Biomedical Engineering Reference
In-Depth Information
2.2.1. Two-phase segmentation
For a bipartitioning problem, only one level set function is needed to represent
the boundaries and regions. We denote
R
1
as regions with negative values of the
level set function and
R
2
those with positive values of the level set function. Their
interface is Γ:
R
1
:
φ<
0
,
1
−
H
(
φ
)
,
|∇
H
(
φ
)
|
Γ:
φ
=0
,
,
R
2
:
φ>
0
,
(
φ
)
.
The energy functional of Eq. (3) incorporating region and edge information is
rewritten in terms of level sets as follows:
E
(
φ, θ
1
,θ
2
)=
λ
1
·
Ω
x
+
λ
2
·
Ω
D
1
·
(1
−
D
2
·
x
+
H
(
φ
))
d
H
(
φ
)
d
(10)
ν
·
g
(
|∇
I
|
)
|∇
H
(
φ
)
|
d
x
,
Ω
where
D
1
=
D
(
p
(
N
(
x
))
θ
1
)) is the J-divergence between the probability
distributions of region
R
1
and the local region.
D
2
=
D
(
p
(
N
(
x
))
p
(
R
1
|
p
(
R
2
|
θ
2
)).
λ
i
is the weight of the energy based in region
R
i
and
ν
is the coefficient of the
geodesic length constraint.
2.2.2. Multiphase segmentation
For multiple region segmentation, we follow the method introduced in [27],
where the authors present an elegant way of representing
n
regions with log
2
n
level set functions. Their multiphase level set model has no vacuum or overlap
among the phases, in contrast with other multiphase models, where one region
is represented by one level set function. Using the level set representation [11],
Eq. (3) can be rewritten as
s
E
(
φ
1
,φ
2
,...,φ
n
)=
λ
i
·
D
i
·
G
i
(
φ
1
,φ
2
,...,φ
n
)
d
x
Ω
i
=1
n
+
ν
j
·
g
(
|∇
I
(Γ
j
)
|
)
|∇
H
(
φ
j
)
|
d
x
,
(11)
Ω
j
=1
where
n
is the number of level set functions and
s
is the number of regions. They
satisfy
n
n
.
G
i
is the characteristic function representing the region
R
i
with
n
level set functions. They satisfy
G
i
(
x
)=1
,
≤
s
≤
2
x
∈
if
R
i
,
(12)
G
i
(
x
)=0
,
otherwise.
