum_{\chi_{i}\in X}[\text{{pred} }_{i}=s\wedge l_{i}=s]}{\sum_{\chi_{i}\in X}[\text{{pred}}_{i}=s\lor l_{i}=s]}, \tag{2}\]  
\[\text{OA}=\frac{\sum_{\chi_{i}\in X}[\text{{pred}}_{i}=l_{i}]}{n}, \tag{3}\]  
\[\text{mACC}=\frac{1}{S}\sum_{s=1}^{S}\frac{\sum_{\chi_{i}\in X}[\text{{pred} }_{i}=s\wedge l_{i}=s]}{\sum_{\chi_{i}\in X}[l_{i}=s]}, \tag{4}\]  
where \(S\) represents the total number of classes, \(\text{{pred}}_{i}\) represents the predicted label of point \(\chi_{i}\), \([\cdot]\) represents a Boolean function that outputs 1 if the condition within \([\cdot]\) is true and 0 otherwise. As can be seen, the experimental group achieved the better mIoU of 81.6%. To explore the role of the boundary, the change in entanglement between boundary point and its neighboring points is analyzed. The detailed steps are as follows:  
**Partition the neighboring points into four collections. ** For a center point \(\chi_{i}\in B_{l}\), its neighboring points are partitioned into four collections from the perspective of two factors: whether they are on the boundary, and whether their categories are the same as \(\chi_{i}\). The four collections are as follows:  
1. \(