Biomedical Engineering Reference
In-Depth Information
term gives
n
+
1
ij
n
n
n
ij
,
0)
2
ij
−
t
(max( sign(
F
ij
)
D
−
x
φ
ij
,
−
sign(
F
ij
)
D
+
x
φ
φ
=
φ
n
ij
,
0)
2
)
1
/
2
n
+
max( sign(
F
ij
)
D
−
y
φ
ij
,
−
sign(
F
ij
)
D
+
y
φ
D
+
x
D
−
x
φ
ij
D
0
y
φ
ij
+
D
+
y
D
−
y
φ
ij
D
0
x
φ
ij
−
2
D
0
x
D
0
y
φ
ij
D
0
x
φ
ij
D
0
y
φ
ij
(
D
0
x
φ
ij
)
2
+
.
+
(
D
0
y
φ
ij
)
2
(4.18)
Here, the difference operators,
D
0
x
,
D
0
y
, are the central finite difference opera-
tors defined by
D
0
x
φ
i
,
j
=
φ
i
+
1
,
j
−
φ
i
−
1
,
j
2
x
D
0
y
φ
i
,
j
=
φ
i
,
j
+
1
−
φ
i
,
j
−
1
2
y
(4.19)
,
.
4.2.3 The Fast Marching Method
An interesting method related to the level set method is the fast marching
method, which was introduced by Sethian [105, 106]. While the fast march-
ing method is used for some subsidiary algorithms within the general level set
method, this method is interesting in its own right. The fast marching method
solves a subclass of the problems normally solved with the level set method, but
it does so much more quickly.
Like the level set method, the fast marching method also uses an implicit rep-
resentation for an evolving interface, but for the fast marching method, the em-
bedding function carries much more information. For the fast marching method,
the entire evolution of the interface is encoded in the embedding function, not
just a single time slice. In other words, the location of the interface at time
t
is
given by the set
(
t
)
={
x
:
φ
(
x
)
=
t
}
.
(4.20)
As a result, in the fast marching method, the embedding function,
φ
, has no time
dependency.
The embedding function,
φ
, is constructed by solving a static Hamilton-
Jacobi equation of the form
F
∇
φ
=
1
,
(4.21)
where
F
is again the speed of the interface. What makes the fast marching
method fast is the fact that Eq. 4.21 can be solved with one pass over the mesh.
