Recent Advances in the Level Set Method - Biomedical Image Analysis - page 218

4.2.8 Example Calculations

There are numerous examples of applications for the level set and fast marching
methods. A comprehensive list of the most recent applications will be given in
Section 4.4. Two simple examples which illustrate the basic level set method are
presented here.

Minimal Surfaces with Voids

As noted in the introduction, the original paper on the level set function involved
a speed function which depended on mean curvature. Flow by mean curvature
was also used to compute examples of minimal surfaces [19]. In the present
example, flow by mean curvature is again used, but this time there are void
regions where the surface area contained in the void is not counted in the total
surface area.
It is shown in [123] that the minimal surface in this case will meet the voids
orthogonally. The orthogonality boundary condition can be rewritten in a way
that is familiar. Suppose φ is the evolving surface moving by mean curvature, and
assume ψ is a level set function representing the voids with the surface of the
voids identified by ψ = 0. The orthogonality boundary condition is equivalent
to the surface normals being orthogonal; in other words, we must have

∇ φ ·∇ ψ = 0 .

(4.35)

This equation is reminiscent of the equation for velocity extensions, Eq. 4.32. In
fact, the velocity extension algorithm is used to determine φ inside the voids.
In Fig. 4.9, a surface which passes through five spherical voids is illustrated.
Initially, the surface passes over the central void. As the surface relaxes, it strikes
the center sphere and finally reaches equilibrium on the lower side of the sphere.
The voids are semitransparent so that the results of applying the velocity ex-
tension code to φ can be seen. Also, the shading on the surface indicates the
magnitude of the mean curvature.

Curvature Flow in Hyperbolic Space

In [25], mean curvature flow in hyperbolic space mapped onto the upper half-
space is investigated. In particular, foliations of the space are computed using a