## 2 Methods
### Camera Systems in Orthodontic Measures Evaluation  
Dental casts still play an important role in the diagnosis and in the planning of the treatment of dental disorders. They are the gold standard in the evaluation of the dental arch shape, in measuring dental arch distances, and tooth positions. Associated digital models can simplify their use because they allow access to the patients' records through the computer network, they increase the possibility to share the models with other specialists during the therapy, and they enable accurate measurements for the diagnosis setups using selected image processing methods. While manual measurements of plaster casts can be used, it is possible to create digital models of them [22] and collect these measurements using general image processing methods applied to images from a single camera, as presented in Figure 4. Manually detected rectangular regions \(A\) and \(B\) can be then used for the automatic detection of their extreme values applied to the thresholded image. If we know the camera resolution and the distance between image pixels, it is possible to evaluate the distance between selected teeth during the treatment. The multicamera spatial modelling approach can be used to reduce the errors caused by the use of a single camera. Figure 5 presents the system of two cameras located at a selected distance \(c\) to follow a specific point \(C\) on a plaster cast in 3D space. Cameras \(R1\) and \(R2\) together with the object \(C\) form a triangle that can be used to determine its spatial coordinates. The angular and spatial resolution of both cameras in the selected coordinate system was evaluated during the calibration stage [22]. Corresponding objects \(C1,C2\) detected by both cameras can be then specified by their coordinates \([x_{C1},y_{C1},z_{C1}]\) and \([x_{C2},y_{C2},z_{C2}]\) and their distance can be evaluated using relation  
\[D{=}\sqrt{(x_{C2}{-}x_{C1})^{2}{+}(y_{C2}{-}y_{C1})^{2}{+}(z_{C2}{-}z_{C1})^{ 2}}. \tag{2}\]  
Figure 4: Evaluation of distances between selected teeth presenting (**a**) the front image, (**b**) the thresholded gradient image and selected regions \(A\) and \(B\) for their extrema detection, and (**c**) the image contour plot and evaluation of the distance between selected front teeth. Figure 5: The determination of the distance between specified objects in space: (**a**) its principle using two spatially distributed cameras \(A\) and \(B\), and (**b**,**c**) its use for plaster cast images. Given that the process of the treatment in dentistry includes also time evolution aspects, it is often necessary to compare the dental arch parameters changes. The dental plane evaluated from the spatial coordinates of the teeth tops from separate observations can be then used [22] to define its parameters  
\[a_{1}\ x+a_{2}\ y+a_{3}\ z+a_{4}=0 \tag{3}\]  
using the least square method and the selected Cartesian coordinate system. Coefficients \(\{a_{i}\}_{i=1}^{4}\) were then used [22] to rotate all of the objects around individual coordinates into the horizontal plane, which allows digital comparison of individual teeth positioning. Figure 6a,b present locations of the centres of individual teeth in 3D space as evaluated by the proposed method [22] and the plane of their positions in the selected Cartesian coordinate system. The teeth centers specified in dental planes and rotated into the horizontal position are presented in Figure 6c, together with the dental arch curve. This enables locations of the teeth to be evaluated during treatment from the spatial coordinates that have been obtained. This transform allows us to detect changes in the dental arch with a stereo matching algorithm [38; 39]. Corresponding pairs of images obtained before and after the dental treatment enable us to use 2D registration to evaluate the results of the operations [40]. The use of camera systems is closely related to the appropriate illumination [23] and separation of overlapping regions.