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¼
p
O
X
i
;
p
D
X
i
;
p X
i
;
X
j
p
S
X
i
;
X
j
X
j
X
j
ð
4
Þ
The three items in (
4
) specify the size, the orientation and the distance con-
straints as detailed below:
X
j
is used to set constraint on the sizes
(radius and height of the vertebral body) of the neighboring components and is
de
: p
S
X
i
;
Size constraint p
S
X
i
;
X
j
ned as
j
j
j
j
r
i
r
j
h
i
h
j
ðx
r
þx
h
Þ
j
j
j
j
¼
r
i
þ
r
j
h
i
þ
h
j
e
p
S
X
i
;
X
j
ð
5
Þ
j
i
j
j
which means that neighboring components should have similar sizes and that
the strength of the constraint should decay with the order distance between these
two components.
:wede
Orientation constraint p
O
X
i
;
X
j
ne
¼
x
0
ð
a
i
ð
h
i
Þ
a
j
ð
h
j
ÞÞ
j
e
p
O
X
i
;
X
j
ð
6
Þ
i
j
j
to ensure that the neighboring vertebral bodies should have similar orientations,
in which a
i
ðh
i
Þ
de
nes the orientation of a vertebral body template as shown in
Fig.
2
.
:
Distance constraint p
D
X
i
;
X
j
for direct neighboring nodes V
i
;
V
j
,
i.e.,
j
ne constraints on the spatial distance between their
vertebral body centers. Without losing any generality, for the case j
¼
i
þ
i
j
j ¼
1, we also de
1we
as
de
ne p
D
X
i
;
X
j
8
<
e
x
D
d
C
;
ij
d
h
;
ij
4
;
d
C
;
ij
\
d
h
;
ij
¼
ð
Þ
ij
þ
h
l
i
þ
h
u
j
d
h
p
D
X
i
;
X
j
;
ð
7
Þ
e
x
D
d
C
;
ij
:
2
5
3
;
4
d
h
;
ij
[
d
C
;
ij
[
4
d
h
;
ij
d
h
;
ij
0
;
elsewhere
, d
h
;
ij
¼
where d
C
;
ij
¼
C
i
C
j
h
i
þ
h
j
; C
i
and C
j
are the centers of
the
with respect
neighboring nodes Vi
i
and V
j
, respectively. The value of p
D
X
i
;
X
j
to the value of d
C
;
ij
=
d
h
;
ij
is shown in Fig.
3
. It can be observed that p
D
X
i
;
X
j
obtains a high value in the regions 0
1.25. Intuitively this
constraint means that direct neighboring components should either be closely
connected side-by-side (around the region when d
C
;
ij
=
d
h
;
ij
is in the range of
0.75
-
0.25 and 0.75
-
-
1.25 as shown in Fig.
3
) or merge to one object (around the region when
d
C
;
ij
=
0.25 as shown in Fig.
3
). This makes our
graphical model capable of automatically adjusting the con
d
h
;
ij
is in the range of 0.0
-
guration of the
