Biomedical Engineering Reference
In-Depth Information
usually based on formulations such as (11.9) [7-9,30,31,48-50] where all deriva-
tives are expanded to get curvature and advection terms. Then, e.g., Eq. (11.2)
for
ε
=
1 is written in the form
u
t
=
g
0
(1
+
u
x
2
)
u
x
1
x
1
−
2
u
x
1
u
x
2
u
x
1
x
2
+
(1
+
u
x
1
)
u
x
2
x
2
1
+
u
x
1
+
u
x
2
+
g
x
1
u
x
1
+
g
x
2
u
x
2
,
where
u
s
means partial derivative of a function
u
with respect to a variable
s
and
x
1
and
x
2
are spatial coordinates in the plane. In this form, it is not clear (reader
may try) which terms to take from previous and which on the current time
level, having in mind the unconditional stability of the method. Fully implicit
time stepping would lead to a difficult nonlinear system solution, so the explicit
approach is the one straightforwardly utilizable. In spite of that, the basic for-
mulation (11.2) leads naturally to convenient semi-implicit time discretization.
Let us recall the usual criterion on numerical schemes for solving partial dif-
ferential equations: numerical domain of dependence should contain physical
domain of dependence. In diffusion processes, in spite of advection, a value of
solution at any point is influenced by any other value of solution in a computa-
tional domain. This is naturally fulfilled by the semi-implicit scheme. We solve
linear system of equations at every time step which, at every discrete point, takes
into account contribution of all other discrete values in computational domain.
11.4 Discussion on Numerical Results
This section is devoted to the discussion on further numerical experiments
computed by the semi-implicit co-volume level set method. In Section 11.2 we
already discussed some examples which have been used mainly to illustrate
the advection-diffusion mechanism of the segmentation equation (11.2) and the
role of parameter
ε
in closing the gaps. In the sequel we will discuss the role of
additional model parameters as well as all aspects of our implementation. We
also compare the method with different approaches to confirm efficiency of our
numerical scheme.
For a given discrete image
I
0
with
n
1
,
n
2
, the number of pixels in the vertical
and horizontal directions, respectively, we define space discretization step
h
=
1
n
1
. It means, we embed the image into a rectangle [
−
0
.
5
n
n
1
,
0
.
5
n
n
1
]
×
[
−
0
.
5
,
0
.
5].
If one wants to use
h
=
1 (which would correspond to pixel size equals to 1),
