## 4 Experiments and Results
### Results on Bound57  
In our experiments, we test the performance of the proposed PcBD network in three steps: the outlier-removal results, the boundary detection results, and the boundary de-noising results. For outlier-removing, we evaluated the effectiveness of our proposed method by comparing it with several traditional and state-of-the-art point cloud de-noising approaches, including Radius Outlier Removal (ROR) [47], Statistical Outlier Removal (SOR) [48], DBSCAN [49], PointCleanNet [22], DMRde-noise [24], Score de-noise [26], and PD-LTS [50]. For ROR, SOR, and DBSCAN, we directly write the codes based on the algorithm principles and test them on the dataset. We set the input parameters of these geometric methods as the average point-to-point distance of the input point cloud multiplied by a set fixed value, which is consistent with the setting process of general experiments. For DMRde-noise and Score de-noise, we train the networks on the proposed dataset Bound57 using their respective open-source code, best hyperparameters, and the loss functions reported in their respective papers. We use the ground-truth boundary labels to extract the boundary points of the output point cloud from the comparison methods,calculate the CD loss between the ground-truth boundary point cloud, and add it to the loss function of the comparison methods. This allows us to compare the smoothing effects of different methods on the target boundary point cloud. The PointCleanNet and PD-LTS methods have been verified in the original articles to have the ability to remove outlier points, so we utilize their provided pre-trained models so that they can only be trained on local patches sampled from several target point clouds, which is not suitable for our dataset. The outlier-removal results of some categories and the overall average are shown in Table 1, as we use the CD loss of the cleaned point cloud to the extracted outlier-removed ground truth as a judging standard. It should be noted that the unlike our proposed PcBD which firstly removes outliers then de-noises the projection boundary, the output point clouds of the compared deep-learning-based methods (besides PointCleanNet) are both outlier-removed and de-noised, so it is unfair to judge their outlier-removing abilities by comparing their outputs with the un-de-noised ground truths. However, we could still extract some effective information from the table. For example, for traditional methods (ROR, SOR, and DBSCAN), their outlier-removing effects vary greatly in different catalogs, while PcBD, Score de-noise and PD-LTS are relatively stable. We make a visual comparison in Figure 10 to more intuitively show the effects of different methods on the outlier points. It can be seen that the geometric methods such as ROR and SOR suffer from the differences in density in different local regions, especially the lower half of the piano, which has been completely cleared due to its low density. The PointCleanNet predicts too many outliers on the proposed dataset, resulting in the output point cloud being too sparse. The Score de-noise and PD-LTS have some point cloud shrinkage problems, and although they smooth the point cloud underlying surfaces, some outliers close to the surface are still not filtered out. Compared with the above methods, PcBD accurately predicts almost all of the outliers, and can also use 3D structural features to distinguish target points from outliers in some relatively sparse areas (such as bicycle handlebars and spokes). To evaluate the boundary detection performance of our method, we compared the boundary outputs with several approaches, including traditional and some improved novel methods. The \(\alpha\)-Shapes [33] was chosen as a classic geometric method, providing a baseline for simple and robust boundary estimation. We chose the Adaptive \(\alpha\)-Shapes [34] as it automatically calculates the parameter \(\alpha\). The Grid-Contour [51] and the normal-vector-based method proposed in [52] are also tested as a complement to geometric methods.