## 3 Methods
### Smoothing  
As was discussed in Section 3.2, simply applying an outlier-removal process is not enough to correct errors in ToF camera imaging. It is time to deal with the systematic errors, which manifest themselves as random ranging noise on the point cloud. Since we are only focusing on the projection boundary, we only design a smoothing model for the output boundary prediction \(\mathcal{P}_{B}\), effectively reducing the computing power requirement. The smoothing process not only de-noises the boundary point cloud, but also corrects the inherent limitation of ToF cameras, that is, the incompletion of the target edge, which always occurs when the target has non-planar edges, especially in regions where the surface normal deviates significantly from the camera's viewpoint, as is shown in Figure 5.  
Point cloud de-noising methods based on deep learning can be roughly divided into two types: predicting displacement vectors for each point [22; 25], or directly fitting  
Figure 4: The Boundary-Detecting block, with detailed structure of the proposed 2D detection process and the Cross Transformer. underlying surfaces [23; 24; 26]. Our PcBD model follows the first method to predict a replacement vector \(\mathcal{V}_{B}\) to smooth the predicted boundary. It has been argued that shrinkage and outliers may cause inaccuracy in the prediction of displacement vectors, however our input \(\mathcal{P}_{B}\) has been outlier-removed, and we will introduce how to eliminate the effects of shrinkage by combining 2D and 3D features below. On the other hand, the underlying-surfaces-based methods is only suitable for de-noising tasks for the whole point cloud, they themselves have a fixed shortcoming of insufficient noise removal on the edge of the point cloud, which happens to be the part we are most concerned about. To precisely predict the proper projection boundary to fit the spatial relationships, we design a smoothing block as Figure 6 shows. We first project the predicted three-dimensional boundary \(\mathcal{P}_{B}\) to a two-dimensional boundary \(\mathcal{P}_{B2D}\), in order to make kNN neighbor connections with the projected 2D shape. We find the k-nearest neighbors of the projected boundary in the projected point cloud and group relevant features including the 3D point-wise feature \(\mathcal{F}_{C}\), the 2D point-wise feature \(\mathcal{F}_{2D}\) and the 3D outlier-removed 3D point cloud \(\mathcal{P}_{C}\) using the kNN neighbor information. The shapes of the corresponding outputs are as shown in Figure 6. Each boundary point in the 2D projection now has an associated neighborhood that includes 3D coordinates and point-wise features from 3D structures and 2D shapes. This combination enables the model to refine the boundary smoothing, while maintaining the geometric integrity of the underlying projection hull shape. We then send the combined kNN features and coordinates into a PAConv block to integrate 3D information with the 2D boundary. This process enhances the network's ability to understand the extracted boundary from a higher dimensional perspective. Through our design, the PcBD model will not only try to smooth the two-dimensional shape, but also makes the detailed 3D local structure more reasonable. We reshape the output feature to shape \(N\times(k\cdot C)\), so that the subsequent blocks can operate on each point based on its multi-dimensional description. A Point Transformer block follows next, to adaptively focus  
Figure 5: Illustration of the inherent limitation of ToF cameras when detecting projection boundaries. Figure 6: The Smoothing block. on relevant points in the neighborhood of each boundary point using self-attention. This is particularly useful for a reasonable allocation of the role of 2D and 3D features in the smoothing process. At last, the output of the Point Transformer is passed through a simple MLP layer, leading to a moving vector \(\mathcal{V}_{B}\) of the size \(N\times 3\).