## 3 Methods
### Feature-Extracting & Outlier Removal  
Since our final goal is to predict the two-dimensional boundary of the projected target point cloud, the Feature-Extracting and Outlier Removal block only cleans the spatial outlier points to save hardware space, and we will deal with systematic error noise in the Bound-Detecting block, which will be introduced in Section 3.4. Figure 3 illustrates the complete structure of the Feature-Extracting Block as well as some detailed modules within it. The input includes a 3D point cloud PC and the corresponding local reference frame vectors LRF, consisting of a \((3+9)\)-dimensional geometric descriptor for each point. The **SA** block follows the idea of PointNet++ [15], but extends it by incorporating LRFs and relative angles into the feature input. The output from the SA block is passed into an an advanced Point Transformer layer **PT**[7] as we integrate the normal vector encoding into the positional encoding in the attention query to further refine the neighborhood structure representation. The down-sampled features are propagated back to the original point cloud resolution through two Feature Propagation **FP** blocks. Finally, the extracted 3D feature \(\mathcal{F}\) is processed by a Multi-Layer Perceptron (MLP) with one output channel followed by a sigmoid activation function to predict the outlier labels \(\mathcal{L}_{O}\). The detailed design principals and structural details are introduced as follows:  
It is not enough to calculate the relevant geometric information only based on the three-dimensional coordinates of a point cloud. Noisy point clouds may contain a lot of invalid information, so it is necessary to extract more stable embedded connections from three-dimensional coordinates. Deep-learning-based methods can not only process three-dimensional point cloud data, but many claim to accept data inputs that contain more dimensions, such as six-dimensional point clouds that also contain normal vectors. However, in practical applications, most researchers want to directly input the three-dimensional point cloud acquired by the sensor into the model to obtain calculation results without additional operations. We hope that the normal vectors can be internally calculated in the proposed model, thereby improving the robustness of the model under high noise without extra process. Since its introduction, the SHOT descriptor [6] has become one of the most accurate point cloud descriptors and has greatly improved the accuracy of point cloud registration. Its main contribution is the construction of the local reference frames and how it splits the neighborhood of a point and builds a histogram within each split area. In our proposed network, the input point cloud \(\mathcal{P}\) of size \(N\times 3\) will be sent to build the local reference frames for each point [6]:  
\[\mathbf{M}=\frac{1}{\sum_{i:d_{i}\leq R}(R-d_{i})}\sum_{i:d_{i}\leq R}(R-d_{i} )(\mathbf{p}_{i}-\mathbf{p})(\mathbf{p}_{i}-\mathbf{p})^{T}=\mathbf{U}\mathbf{ \Sigma}\mathbf{V}^{T} \tag{4}\]  
where \(\mathbf{V}=\{\mathbf{x},\mathbf{y},\mathbf{z}\}\). Then, the algorithm defines \(M(k)\) as the subset of points within the support whose distance from the feature point is among the k-closest to the median distance:  
Figure 3: The Feature-Extracting Block, with detailed structures of the advanced Set-Abstraction and Feature Propagation modules. Note that k is the kNN-sampling neighbors, C, C1, and C2 are number of channels, N is the input number of points of the SA module, and N1 is the output number of points. The Feature Propagation module takes both the input and output features of an SA module as its input. \[M(k)\triangleq\big{\{}i:|m-i|\leq k,\quad m=\operatorname{arg median}_{j}d_{j}\big{\}} \tag{5}\]  
And the spatial splitting sets are defined as:  
\[