Information Technology Reference
In-Depth Information
ʴ
ʵ
, all pixels of a distance (measured from the nearest point on the
boundary) greater than
Due to
are not considered in the energy optimization problem
which reduces the computational time of our problem (Narrow-banding effect).
After applying the gradient descent method [
26
], it is clear that:
ʵ
2
Z
X
h
i
d
d
dt
s
i
¼
r
t
t
f
r
s
i
A
d
e
ðU
p
; U
f
Þ
r
s
i
S
U
p
rU
X;
2
Z
h
i
d
d
dt
h
i
¼
d
e
ðU
p
; U
f
Þ
r
t
t
rU
p
r
h
i
A
X;
ð
58
Þ
X
2
Z
d
d
dt
T
i
¼
r
t
t
f
r
T
i
A
d
e
ðU
p
; U
f
Þ
rU
X;
X
h
x
; h
y
and
where s
i
2f
s
x
;
s
y
g
,
h
i
2
T
i
2fT
x
; T
y
g
of
the transformation
A. Regarding to the weighting coef
s, and similar to [
26
], the energy
function is a quadratic function of this weights, which leads to a closed-form when
the derivatives with respect to the weights are zeros:
cients w
n
'
W
w ¼
K;
ð
59
Þ
where
Λ
is a column vector of size N and
Ψ
is and N
×
N matrix. Their elements are
calculated as follows [
42
]:
Z
d
X;
d
e
ðU
p
; U
f
Þ
S
U
f
Uð
A)
t
U
i
Uð
A)
K
i
¼
½
ð
60
Þ
X
Z
t
d
d
e
ðU
p
; U
f
Þ U
j
ð
ÞUð
U
i
ð
ÞUð
W
ij
¼
A
A
Þ
A
A
Þ
X;
ð
61
Þ
X
[1, N]. Using unique training shapes (with variability not
identical) guarantees that
∀
(
i
,
j
)
∈
[1, N]
×
Ψ
is a positive de
nite matrix avoiding singularity.
2.6.3 Experimental Results
We tested our algorithmon 500 CT slices/25VBs which are obtained from15 different
patients. The goal is to segment the VB region correctly. The segmentation accuracy
and robustness of our framework are tested on the phantom named as the ESP as well
as the clinical datasets. All algorithms are implemented using Matlab
®
7.
1
