## 4 Experiments and Results
### Results on Bound57
As above, we adaptively set the input parameters of these methods to a value related to the input density to make sure they achieve the best results. We applied these methods on the outlier-removed ground truth to eliminate the effects of outliers, and the results are illustrated in Table 2. Most deep-learning-based methods on point cloud boundary detection focus on 3D semantic edges [37; 53; 54; 55], so we can't directly compare the performance. However, we \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Methods/Results** & **Avg.** & **Airplane** & **Bench** & **Bicycle** & **Lamp** & **Microphone** & **Piano** & **Stove** & **Watercraft** \\ \hline DMRde-noise [24] & 14.43 & 13.83 & 13.77 & 14.29 & 16.48 & 17.46 & 14.10 & 15.26 & 12.49 \\ PointCleanNet [22] & 5.02 & 4.39 & 4.68 & 5.72 & 4.48 & 5.93 & 6.13 & 2.79 & 5.81 \\ Score de-noise [26] & 3.90 & 3.73 & 3.89 & 4.33 & 3.74 & 3.56 & 3.87 & 3.98 & 3.83 \\ PD-LTS-Heavy [50] & 3.79 & 3.17 & 3.92 & 4.39 & 3.13 & 3.22 & 3.85 & 4.08 & 3.62 \\ PD-LTS-Light [50] & 3.15 & 2.65 & 3.45 & 4.18 & 2.61 & 2.66 & 3.17 & 3.35 & 3.10 \\ DBSCAN [49] & 2.31 & 0.81 & 1.54 & 1.30 & 2.00 & 3.00 & 2.89 & 2.46 & 1.11 \\ SOR [48] & 1.38 & 0.58 & 0.96 & 0.98 & 1.10 & 1.07 & 1.70 & 1.77 & 0.73 \\ ROR [47] & 0.95 & 0.26 & 0.61 & 0.58 & 0.58 & 0.56 & 1.02 & 1.42 & 0.46 \\
**PcBD (Ours)** & **0.10** & **0.07** & **0.13** & **0.23** & **0.05** & **0.04** & **0.11** & **0.09** & **0.10** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Outlier-removal results on Bound57 under Chamfer-L1 distance (\(\times 10^{3}\), in millimeters). show the potentials to fulfill 3D boundary-detection tasks of PcBD in Section 3.4 as a proof of the advanced nature of our network. A visual comparison of different 2D boundary-detecting methods is made in Figure 11. When \(\alpha\)-Shapes faces an input point cloud whose basic density does not match the set \(\alpha\) value, the predicted projection boundary may be missing some parts. Other prediction methods may output holes caused by noise inside the point cloud. Compared with these methods, PcBD also compares and establishes connections between the three-dimensional features of the input point cloud, so that it can accurately predict the projection boundary while avoiding identifying internal areas as boundaries. On the other hand, even if it has not been smoothed, the predicted projection boundary of PcBD is already smoother than the above methods. The improvement of PcBD across different shapes suggests that point-wise features with 3D context and attention across 2D and 3D features significantly outperform handcrafted 2D descriptors. Notably, PcBD's performance does not rely on pre-determined parameters, but instead learns to interpret structural continuity via cross-dimensional attention. This allows the model to distinguish real boundaries from internal gaps caused by noise or projection sparsity. To evaluate the boundary smoothing, or the de-noising performance of our method, we compared Bilateral Filter [10], Iterative guidance normal filter [56], Weighted Multi-Proj [57], Sparse Regularization-Based de-noising [58], Moving Least Squares (MLS) [59], and Adaptive MLS [60] as traditional geometric approaches, while the PointCleanNet [22], \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Methods/Results** & **Avg.** & **Bag** & **Bowl** & **Car** & **Chair** & **Mug** & **Printer** & **Telephone** \\ \hline Normal-based [52] & 38.78 & 45.26 & 61.28 & 39.80 & 34.60 & 55.84 & 40.78 & 33.62 \\ Ada \(\alpha\)-Shapes [34] & 25.18 & 25.77 & 55.62 & 25.27 & 19.44 & 47.28 & 46.39 & 19.07 \\ \(\alpha\)-Shapes [33] & 23.02 & 27.11 & 48.50 & 19.79 & 16.62 & 29.50 & 50.12 & 27.62 \\ Grid-Contour [51] & 17.30 & 13.36 & 32.65 & 16.40 & 14.89 & 31.14 & 29.20 & 20.21 \\
**PcBD (Ours)** & **7.24** & **10.55** & **14.20** & **7.08** & **7.72** & **9.54** & **6.42** & **3.23** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Boundary-detecting results on Bound57 under Chamfer-L1 distance (\(\times 10^{3}\), in millimeters).