# Semantic Segmentation Method for Road Intersection Point Clouds Based on Lightweight LiDAR
## 2 Materials and Methods
### Point Cloud Registration and Stitching  
#### 2.2.1 Principles of Point Cloud Registration and Stitching  
The essence of point cloud registration and stitching is to perform coordinate transformation on original data, unifying the local point clouds into an overall coordinate system [29]. Let P and Q respectively represent the source and target point clouds to be registered. The process of rigid transformation in a three-dimensional space can be described as \(\text{Q}=\mathbf{R}\times\text{P}+\mathbf{T}\), where \(\mathbf{R}\) is the rotation matrix and \(\mathbf{T}\) is the translation matrix. It should be noted that the stitching based on rotation and translation is based on the recognition of feature points. Let the source point set be \(\text{P}=\{\text{p}_{\text{i}}|\text{p}_{\text{i}}\in\text{R}^{3},\text{i}=1, 2,\cdots,\text{ n}\}\) and the target point set be \(\text{Q}=\{\text{q}_{\text{i}}|\text{q}_{\text{i}}\in\text{R}^{3},\text{i}=1, 2,\cdots,\text{ m}\}\). Any pair of feature points \(\text{p}_{\text{i}}\) and \(\text{q}_{\text{i}}\) should satisfy the following rigid transformation relationship:  
\[\text{q}_{\text{i}}=\mathbf{R}_{3\times 3}\times\text{p}_{\text{i}}+\mathbf{T}_{3 \times 1} \tag{1}\]  
The rotation matrix \(\mathbf{R}\) and translation matrix \(\mathbf{T}\) are respectively represented by the following equations:  
\[\mathbf{R}_{3\times 3}= \begin{bmatrix}1&0&0\\ 0&\cos\alpha&\sin\alpha\\ 0&-\sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}\cos\beta&0&-\sin\beta\\ 0&1&0\\ \sin\beta&0&\cos\beta\end{bmatrix}\begin{bmatrix}\cos\gamma&\sin\gamma&0\\ -\sin\gamma&\cos\gamma&0\\ 0&0&1\end{bmatrix}  \tag{2}\]  
\[\mathbf{T}_{3\times 1}= \begin{bmatrix}\text{t}_{\text{x}}&\text{t}_{\text{y}}& \text{t}_{\text{z}}\end{bmatrix} ^{\text{T}} \tag{3}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are the rotation angles of the x, y, and z axes in the local spatial coordinate system of the point cloud, respectively, and t\({}_{\text{x}}\), t\({}_{\text{y}}\), and t\({}_{\text{z}}\) are the translation changes of the coordinate axes along their respective directions. It is necessary to solve for the optimal rotation and translation parameters, that is, \(\alpha\), \(\beta\), \(\gamma\), t\({}_{\text{x}}\), t\({}_{\text{y}}\), and t\({}_{\text{z}}\). To determine a unique solution, at least six linear equations must be established, meaning at least three sets of non-collinear feature point pairs must be found between the source and target point clouds to solve the system of equations. Additionally, selecting as many non-collinear feature point pairs as possible can effectively improve the computational accuracy of the system of equations, enhancing the point cloud registration effect. To maximize the overlap of identical areas between the source and target point clouds, the model constraint can be expressed as the sum of the minimal distances between each set of feature points. Let the rigid transformation matrix be (\(\mathbf{R}\), \(\mathbf{T}\)) and establish a target \begin{table}
\begin{tabular}{c c} \hline
**Subject** & **Parameters** \\ \hline Product dimensions & 88 \(\times\) 69 \(\times\) 76 mm \\ Weight & Approx. 760 g \\ Laser wavelength & 90 m @ 10\% reflectivity \\ Detection range & 130 m @ 20\% reflectivity \\  & 260 m @ 80\% reflectivity \\ Field of view (FOV) & 38.4\({}^{\circ}\) (circular) \\ Distance accuracy & 2 cm \\ Angle accuracy & \textless{}0.1\({}^{\circ}\) \\ Beam divergence angle & 0.28\({}^{\circ}\) (vertical) \(\times\) 0.03\({}^{\circ}\) (horizontal) \\ Data acquisition rate & 10,000 pts/s \\  & 20\% @ 0.1 s \\ Field of view coverage rate & 68\% @ 0.5 s \\  & 93\% @ 1 s \\ \hline \end{tabular}
\end{table} 
Table 1: Livox Mid-40 LiDAR sensor main technical parameters.