Biomedical Engineering Reference
In-Depth Information
Figure 11.9:
Image of a solid circle.
point-of-view surface
u
0
is plotted on the top left. The subsequent evolution is
depicted in the next subfigures. First, isolines which are close to the edge, i.e.
in the neighborhood of the solid circle where the advection term is nonzero,
are attracted from both sides to this edge. A small shock (steep gradient) is
formed due to accumulation of these level lines (see Fig. 11.10 (top right)).
In the regions outside the neighborhood of the circle, the advection term is
vanishing and
g
0
≡
1, so only intrinsic diffusion of level sets plays a role. This
means that all inside level sets are shrinking and finally they disappear. Such
a process is nothing else but a decrease of the maximum of our segmenta-
tion function until the upper level of the shock is achieved. It is clear that a
flat region in the profile of segmentation function inside the circle is formed.
Outside of the circle, level sets are also shrinking until they are attracted by
nonzero velocity field and then they contribute to the shock. In the bottom left
of Fig. 11.10, we see the shape of segmentation function
u
after such evolution,
in the bottom right there are isocontours of such function accumulated on the
edges. It is very easy to use one of them, e.g., (max(
u
)
+
min(
u
))
/
2, to get the
circle.
The situation is not so straightforward for the highly nonconvex image de-
picted in Fig. 11.11. Our numerical observation leads to formation of steps in
subsequent evolution of the segmentation function, which is understandable,
because very different level sets of initial surface
u
0
are attracted to different
parts of the boundary of “batman.” Fortunately, we are a bit free in choosing
the precise form of diffusion term in the segmentation model. After expansion
of divergence, Eqs. (11.2) and (11.8) give the same advection term,
∇
g
0
·∇
u
(cf.
Eq. (11.9)), so important advection mechanism which accumulates segmenta-
tion function along the shock is the same. However, diffusion mechanisms are a
