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:

int(v) = j j T (w)

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 = (<p, 0). The surface tension parameter t is a user-supplied parameter in the range 0..1, varying the behavior of the surface between a thin plate (t = 0) and a membrane (t = 1). When endowing the surface with a mass ¡x and embedding it into a viscous medium the corresponding energy terms are the same as for the traditional Snake:

The Euler-Lagrange differential equation for constant parameters \x,y,t can be formulated d P

where v stands for either x, y, or 2. These coupled differential equations can be solved numerically as in two dimensions.

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