Statistical Modeling of Normal Variants
Although many statistical modeling methods for continuous variations have been introduced, few methods for discrete normal variants have been reported. Among them, Mori et al.  reported a graph-based method to identify the branching pattern of the abdominal arteries. Another work by Hanaoka et al. proposed a method to detect anomalies of the number of vertebrae .
Fig. 2.36 Pancreas divisum. The main pancreatic duct continues to the papilla duodeni minor (arrow), not to the papilla duodeni major (arrowhead) (MR cholangiopancreatography, frontal view)
Fig. 2.37 Aberrant right hepatic artery. The right hepatic artery (arrow) arises from the superior mesenteric artery (arrowhead) instead of the celiac artery (Conventional angiography, frontal view)
However, as described above, it is difficult to represent both normal and variant subjects with a single model where these variants change the topology of the target organ. The application of a PDM to such a situation is an example. To build a PDM, point-to-point correspondence must be established. However, the points that
- 2 Fundamental Theories and Techniques
Fig. 2.38 A large cavum septum pellucidum (T2-weighted MR image, axial cross-section)
are defined on an abnormal extra structure have no corresponding points in normal subjects.
One possible solution is to build two models for normal and variant subjects. However, collecting a sufficiently large number of variant subjects is usually not practical. Otherwise, level-set methods  can be used to build a statistical shape model with topology changes. Another promising approach is multi-atlas-based methods . Because variant and normal subjects can be combined to form a multi-atlas in multi-atlas segmentation methods, it can handle variants with high frequency without explicitly considering the existence of an abnormality.
Up to now, no general method has been introduced to build an SSM that can represent both continuous (e.g., the position, dimensions, or pose) and discrete (e.g., the branching pattern, number of objects, or topology) properties of the shape variety. Future development of new modeling methodology is needed.