Biomedical Engineering Reference
In-Depth Information
all considerations can be changed accordingly. We prefer the above definition
of spatial discretization step, because it is closer to standard approaches to
numerical solution of PDEs.
First we give some CPU times overview of the method. Since we are inter-
ested in finding a “steady state” (see discussion in Section 11.2) of the evolution
in order to stop the segmentation process, the important properties are the
number of time steps needed to come to this “equilibrium” and a CPU time for
every discrete time step. We discuss CPU times in the experiment related to
segmentation of the circle with a gap given in Fig. 11.12 (left), computed using
ε
=
10
−
2
(see Fig. 11.14 (top left)). The testing image has 200
×
200 pixels and
the computational domain
corresponds to the whole image domain. Since
for the boundary nodes we prescribe Dirichlet boundary conditions, we have
M
=
198
×
198 degrees of freedom. As the criterion to recognize the “steady
state,” we use a change in L
2
norm of solution between subsequent time steps,
i.e., we check whether
(
u
p
−
u
n
−
1
h
2
)
2
<δ
p
p
with a prescribed threshold
δ
. For the semi-implicit scheme and small
ε
(then the
downwards motion of the “steady state” is very slow) a good choice of threshold
is
δ
=
10
−
5
.
Reasonable time steps for our semi-implicit method are of order (10
h
)
2
, e.g.,
for the discussed example very good results regarding CPU times and precision
have been obtained for
τ
∈
[0
.
001
,
0
.
01]. Since by a classical criterion the pre-
cision of numerical schemes for parabolic equations is optimal for
τ
≈
h
2
,we
have also computed such a case. But, no significant difference due to precision
has been observed, only much longer CPU time was necessary. In our example
τ
=
5
×
10
−
3
and 20 time steps yield the segmentation result (using threshold
δ
=
10
−
5
). On 2.4 GHz Linux PC, the overall CPU time for this segmentation was
4.93 sec (i.e., approximately 0.25 sec for one time step including construction
of coefficients and solving the linear system). This CPU time was obtained with
TOL
=
10
−
3
. Since we are mainly interested in “equilibrium,” one can also decide
that such precision is not necessary in every discrete time step. With increasing
TOL fewer numbers of SOR iterations are needed. Another way is to prescribe
a fixed number (but not too small) of iterations in every time step, e.g., ten
