Biomedical Engineering Reference
In-Depth Information
Figure 11.16: Isolines of the segmentation function in the segmentation of the
noisy circle (Fig. 11.12 (right)) are shown in time steps 0, 50, 100, and 200. Since
the gap is not so big we have chosen
ε
=
10
−
1
(color slide).
n
=
1
,...,
N, we look for a function u
n
, solution of the equation,
g
0
u
n
−
u
n
−
1
τ
∇
u
n
|∇
u
n
−
1
1
|∇
u
n
−
1
(11.11)
=∇·
.
|
|
A digital image is given on a structure of pixels with rectangular shape, in
general (red rectangles in Fig. 11.18). Since discrete values of
I
0
are given in
pixels and they influence the model, we will relate spatially discrete approxi-
mations of the segmentation function
u
also to image pixels, more precisely, to
their centers (red points in Fig. 11.18). In every discrete time step of the method
(11.11), we have to evaluate gradient of the segmentation function at the previ-
ous step
|∇
u
n
−
1
|
. For that goal, it is reasonable to put a triangulation (dashed
black lines in Fig. 11.18) inside the pixel structure and take a piecewise linear
approximation of the segmentation function on this triangulation. Such an ap-
proach will give a constant value of the gradient per triangle, allowing simple
and clear construction of fully discrete system of equations. This is the main
feature of the co-volume [25, 56] and finite element [13-15] methods in solving
mean curvature flow in the level set formulation.
