# Full description

With the development of cone beam computed tomography(CBCT), dental CT images, which can provide complete tooth crowns and roots, are becoming more widely used for 3D tooth visualization and dental treatment. However, segmentation and manipulation of individual teeth from CT images are challenging issues due to the following image characteristics:

• The tooth touches adjacent teeth and have common boundaries.
• The tooth is very close to jaw bones.
• The tooth contour has topological change
• Cone beam CT may be very noisy
• CT volume data are large size data

We will segment the tooth in each slice. and each slice is a 2D image. To overcome upper problems, we analysis the tooth characteristics and divide the tooth into 3 parts which are crown, middle and root parts.
Crown parts have common boundaries. Middle parts is very close to jaw bone. Root parts suffer from  opological changes. Therefore, we should make different methods to separate each part. In 3D CT volume, we find initial position of the tooth. Initial contours may not accurate. So to find exact contour we apply level set method to the initial contours. And the contour between crown and middle part of the tooth are relatively well separated. We start from it, to segment root parts we apply level set method downward and to segment crown part we apply level set method upward.

# Tooth Contour Initialization

We propose a method for the automatic tooth contour initialization. In contrast to the 3D level set method, this initialization is quite easy and accurate. We make use of the panoramic image which can be generated from the CT images. To this end, the dental arch curve is fitted automatically as following: Observe from the original images, one middle slice can be selected. The simple thresholding method can generate the binary image of the jaw bone with the soft tissue removed. Then, the thinning operator is applied on the binary image to get the skeleton of the jaw bone. Next, a fourth order polynomial curve is fitted on the skeleton to get the dental arch curve.

# Root Segmentation:Single Level Set

We propose the single level set based tracking method for the root segmentation. The succeeding slice other than the initial slice uses the segmentation result from the previous slice as the initial contour. Thus, some parts of the contour are inside the current tooth region, whereas some are not. Therefore, adaptive coefficient should be applied. Either positive or negative may be possible. To this end, we propose the following energy functional with shape and intensity prior for tooth segmentation and explore the details in this section. \begin{cases} E_s(\phi) &= \mu \int_{\Omega} {\frac{1} {2} (|\Delta \phi|-1)^2 dx dy} +\\ &\lambda \int_{\Omega} { (g + \beta d^2) \delta(\phi)|\Delta \phi| dx dy} + \nu \int_{\Omega} {\log \left( \frac{p_B(I(x,y))} {p_A(I(x,y))} \right) H(-\phi)dx dy}\\ d &= \phi_{prior} \end{cases}

Crown Segmentation: Coupled Level Set

Two level set functions are used to represent all the teeth.

To avoid the overlap of two level set functions, the new energy term is proposed:

$E_R = \int_{\Omega}{H(-\phi_1)H(-\phi_2)}dx dy$
The final energy functional is as follows:

\begin{align} E_c(\phi_1,\phi_2) &= E_S(\phi_1)+E_S(\phi_2)+\alpha E_R\\ \frac{\partial \phi_1}{\partial t} &= \mu \left[\Delta\phi_1 – \operatorname{div} \left( \frac{\nabla \phi_1}{|\nabla \phi_1|}\right) \right]+\\ &\delta(\phi_1)\left( \lambda \operatorname{div} \left( \left( g+\beta d_1^2 \right) \frac{\nabla \phi_1}{|\nabla \phi_1|}\right)+\nu \log \left( \frac{p_{B_1}}{p_{A_1}} \right) + \alpha H(-\phi_2)\right)\\ \frac{\partial \phi_2}{\partial t} &= \mu \left[ \Delta\phi_2 – \operatorname{div}\left(\frac{\nabla \phi_2}{|\nabla \phi_2|}\right) \right]+\\ &\delta(\phi_2)\left(\lambda \operatorname{div}\left( \left(g+\beta d_2^2\right)\frac{\nabla \phi_2}{|\nabla\phi_2|}\right)+\nu \log\left(\frac{p_{B_2}}{p_{A_2}}\right)+\alpha H(-\phi_1)\right) \end{align}
Improved Level Set Method
Integration of gradient direction into the level set method
The boundary is the valid tooth boundary only when the gradient direction is consistent with the normal direction of the negative level set function. Equation represents the relationship of the two directions:

$\cos(\theta) = – \frac{\nabla\phi \cdot \nabla I_G}{|\nabla\phi||\nabla I_G|}$
$$\theta$$ is the angle between gradient direction and the normal direction. $$I_G$$ is the smoothed image.

