Biomedical Engineering Reference
In-Depth Information
The central point
φ
4
denotes the current pixel. Using the central difference
approximations, the derivatives of the level set function
φ
are defined as
D
x
=
φ
5
−
φ
3
)
/
2
D
y
=(
φ
7
−
φ
1
)
/
2
D
xx
=
φ
5
−
2
φ
4
+
φ
3
)
/
2
D
yy
=(
φ
7
−
2
φ
4
+
φ
1
)
/
2
D
xy
=
φ
2
−
φ
8
−
φ
0
+
φ
6
)
/
4
D
x
D
y
=
φ
5
−
φ
4
=
φ
7
−
φ
4
D
x
D
y
=
φ
4
−
φ
3
=
φ
4
−
φ
1
D
+
y
x
D
−y
x
=
φ
8
−
φ
6
)
/
2
=(
φ
2
−
φ
0
)
/
2
D
+
x
y
D
−x
y
=
φ
8
−
φ
2
)
/
2
=(
φ
6
−
φ
0
)
/
2
(24)
is obtained by [
D
x
+
D
y
]
1
/
2
. The individual terms of
T
1
and
T
2
are then calculated as:
The stretch of
|∇
φ
|
T
1
=
κ
[
D
x
+
D
y
]
1
/
2
,
(25)
where the curvature
κ
can be obtained by
D
xx
(
D
y
)
2
−
2
D
y
D
x
D
xy
+
D
yy
(
D
x
)
2
(
D
x
+
D
y
)
3
/
2
κ
=
,
(26)
and
T
2
can be calculated as
T
2
= max(
F
i,j
,
0)
∇
+
+ min(
F
i,j
,
0)
∇
−
,
(27)
where
∇
+
[max(
D
x
,
0)
2
+ min(
D
x
,
0)
2
+ max(
D
y
,
0)
2
+ min(
D
y
,
0)
2
]
1
/
2
=
∇
−
[max(
D
x
,
0)
2
+ min(
D
x
,
0)
2
+ max(
D
y
,
0)
2
+ min(
D
y
,
0)
2
]
1
/
2
(28)
=
To certify the stability of the evolution, the size of the time step is restricted
following the Courant Friedrich Levy (CFL) condition ∆
t
≤
1
/
max(
g
I
(
αT
1
+
βT
2
)) (Section 4.3). This requires that the curve cannot move more than one
grid point at each time step. Here the maximum is calculated in the whole image
domain.
4.
AUGMENTING THE SPEED FUNCTION
Many challenges in interface problems arise from the production of an ap-
propriate model for the speed term [4]. As mentioned before, the construction of
