## 2 Materials and Methods
### Research Methodology
#### 2.2.2 Phase 2: Point Clouds Analysis  
At this phase, the point clouds were segmented in order to isolate the object of experimentation. If segmentation was not carried out, the values of the mathematical descriptors could be altered in the subsequent processing phase. After the segmentation step, the point clouds were analysed by mathematical descriptors. Indeed, the geometric features of a point cloud can be described by applying a principal component analysis (PCA) to the covariance matrix \(C\) of the neighbourhood \(p_{i_{j}}\) of a point \(\overline{p}\) of a set of points \(N_{p}\). The covariance matrix \(C\) (Equation (1)) can be defined as a three-dimensional tensor containing the geometric information of a set of points \(N_{p}\) in the neighbourhood \(p_{i_{j}}\)[17, 18]. \[C= \begin{bmatrix}p_{i_{1}}&-\overline{p}\\ \dots&\dots\\ p_{i_{k}}&-\overline{p}\end{bmatrix} ^{T} \begin{bmatrix}p_{i_{1}}&-\overline{p} \\ \dots&\dots\\ p_{i_{k}}&-\overline{p}\end{bmatrix} ,\;i_{j}\in N_{p} \tag{1}\]  
Figure 4: Detail of the room scanned in the third test. where \(\overline{p}\) is the centroid of a neighbourhood, \(p_{ij}\), of a set of points \(N_{p}\) (Figure 6). Considering an eigenvector problem, it can be written:  
\[C\cdot v_{l}=\lambda_{l}\cdot v_{l},\;l\in\{1,\;2,\;3\} \tag{2}\]  
The covariance matrix \(C\) is symmetrical and positive semidefinite, and using a PCA, it is possible to extract the three eigenvalues \(\lambda_{1},\;\lambda_{2},\;\lambda_{3}\)[19] from the matrix. The eigenvalues locally describe the 3D structure of the point set \(N_{p}\) and quantify its variation along the direction of the corresponding eigenvector \(v_{1},\;v_{2},\;v_{3}\) (Figure 6). Through mathematical operations between the three eigenvalues, it is possible to calculate certain form-specific mathematical descriptors called covariance-features or eigenfeatures that are capable of describing certain geometric characteristics of the point cloud. For example, some descriptors found in the literature are linearity, planarity, anisotropy, omnivariance, eigentropy and surface variation and sphericity [17; 18; 20]. In this research, we used the eigenfeatures of planarity, primivariance and surface variation to analyse point clouds based on their shape (eigenfeature analysis). The equations of the three latter descriptors are:  
\[\text{Planarity }P_{\lambda}=\frac{\lambda_{2}-\lambda_{3}}{\lambda_{1}} \tag{3}\]  
\[\text{Omnivariance }O_{\lambda}=(\lambda_{1}\cdot\lambda_{2}\cdot\lambda_{3})^{ \frac{1}{3}} \tag{4}\]  
\[\text{Surface Variation }SV_{\lambda}=\frac{\lambda_{3}}{\lambda_{1}+\lambda_{2 }+\lambda_{3}} \tag{5}\]  
Figure 5: Overview of the Doric columns of Taranto (**a**); remains of the third column used for the fourth test (**b**). Figure 6: Neighbourhood and covariance analysis, images adapted from [18]. Planarity quantitatively describes the tendency of the point cloud to arrange itself along plane surfaces. This eigenfeature can be used to describe point clouds of planar objects or those formed by a combination of planes. For this reason, during data analysis, we used Planarity as a descriptor for the 10 samples in the first test and the room in the third test. Omnivariance quantifies the degree of inhomogeneity of the point cloud in 3 dimensions. This descriptor can be used to describe the point clouds of freeform objects, and in this work, we applied omnivariance to the statue in the second test and to the remains of the column in the fourth test. Surface variation quantitatively describes the variations along the normal to the surface of a point cloud and, as demonstrated in [PERSON] et al. [18], can be used to identify two point clouds side-by-side and overlapping at a certain distance (Figure 7a). Figure 7b shows the descriptor applied to two flat point clouds that are side-by-side with their corners overlapping (the orange ellipse indicates the green overlapping parts identified with surface variation).