Biomedical Engineering Reference
In-Depth Information
represent the step edges of a stratified material [21]. Lower dimensional mani-
folds, such as one-dimensional curves in
R
3
, can be captured by the intersection
of two level surfaces [14]. Multiple distinct regions with interfaces that result in
triple junctions can also be captured using multiple level set functions [17, 108].
Interfaces with boundaries, such as a finite-length crack in a plate, can also be
represented using multiple level sets [107, 115, 117].
In other examples, multiple level surfaces of a single level set function are
used. In [95], the level surfaces for each of the integers represent a different
interface. In other applications, the entire continuous spectrum of level surfaces
are used. For example, in [25], each level surface evolves to a surface of constant
curvature, while in [20], the spectrum of evolving level surfaces is shifted in order
to locate an unstable equilibrium surface.
The fast marching method has also been used in a variety of applications,
resulting in dramatically increased speed in some computationally intensive cal-
culations. For example, see the work on computing multiple travel-time arrivals
in [46].
The range of applications for the level set and fast marching methods is
now very wide, and still growing. Many times, variations of the method are
required to make it fit the problem. In this section, some recent improvements
and variations, which will be of general interest are presented.
4.3.1 Ordered Upwind Methods
In [101, 129], Sethian and Vladimirsky developed a novel extension of the fast
marching method, making it applicable to a significantly wider class of problems.
Recall the fast marching method equation
F
∇
φ
=
1
.
(4.38)
It is important to recognize that this equation assumes that from any point
x
,
the speed,
F
, is the same, independent of the direction the interface is traveling.
In other words, the speed function is isotropic. Sethian and Vladimirsky have
generalized the fast marching method so that the speed function can vary with
direction, i.e. the speed function,
F
, may depend on
∇
φ
. In this case, the speed
function is called anisotropic. The generalized method is called the ordered
upwind method, of which the fast marching method is a special case.
