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sagittal oblique curved-planar cross-sections are obtained by sampling the 3D
image on selected rotated sagittal orthogonal planes, de
ned in the spine-based
coordinate system (section
Sagittal Oblique Curved-Planar Cross-Sections
),
coronal oblique curved-planar cross-sections are obtained by sampling the 3D
image on selected rotated coronal orthogonal planes, de
ned in the spine-based
coordinate system (section
Coronal Oblique Curved-Planar Cross-Sections
),
axial oblique curved-planar cross-sections are obtained by sampling the 3D
image on selected rotated axial orthogonal planes, de
ned in the spine-based
coordinate system (section
Axial Oblique Curved-Planar Cross-Sections
).
The rotation principle is therefore in general the same as in the case of oblique
MPR (Sect.
3.1.2
), however, in the case of oblique CPR, only rotation about one
axis is usually applied for sagittal and coronal cross-sections so that they remain
aligned with the observed spinal anatomy. As a result, by selecting a point p
c
¼
ð
3
S
#
¼
#
p
,
w
¼
w
p
, and
/
¼
/
p
, exactly one sagittal (u ¼ u
c
and
/
¼
/
p
), one coronal (v ¼ v
c
and
/
¼
/
p
) and three axial (w
u
c
;
v
c
;
w
c
Þ
in the spine-based coordinate system
and rotation angles
R
¼
w
c
and
#
¼
#
p
,
w
¼
w
p
or both) oblique curved-
planar cross-sections can be de
ned through p
c
.Ifp
c
is located on the spine curve
c
, the obtained sagittal and
coronal cross-sections show, irrespectively of the shape of the spine, the spinal
anatomy at its midline along its entire length, while the obtained axial cross-section
is arbitrarily inclined against the midline of the spinal anatomy.
ð
i
Þ
, e.g. at point i
¼
i
p
so that p
c
¼
ð
c
x
ð
i
p
Þ;
c
y
ð
i
p
Þ;
c
z
ð
i
p
ÞÞ
Sagittal Oblique Curved-Planar Cross-Sections
The sagittal oblique curved-planar cross-section C
u¼u
c
;/
¼
/
p
is obtained by sampling
the 3D image I on the sagittal orthogonal plane at the selected
fixed coordinate
u
u
c
that is additionally rotated for angle
/
¼
/
p
about axis w of the spine-based
coordinate system:
¼
C
u¼u
c
;/
¼
/
p
ð
v
;
w
Þ
¼I
ð
R
w
ð/
p
Þ
½u
c
;
v
;
w
Þ;
ð
84
Þ
with the center of rotation at point p
c
¼
ð
u
c
;
v
c
;
w
c
Þ
(Eq.
57
). The rotation angle
4
; þ
4
must be on the closed interval
/
p
2
½
, otherwise the cross-section turns into
a coronal oblique curved-planar cross-section. In the image-based coordinate sys-
tem, the sampling plane is represented by a curved surface that is parallel to the
spine curve cðiÞ
ð
i
Þ
¼
ð
c
x
ð
i
Þ;
c
y
ð
i
Þ;
c
z
ð
i
ÞÞ
and follows the axial vertebral rotation
uð
i
Þ
with an offset of
/
¼
/
p
. If the selected
fixed coordinate u
c
is represented as
u
c
$
c
x
ð
i
Þþ
D
x, where
D
x is a
fixed offset in the left or right direction from the
spine curve cðiÞ
ð
i
Þ
from
the origin of the spine-based coordinate system, then the sagittal oblique curved-
planar cross-section C
u
¼
u
c
;/
¼
/
p
that corresponds to the sagittal offset of point p
c
¼
ð
u
c
;
v
c
;
w
c
Þ
can be obtained as:
