Modeling Deformable Surfaces with Level Sets

his article describes how to use level sets to represent and compute deformable surfaces. A deformable surface is a sequence of surface models obtained by taking an initial model and incrementally modifying its shape. Typically, we can parameterize the deformation over time, and thus we can imagine that a surface moves or flows under the influence of a vector field. The surface flow, v, can be determined as a function of spatial position (and time), or it can depend on the shape of the surface itself. The latter is called a geometric flow. Deformable surfaces have been used to solve a variety of problems in image processing, computer vision, visualization, and graphics. In graphics, for instance, deformable surface models have been used to form sequences of shapes that animate the morph-ing of one object into another. They have also been used to denoise or smooth surface models derived from a set of noisy 3D measurements. Of course, whenever you model surfaces, you must choose a surface representation, and there are many options, including triangle meshes, splines, spherical harmonics, or superquadrics. When considering deformable surfaces, the choice of surface representation becomes especially important, because the representation defines the set of possible shapes. Most of the interesting flows quickly push us outside the set of shapes associated with a particular representation. Therefore, the choice of surface representation interacts with the kinds of flows we can compute. The problem becomes even more acute when you consider topology. When using a standard, parametric representation , you'd usually rely on a function S: D a ℜ 3 , where D ⊂ ℜ 2. In this case the topology of the surface (for example, the number of holes or connected pieces) depends on the topology of the domain D. In the context of deformation, this means that a surface cannot change topology without some reparameterization that modifies the domain topology. An alternative strategy for modeling surfaces is to use an implicit representation. An implicit model represents a surface as the set of points that meet a constraint, where the constraint is normally defined in terms of the zero crossings of a scalar function, F : V a ℜ (where V ⊂ ℜ 3). That is, the surface S is the set of points S = {x|F(x) = 0}. In graphics this set of points is often called an isosurface of F, but the mathematical literature more …