Biomedical Engineering Reference
In-Depth Information
(a)
(b)
(c)
(d)
(e)
(f )
Figure 4.10: Change in topology of prescribed level for a notched annulus.
Time steps are (a)
t
=
0, (b)
t
=
0
.
05, (c)
t
=
0
.
1, (d)
t
=
0
.
15, (e)
t
=
0
.
2, and
(f)
t
=
0
.
5. Reprinted with permission from [25].
flow of the form
∂φ
∂
t
=
(
κ
−
φ
)
∇
φ
,
(4.36)
with the boundary condition
1
x
inside
−
1
x
outside
,
φ
(
x
)
=
on the plane
z
=
0
,
(4.37)
where
is a closed curve in the
z
=
0 plane.
In Fig. 4.10, a sample evolution of one of the level curves is shown. One of the
questions addressed in [25] is whether all disks in a foliation are topologically
disks. In Fig. 4.10, the resolving of the topology for a particular leaf in a foliation
is illustrated as it evolves in time. The numerical experiments conducted in [25]
suggest that the answer is that the foliation is of disks, even for very complicated
boundary curves.
4.3 Recent Developments
There have been numerous modifications and additions to the level set method
since its inception. Sometimes multiple level set functions are used in tandem
to produce other effects. For example, multiple level set functions are used to
