## 3 Methodology
### Adaptive Neighborhood Selection  
In different regions of the same point cloud, the point density varies, especially near edges, corners, and areas with complex surface geometry. This spatial variation in density leads to significant challenges in accurately estimating geometric properties such as normal vectors and curvature. Specifically, using a fixed \(k\)-value for the neighborhood selection in such varying-density regions can result in inaccurate normal vector estimations and curvature calculations. For example, in regions with higher point density, the fixed \(k\) value may lead to an overly localized estimate of the normal, which may not fully represent the true surface orientation. Conversely, in sparse areas or near edges, a fixed \(k\) value may include too few points, resulting in unreliable normal vectors and curvature values, thus introducing significant bias in the defect detection process. These inaccuracies can severely impact the precision of surface defect identification and quantitative evaluation. Figure 8 presents a projected point map of a house. At P1, the ground surface is structurally simple and uniform, resulting in negligible changes to normal and curvature estimation regardless of neighborhood size adjustments. At P2, an excessively large neighborhood encompasses two intersecting planes at a corner, leading to significant estimation bias. At P3, an overly large neighborhood fails to capture local variations around the window, while at P4, an overly small neighborhood overlooks details of the eaves. These observations further underscore the importance of adaptive neighborhood selection. Existing studies widely employ entropy, neighborhood self-information, and neighborhood mutual information to quantify nonlinear relationships between features and classes, thereby reducing uncertainty in the output [74; 75; 76; 77]. Although these approaches were not originally designed for point cloud data, the inherent uncertainty and unstructured nature of point clouds make them a subset of complex data. Therefore, these concepts can be effectively applied to point cloud processing for key point extraction. To overcome the limitations imposed by a fixed \(k\) value, we propose an adaptive neighborhood selection  
Figure 8: Projected point map of a structurally complex house. method based on information entropy. This approach adapts to local variations in point cloud density, enabling more accurate estimations of normal vectors and curvature, especially in regions where the point density is low or irregular. The entropy function is used to measure the information content and distribution characteristics of the data, reflecting the complexity and irregularity of the point cloud in each region. By adjusting the neighborhood size dynamically based on the local point distribution, this method ensures that the neighborhood size is optimized for each region, leading to more accurate geometric property estimates and, in turn, improving defect detection accuracy. Specifically, the process involves the following five steps:  
**a. ** The radius search range (\(r_{min},r_{max}\)) and step size \(\Delta_{r}\) are defined based on the point cloud density, with the initial radius \(r_{i}\) set to \(r_{min}\). Specifically, \(r_{min}\) and \(r_{max}\) are determined using a k-nearest neighbor search. Based on previous studies, a smaller \(k\) is preferable in high-density regions, while a larger \(k\) is more suitable for low-density areas. The optimal \(k\) value typically ranges between 9 and 60 [9; 40; 41; 42; 43; 44; 45; 78]. Therefore, in this study, we set the range to 3-80 to encompass common values of \(k\). For \(k=3\), the Euclidean distances from the nearest neighbors to the central point were computed, and the maximum distance across the entire point cloud was assigned to \(r_{min}\). Similarly, for \(k=80\), the maximum distance was assigned to \(r_{max}\).