# Quality Analysis of 3D Point Cloud Using Low-Cost Spherical Camera for Underpass Mapping
## 2 Materials and Methods
### Quality Analysis
#### 2.6.4 Geometric Features  
In this case study, the color and intensity values of point clouds were not used in the geometric analysis. This is particularly important due to significant lighting effects at the start and end of the underpass, which can result in unreliable intensity values [9]. The geometric features of the point cloud were utilized to determine the geometric consistency and structure of the spherical point cloud in relation to the corresponding reference point  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Calibration**} & **RMSE** & **Calibration RMS** & **Number of** & **Number of Tie** \\  & **Re-Projection** & **Residual (pix)** & **Point Clouds** & **Points** \\ \hline Not calibrated & 0.165979 & 0.158 & 42,193,069 & 14,306/29,616 \\ \hline Calibrated & 0.32821 & 1.56 & 11,542,445 & 15,136/29,614 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparing the results of the DSLR Nikon D800 camera with a fisheye lens for different relative orientation and camera calibration solutions.  
cloud. The geometric features of the point clouds were defined based on certain elements of the covariance tensor (\(\Sigma_{i}\)) [13].  
\[\Sigma_{i}=\frac{1}{N}\sum_{n\in\mathbf{p}_{i}^{N}}\left(\mathbf{p}_{i}-\overline{P} \right)\left(\mathbf{p}_{i}-\overline{P}\right)^{T} \tag{1}\]  
where \(N\) is the number of adjacent points of \(\mathbf{p}_{i}\), and \(\overline{P}\) is the center of the adjacent [33]. These elements consist of the eigenvectors (e\({}_{1}\), e\({}_{2}\), e\({}_{3}\)) and eigenvalues (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\)) of the covariance tensor. The combination of these parameters led to the definition of the unique geometric features of point clouds of all sensors used, imported separately in Cloud Compare. These features included curvature, density, verticality, linearity, planarity, and roughness, defined within a neighborhood with respect to each point which relies on the sampling distance. The radius (r) for the neighboring points (\(\mathbf{p}_{i}\)) was determined in the Cloud Compare toolbox and the following results show their present effect on the spherical point clouds. the proper geometric structure was also shown in the laser scanner reference. The mentioned geometric features in this context were calculated with the following equations (Equations (2)-(7)) [30; 31; 34].  
\[\text{Verticality: }1-\mathbf{p}_{i} \tag{2}\]  
\[\text{Curvature: }\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}+\lambda_{3}} \tag{3}\]  
\[\text{Roughness: }\mathbf{p}_{i}-\mu \tag{4}\]  
\[\text{Linearity: }\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}} \tag{5}\]  
\[\text{Planarity: }\frac{\lambda_{2}-\lambda_{3}}{\lambda_{1}} \tag{6}\]  
\[\text{Density: }\frac{N}{\left(\frac{4}{3}\times\pi\times r^{3}\right)} \tag{7}\]  
Each of these geometric features is associated with a specific behavior of a three-dimensional point cloud within the context of the geometric concept. Verticality measures the alignment of points with the vertical axis [33], curvature indicates the change in direction of the surface normal at a point, roughness is defined as the distance between the least-squares best-fitting plane and its neighborhood points contained in the radius (r) [35], and linearity determines how well points fit into a line, which is crucial for edge and feature extraction [34]. Planarity measures how points fit into a plane and is key for modeling surfaces [34], and density refers to the number of points within a given volume and affects the detail captured in the point cloud.