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a
. In this case,
b
and
a
correspond to f
and J, respectively. The transformation can
be written for any point X in the space as T
ð
x
Þ
¼X ¼ Sx
þ
Tr. Now consider
x
2 /
J
and X
2 /
f
.
The registration of the shape model and testing image is done as follows:
(i) First, the average of the position factor (
l
f
) and scale factor (
r
f
) are
obtained using the following equations
T
P
X
xH
ð
/
f
ð
x
ÞÞ
P
X
yH
ð
/
f
ð
x
ÞÞ
h
i
l
f
l
f
x
l
f
y
P
X
H
ð/
f
ð
x
ÞÞ
P
X
H
ð/
f
ð
x
ÞÞ
¼
¼
;
ð
21
Þ
T
P
X
ð
x
l
f
x
P
X
ð
y
l
f
y
h
i
2
H
ð/
f
ð
x
))
2
H
ð/
f
ð
x
))
Þ
Þ
r
f
¼
ðr
f
x
Þ
2
ðr
f
y
Þ
2
P
X
H
ð/
f
ð
x
))
P
X
H
ð/
f
ð
x
))
¼
:
ð
22
Þ
(ii) Obtain the transformation parameters (t
x
;
t
y
;
s
x
;
s
y
) for the shape model,
/
J
,
as
P
X
xH
ð
/
J
ð
x
ÞÞ
P
X
yH
ð
/
J
ð
x
ÞÞ
T
T
l
f
x
P
X
H
ð/
J
ð
x
ÞÞ
l
f
y
P
X
H
ð/
J
ð
x
ÞÞ
Tr
¼
½
t
x
t
y
¼
;
ð
23
Þ
2
3
T
r
f
x
r
P
X
ð
x
l
f
x
Þ
2
H
ð
/
J
ð
x
ÞÞ
0
4
5
P
X
H
ð/
J
ð
x
ÞÞ
s
x
0
S
¼
¼
r
f
x
0
s
y
r
0
s
y
¼
P
X
ð
y
l
f
x
Þ
2
H
ð/
J
ð
x
ÞÞ
P
X
H
ð/
J
ð
x
ÞÞ
ð
Þ
24
(iii) Transform each point x
to the new point X. Hence, the shape model is
registered to the image domain.
(iv) The new probabilistic function at each pixel
2 X
.
Hence, the new transformed pixels will have the same probabilistic value
with corresponding pixels. An example of the registration and
is p
ð
d
X
j
f
X
Þ
¼p
ð
d
x
j
f
x
;
T
Þ
final seg-
mentation results are shown in Figs.
15
and
16
.
2.4.6 Final Energy Minimization Using Three Models: Intensity,
Spatial Interaction, and Shape
Three probabilistic models are used. Before this step, the followings are obtained
already (i) the initial labeling f
that maximizes p
f
Þ
ð
I
j
, (ii) the MGRF model for
f
Þ
f
;
p
ð
, and (iii) the transformed shape prior to maximize pðfÞ,
ð
d
j
T
Þ
. It should be
noted that the transformation step is not an iterative process, and there is a unique
