## 2 Materials and Methods
### Iso-Topological Meslues: Mesh Morphing  
As detailed in the next paragraph, PCA requires the identification of points of correspondence, called landmarks, among all the shapes in the training set. For 3D shapes the identification of these nodes is not trivial [12; 13] and one solution is to create iso-topological meshes, which are meshes having the same number of nodes N, the same connectivity matrices, and where nodes having the same identifying number are located at the same positions on the given geometry [14]. For example, with reference to the mandible, node #851 is located on the left coronoid process for all geometries, node #1235 is located on the right conodyle, and so on. The approach implemented for the definition of corresponding nodes on 3D shapes is based on the mesh morphing technique, which allows a mesh representing a shape (standard mesh) to adapt to another mesh representing a similar shape (target mesh); the standard mesh is built on a given shape which can be freely chosen among all the available ones, and it should possibly represent a geometry that does not include peculiar morphologies. This kind of approach overcomes the correspondence problem for three-dimensional shapes, especially for complex ones, because the morphing of a standard mesh to all the other shapes, defining the training dataset, guarantees the perfect correspondence between nodes and therefore facilitates the identification of landmarks for shape analysis. Moreover, due to the iso-topology among meshes, all nodes can potentially be considered as shape landmarks. The mathematical problem of mesh morphing can be reduced to an interpolation between nodes and in the present study this interpolation was performed using the so-called radial basis functions (RBF) [15; 16]. The RBF approach, based on the known displacement of a limited number of nodes, called control points, allows to interpolate the displacement that the whole set of nodes has to undergo in order to obtain morphing. The basis function \(\kappa\) is \"radial\" in the sense that it depends on the Euclidean distance \(r\) between the set of nodes \(x\), representing the nodes of a standard shape, and the set of control points \(x_{c}\) (\(c=1,\,\ldots,\,n\)), representing the centers of the function:  
\[r=||x-x_{c}||_{2} \tag{1}\]  
Many different radial basis functions exist and can be implemented for the displacement interpolation: both piecewise smooth (linear, cubic, thin plate spline) and infinitely smooth (i.e., Gaussian, inverse quadratic, multi-quadratic) functions can be implemented for morphing purposes. Herein, after some preliminary comparison analysis involving different RBFs, a linear formulation has been found to provide the best response in terms of nodes deformation for the analyzed shapes; this RBF is defined as:  
\[\kappa(r)=r\] \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Mantibles Training Dataset**} & **Femurs Training Dataset** \\ \hline
**Number of CT scans** & 40 & 98 \\
**Age** & 20–96 & 55–90 \\
**Gender** & 32.5\% female & 100\% female \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of mandibles and femurs database. Nodes belonging to the standard mesh (\(x\)) are moved to a new position (\(x_{\text{new}}\)) following the interpolation provided by the RBF as follows:  
\[x_{\text{new}}=x+\sum_{c=1}^{n}\kappa(r)w_{c}=x+\sum_{c=1}^{n}||x-x_{c}||_{2}w_{c} \tag{3}\]  
where the coefficients \(w_{c}\) are computed according to the constraint that the new position of the control points (\(x_{c,\text{new}}\)) on the standard mesh has to coincide with the corresponding control points on the target shape (\(x_{c,\text{target}}\)):  
\[x_{c,\text{new}}=x_{c,\text{target}} \tag{4}\]  
\[x_{c,\text{target}}-x_{c}=\sum_{c=1}^{n}rw_{c} \tag{5}\]  
The implemented process for the generation of iso-topological meshes was built following the following steps:  
1.