# Robust Segmentation of Partial and Imperfect Dental Arches
## 3 Methods
### Registration Based on Oriented Bounding Box (OBB)  
After aligning the oriented bounding box (OBB) of each labeled large-span arch in GCS, we obtained the average teeth centroids for each tooth position in GCS as illustrated in Figure 2. The arch was positioned in the x-y plane with all teeth positioning toward the z-positive direction.  
We registered both full and partial arches with the guidance of these average teeth centroids and range labels. Teeth centroid information, considered as a part of the registration method, was consistently applied to process data during training and inference. The registration method can be described in 4 steps:  
1. [label=(0)]
2. Bring the center of mass of each input mesh to origin;
3. Compute the oriented bounding box (OBB) of the centered mesh with the class vtkOBTree of the VTK library [37]. More specifically, OBB is constructed by finding the mean and covariance matrices of the cells (and their points) that define the mesh. The eigenvectors of the covariance matrix are extracted, offering a set of orthogonal vectors that define the tightest-fitting OBB;
4. Align the obtained OBB in the x-y plan of GCS in a way that the largest edge follows the x-axis (the width of an arch), the second largest edge is in the y-axis (the height of an arch), and the smallest edge is in z-axis, the height of teeth on the arch;
5. Compute the required translation and rotation in the x-y plan using the estimated center of input arch, the centroids of teeth computed in Figure 2, and the start and end teeth labels specified by the user. The centroids between the start and end labels influence the translation and the two centroids of start and end labels influence the  
Figure 2: Determining teeth centroids of segmented arches. (**a**) Teeth centroids of upper arches, (**b**) Average upper teeth centroids, (**c**) Teeth centroids of lower arches, (**d**) Average lower teeth centroids. Different colors correspond to different teeth types.  
rotation. Then, position the arch in the GCS using the constructed transformation matrix.  
Figure 3 illustrates the steps of our registration method with a lower partial arch for which start tooth label1 = 43 and end tooth label2 = 46. In Step 3, the first principal axis of the arch OBB was aligned with the x-axis. While it is relatively easier to register full arches (with more than 8 teeth), there exist issues with registration of partial arches. In short partial arches (containing 3 or 4 teeth), it is challenging to identify the lingual buccal side. Therefore, for partial arch registration in the GCS, we relied on both the first and last labels provided by the user. The die mesh, if available, could also be used to enhance the lingual buccal identification. This could be achieved if the label of the provided die is the same as the first or last labels or close to either of them. The centroids shown in Step 4 are added for demonstration purposes. The die mesh is not only helpful for registering and segmenting arches but is also the essential part used to define a context with a pair of aligned master and antagonist arches for dental crown design purposes.  
Figure 4 shows a set of upper arches before and after registration. The OBB-based arch registration is the most important pre-processing step to bring different types of arches (full and partial) to a common GCS. Figure 5 illustrates lingual buccal ambiguity for certain partial arches. In both left sub-figures, we put a registered full lower arch as a reference. For this example, the die mesh is at the middle of the partial arch, so it is not helpful. This example shows that the registration has an important impact on the segmentation result, especially for partial arches.  
Figure 3: Illustration of the OBB-based registration method for a lower partial arch. (**a**) Step 1: center mesh, (**b**) Step 2: compute OBB, (**c**) Step 3: align OBB to x–y plan, (**d**) Step 4: map mesh to GCS.