Biomedical Engineering Reference
In-Depth Information
snakes with time is guided by differential equations. These equations are de-
rived from the energy minimization concept to describe the change of snakes
with time. The output obtained using snakes depends highly on the initializa-
tion. It was found that initial curve has to lie close to the final solution to obtain
required results. The initialization is relatively easy in the case of 2D images but
in the 3D case it is very difficult. Also the topology change of the solution needs
a special regulation to the model.
Level sets were invented to handle the problem of changing topology of
curves. The level sets has had great success in computer graphics and vision.
Also, it was used widely in medical imaging for segmentation and shape re-
covery. It proved to have advantages over statistical approaches followed by
mathematical morphology. In the following section we will give a brief overview
on level sets and its application in image segmentation.
9.5.1 Level Set Function Representation
Level sets was invented by Osher and Sethian [52] to handle the topology changes
of curves. A simple representation is that a surface intersects with the zero plane
to give the curve. When this surfaces changes the curve changes. The surface
can be described by the following equation:
φ
(
x
,
t
)
>
0if
x
∈
,
φ
(
x
,
t
)
<
0if
x
∈
,
and
φ
(
x
,
t
)
=
0if
x
∈
,
(9.49)
where
φ
represents the surface function,
denotes the set of points where
the function is positive, and
represents the set of points at which the func-
tion is zero. In Fig. 9.21, an example of a surface and its intersection with the
zero plane is shown. This intersection is called the front. The surface changes
with time, resulting in different fronts. So the level set function is positive at
some points, negative at other points, and zero at the front
. The time as ex-
tra dimension is added to the problem to track the changes of the front. The
topology changes of the curve are handled naturally by this presentation as we
see from Fig. 9.22. The first row represents the surface and the zero plane at
different time samples and the second row represents the resulting curves. The
front is initially two ellipses, then the two ellipses merge to make a closed curve
and it changes and so on. This representation allows the front to merge and
break.
