Biomedical Engineering Reference
In-Depth Information
follows:
κ
=(
φ
xx
+
φ
yy
)
φ
z
+(
φ
xx
+
φ
zz
)
φ
y
+
φ
zz
+
φ
yy
)
φ
x
−
2
φ
x
φ
y
φ
xy
−
2
φ
x
φ
z
φ
xz
−
2
φ
z
φ
y
φ
zy
)
/
(2(
φ
x
+
φ
y
+
φ
z
)
3
/
2
)
.
(8)
With this representation, a single level set either contracts until vanishing or ex-
pands to cover all the space. To stop the evolution at the edge,
F
can be multiplied
by a value that is a function of the image gradient [38]. However, if the edge
is missed, the surface cannot propagate backward. Hence, relying mainly on the
edge is not sufficient for an accurate segmentation and other information from the
image should be used.
The segmentation partitions the image into regions, each belonging to a certain
class. In our approach a separate level set function is defined for each class and
automatic seed initialization is used. Given parameters of each class, the volume
is initially divided into equal non-overlapped sub-volumes. For each sub-volume,
the average gray level is used to specify the most probable class with the initial
parameters estimated by SEM. Such initialization differs from the one in [39],
where only the distance to the class mean is used. Then, a signed distance level
set function for the associated class is initialized. Therefore, selection of the
class parameters is very important for successful segmentation. The probability
density function of each class is embedded into the velocity term of each level set
equation. The parameters of each of these density functions are re-estimated at
each iteration. The automatic seed initialization produces initially non-overlapped
level set functions. The competition between level sets based on the probability
density functions stops the evolution of each level set at the boundary of its class
region.
Tracking the curve/surface with time needs a solution of the associated partial
differential equations. Dealing with non-smooth data represents a challenge in this
problem. More sophisticated numerical techniques are required. The details of
the numerical solution used for getting the front at any time are presented below.
4.2.5. Numerical implementation for interface evolution
A standard numerical approach to modeling moving fronts results from dis-
cretizing the Lagrangian description of the problem (Eq. (7)) with a set of discrete
marker particles lying on the moving front and whose positions at any time are
used to reconstruct the front. This technique, known as the
marker particle tech-
nique
or the
strings method
, has several drawbacks, including amplification of the
errors in the computed particle positions due to the curvature term. The absence
of a smoothing curvature (
viscous
) term leads to development of singularities in
the propagating front. In addition, managing the topological changes becomes
very complex as the front breaks and/or merges. Finally, bookkeeping of remov-
