Information Technology Reference
In-Depth Information
M
z¼z
c
;a
¼
a
p
;b
¼
b
p
;c
¼
c
p
ð
x
;
y
Þ
¼
I
ð
R
ða
p
; b
p
; c
p
Þ
½
x
;
y
;
z
c
Þ;
ð
69
Þ
with the center of rotation at point p
c
¼
ð
x
c
;
y
c
;
z
c
Þ
(Eq.
57
). However, in the case of
3D spine images, the value of such generalized oblique multi-planar cross-sections
is questionable since the arbitrarily de
ned sampling plane may not include any
spinal anatomy.
A more valuable result can be achieved by de
ning the sampling plane according
to the spine as the observed anatomical structure. By selecting three non-collinear
points p
1
¼
ð
x
1
;
y
1
;
z
1
Þ
, p
2
¼
ð
x
2
;
y
2
;
z
2
Þ
and p
3
¼
ð
x
3
;
y
3
;
z
3
Þ
on the spine, which
may be located on the spine curve cðiÞ,
ð
i
Þ
, e.g. at points i
¼
i
1
, i
¼
i
2
and i
¼
i
3
so that
p
1
¼
ð
c
x
ð
i
1
Þ;
c
y
ð
i
1
Þ;
c
z
ð
i
1
ÞÞ
, p
2
¼
ð
c
x
ð
i
2
Þ;
c
y
ð
i
2
Þ;
c
z
ð
i
2
ÞÞ
and p
3
¼
ð
c
x
ð
i
3
Þ;
c
y
ð
i
3
Þ;
c
z
ð
i
3
ÞÞ
, respectively, a sampling plane P can be uniquely de
ned in the image-based
coordinate system and used to rede
ne the rotation matrix R (Eq.
55
). The unit normal
vector
n
P
of plane P is:
n
P
n
kk
;
n
P
¼½
^
n
Px
; ^
n
Py
; ^
n
Pz
¼
n
P
¼
ð
p
1
p
3
Þð
p
2
p
3
Þ:
ð
70
Þ
^
^
To rede
ne the rotation matrix R, two unit vectors
e
1
and
e
2
have to be addi-
ned that are, including n
P
, mutually orthogonal. We have one degree of
freedom for the selection of e
1
, e.g.:
tionally de
^
n
P
^
e
1
¼
0
e
1
e
kk
;
ð
Þ
71
^
e
1
¼½
^
e
1x
; ^
e
1y
; ^
e
1z
¼
e
1
¼½
^
n
Pz
; ^
n
Pz
; ^
n
Px
þ ^
n
Py
;
which is then used to determine e
2
:
^
e
2
¼½
^
e
2x
; ^
e
2y
; ^
e
2z
¼
^
e
1
^
n
P
:
ð
72
Þ
In 3D spine images, normal spines are usually aligned with sagittal orthogonal
planes, while scoliotic spines are usually aligned with coronal orthogonal planes. As a
result, in
P
(Eq.
70
) represents, in the coordinate system of plane P, the unit vector
e
Px
¼½
;
;
P
in the case of normal spines, and the unit vector e
Py
¼½
;
;
P
in the
1
0
0
0
1
0
^
case of scoliotic spines. On the other hand,
e
1
(Eq.
71
) is selected so that it always
^
^
represents the unit vector
e
2
(Eq.
72
) represents the remaining
unit vector in the coordinate system of plane P. As a result, the rotation matrix is in the
case of normal spines rede
e
Pz
¼½
0
;
0
;
1
P
, while
ned as R
n
and used to obtain the generalized oblique
multi-planar cross-section M
p
1
;
p
2
;
p
3
:
2
3
^
n
Px
^
e
2x
^
e
1x
4
5
;
M
p
1
;
p
2
;
p
3
ð
R
n
R
n
y
;
z
Þ
¼
I
ð
½
x
j
;
y
;
z
Þ;
¼
^
n
Py
^
e
2y
^
e
1y
ð
73
Þ
^
n
Pz
^
e
2z
^
e
1z
