Biomedical Engineering Reference
In-Depth Information
2.2.2 General HSC Segmentation Framework
We now present an HSC framework for performing segmentation of medical
images, shown graphically in Fig.
4
. Generally, the active contour segmentation
problem can be written as:
Z
þ l
Z
2
dx
min
c
E
ðcÞ¼
g
ðc;
I
Þ
dx
O
jjrHðcÞjj
(1)
O
8
<
1
;
x
>E
0
;
x
< E
Hð
x
Þ¼
(2)
:
1
2
x
E
þ
1
p
px
E
1
þ
sin
;
other
8
<
1
;
x
¼
0
0
;
j
x
j<E
dð
x
Þ¼
(3)
:
1
2
E
px
E
1
þ
cos
;
other
:
in
(
5
) is the curvature, and
e
is a scalar that determines the smoothness of the
approximation to the Heaviside function,
Here,
c
is the level set function,
I
the image data,
l
the curvature penalty,
k
, and the approximation to the Dirac
delta function,
d
. The evolution equation for the level set function is
H
c
t
¼ dðcÞ
g
c
ðc;
I
Þþlk
¼
G
ðc;
I
Þ
(4)
k ¼r
rc
jjrcjj
:
(5)
If we define the zero level set of
c
∗
to be the ground truth segmentation result, a
“good” function
g
(
c
,
I
) is one that will cause
c
to converge to
c
∗
when initialized
within some small region around
c
∗
(i.e.,
c
∗
is a local minimum of
E
(
c
)). In other
words, a
g
(
c
,
I
) is “good” if there is some
c
0
that converges to
c
∗
and for some
positive function
e
satisfies
c
ð
x
ÞEð
x
Þc
0
ð
x
Þc
ð
x
ÞþEð
x
Þ
(6)
with
e
, defining the region of convergence. Unfortunately, both
c
and
e
are unknown,
and the function
g
(
c
,
I
) may have certain exceptional regions where it is not
discriminative between the object and background. To remedy these problems, the
user is present in the segmentation process to guide the automatic algorithm.
