## 4 Experiments and Results
### Training  
We implemented our network on Pytorch ([[https://pytorch.org/](https://pytorch.org/)]([https://pytorch.org/](https://pytorch.org/))) and trained it on NVIDIA A100 GPU (Santa Clara, CA, USA). We set the learning rate to 0.0002 and exponentially decay the learning rate by 0.95 after every 10 epochs. During the training process, we down-sample the input point clouds along with their associated outlier and boundary labels to a fixed size of 4096 points using shared indexing. Similarly, the ground truth for the 2D projection is also down-sampled to 4096 points to maintain consistency with the input size. This ensures that the network is trained on uniform point sets, optimizing both computational efficiency and learning efficacy. Our loss function \(L\) is designed to optimize three key aspects of the network's output: accurate classification of outlier and boundary points, precise reconstruction of cleaned and boundary point clouds, and effective adjustment of boundary points to their target positions. It combines the binary cross-entropy loss (BCE) [44] for label predictions with Chamfer Distance (CD) [45] for point cloud alignment:\[L_{\text{CD}}(\mathcal{P},\mathcal{G})=\frac{1}{N}\sum_{x_{i}\in\mathcal{P}}\min_{g \in\mathcal{G}}\|x_{i}-g\| \tag{13}\]  
\[L_{\text{BCE}}(\mathcal{L},\mathcal{L}^{\prime})=-\frac{1}{N}\sum_{i=1}^{N} \big{[}\mathcal{L}_{i}\log(\mathcal{L}^{\prime}_{i})+(1-\mathcal{L}_{i})\log( 1-\mathcal{L}^{\prime}_{i})\big{]} \tag{14}\]  
\[L =\big{[}L_{\text{BCE}}(\mathcal{L}_{O},\text{gt}_{\mathcal{L}_{O} })+L_{\text{CD}}(\mathcal{P}_{C},\text{gt}_{\mathcal{P}_{C}})\big{]}\] \[+\big{[}L_{\text{BCE}}(\mathcal{L}_{B},\text{gt}_{\mathcal{L}_{B }})+L_{\text{CD}}(\mathcal{P}_{B},\text{gt}_{\mathcal{P}_{B}})\big{]} \tag{15}\] \[+\big{[}L_{\text{CD}}(\mathcal{B},\text{gt}_{\mathcal{B}})\big{]}\]  
Here, \(\mathcal{L}\) and \(\mathcal{L}^{\prime}\) are two input labels to calculate BCE loss, \(\text{gt}_{\mathcal{L}_{O}}\) and \(\text{gt}_{\mathcal{L}_{B}}\) are the corresponding ground-truths of predicted outlier labels \(\mathcal{L}_{O}\) and boundary labels \(\mathcal{L}_{B}\), while \(\text{gt}_{\mathcal{P}_{C}}\) and \(\text{gt}_{\mathcal{P}_{B}}\) refer to the outlier-removed point cloud and the projected boundary extracted from ground-truth labels. \(\text{gt}_{B}\) is the ground-truth boundary from the dataset. Given the input point cloud \(\mathcal{P}\) and ground-truth \(\mathcal{G}\), Chamfer distance is widely used in 3D vision tasks [38; 46] to measure the similarity between two point clouds, particularly when the correspondences between points are not known. It penalizes the average squared distances between each point in one point cloud to its nearest neighbor in the other one. In our implementation, we adopt the \(L_{1}\) variant of Chamfer distance as in Equation (13), where each distance term \(\|x_{i}-g\|\) denotes the point-wise Euclidean distance without squaring. Compared with the squared \(L_{2}\) formulation, the \(L_{1}\) Chamfer distance is less sensitive to outliers and yields more stable gradients, especially in regions with sharp geometric boundaries or sparse correspondences. In the training step, we calculate the CD loss between the outlier-removed prediction \(\mathcal{P}_{C}\) and the corresponding ground-truth with the BCE loss between the outlier label prediction \(\mathcal{L}_{O}\) and the corresponding ground-truth to evaluate the Feature-Extracting & Outlier-Removal block; we calculate the CD loss between the 2D boundary prediction \(\mathcal{P}_{B}\) and the corresponding ground-truth with the BCE loss between the boundary label prediction \(\mathcal{L}_{B}\) and the corresponding ground truth to evaluate the Boundary-Detecting block. Finally, we calculate the CD loss between the smoothed 2D boundary prediction \(\mathcal{B}\) and the corresponding ground truth to evaluate the Smoothing block.