## 2 Method
### Data Analysis Procedure
#### 2.2.3 3D Shape Based Analysis of Face Variation  
In order to analyse the shape variation of the 3D face images, a morphological correspondence between each individual face image needs to be realized. Meshes of individual scans contain a varying amount of 3D data points that are distributed in different ways, which makes it impossible to compare them directly. By creating a correspondence, meshes are converted in such a way that the data points are approximately uniformly distributed over the shape, creating meshes that have the same number of data points and connectivity (triangles), with comparable landmark locations. This is referred to as homologous meshes.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Measurement** & **Definition** \\ \hline Face length & Straight-line distance between the sellion (s) and menton (me) landmarks \\ Sellion-pogonion length & Straight-line distance between the sellion (s) and pogonion (pg) landmarks \\ Nose bridge length & Straight-line distance between the sellion (s) to promasale (prn) landmarks \\ Mouth width & Straight-line distance between the left and right chelion landmarks (ch-ch) \\ Nose tip protrusion & Straight-line distance between the the subnasale (sn) and the promasale (prn) landmarks \\ Nasal root breadth & The horizontal breadth of the nosals root spanning from the left nasal root point to the right (nrp-nrp) \\ Nose breadth & Straight line distance between the left and right alare landmarks (al-al) \\ Inter-pupillary distance & The straight-line distance between the centre of the left and the centre of the right pupil \\ Face width & Horizontal breadth between the left and right zygion landmarks (zy-zy) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Definitions of the nine selected measurements for the measurements based PCA.  
In this study, the so-called non-rigid template registration method [43] is used using Wrap 3.4 software (Russian3 dscanner, Moscow, Russia). First, a high-quality individual scan was selected to serve as the template mesh. This scan was then processed to be topologically equivalent to a disc (no surface handles and a single boundary at the neck) and with a uniform distribution of vertices. The obtained template was then used to create these homologous meshes by deforming the template mesh towards each individual face scan (target scan) as is illustrated in Figure 3. The annotated landmarks were used to steer the registration by forcing an exact match. Finally, the homologous meshes were spatially aligned (limited to translation and rotation) via partial Generalized Procrustes Analysis [44].  
A PCA was then conducted on the 3D coordinates of the homological meshes in order to analyse and visualize the 3D shape variation of the children's faces. The principal component analysis was conducted on the \(x\)-, \(y\)-, and \(z\)-values of each vertex of the 3D mesh data using a custom script in Python. The average head mesh was processed in ParaView to extract the relevant facial area through clipping with two interactively positioned planes. The resulting facial region of interest of the average shape was then warped to each individual face through the homology, effectively resulting in corresponding regions of interest for all subjects in the database, forming the input for the facial PCA analysis. Face shape modes that represent the shape variation are visualized along the first 7 Principal Components (PC's).