## 3 Methodology
### Adaptive Neighborhood Selection
Regarding the step size \(\Delta_{r}\), a smaller \(\Delta_{r}\) improves accuracy but increases computational cost, whereas a larger \(\Delta_{r}\) reduces accuracy while enhancing efficiency. To balance precision and computational performance, we fine-tuned \(\Delta_{r}\) by setting it to 1-2 times the last significant digit of \(r_{min}\). **b. ** All points within a boundary sphere of radius \(r_{min}\) are searched, and an improved weighted covariance matrix \(\mathbf{C}\) is constructed [79], which can be expressed as:  
\[\mathbf{C}=\sum_{i=1}^{k}\tau_{i}(p_{i}-\bar{p})(p_{i}-\bar{p})^{T} \tag{5}\]  
where \(p_{i}\) represents the coordinates of each point, \(\bar{p}\) denotes the weighted average of these coordinates, and \(\tau_{i}\) signifies the weighting coefficient, with points closer to the center receiving higher coefficients. \(\tau_{i}\) and \(\bar{p}\) can be expressed as follows:  
\[\tau_{i}=exp(-\frac{||p_{i}-\mu_{d}||^{2}}{2\sigma_{d}{}^{2}}) \tag{6}\]  
\[\bar{p}=\frac{\sum_{i=1}^{k}\tau_{i}\cdot p_{i}}{\sum_{i=1}^{k}\tau_{i}} \tag{7}\]  
where \(\mu_{d}\) denotes the average of the coordinates of each point, while \(\sigma_{d}\) represents the standard deviation of the distances from the points to the center. Subsequently, a singular value decomposition (SVD) [80] is employed to compute the eigenvalues of the covariance matrix \(\mathbf{C}\), which are then arranged in descending order as \(\lambda_{0}\geq\lambda_{1}\geq\lambda_{2}>0\). **c.** The eigenvalues are used to compute three metrics of the structure tensor: \(L_{\lambda}\), \(P_{\lambda}\), and \(S_{\lambda}\). \(L_{\lambda}\) represents the linearity metric, indicating whether the point cloud predominantly extends along one direction. \(P_{\lambda}\) is the planarity metric, reflecting whether the points are mainly distributed within a plane. \(S_{\lambda}\) is the scattering metric, describing whether the points are uniformly distributed in three-dimensional space. These metrics can be expressed as follows:  
\[\left\{ \begin{aligned} L_{\lambda}&=\frac{\sqrt{ \lambda_{0}}-\sqrt{\lambda_{1}}+\epsilon}{\sqrt{\lambda_{0}}}\\ P_{\lambda}&=\frac{\sqrt{\lambda_{1}}-\sqrt{\lambda_{2 }}+\epsilon}{\sqrt{\lambda_{0}}}\\ S_{\lambda}&=\frac{\sqrt{\lambda_{2}}+\epsilon}{ \sqrt{\lambda_{0}}}\end{aligned} \right. \tag{8}\]where \(\epsilon=1\times 10^{-10}\) is a small constant introduced to avoid division by zero. Based on [PERSON]'s entropy theory [81], the entropy function is constructed using the aforementioned metrics and can be expressed as:  
\[E_{p}=-(L_{\lambda}ln(L_{\lambda})+P_{\lambda}ln(P_{\lambda})+S_{\lambda}ln(S_{ \lambda})) \tag{9}\]  
**d.** Adjust the search radius by setting \(r_{i}=r_{i}+\Delta_{r}\) and repeat steps **b** to **d** until \(r_{i}>r_{max}\). **e.** Select the radius that minimizes \(E_{p}\) as the optimal radius for the point, as the entropy function reflects the complexity of the point cloud. Higher entropy values indicate greater complexity and lower correlation between points in the neighborhood, while lower entropy values suggest a simpler structure with stronger correlations between points.