## 2 Methods
### Scanning Technologies in Spatial Model Construction  
The scanning technologies that are used in dentistry [41; 42] include triangulation using laser light, parallel confocal imaging, or fringe interferometry based upon projection of light patterns. Figure 6: The spatial location of teeth centers detected by the two-camera system and used for evaluation of the corresponding teeth distances, together with their dental planes before and after rotation into the horizontal position for examination (**a**) before the treatment, (**b**) after the treatment, and (**c**) for the location of teeth centers rotated to the horizontal plane and the dental arch evaluated by the mean square method. Data acquisition systems mostly include sensors located in a handheld camera on a wand that collects surface data inside the mouth cavity, as presented in Figure 2a, and a communication system that transmits these data into a computer for further processing. Either laser or white light is used to illuminate the scanned area and its reflection is acquired by specific sensors. Signal and image processing methods are then used to analyse and visualise the resulting 3D model. The widely used Trios intra-oral scanning device uses advanced parallel confocal imaging. In principle, this device [43; 44; 45; 46] captures sets of 2D images of the oral cavity and then combines them into a spatial model using optical sectioning technology [41] and geometrical modelling. STL data exported from the Trios intra-oral scanner into the Matlab and Comsol computational environments were used to construct and analyze digital models of the lower and upper dental arches. The resolution of the Trios scanner [47] was 41.2 points/mm\({}^{2}\) and each model was constructed from about 300,000 vertices. Algorithms have also been developed to process the point cloud datasets acquired by the intra-oral scanner, including the selected signal processing methods, imaging technologies [32; 41], and mathematical methods to construct the surface areas. Delaunay triangulation and Voronoi diagrams form the basis of the computational geometry used in this area. Figure 8: Principle of the separation of overlapping regions presenting (**a**) the original image acquired with a combined illumination, (**b**) area selection, (**c**) region growing segmentation, (**d**) the convex region estimation, and (**e**) the separation curve construction. Figure 7: The segmentation process presenting (**a**) the original image and the circular Hough transform and (**b**) the segmentation using the region growing method and the convex hull. Specific methods of data processing presented in [34] included the transform of recorded datasets into the selected 3D coordinate system, forming a matrix **A** and using the chosen grid size of the plane \([x,z]\), as presented in Figure 9. The evaluated data were then de-noised by a median filter as a nonlinear digital filtering technique using the moving window and a mask of 9 by 9 elements, resulting in a new matrix **B**. The selection of the contour level density was then applied to detect specific areas of the individual teeth. These regions of interest were then used to estimate the dental arch parameters. The coordinates of evaluated tops of each tooth were used [34] for parabolic approximation of the dental arch by a function  
\[f(x)=c_{1}\ x^{2}+c_{2}\ x+c_{3} \tag{4}\]  
for \(x\) in a range of minimum and maximum values on the horizontal axes. The coefficients of this function formed further measures associated with each dental arch. Image rigid registration methods that are used for acquired 3D dental arch data can be then applied to study the reproducibility of the proposed method in the analysis of dental arches and to follow the time evolution of observed dental measures during the treatment. The results presented in [34] used datasets acquired by an intra-oral scanner and recorded in the STL format, which were analyzed and processed in the MATLAB (ver. R2019a, The MathWorks, Inc., Natick, MA, USA) and COMSOL Multiphysics (ver.