# Point Cloud Registration Algorithm Based on Adaptive Neighborhood Eigenvalue Loading Ratio
## 2 Methods
### Feature Point Extraction Based on Adaptive Neighborhood Eigenvalue Loading Ratio  
The nearest neighbor points for the preprocessed source point cloud \(P_{oi}\) and the target point cloud \(Q_{oj}\) within the spherical neighborhood are searched for. The point cloud resolution calculated in the previous section serves as the adaptive basis for determining the radius of the spherical neighborhood in this section. All point numbers within the spherical neighborhood are recorded. Then, Formula (6) is used to compute the covariance matrix [25]\(\text{Cov}(P_{oi})\) of the adaptive spherical neighborhood for the source point cloud \(P_{oi}\), for example:  
\[\text{Cov}(P_{oi})=\frac{\sum\limits_{p_{i}\in N_{i}}W_{Pi}(P_{i}^{\prime}-P_ {oi})(P_{i}^{\prime}-P_{oi})^{\text{T}}}{m} \tag{6}\]  
where \(m\) refers to the number of points contained in the point sphere field, and \(W_{pi}\) denotes the weights, which are calculated using Formula (7):  
\[W_{Pi}=\frac{1}{\parallel P_{i}^{\prime}-P_{oi}\parallel}(P_{i}^{\prime}\in N _{i}) \tag{7}\]  
The point cloud covariance matrix corresponding to each point is obtained by using Formulas (6) and (7). Then, three eigenvalues are derived through singular value decomposition (SVD), which are arranged in descending order according to their magnitude to yield \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\). \(\lambda_{1}\) and \(\lambda_{2}\) primarily reflect the extent of variation in the directions of the corresponding eigenvectors of the principal plane, while \(\lambda_{3}\) represents the magnitude of variation in the direction perpendicular to this plane. In this study, the ratios \(\lambda_{1}\)/\(\lambda_{2}\) and \(\lambda_{1}\)/\(\lambda_{3}\) termed the eigenvalue loading ratios, are utilized for feature extraction and feature point matching. Regarding the neighborhood, the size relationship between \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) is mainly related to the neighborhood's characteristics. To be specific, if \(\lambda_{1}\) is significantly larger than \(\lambda_{2}\), while \(\lambda_{2}\) and \(\lambda_{3}\) are approximately equal, the local region of the point cloud at the point exhibits a linear distribution. Conversely, if \(\lambda_{1}\) and \(\lambda_{2}\) are similar and greatly larger than \(\lambda_{3}\), the local region displays a planar distribution. If \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are close in value, the region shows a 3D geometric form [26]. The specific division of the point neighborhood is depicted in Figure 2. Since the relative position of each point remains constant during a rigid-body transformation of the point cloud, the corresponding eigenvalue loading ratio remains unchanged with the constant extent of the point neighborhood. In this study, this property is used to conduct feature extraction and pinpoint matching for the point cloud. The algorithm proposed in the literature [26] extracts feature points on a plane and assesses whether the eigenvalues of the local neighborhood's covariance matrix corresponding to these feature points satisfy the conditions \(\lambda_{1}\approx\lambda_{2}\) and \(\lambda_{2}>\lambda_{3}\). However, the feature points extracted are not distinct, and the neighborhood search is relatively time-consuming. In view of that, the algorithm proposed in this study extracts feature points along the surface contour and determines whether the eigenvalues of the adaptive spherical neighborhood's covariance matrix corresponding to these feature points meet a specific curvature ratio of \(\lambda_{3}\)/\(\lambda_{1}\). When the matrix eigenvalues satisfy \(\lambda_{1}\approx\lambda_{2}\) and \(\lambda_{2}\approx\lambda_{3}\), taking \(\lambda_{3}\)/\(\lambda_{1}\) and such eigenvalue loading ratios as the curvature, as the curvature approaches 1, the neighborhood features are closer to the curved surface, which also means that the features are more pronounced. Consequently, if the curvature at a point fulfills Formula (8), the point is identified as a feature point.