Biomedical Engineering Reference
In-Depth Information
Figure 6.
Numbering scheme for derivative computation.
3.4. Numerical Implementation
In image segmentation case, the previous discussion in Section 3.2.2 shows
that the speed function usually includes: (1) a curvature-based speed term
F
(
κ
);
(2) a constant speed term (
F
A
), where the speed can be expressed as a function
F
A
(
x, y
) of the location (
x, y
); (3) a stopping criterion
g
I
based on the image gra-
dient to extract the boundary (Eq. (21)). The level set partial differential equation
for this curve evolution is:
∂φ
∂t
=
g
I
(
ακ
|∇
φ
|
+
βF
A
(
x, y
)
|∇
φ
|
)
,
(22)
where
α
and
β
are constants;
F
A
is a constant speed term;
g
I
is also constant
during the propagation.
Consider the implementation of this PDE within the image domain. First,
define the function
φ
(
t
) in the narrow band area. We use the Euclidean distance
from point (
x, y
) to the zero level set as the value of the signed distance. The sign
is negative if the point is inside the curve, and positive if (
x, y
) is outside the curve.
All the calculations are carried out only within the narrow band area.
The evolution equation can now be represented as
φ
(
t
+∆
t
)=
φ
(
t
)+
g
I
(
αT
1
+
βT
2
)∆
t.
(23)
First, we describe the finite-difference derivatives required for level set update and
curvature computation in
T
1
and
T
2
. For the convenience of representation of
derivatives, the neighborhood of
φ
is specified with the numbering scheme shown
in Figure 6.
