# Boundary-Inner Disentanglement Enhanced Learning for Point Cloud Semantic Segmentation
## 3 Methods
### Boundary-Inner Entanglement Measurement
\begin{array}{l}\text{C}_{i}^{1}=\{\chi_{k}\in N_{i}\mid l_{k}=l_{i}\ \wedge\ \chi_{k}\in B_{l}\}\text{: boundary points within the same category;}\\ \text{C}_{i}^{2}=\{\chi_{k}\in N_{i}\mid l_{k}=l_{i}\ \wedge\ \chi_{k}\
otin B_{l}\}\text{: inner points within the same category;}\\ \text{C}_{i}^{3}=\{\chi_{k}\in N_{i}\mid l_{k}\
eq l_{i}\ \wedge\ \chi_{k}\in B_{l}\}\text{: boundary points in a different category;}\\ \text{C}_{i}^{4}=\{\chi_{k}\in N_{i}\mid l_{k}\
eq l_{i}\ \wedge\ \chi_{k}\
otin B_{l}\}\text{: inner points in a different category.}\\ \end{array} \)
2. **Measure the entanglement between boundary point to its each collection. ** Inspired by [46; 47], the soft nearest loss without negative logarithm function is used to measure the entanglement. For a boundary center point \(\chi_{i}\in B_{l}\), its entanglement with \(C_{i}^{j}\) is defined by \(P_{i}^{j}\): \[P_{i}^{j}=\frac{\sum_{\chi_{k}\in N_{i}\wedge\chi_{k}\in C_{i}^{j}}exp(-\parallel f _{i}-f_{k}\parallel)}{\sum_{\chi_{k}\in N_{i}}exp(-\parallel f_{i}-f_{k} \parallel)},\ 0\leq P_{i}^{j}\leq 1,\] (5) where \(\parallel\cdot\parallel\) represents the \(L_{2}\) Euclidean distance. A larger \(P_{i}^{j}\) denotes stronger entanglement between point \(\chi_{i}\) and its neighboring collections \(C_{i}^{j}\). \(B_{l}\) is generated from the ground truth of the input point cloud, and the boundary feature is a kind of low-level local feature that can be extracted by LAO. Therefore, \(f_{i}\) refers specifically to the internal representation learned by the LAO in the first encoding stage, where the point cloud has not yet been subsampled. In the LAO, greater affinity between the center point \(\chi_{i}\) and its neighboring point results in a greater corresponding convolution weight and denotes more similar transformed features. Entanglement essentially represents feature the similarity between point pairs. Therefore, the metric \(P_{i}^{j}\) can be intuitively described as the degree of attention between point \(\chi_{i}\) and its neighboring collections \(C_{i}^{j}\). **Compare the measuring results. ** The measuring results are plotted in Figure 2. Comparing the plots by column, in the control group (Figure 2a), \(P_{i}^{1}>P_{i}^{2}>P_{i}^{3}>P_{i}^{4}\) on average, indicating that the boundary point is more entangled with its neighboring points in the same category, whereas, in the experimental group (Figure 2b), \(P_{i}^{1}>P_{i}^{3}>P_{i}^{2}>P_{i}^{4}\) on average, indicating that the boundary point is more entangled with its neighboring points on the boundary. Comparing the plots by row, \(P_{i}^{2}\) and \(P_{i}^{4}\) decreased greatly in the experimental group. It was speculated that the boundary acts as a barrier in the LAO, where it prevents the boundary point from focusing on neighboring points in inner areas. Due to the role of the boundary, relatively more attention is paid to neighboring collections \(C_{i}^{1}\) or \(C_{i}^{3}\) (boundary points when put together), such that boundary feature is preserved and, subsequently, the overall performance is improved. In summary, the entanglement between boundary point \(\chi_{i}\) and its neighboring collections \(C_{i}^{2}\) or \(C_{i}^{4}\) (inner points when put together) are weakened greatly after utilizing boundary information. This shows that reducing boundary-inner entanglement is beneficial for semantic segmentation accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Input** & **OA (\%)** & **mACC (\%)** & **mIoU (\%)** \\ \hline Control group & (x, y, z, r, g, b) & 88.7 & 85.9 & 74.4 \\ Experiment group & (x, y, z, r, g, b, @boundary) & **93.0** & **90.5** & **81.6** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The semantic segmentation results of S3 DIS Area1 on RandLA-Net [28]. **Bold** font in the table body denotes the best performance.