Information Technology Reference
In-Depth Information
2
4
3
5
;
cos
b
0
sin
b
R
y
ðbÞ
¼
0
1
0
ð
53
Þ
sin
b
0
cos
b
2
4
3
5
;
cos
c
sin
c
0
R
z
ðcÞ
¼
sin
c
cos
c
ð
Þ
0
54
0
0
1
and the composition of extrinsic rotations about axes x (
(first), y (second) and z (last)
can be represented by the rotation matrix R
ða; b; cÞ
:
R
ða; b; cÞ
¼R
z
ðcÞ
R
y
ðbÞ
R
x
ðaÞ:
ð
55
Þ
3
For an arbitrary point p
¼
ð
x
;
y
;
z
Þ
in the image-based coordinate system
I
, its
R
location p
0
¼
ð
x
0
;
y
0
;
z
0
Þ
after rotation is obtained by:
p
0
¼
ð
x
0
;
y
0
;
z
0
Þ
¼
R
ða; b; cÞ
½
x
;
y
;
z
ð
56
Þ
¼
R
ða; b; cÞ
p
;
with the center of rotation at the origin p
0
¼
ð
0
;
0
;
0
Þ
of the image-based coordinate
system.
2
If the center of rotation is at point p
c
¼
ð
x
c
;
y
c
;
z
c
Þ
, then the location of p
after rotation is:
p
0
¼
ð
x
0
;
y
0
;
z
0
Þ
¼
ð
R
ða; b; cÞ
½
x
x
c
;
y
y
c
;
z
z
c
Þ þ
½
x
c
;
y
c
;
z
c
ð
57
Þ
¼
ð
R
ða; b; cÞ
p
p
c
ð
ÞÞ
p
c
;
3
I
By selecting a point p
c
¼
ð
x
c
;
y
c
;
z
c
Þ
in the image-based coordinate system
R
and rotation angles
a
¼
a
p
,
b
¼
b
p
,
and
c
¼
c
p
,
three different
sagittal
, coronal
x
¼
x
c
and
b
¼
b
p
;
c
¼
c
p
or both
y
¼
y
c
and
a
¼
a
p
;
c
¼
c
p
or both
and axial (z
¼
z
c
and
a
¼
a
p
,
b
¼
b
p
or both) oblique multi-planar cross-sections
can be de
ned through p
c
, which also represents the center of rotation. If p
c
is
located
on
the
spine
curve
c
ð
i
Þ
,
e.g.
at
point
i
¼
i
p
so
that
, the obtained cross-sections show, depending on the
shape of the spine, parts of the spinal anatomy.
p
c
¼
c
x
ð
i
p
Þ;
c
y
ð
i
p
Þ;
c
z
ð
i
p
Þ
Sagittal Oblique Multi-planar Cross-Sections
The sagittal oblique multi-planar cross-section M
x¼x
c
;b
¼
b
p
;c
¼
c
p
is obtained by sam-
pling the 3D image I on the sagittal orthogonal plane at
the selected
xed
2
It is assumed that
p is a column vector. If p is a row vector, vector transpose operation is
required, therefore Eq.
56
turns into p
0
¼
ð
T
T
.
