Biomedical Engineering Reference
In-Depth Information
functional:
inside
C
|
I
−
C
0
|
outside
C
|
I
−
C
1
|
2
d
+
2
d
,
E
(
C
)
=
(2.28)
where (
C
0
,
C
1
) are the average intensity values of
I
inside and outside the curve
C
. With this functional, the boundary between the two regions is defined by its
minimum state. They further combined this homogeneity-based fitting term with
regularizing terms that put constraints on the length and the area of the curve
with the following functional:
inside
C
|
I
−
C
0
|
2
d
+
λ
1
2
d
E
(
C
0
,
C
1
,
C
)
=
λ
0
outside
C
|
I
−
C
1
|
(2.29)
+
µ
length(
C
)
+
µ
Area(
C
)
.
Details for the mathematical definitions of the length and the area of the bound-
ary curve
C
can be found in [40].
In a level-set framework implementation, the functional (2.29) is expressed
as:
E
(
C
0
,
C
1
,φ
)
=
λ
0
2
H
(
φ
)
d
+
λ
1
2
(
1
−
H
(
φ
))
d
|
I
−
C
0
|
|
I
−
C
1
|
H
(
φ
)
d
(2.30)
δ
(
φ
)
|∇
φ
|
d
+
ν
+
µ
Advantages of this method include the possibility of segmenting objects with
discontinuous edges and robustness of the method to arbitrary initialization,
avoiding the problem of local minima at spurious edge locations or leakage of
the model at missing edge locations. The initial work from these authors have
generated many applicative research works for segmentation of medical images,
starting with works from the authors themselves in [37] with illustration of their
method on brain MRI, three-dimensional ultrasound.
A simultaneous and parallel effort to the work of Chan and Vese, from Tsai
et al.
[42] proposed a reformulation of the Mumford-Shah functional from a
curve evolution perspective using a gradient flow formulation and a level set
framework implementation. Recent works applying this segmentation method
to three-dimensional cardiac ultrasound include Angelini
et al.
[43], and Lin
et al.
[44].
We note two powerful extensions of this region-based implicit deformable
model for applications to medical images:
