Biomedical Engineering Reference
In-Depth Information
Four evolution equations are derived for the Euler-Lagrange system for tem-
poral derivatives of the level set function
φ
and the rigid registration parame-
ters
(
µ,
R
,
T
)
with detailed numerical implementation described in [68]. In this
paper the authors also report on experiments performed with this method to
segment the endocardial borders of the left ventricle on an ultrasound image
and segment the corpus callosum on misaligned functional MRI images in a time
series.
In [60], the authors proposed a second functional for combining registration
and segmentation in an implicit deformable model framework where the image
gradient term is replaced by an homogeneity measure. Their approach is derived
from the Mumford-Shah functional [36] in a similar fashion as described in the
previous section of region-based level set methods. They proposed the following
functional:
H
(
φ
)
I
−
S
+
2
+
α
(1
−
H
(
φ
))
I
−
S
−
2
E
(
φ,
c
0
,
c
1
,µ,
R
,
T
)
=
α
(2.33)
+
β
H
(
φ
)
∇
S
+
+
β
(1
−
H
(
φ
))
∇
S
−
+
δ
(
φ
)
d
2
(
µ
Rx
+
T
)
|∇
φ
|
d
with
S
+
,
S
−
smooth approximations of the image
I
on, respectively,
{
x
/φ
(
x
)
>
0
}
and
{
x
/φ
(
x
)
<
0
}
and (
α,β
) a set of positive parameters.
In a similar effort to combine registration and segmentation Paragios
et al.
[69] proposed a level-set approach for knowledge-based registration and seg-
mentation of the left ventricle. In their method a level set framework was used
to perform simultaneous segmentation of the epicardial and the endocardial sur-
faces of the myocardium muscle via coupling two level set functions (
φ
0
,φ
1
).
The proposed functional is generalized as:
E
(
φ
0
,φ
1
,
A
0
,
A
1
)
=
α
E
G
(
φ
0
,φ
1
)
+
β
E
R
(
φ
0
,φ
1
)
+
γ
E
A
(
φ
0
,φ
1
)
(2.34)
+
δ
E
S
(
φ
0
,φ
1
,
A
0
,
A
1
)
.
This functional integrates four components:
1.
E
G
is a constraint on the regularity of the contour via minimization of its
length.
2.
E
R
is an intensity-based region component. This component identifies the
partition of the image into regions that maximizes the posterior segmen-
tation probability given a priori gray level histogram distribution for the
endocardium, the epicardium and the background;
