# Research on a Point Cloud Registration Method Based on Dynamic Neighborhood Features
## 2 Methods
### Dynamic Neighborhood Calculation
In regions with high curvature, the geometric changes on the point cloud surface are significant and the wavelet transform can extract prominent high-frequency information, resulting in larger detail coefficient values. These high-frequency components reflect the complexity and detailed variations in local geometric features. In such instances, the dynamic neighborhood should be appropriately reduced to ensure accurate capture and preservation of these detailed features. Conversely, in regions with low curvature, the geometric changes on the point cloud surface are gradual and the wavelet transform extracts less high-frequency information, leading to smaller detail coefficient values [24]. The geometric features in these regions are relatively uniform, with no significant detail variations. In these cases, the dynamic neighborhood can be appropriately enlarged. This approach captures more local geometric information while enhancing the stability of feature computation. Therefore, the size of the dynamic neighborhood needs to comprehensively consider the influence of point cloud density, local curvature, and detail coefficients. In dense or high-curvature regions, the point cloud information is rich and the geometric features are complex, requiring a smaller dynamic neighborhood to preserve details. In sparse or low-curvature regions, the point cloud information is insufficient and the geometric features are simple, necessitating a larger dynamic neighborhood to ensure computational stability [25]. As a complement to local curvature, the detail coefficients further refine the neighborhood adjustment strategy, ensuring accurate capture of geometric features in point cloud data across multiple scales. The following details the computational steps involved in determining the dynamic neighborhood. First, the principal direction is computed for each local feature point \(P_{c}\) within each small grid cell. The covariance matrix T is constructed using the other points within the same grid cell, The principal directions and curvature information at a point can be calculated through the covariance matrix, as shown in Equation (8) which is as follows:  
\[T=\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}\left(p_{i}-\overline{p}\right)\left(p_{i}- \overline{p}\right)^{T} \tag{8}\]  
The covariance matrix \(T\) is then subjected to eigenvalue decomposition, yielding eigenvalues \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) (assuming \(\lambda_{1}>\lambda_{2}>\lambda_{3}\)), with corresponding eigenvectors \(\mathbf{v}_{1}\), \(\mathbf{v}_{2}\), and \(\mathbf{v}_{3}\). From this, the dominant principal direction \(\mathbf{v}_{1}\) and the secondary principal direction \(\mathbf{v}_{2}\) of the local feature point \(P_{c}\) are obtained. Similarly, the eigenvalues \(\lambda_{i_{1}}\), \(\lambda_{i_{2}}\), and \(\lambda_{i_{3}}\) (assuming \(\lambda_{i_{1}}>\lambda_{i_{2}}>\lambda_{i_{3}}\)) for each of the other points within the same grid cell can be computed by constructing a covariance matrix \(T_{i}\). The curvature \(C_{i}\) of the other points within the grid cell is calculated as shown in Equation (9), which is as follows:  
\[C_{i}=\frac{\lambda_{i_{3}}}{\lambda_{i_{1}}+\lambda_{i_{2}}+\lambda_{i_{3}}} \tag{9}\]  
The average curvature \(\overline{C_{i}}\) of the points within the grid cell is obtained from Equation (10), which is as follows:  
\[\overline{C_{i}}=\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}C_{i} \tag{10}\]  
The point cloud density \(\rho_{i}\) within a grid cell refers to the number of points per unit volume, as shown in Equation (11) which is as follows:  
\[\rho_{i}=\frac{N_{j}}{m_{1}\cdot m_{2}\cdot m_{3}} \tag{11}\]  
The curvature-based wavelet detail coefficient \(W_{i}\) reflects local shape variations. The following outlines the computation of \(W_{i}\). The curvature \(C_{i}\) of each point within the grid cell is used as the input signal \(x\) for wavelet transformation.