# Accurate Extraction of Cableways Based on the LS-PCA Combination Analysis Method
## 2 Methodology
### Preprocessing of Ropeway Data
#### 2.2.2 Preliminary Segmentation of the Ropeway  
After elevation filtering, the non-ground points include towers and some high-ground objects in addition to ropeway points. The tower consists of regular polygonal metal patches, while the point cloud in the local area has a regular surface pattern. The cableway can be described as a flexible cable chain. There is a linear distribution of point clouds in the local area, while vegetation points are irregular and scattered. Therefore, the cableway points can be extracted based on the geometric distribution of the point cloud space. By examining the three eigenvalues \(\lambda_{1},\lambda_{2}\) and \(\lambda_{3}(\lambda_{1}\geqslant\lambda_{2}\geqslant\lambda_{3})\) of the point set covariance matrix in the neighborhood of \(R\), we can determine the geometric distribution characteristics of spatial points. When \(\lambda_{1}\cong\lambda_{2}\gg\lambda_{3}\), the target point shows a planar distribution feature, such as roads and towers, etc.; when \(\lambda_{1}\geqslant\lambda_{2}\cong\lambda_{3}\), the target point shows a linear distribution feature, such as cableway and road boundary, etc. When \(\lambda_{1}\cong\lambda_{2}\cong\lambda_{3}\), the target point can be divided into scattered and irregularly distributed point clouds, such as vegetation.  
If a fixed neighborhood radius is used to estimate the local geometric features of points, it will result in large calculation errors, which will affect the segmentation results of the target points due to the uneven distribution of density in the point cloud. In order to determine the optimal neighborhood radius for solving the eigenvalue of each laser point,  
Figure 3: An overview of the elevation filtering process on raw point cloud data: (**a**) normalized point clouds; (**b**) elevation-point-figure maps; and (**c**) elevation filtering effects (white and color).  
the probability \((a_{\rm 1D},a_{\rm 2D},a_{\rm 3D})\) dimension characteristics of the three dimensions to which the target point belongs are defined as follows:  
\[(a_{\rm 1D},a_{\rm 2D},a_{\rm 3D})=\left(\frac{\sqrt{\lambda_{1}}-\sqrt{\lambda_{2} }}{\sqrt{\lambda_{1}}},\frac{\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}}{\sqrt{ \lambda_{1}}},\frac{\sqrt{\lambda_{3}}}{\sqrt{\lambda_{1}}}\right) \tag{1}\]  
In Formula (1), \(a_{\rm 1D}+a_{\rm 2D}+a_{\rm 3D}=1\), Shannon entropy defines the amount of information contained in the neighborhood of this point as follows:  
\[E_{f}=-a_{\rm 1D}\,\text{Ln}(a_{\rm 1D})-a_{\rm 2D}\,\text{Ln}(a_{\rm 2D})-a_{ \rm 3D}\,\text{Ln}(a_{\rm 3D}) \tag{2}\]  
As described in Formula (2), \(E_{f}\) represents the amount of entropy contained in the neighborhood of the point, and the smaller this value, the less information the neighborhood contains, which means the more singular the dimensional feature of the point. The radius \(R\) with the minimum value of the function \(E_{f}\) is the best neighborhood radius for this point when the radius interval \([R_{\rm min},R_{\rm max}]\) and radius increment \(\triangle R\) are set. The eigenvalue of each point is calculated using the adaptive neighborhood radius, and the preliminary extraction of the cableway point cloud is based on the relationship between the eigenvalues. The neighborhood radius interval should be selected based on the spatial range of the points showing the three characteristics. Radius increments should be based on the volume of point cloud data and the performance of the computer. The smaller the increment, the more time-consuming the process will be. In most cases, the radius increment for the filtered cableway point cloud extraction is 0.5, and the neighborhood radius interval is \([1.4,2.8]\).