Since their introduction by Kass et al. (Kass 1988), a variety of deformable models have been proposed and used in a numerous areas such as image segmentation, shape modeling, and object tracking. Existing deformable models can be classified as either parametric deformable models (PDMs) (Kass, 1988; Cohen, 1991; Xu, 1998; McInery, 1996; Cottes, 1991, 2001; Shen, 2001; Liu, 2005) or geometric deformable models (GDMs) (Malladi, 1995; Caselles, 1997; Yezzi, 1997; Sethian, 1999; Han, 2003) according to their representation and implementation. In particular, PDMs are represented explicitly as parameterized curves or surfaces in a Lagrangian formulation. GDMS, on the other hand, are represented implicitly as level sets of higher-dimensional level set functions. That is, to model a contour within a 2-D space, or a surface within a 3-D space, we define a continuous function (a level set function) which takes a certain value (e.g. zero) on the contour or surface (referred to as the “level set”), and a non-zero value elsewhere. The level set function evolves according to an Eulerian formulation (Sethian 1999).
GDMs are superior to PDMs in the following aspects:
Model representation and construction. GDMs are completely intrinsic and independent of the parameterization of the evolving contour. Thus, there is no need to add or remove nodes from the initial parameterization or adjust the spacing of the nodes. For PDMs, particularly, in the 3D cases, it is not a trivial task to generate parameterized representations for surface models.
Model property calculation. The intrinsic geometric properties of a geometric deformable model, such as its normal and curvature, can be easily determined from the level set function.
Shape adaptation. In GDMs, the propagating contour or surface can automatically change its topology (e.g., merge or split) without requiring an elaborate mechanism to handle such changes.
Self-intersection detection. When the topology preserving transformations are required for a deformable model, self-intersections can become a problem. In GDMs, digital topology methods (Bertrand 1994) are available to detect self-intersections, while in PDMs, the computational demands related to self-intersection detection are usually very high (MacDonald 2000).
On the contrary, PDMs have the following merits over GDMs:
Model constraints. Since PDMs are represented globally, it is convenient to add high level constraints on shape and topology. These constraints are particularly important when local features of the model and image are liable to deform the model in unexpected ways.
Leakage prevention. Leakage from the boundary is less common in PDMs, whereas it is sensitive to boundary gaps and weak edges in GDMs.
Convergence detection. The deformation procedures in PDMs have an explicit stop condition which is the minimization of some kinds of energy function, whereas in GDMs, the procedure usually stops when the certain number of iterations reaches a predetermined maximum, but no simple and fast methods are available to automatically estimate a suitable maximum number of iterations.