Biomedical Engineering Reference
In-Depth Information
Figure 9.23: (a) The plot of the heaviside and delta functions for a specific
value of
α
, (b) the narrow band points, (c) the level set function, (d) applying
the heaviside step function, and (e) applying the delta function.
In Fig. 9.23(a), the two functions are plotted for
α
=
0
.
5. The value of
α
is always
taken to be 1
.
5
x
to make the band equal to 3
x
where
x
is the mesh size,
which is always 1. The enclosed area (
A
) and the length of the interface or front
(
L
) are calculated as follows:
A
=
H
α
(
φ
)
dxdy
,
(9.65)
D
D
δ
α
(
φ
)
|∇
φ
|
dxdy
,
L
=
(9.66)
where
D
is the domain. A proof of Eq. 9.66 to be the length of the front is found
in [53].
In Fig. 9.23(b), the red line represents the front and the yellow area represents
the points around the front where this area is called the narrow band. In (c), (d),
and (e) an example of a level set function and application of the heaviside step
and delta functions are shown.
9.5.5 Narrow Banding and Speed of the Solution
Solving the PDE of the level set function requires numerical processing at each
point of the image domain which is a time consuming process. Only we are
