Biomedical Engineering Reference
In-Depth Information
Eq. 4.3 gives the level set evolution equation,
∂φ
∂
t
+
F
∇
φ
=
0
.
(4.5)
This is the key evolution equation that was introduced in [85]. Through this
equation, the motion of the interface
(
t
) is captured through Eq. 4.5 so that at
any time
t
,
(
t
)
={
x
:
φ
(
x
,
t
)
=
0
}
.
(4.6)
One key observation about Eq. 4.5 is that we have implicitly assumed that
the function
F
is known over the entire domain of
φ
. Very often, this is not the
case, and
F
is only defined on the interface itself. However, this problem can be
solved by using velocity extensions, which will be discussed in Section 4.2.5.
4.2.2 Numerical Implementation of the
Level Set Method
As noted in the introduction, the second critical part of the paper by Osher and
Sethian was the use of methods borrowed from hyperbolic conservation laws for
discretizing the level set equation Eq. 4.5. This concept was generalized in [103],
where numerical flux functions designed for hyperbolic conservation laws were
used to solve Hamilton-Jacobi equations of the form
∂φ
∂
t
+
H
(
∇
φ
)
=
0
.
(4.7)
Here, the function
H
(
∇
φ
) is called the Hamiltonian, and it is a function of the
gradient of
φ
. There is a rich history of numerical methods for hyperbolic con-
servation laws. An excellent review of numerical methods for hyperbolic con-
servation laws can be found in [75].
In the case of the level set method, the Hamiltonian is given by
H
(
∇
φ
)
=
F
∇
φ
.
(4.8)
A first-order numerical Hamiltonian for solving Eq. 4.7 is given by Godunov's
method, where
n
+
1
ij
n
ij
−
t
(max( sign(
F
ij
)
D
−
x
φ
n
ij
,
−
sign(
F
ij
)
D
+
x
φ
ij
,
0)
2
n
φ
=
φ
ij
ij
,
0)
2
)
1
/
2
+
max( sign(
F
ij
)
D
−
y
φ
,
−
sign(
F
ij
)
D
+
y
φ
(4.9)
.
