Biomedical Engineering Reference
In-Depth Information
Now the question becomes,
What is the minimum number of colors, thus the
minimum number of level sets needed?
The
Four-Color Theorem
(FCT), states
that every planar graph is four-colorable while insuring that neighboring vertices
do not share the same color [29--31]. Since our neighborhood graph is a planar
graph, the
N
--level set formulation of Zhang et al. [25] can be effectively reduced
to a 4-level set formulation. Assignment of
N
objects to
k
N
level sets (
k
=
4 for our case) is justified by the assumption that objects (in our case cells) have
similar, or nearly similar characteristic features. In this case objects/cells can be
randomly assigned to any of the four level sets. For a wide variety of applications
(including our application of cell segmentation), the upper limit of the chromatic
number for planar graphs is four. However, we do acknowledge that there may
exist other planar graphs having even smaller chromatic numbers.
In order to evolve the four level sets, we propose minimizing the
N
<<
−
coupled
level sets energy functional shown in (3.17), with
N
4. In addition, we replace
the length minimizer term [third term in (3.17)] with its geodesic equivalent, and in-
corporate additional constraints in our proposed energy functional
E
fls
(
=
c
in
,
c
out
,
F
)
as shown below:
4
i
=
1
c
in
)
2
H
E
fls
(
c
in
,
c
out
,
F
)=
m
in
W
(
I
−
(
f
i
)
d
y
Homogeneity of Foregrounds
m
out
4
i
=
1
4
i
=
1
2
−
−
W
|
)
∇
f
i
)
|
+
W
(
I
c
out
)
(
1
H
(
f
i
))
d
y
+
n
g
(
I
H
(
d
y
∀
i
:
H
(
f
i
)
<
0
Geodesic Length Minimizers
Homogeneity of Background
h
4
2
d
y
4
i
=
1
4
å
1
2
(
|∇
i
=
1
|−
+
g
H
(
f
i
)
H
(
f
j
)
d
y
+
f
i
1
)
(3.18)
W
W
j
=
i
+
1
Implicit Redistancing
Energy-based Coupling
where
g
(
I
)
is an edge indicator function given by [32]
a
g
(
I
)=
1
+
b
|∇
G
s
I
|
. |∇
|
with constants a
can also be replaced with a regularized version of
our adaptive robust structure tensor (ARST) [33]. The last term in (3.18) enforces
the constraint of
,
b
G
s
I
|∇
|
=
1, thus helping us avoid explicit redistancing of level sets
during the evolution process [34], with
h being a constant. As noted by Kim and
Lim, addition of a geodesic length term improves the performance of an energy-
only Chan and Vese level set formulation by preventing detection of regions with
weak edge strengths in [35]. This property was also exploited by Dufour et al. in
3D cell segmentation [27].
Gradient-descent minimization can be used to iteratively solve the level set
functions f
i
as
f
i
Dt
¶f
i
(
k
)
f
i
(
k
+
1
)=
f
i
(
k
)+
(3.19)
¶
k
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