Biomedical Engineering Reference
In-Depth Information
the energy function
F
(
c
1
,c
2
,C
) can be rewritten as
F
(
c
1
,c
2
,
Φ) =
µ
Ω
dxdy
+
ν
Ω
δ
(Φ(
x, y
))
|∇
Φ(
x, y
)
|
H
(Φ(
x, y
))
dxdy
+
λ
1
inside(
C
)
|
c
1
|
2
H
(Φ(
x, y
))
dxdy
u
0
(
x, y
)
−
+
λ
2
outside(
C
)
|
c
2
|
2
(1
−
u
0
(
x, y
)
−
H
(Φ(
x, y
)))
dxdy
(26)
where
c
1
(Φ) and
c
2
(Φ) are defined as
Ω
u
0
(
x, y
)
H
(Φ(
x, y
))
dxdy
Ω
c
1
(Φ)
=
,
H
(Φ(
x, y
))
dxdy
Ω
u
0
(
x, y
)(1
−
H
(Φ(
x, y
)))
dxdy
c
2
(Φ)
=
(27)
Ω
(1
−
H
(Φ(
x, y
)))
dxdy.
Finally, the corresponding level set equation that is minimizing the energy can
be solved by the following equation:
∂
∂t
=
δ
(Φ)[
µdiv
(
∇
Φ
c
1
)
2
+
λ
2
(
u
0
−
c
2
)
2
]
.
)
−
ν
−
λ
1
(
u
0
−
(28)
|∇
Φ
|
The level set equation could be solved iteratively using time step ∆
t
. However,
there are inherent time step requirements to ensure the stability of the numerical
scheme via the CFL condition. In Chan and Vese's approach, the time step can be
set based on the following equation:
min(∆
x,
∆
y,
∆
z
)
∆
t
≤
.
(29)
(
|
µ
|
+
|
ν
|
+
|
λ
0
+
λ
1
|
)
4.4.3. Applications of the region-based active contour
One example using the region-based active contour proposed by Chan and
Vese in 2001 is demonstrated Figure 15. The original image to be segmented was
badly noised, with the initial contour labeled in red and the final segmentation
based on Eq. (23). In this example, the initial contour was not strictly placed
inside or outside the object to be segmented. What is more, the 2D curvature was
taken as the approximation for div(
∇
Φ
|∇
Φ
|
).
The authors of [59]presented an implementation and validation of a 3D de-
formable model method based on Chan and Vese's energy functional for segmenta-
tion of 3D real-time ultrasound. The clinical study showed superior performance
of the deformable model in assessing ejection fraction (EF) when compared to
MRI measures. It also showed that the three-dimensional deformable model im-
proved EF measures, which is explained by a more accurate segmentation of small
