## Deformable Surface Models

The basic concepts of Snakes—minimization of an energy term through optimization—can easily be generalized to three dimensions. Additional effort is required only to handle the parameterization problem adherent to 2-D manifolds.

In contrast to 2-D active contour models, where arc length provides a natural parameterization, 2-D manifolds as used for 3-D deformable models pose a complex, topology, and shape-dependent parameterization problem. Parameterizing a surface effectively is difficult because there is no easy way to distribute the grid points evenly across the surface.

A generalized deformable surface model is defined as v(w, t) = (x(w, t), y(w, t), z(w, t))

where u e ^ c is a suitable parameterization, t the current time, and x, y and 2 are the corresponding coordinate functions of the surface. Analogous to the 2D case, the surface deforms itself so as to minimize its image potential energy. Instead of the elastic rod model, the thin plate under tension model is employed to regulate the model's shape during energy minimization. Thus, the term Ent has the following form:

 d v + d v 2 + P (w) d 2v + 2 d 2v + d 2v 2 d0 de d02 d0de de2 where p(u) = 1 — t(u) for convenience and u = (
0 0