Biomedical Engineering Reference
In-Depth Information
intervention. The skull-stripped brain is then fed into the first stage for WM/GM
segmentation, followed by GM/CSF segmentation in the next stage.
In the first stage, WM/GM segmentation, we propose a unified variational
formula that integrates region and boundary information. Specifically, We modify
the Chan-Vese model [58] to describe regional homogeneity of brain tissues, and
present a new maximum a posteriori (MAP) framework with which to statistically
detect coupled boundaries. The zero level set, which represents the WM/GM
surface, is initiated in the WM and evolves until it reaches the WM/GM boundary.
In the second stage the WM/GM surface obtained in the first stage moves out
from the WM/GM boundary to fit the GM/CSF boundary. The prior information
of approximately constant thickness is also utilized as a constraint.
5.2.1. Define region information
For the sake of clarity of explanation, we first review the original Chan-Vese
model [58], which was proposed for segmenting bimodal objects.
LetΩ
⊂
R
2
be a bounded and open region,
C
its surface represented by the
level set function Φ such that
>
0,if
x
is inside
C,
Φ(
x
)
=0,if
x
is on surface
C,
<
0
,
if
x
is outside
C.
Thus, the Mumford-Shah model can be formulated as the minimization of the
following functional:
E
=
λ
1
Ω
|
c
1
|
2
H
(Φ (
x
))
d
I
0
(
x
)
−
x
+
λ
2
(26)
c
2
|
2
(1
−
Ω
|
I
0
(
x
)
−
H
(Φ (
x
)))
d
x
+some regularizing terms,
where
H
is the Heaviside function,
c
1
and
c
2
are the average intensities inside and
outside
C
, respectively, and
λ
1
and
λ
2
are two weighting parameters. The first and
second terms encode the intensity variations inside and outside
C
, respectively.
This energy functional is thus minimized at the point where both regions inside and
outside the surface are most uniform, namely, the boundary of these two regions.
By taking the Euler-Lagrange equation of (26) and minimizing it using the
gradient descent method, we obtain the following governing partial differential
equation evolving the surface to the minimum of energy functional:
∂
∂t
c
1
)
2
c
2
)
2
=
−
λ
1
(
I
0
(
x
)
−
λ
2
(
I
0
(
x
)
−
δ
(Φ)
inside force
outside force
+some regularizing terms.
(27)
