Biomedical Engineering Reference
In-Depth Information
inner edges of the circle) are given by two large gaps in histogram. In the top
right there is a histogram of the segmentation function given by the explicit
scheme after 500 time steps, and then after 1000 (bottom left) and 5000 (bottom
right) time steps. We see that, due to necessity of small time step, the formation
of the piecewise flat solution is very slow for explicit scheme. Although after
1000 time steps one can see the formation of two gaps which could be already
used for detection of “final” segmentation result, the CPU time for 1000 steps
of explicit scheme is 49.5 sec, which is ten times longer than for semi-implicit
scheme. If we would like to obtain a similar histogram as plotted in the top left
using an explicit scheme, we would need 100 times longer CPU time as in the
case of semi-implicit scheme.
In all computations presented above, we have used
g
(
s
)
=
1
1
+
Ks
2
,
K
=
1. In
experiments without noise there is no significant difference by changing
K
.We
get the same behavior of the method changing
K
from 0.1 to 10. It is understand-
able because the function
g
plays a role only along edges and its more (
K
>
1) or
less (
K
<
1) quickly decreasing profile governs only speed by which level sets
of solution are attracted to the edge from a small neighborhood. Everywhere
else only pure mean curvature motion is considered (
g
=
1).
The situation is different for noisy images, e.g., depicted in Fig. 11.12 (right)
and Figs. 11.16 and 11.17. The extraction of the circle in noisy environment takes
a longer time (200 steps with
τ
=
0
.
01 and
K
=
1) and it is even worse for
K
=
10.
However, decreasing the parameter
K
gives stronger weight to mean curvature
flow in noisy regions, so we can extract the circle fast, in only 20 steps with the
same
τ
=
0
.
01. In the case of noisy images, also the convolution plays a role. For
example, if we switch off the convolution, the process is slower. But decreasing
K
can again improve the speed of segmentation process. In our computations
we either do not apply convolution to
I
0
or we use image presmoothing by
m
×
m
pixel mask with weights given by the Gauss function normalized to unit sum.
We start all computations with initial function given as a peak centered in
a “focus point” inside the segmented object, as plotted, e.g., in Fig. 11.10 (top
left). Such a function can be described for a circle with center
s
and radius
R
by
u
0
(
x
)
=
1
|
x
−
s
|+
v
1
v
gives maximum of
u
0
.
, where
s
is the focus point and
1
R
+
v
Outside the circle we take value
u
0
. If one needs zero Dirichlet
boundary data, e.g., due to some theoretical reasons (cf. [11, 49]), the value
1
R
+
v
can be subtracted from the peak-like profile. If the computational domain
corresponds to image domain, we use
R
=
0
.
5. For small objects a smaller
R
equal to
