Biomedical Engineering Reference
In-Depth Information
level a corresponding signed distance level set function is initialized for each
window. After segmenting the volume, a connectivity filter [63] is used to exploit
the fact that the vascular system is a tree-like structure and makes use of the
3D computer graphics region-filling algorithm to extract the vascular tree. The
used algorithm with MRA data volumes is evaluated using a phantom, showing
a good accuracy. The algorithm is applied to different types of MRA data sets,
showing good results. This approach can be extended to be not dependent only
on the gray level, but also on the geometrical features of the segmented areas,
leading to more accuracy.
9.6.1 Level Sets and Segmentation
Consider an image of
c
classes. We assign a level set function
φ
i
for each class.
From the definition of the level set function in Eq. 9.49,
is the class and
is
the interface of the class. Class interface denotes the boundary line between the
class and the other classes. From the following equations,
F
1
is the partitioning
condition as follows,
c
F
1
=
λ
i
2
(
H
α
(
φ
i
)
−
1)
2
dx
,
where
λ
i
∈
R
+
,
∀
i
∈
[1
,
c
]
.
(9.69)
i
=
1
The partitioning condition penalizes the vacuum points and prevents the over-
lapping between regions:
e
i
c
H
α
(
φ
i
)
(
u
0
−
u
i
)
2
σ
i
2
F
2
=
dx
,
where
e
i
∈
R
,
∀
i
∈
[1
,
c
]
.
(9.70)
i
=
1
F
2
is the data term condition with mean
u
i
and variance
σ
i
2
where
u
0
is the data
value.
i
=
1
γ
i
c
F
3
=
δ
α
(
φ
i
)
|∇
φ
i
|
dx
,
where
γ
i
∈
R
,
∀
i
∈
[1
,
c
]
.
(9.71)
F
3
is the sum of interfaces length between classes. The summation
F
1
+
F
2
+
F
3
is minimized with respect to
φ
to get the following equation:
K
−
γ
i
div
∇
φ
i
t
|
φ
i
t
e
i
(
u
0
−
u
i
)
2
σ
i
2
φ
i
t
+
1
=
φ
i
t
−
t
δ
α
(
φ
i
t
)
H
α
(
φ
i
t
)
−
1
+
λ
i
.
|
i
=
1
(9.72)
