Biomedical Engineering Reference
In-Depth Information
2.2.3 Selecting an HSC Segmentation Energy
In (
6
), the segmentation problem has been reduced to providing an initialization
c
0
ð
within the region of convergence ; however, both
c
0
and
e
are unknown.
From the discussion of Sect.
2.1
, we recognize that the energy used for segmenta-
tion must be local since the organ (in our case, bone) has statistics that change
drastically spatially. While it is true that intensities of an organ's tissue do not fall
in a narrow range globally, locally, the bone appears approximately homogeneous
with a clear separation between the object of interest and the background; hence, we
select a local variant of the Chan-Vese energy as presented in [
5
] and segment
based on first moments. An added benefit of using this local energy is that user input
in one region will have no effect on a remote region of the contour, which is
intuitive for the user. The functional presented in [
5
] has been modified to incorpo-
rate the novel HSC segmentation term (the 2nd integral in the functional), which
makes the dynamics of the evolving curve depend on the user input as it is being
provided interactively. The level set formulation is presented in (
11
)-(
15
).
x
Þ
Z
Z
min
c
E
cv
ðcÞ¼
dcð
x
Þ
F
ðcð
y
Þ;
I
Þ
dydx
O
x
O
y
Z
K
U
2
2
U
2
dx
(11)
þ
O
ðc
U
Þ
þ l
Z
2
dx
O
jjrHðcÞjj
(12)
2
2
F
¼Bð
x
;
y
Þ Hðcð
y
ÞÞ ð
I
ð
y
Þ
u
l
Þ
þð
1
Hðcð
y
ÞÞÞ ð
I
ð
y
Þ
v
l
Þ
R
O
y
H cð
ð
y
Þ
Þ
I
ð
y
Þ
dy
R
O
y
H cð
u
l
¼
(13)
ð
y
Þ
Þ
dy
R
O
y
ð
1
Hc y
ð
ðÞ
Þ
Þ
I
ð
y
Þ
dy
R
v
l
¼
(14)
ð
1
Hc y
ð
ðÞ
Þ
Þ
dy
O
y
(
1
if
jj
x
y
jj<
r
Bð
x
;
y
Þ¼
(15)
0
if
other
:
Here,
c
is the level set function,
I
is the image,
K
U
is the gain (scalar) on the user
control term,
U
(
9
) is the integral of the user input,
l
is the curvature penalty term,
and
(
2
) and
d
(
x
)(
3
) are smooth approximations to the Heaviside and the Dirac
delta functions, respectively. Using the calculus of variations, we find the dynamics
equation, (
16
), to minimize the functional in (
11
).
Hð
x
Þ
