# Full description

For contour based surface reconstruction, in each slice, a 2D virtual image is created by calculating the signed distance function of the contours, and all the virtual images of all slices are stacked up to create the virtual 3D volumetric data. Before generating the signed distance function, the contours may need to be smoothed. In this work, contour parameterization method proposed in [102] and the bi-directional smoothing methods in [101] are applied. Then, the surface is reconstructed by setting the threshold value as 0 using marching cubes method.

# Definition of the Coordinate Systems

For orthodontic treatment, we need to obtain the features for each tooth such as tooth orientation, rotation angle, tooth size, etc. Thus, the local coordinate reference frame should be constructed corresponding to the global coordinate of the occasional plane in ideal occlusion. We fit the center points of resistance of each tooth with a fourth order polynomial curve to simulate the virtual arch. The curve equation and the first derivative of the curve are given by:
\begin{align}y &= a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4\\ y^’ &= a_1 + 2 a_2 x + 3 a_3 x^2 + 4 a_4 x^3\end{align}
Local coordinate system: Regarding to the local coordinate system of each tooth, the z axis is the long axis of the tooth. The x axis is the mesiodistal axis which follows the tangent direction of the dental arch at the position to which the center of resistance is closest. The y axis is the faciolingual axis and points to the internal side of the dental arch. For better illustration some of the drawings are displayed in the translucent mode in this work. With the definition of the coordinate systems we use PCA method to determine the local system and align it along the dental arch for each tooth.

PCA method: Principal component analysis (PCA) is a classical statistical technique which analyses the covariance

structure of multivariate data.

Different bounding boxes of the point cloud. (a) The axis aligned bounding box. (b) The bounding box with minimal area. (c) The bounding box given by the major axis and the minor axis with PCA method.

(a) The center line of the tooth contours in each slice is used to determine the major z axis. (b) Difference of the minor axes generated by using all the tooth contour points and only the crown contour points.

One root tooth. (a) The Cartesian coordinate system. (b) The bounding box of the tooth with minimal volume. (c) The axes computed by the one step PCA method. (d) The axes computed by the proposed two-step PCA method.

Two-root tooth. (a) The Cartesian coordinate system. (b) The bounding box of the tooth with minimal volume. (c) The axes computed by the one step PCA method. (d) The axes computed by the proposed two-step PCA method.

# Tooth Alignment

(a) Local coordinate system determined by two-step PCA method. (b) Align the x axis of the tooth consistent with the tangent direction of arch.

# Tooth Movements

(a) shows the original position and the orientation of the tooth. Figure 4.9 (b) shows the tooth rotation around the z axis. Figure 4.9 (c) shows the tooth tipping and Figure 4.9 (d) shows the tooth torque.