## 3 Methods
### Overview  
The overall structure of PcBD contains 3 main blocks: the Feature-Extracting block with an Outlier-Removal block to remove the outlier points, the Boundary-Detecting block to predict the projection boundary, and the Smoothing block to smooth the predicted boundary curves. Given an initial input point cloud \(\mathcal{P}=\{\mathbf{p}_{i}\}_{i=0}^{N}\) where \(N\) denotes the number of points, the Feature-Extracting block calculate the local reference frames (LRF) of every point to enhance the perception ability of the model with regard to geometric structures. The local reference frames contain three orthogonal vectors as the 3-dimensional axes of the local coordinate system for each point. We optimize some point cloud processing units such as PointNet++ [15] and PAConv [16] to satisfy our need to understand geometry connections and fit the point cloud with local reference frames. The output point-wise feature \(\mathcal{F}=\{\mathbf{f}_{i}\}_{i=0}^{N}\) will be of size \(N\times C\). The Outlier-Removal process in the Feature-Extracting block takes the extracted feature \(\mathcal{F}\) and outputs outlier labels \(\mathcal{L}_{O}\) of size \(N\times 1\) to mark the spatial outlier points with scores. To keep the data contiguous and remain the tensor shape, we replace the predicted outlier points with the first non-outlier point, and apply the same operation to the point-wise feature \(\mathcal{F}\) to generate an outlier-removed feature \(\mathcal{F}_{C}\): threshold  
\[ \begin{cases}\mathcal{P}_{C}=\{\mathbf{p}_{i}\}_{\mathcal{L}_{O_{i}}<\theta_{ O}}\cup\{\mathbf{p}_{0}\}_{N-|\{\mathcal{L}_{O}<\theta_{O}\}|}\\ \mathcal{F}_{C}=\{\mathbf{f}_{i}\mid\mathbf{p}_{i}\in\mathcal{P}_{C}\}\end{cases}  \tag{1}\]  
where \(\theta_{O}\) is the threshold of noise label, and \(\mid\mid\) represents the length of a set. In this way, the number of points remains the same, allowing us to utilize a further process. We then project the outlier-removed point cloud \(\mathcal{P}_{C}\) along the \(Z\) axis to size \(N\times 2\) before the Boundary-Detecting block. To detect the projection shape, we send the projected 2D point cloud through a 2D-Detection block to extract 2D shape information \(\mathcal{F}_{2D}\). To recover the missing 3D structural data, we employ a Transformer to perform cross-attentionbetween 3D and 2D features \(\mathcal{F}_{C}\) and \(\mathcal{F}_{2D}\). The output feature \(\mathcal{F}_{B}\) contains both 2D geometry and 3D structural data. The boundary labels \(\mathcal{L}_{B}\) of size \(N\times 1\) are predicted through an MLP layer to mark projection boundary points with scores. In the same way as before, the projection-bound point cloud is generated by replacing non-boundary points with the first boundary point:  
\[\mathcal{P}_{B}=\left\{\mathbf{p}_{j}\right\}_{\mathcal{L}_{B_{j}}<\theta_{B}}+ \left\{\mathbf{p}_{0}\in\left\{\mathbf{p}_{j}\right\}\right\}_{N-\text{len} \left(\left\{\mathcal{L}_{B}<\theta_{B}\right\}\right)} \tag{2}\]  
The smoothing block takes the predicted boundary points \(\mathcal{P}_{B}\) with the point-wise 3D and 2D features \(\mathcal{F}_{C}\) and \(\mathcal{F}_{B}\), and detects the edges of the predicted boundary by kNN-sampling and grouping between 3D and 2D features. The block finally outputs a three-dimensional replacement vector \(\mathcal{V}_{B}\) of size \(N\times 3\) to cancel out the noise and fix the inherent limitation of ToF cameras (see Section 3.4):  
\[\mathcal{B}=\mathcal{P}_{B}+\mathcal{V}_{B} \tag{3}\]  
To summarize the entire processing flow of our PcBD framework, the overall Algorithm 1 is provided. Through our design, the proposed PcBD can be directly utilized in real-world experiments to process raw scan data and output an outlier-removed target with smoothed projection boundary \(\mathcal{B}\), which fits a large section of experiment needs. Detailed structures are discussed in Sections 3.2-3.4.