Knowledge of oral anatomy is very important in oral and maxillofacial diagnosis and treatment as well as in orthodontic care. In addition, dental images can be used for diagnosis of systemic diseases. It has been suggested that mandibular cortical width (MCW) measured on dental panoramic radiographs (DPRs) is significantly associated with bone mineral density [69, 159], which, when significantly reduced, may indicate osteoporosis. Measurement of MCWs on images obtained for dental examination purposes may provide the beneficial information of disease risk with no additional cost. In this section, a few oral segmentation methods using CA and statistical models are briefly introduced.
Model-based segmentation of the mandible using cone beam computed tomography (CBCT) was proposed by Antila et al.  for the purposes of aiding dental and maxillofacial surgery planning and reconstructing panoramic radiographs from CBCT images. First they created a mean statistical mandibular model surface S using manual mandible outlines from nine dental CT reconstructions of the mandible and 31 MR scans of the head (Fig. 3.36). A parabolic approximation for the 3D centerline C of the mandibular arch surface was used to fix the coordinate system, x1, x2, x3, and o, corresponding to the orthonormal basis and the origin, respectively, of the model (Fig. 3.36). This coordinate system was used for the affine transformation when applying the model. In the segmentation stage, global and local affine transformations were applied to the model to capture the rough appearance of the mandible, followed by an elastic deformation to refine the segmentation result. The global transformation of C consisted of 3D rotation and translation and x1-x2 scaling, which adjusted the size and shape of the mandibular arch. The local transformation of S consisted of rotation about the tangential vectors of C and 2D scaling in the normal planes spanned by x3 and bj, which adjusted the orientation and shape of the surface cross section. For the elastic deformation, control points were placed on the surface S and were iteratively adjusted by energy minimization controlled by the surface outward normal gradient v»VI and value
Fig. 3.36 A mean statistical mandible model surface S computed from manually delineated mandible outlines of nine dental CTs and 31 magnetic resonance (MR) scans of the head 
I of the grayscale intensity. With a small number of test cases, segmentation was successful in comparison with the manual references.
Other methods introduced here are based on or are modifications of the active shape model (ASM) . The model can be built using feature (landmark) points of the training cases and is represented by
where x is the mean shape, P is a set of orthogonal modes of variation, and b is a set of shape parameters. The mean shape can be computed by n landmark points concatenated into a 2n vector x = (xi, x2, ..., xn, yi, y2, ..., yn) of each training case. P is the matrix of t most significant eigenvectors computed using principal component analysis (PCA), whereas b is the corresponding weight vector.
Allen et al.  applied the ASM in determining the inferior and superior borders of mandibular cortical bone for the purpose of MCW measurement on DPRs. The model was built using 200 manually marked points with 50 equally spaced points on each of the upper and lower margins of cortical bone on the right and left sides of mandibles. Figure 3.37 illustrates the mean model and the shape variation realized by varying the weight b1 by ± 3a. During the application of the model to a test case, an iterative search was conducted by updating the landmarks on the basis of the gradient along the normal to the boundary at each point. Detection of the mandibular borders was relatively successful using ASM; however, the method had some limitations in having lateral shifts along the borders.
Motivated by creating a model not only describing the shape but also the texture, an active appearance model (AAM) was introduced by Cootes et al. .
S. Hanaoka et al.
Fig. 3.37 A mean model of inferior and superior borders of mandibular cortical bone with its shape variation realized by varying the weight b1by ± 3o 
Fig. 3.38 Semiautomatic landmarking steps to extract jaw tissues based on the AAM 
The appearance model is represented by
where the upper equation describes the shape as in ASM and the lower one describes the texture, c is the set of parameters controlling the shape and texture, and g and Pg are the mean texture vector and the matrix describing the mode of variation in texture, respectively.
The AAM was used for automatic segmentation of jaw tissues in CT by Rueda, et al.  for possible utility in oral implant surgery planning. Their aim was to automatically segment cortical bone, trabecular bone, and the mandibular canal on a cross-sectional view (Fig.3.38). The model was constructed using a training set with 87 landmarks. A semiautomatic annotation is processed in five steps: (1) thresholding to find the external contour of the cortical bone (Fig. 3.38a), (2) defining five points of high curvature on the contour (Fig. 3.38b), (3) finding the contour of the trabecular core (Fig. 3.38c), (4) locating the dental nerve in the center of the mandibular canal (Fig. 3.38d), and (5) selecting the radius of the canal (Fig. 3.38e). Using these results, landmarks are placed equally, including
Fig. 3.39 Model landmarking for cortical bone, trabecular bone, mandibular canal, and mandibular nerve 
30 and 28 landmarks for the double contour of the cortical bone, 10 and 10 for the double contour of trabecular core, 8 on the mandibular canal, and 1 on the mandibular nerve inside the canal (Fig. 3.39). After the mean shape is extracted, a piecewise affine warp is applied, and the intensity is sampled from the shape- normalized images. These samples are normalized so that the effect of global intensity variation is reduced, and the texture (gray-level) vector is obtained. In the segmentation process, the model is used as an initial template, and a principal component multivariate linear regression model is used to generate new images to fit the test image. With AAM, segmentation of the cortical bones was generally successful, while segmentation of the trabecular bone was more difficult.
To improve on the AAM, Cristinacce and Cootes  proposed a constrained local model (CLM). The joint shape and texture model has the same form as the AAM (Eqs. 3.9 and 3.10). During the iterative search, a set of templates (patches) is generated from the model at the feature points. Based on the correlation between the current templates and a test image, a new set of feature points is predicted where a new set of templates will be generated. The CLM method was tested on different types of image datasets, including MR slices of brain, photographs of human faces, and DPRs . For detecting mandibular contour on DPRs, while 78 manual points along the mandible were used for the AAM, only 22 points were used for the CLM, and it resulted in more stable performance.
Muramatsu et al.  have also proposed a model-based method in delineation of mandibular contour on DPRs for the purpose of automatic MCW measurement. In this method, manual contour points from the training cases (Fig. 3.40a) are used to create a mask for the lower mandibular border (Fig. 3.40b). This mask is used for detecting candidate edges with specific directions anticipated for the mandible. The individual manual contours of the training cases also serve as models from which the most similar model for a test case is selected on the basis of the similarity score with the detected edges. Using the selected model as an initial
Fig. 3.40 A model-based method in delineation of mandibular contour on DPRs for the purpose of automatic MCW measurement . (a) Manual contour points from the training cases, (b) a probabilistic template for the lower mandibular border
control point, the final contour is determined by fitting the points with the active contour models . Using the proposed method, the mandibular contours of the test cases were successfully determined with a small number of minor partial failures in the mandibular angle.