## 3 Methods
### Feature-Extracting & Outlier Removal
\begin{split} S_{x}^{+}&\triangleq\big{\{}i:d_{i} \leq R\wedge(\mathbf{p}_{i}-\mathbf{p})\cdot\mathbf{x}^{+}\geq 0\big{\}}\\ S_{x}^{-}&\triangleq\big{\{}i:d_{i}\leq R\wedge( \mathbf{p}_{i}-\mathbf{p})\cdot\mathbf{x}^{-}>0\big{\}}\\ \widetilde{S}_{x}^{+}&\triangleq\big{\{}i:i\in M(k) \wedge(\mathbf{p}_{i}-\mathbf{p})\cdot\mathbf{x}^{+}\geq 0\big{\}}\\ \widetilde{S}_{x}^{-}&\triangleq\big{\{}i:i\in M(k) \wedge(\mathbf{p}_{i}-\mathbf{p})\cdot\mathbf{x}^{-}>0\big{\}}\end{split}  \tag{6}\]  
And the LRF is defined as:  
\[ \begin{split}\mathbf{x}&=\begin{cases}\mathbf{x}^{+},&|S_{x}^{+}|>|S_{x}^{-}|\\ \mathbf{x}^{-},&|S_{x}^{+}|<|S_{x}^{-}|\\ \mathbf{x}^{+},&|S_{x}^{+}|=|S_{x}^{-}|\wedge\left|\widetilde{S}_{x}^{ +}\right|>\left|\widetilde{S}_{x}^{-}\right|\\ \mathbf{x}^{-},&|S_{x}^{+}|=|S_{x}^{-}|\wedge\left|\widetilde{S}_{x}^{ +}\right|<\left|\widetilde{S}_{x}^{-}\right|\end{cases}\\ \mathbf{z}&=\mathbf{z}^{+}/\mathbf{z}^{-}\quad\text{(same as above)}\\ \mathbf{y}&=\mathbf{x}\times\mathbf{z}\end{split}  \tag{7}\]  
In the equations, \(R\) denotes the neighborhood radius used to construct LRFs, \(d_{i}\) is the Euclidean distance, \(\mathbf{p}\) is the query point, and \(\mathbf{p}_{i}\) represents its neighboring points such that \(d_{i}=\|\mathbf{p}_{i}-\mathbf{p}\|\leq R\). Singular value decomposition (SVD) of \(\mathbf{M}\) yields \(\mathbf{M}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{T}\), where the columns of \(\mathbf{U}\) correspond to the principal axes of the local neighborhood. The first column \(\mathbf{x}^{+}\) is taken as the initial \(x\)-axis direction of the LRF. To eliminate directional ambiguity, sets \(S_{x}^{+}\) and \(S_{x}^{-}\), and refined subsets \(\tilde{S}_{x}^{+}\) and \(\tilde{S}_{x}^{-}\) (restricted to a median-based neighborhood \(M(k)\)) are constructed to decide the final sign of \(\mathbf{x}\). The final LRF is defined by \(\{\mathbf{x},\mathbf{y},\mathbf{z}\}\), as in Equations (5)-(7). We make full use of the accurate establishment of the point cloud neighborhood coordinate system proposed by the SHOT descriptor, as we use the calculated \(Z\)-axis as the normal vector to make the point cloud to size \(N\times 6\), and send the calculated \(\mathrm{X}\) and \(\mathrm{Y}\) axes together into the Set Abstraction block proposed in [15]. The original SHOT descriptor utilized a complex way to describe the neighborhood of points which is extremely time-consuming. Since we are using a neural network model, we hope to hand this part of the work to the network, so we input the point clouds, calculated 3-axis vectors after kNN-sampling into the Feature-Extraction module for learning. The original Set Abstraction block in PointNet++ firstly kNN-samples the input point cloud \(\mathcal{P}\) with the point-wise feature. Then, the sampled point feature will be sent into a convolutional layer followed by a max-pooling layer, while in the proposed PcBD, the Set Abstraction block takes the calculated 3-dimentional axis as additional input, and applies the grouping operation with the kNN-sampling index as the input feature, as is shown in Figure 3. We also calculate the angles between the neighborhood normal vectors and the local reference of the center point, allowing the network to better learn the distribution of neighborhood surfaces. To handle such comprehensive input, we make use of a recently proposed position adaptive convolutional layer, the PAConv layer [16] to process the combination of 3D coordinates, local reference frames and relative angles. The PAConv block was proposed to replace regular convolutional layers as it attempts to learn 3D structures well with a dynamic kernel called ScoreNet, and has been proved to be efficient and easy to be plugged-in in existing CNN-based models. We make several adjustments on the proposed blocks to be suitable for our shape of inputs, and we also calculate the input feature in the PAConvblock with the distances of the kNN neighbors as a weight to enhance the network's ability to grasp the underlying geometric structure.