SSM construction and registration require details of the following issues:

Shape representation

The method of representation of a target to be extracted from given images needs to be determined. In addition to a point distribution model (PDM) used in [2], other methods include medial representations (m-reps) [3], spherical harmonics (SPHARM) boundary description [4], nonuniform rational basis splines (NURBS) [5, 6], or a set of characterizing shape descriptors [7]. For example, the m-reps represent a target with its medial lines and the diameters on each point of the medial lines. Representation of objects is described in Sect. 2.3.2.

Construction of Statistical Shape Model

For the construction of SSMs using training images, each of the labeled regions or their boundaries is represented with one of the above methods. For example, each of the boundaries of the labeled regions can be represented using a PDM. It is necessary to determine (1) a mathematical model to describe the space and variability of anatomical structures, (2) a computable distance metric to measure the difference between shapes, and (3) statistical analysis tools for the shapes. Representation of objects is described further in Sect. 2.3.2. A diffeomorphism-based framework provides one of the most mathematically fundamental procedures for handling these issues, as will be described in Sect. 2.3.4: Using this framework allows generation of an invertible function that maps one boundary surface of an organ in one training image to another surface in a different training image. Such correspondence is vital

Fig. 2.2 Different approaches for the registration between two shapes

to SSM construction, and many strategies other than the diffeomorphism-based framework have been proposed for making this one-to-one correspondence. These strategies can be classified into five categories (see also Fig. 2.2).