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components Tr. Obtain the transformation parameters (t x ; t y ; s x ; s y ) for each
training shape, / ,as
P X xH ð / ð x ÞÞ
P X yH ð / ð x ÞÞ
T
T
P X H ð/ð x ÞÞ
P X H ð/ð x ÞÞ
Tr
¼
½
t x
t y
¼
l x
l y
;
ð
14
Þ
2
3
T
r
r x
P X ð x l x Þ 2 H ð / ð x ÞÞ
0
4
P X ð H x ÞÞ
5
s x
0
S
¼
¼
ð
15
Þ
r y
r
0
s y
0
P X ð y l y Þ 2 H ð/ð x ÞÞ
P X
H
ð/ð
x
ÞÞ
3. The transformation will be in the form T
ð
x
Þ ¼
X
¼
Sx
þ
Tr, where X is the
transformed point of x.
Note: In 2D case, the rotation parameter for the VB shape registration is not
necessary since VB shape does not show important variation in different rotation.
2.4.3 Training stage (Obtaining Probabilistic Shape Model)
1. Segment training images manually.
2. Align segmented images.
3. Generate shape variation. Intersection of training shape is accepted as an object
volume. The rest of the volume is accepted as variability volume except the
background region.
4. Obtain the probabilities of the object and background in the variability volume
of the shape model.
A new probabilistic shape model is formed using the training shapes as shown in
Fig. 11 a. All registered training shapes are combined as shown in Fig. 11 b. The
shape prior represented as
R ¼ O[B[V
is generated. The proposed shape model
functions are de
ned as follows:
\
N
O ¼
H(
/ i Þ;
ð
16
Þ
i¼1
\
N
B ¼
H
ð/ i Þ;
ð
17
Þ
i¼1
[
\
N
N
V ¼
H
ð/ i Þ
H
ð/ i Þ;
ð
18
Þ
i
¼
1
i
¼
1
where / i represents any training shape.
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