Biomedical Engineering Reference
In-Depth Information
(9.59)
φ
xx
(
x
,
y
,
t
)
=
(
φ
(
x
+
2
x
,
y
,
t
)
−
2
φ
(
x
,
y
,
t
)
+
φ
(
x
−
2
x
,
y
,
t
))
/
(2
x
2
)
,
(9.60)
φ
yy
(
x
,
y
,
t
)
=
(
φ
(
x
,
y
+
2
y
,
t
)
−
2
φ
(
x
,
y
,
t
)
+
φ
(
x
,
y
−
2
y
,
t
))
/
(2
y
2
)
.
(9.61)
φ
y
(
x
,
y
,
t
)
=
(
φ
(
x
,
y
+
y
,
t
)
−
φ
(
x
,
y
,
t
))
/
y
,
There are different numerical techniques used for this problem and the details
are given in [52]. The solution is very sensitive to the time step. Time step is
selected based on the Courant-Friedrichs-Levy (CFL) restriction. It requires the
front to cross no more than one grid cell at each time step
t
. This calculation
will give the maximum time step that guarantees stability. From Eq. 9.62, we
maximize the denominator and minimize the nominator to get the best value
of the time step. The time step is calculated at each iteration of the process to
maintain the stability of the solution:
y
)
1
/
2
F
(
|
φ
x
|
/
x
+|
φ
y
|
/
y
)
x
(
φ
+
φ
t
≤
(9.62)
9.5.4 Tracking the Front
Now, the solution is to find the front iteratively at different time steps. We get the
front by intersecting the surface with the zero plane. We need to track this front
by getting the length of the front or getting the area enclosed. This information is
very important in the segmentation problem as we will see in the next sections.
Simply the enclosed area contains all the points at which the level set function
is greater than or equal to zero and the points of the front are the points at which
the level set function is zero. Applying the heaviside step and delta functions
is very useful in getting the area and the front respectively. For numerical im-
plementation, it is desirable to replace the heaviside and the delta functions by
some counterparts. Approximations of these two functions are used to handle
smoothness problem as follows:
1
,
if
|
φ
|
>α
H
α
(
φ
)
=
(9.63)
,
0
.
5(1
+
α
+
1
π
sin(
π
α
))
if
|
φ
|≤
α
0
,
if
|
φ
|
>α
δ
α
(
φ
)
=
(9.64)
.
1
2
α
(1
+
cos(
π
α
))
,
if
|
φ
|≤
α