To deal with the topological change usually large shrinking force is needed. Thus for the tooth contour splitting from one to several pieces, larger shrinking force than expanding force should be applied.
Moreover too big expanding coefficient will force the contour invade into the surrounding object such as adjacent tooth or bones.
Therefore we use different scheme for the shrinking coefficient and expanding coefficient as following $\nu = \begin{cases} \nu_s, &\log\left(\frac{p_B(I)}{p_A(I)}\right) \geq 0\\ \nu_\varepsilon, &\log\left(\frac{p_B(I)}{p_A(I)}\right) < 0 \end{cases}$
where $$\nu_s$$ and $$\nu_\varepsilon$$ represent the shrinking coefficient and expanding coefficient respectively.

# References

SCI/SCIE Indexed Journal:

▷ HuiGao, OksamChae, “Individual tooth segmentation from CT images using level set method with shape and intensity prior”, Pattern Recognition, Volume 43, Issue 7, July. 14. 2010, Pages 2406-2417 ISSN:0031-3203 IF :2.554 (SCI)

▷ Xiaoling Wu, Hui Gao, Hoon Heo, Oksam Chae, Jinsung Cho, Sungyoung Lee and Young-Koo Lee, “Improved B-Spline Contour Fitting Using Genetic Algorithm for the Segmentation of Dental CT Image Sequences”, The Journal of Imaging Science and Technology, vol. 51, no. 4; p. 328-336, July/August 2007. (SCI)

▷ Yonghak Ahn, Oksam Chae “Marginal Bone Destructions in Dental Radiography Using Multi-template Based on Internet Services”, Springer- Verlag Lecture Notes in Computer Science, Vol. 3984, pp. 1001-1009, May 2006 (SCIE)
▷ Hoon Heo, M. Julius Hossain, Jungheon Lee, Oksam Chae, “Visualization of Tooth for 3-D Simulation”, Springer- Verlag Lecture Notes in Artificial Intelligence, Volume 3398, pp. 675-684, October 2004 (SCIE)

International Conference:

▷ Hui Gao and Oksam Chae, “Automatic Tooth Region Separation for Dental CT Images”, IEEE Third International Conference on Convergence and Hybrid Information Technolog, Busan, Korea, pp. 897-901, (Nov. 2008)
▷ Hui Gao and Oksam Chae, “Touching Tooth Segmentation from CT Image Sequences Using Coupled Level Set Method”, IET International Conference on Visual Information Engineering, Xi’an, China, pp. 382-387, (Jul. 2008)
▷ Gao Hui, M. Julius Hossain, Jim X. Chen, and Oksam Chae “Visualization of Tooth for Non Destructive Evaluation”, 2nd International Conference on Advanced Nondestructive Evaluation (ANDE), Busan, Korea, October, 2007
▷ Hoon Heo, Yonghak Ahn, Oksam Chae “B-spline Contour Fitting in Image Sequences using Genetic Algorithms”, Proceedings of the International Conference on Imaging Science, Systems and Technology (CISST), pp. 118-123, Las Vegas, USA, June 2004
▷ Hoon Heo, Oksam Chae “Segmentation of tooth in CT images for the 3D reconstruction of teeth”, Proceedings of IS&T/SPIE Symposium on Electronic Imaging, Vol. 5298, pp. 455-466, San Jose, USA, January 2004
▷ Hoon Heo, Oksam Chae “The Segmentation of Individual tooth in CT Images by Adaptive Optimal Threshold”, Proceedings of International Symposium on Information Science & Electrical Engineering(ISEE), pp. 97-100, Japan, Nov 2003