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where matrix R
z
ðuð
i
ÞÞ
(Eq.
54
) represents the axial vertebral rotation for angle
uð
i
Þ
about axis z of the image-based coordinate system. In this case, the axial vertebral
rotation
uð
ned in transverse planes that are orthogonal to axis z of
the image-based coordinate system (i.e.
uð
i
Þ
has to be de
i
Þ
¼
u
z
ð
i
Þ
, Eq.
13
). As a result, the
resulting cross-sections are no longer de
ned on the basis of the spine-based
coordinate system, and therefore the spine curve is no longer represented by a
straight line.
Coronal Orthogonal Curved-Planar Cross-Sections
Coronal orthogonal curved-planar cross-sections are obtained by sampling the 3D
image on selected coronal planes that are orthogonal to axis v of the spine-based
coordinate system. The coronal orthogonal curved-planar cross-section C
v¼v
c
is
therefore obtained by selecting a
v
c
, and sampling the 3D
image I along coordinates u and w in the spine-based coordinate system:
fixed coordinate v
¼
ð
;
Þ
¼
ð
;
v
c
;
Þ:
ð
Þ
C
v¼v
c
u
w
Iu
w
78
In the image-based coordinate system, 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
. If the selected
fixed coordinate v
c
is
represented as v
c
$
c
y
ð
i
Þþ
D
y, where
y is a
fixed offset in the anterior or pos-
D
ð
Þ
terior direction from the spine curve cðiÞ
i
that corresponds to the coronal offset of
point p
c
¼
ð
u
c
;
v
c
;
w
c
Þ
from the origin of the spine-based coordinate system, then
the coronal orthogonal curved cross-section C
v¼v
c
can be obtained as:
C
v¼v
c
ð
x
;
c
z
ð
i
ÞÞ
¼
I
ð
R
t
ð
i
Þ
ðuð
i
ÞÞ
R
y
ðbð
i
ÞÞ
½
x
;
c
y
ð
i
Þþ
D
y
;
c
z
ð
i
ÞÞ;
ð
79
Þ
where matrix R
t
ð
i
Þ
ðuð
i
ÞÞ
(Eq.
19
) represents the axial vertebral rotation for angle
ned by t
uð
i
Þ
about axis de
ð
i
Þ
(i.e.
uð
i
Þ
¼
u
w
ð
i
Þ
, Eq.
14
), and matrix R
y
ðbð
i
ÞÞ
ð
t
x
ð
Þ=
t
z
ð
(Eq.
53
) represents the rotation for angle
bð
i
Þ
¼
arctan
i
i
ÞÞ
about axis y of
the image-based coordinate system, considering that t^ðiÞ
Þ
¼ t
x
ð
Þ;
t
y
ð
Þ;
t
z
ð
ð
i
i
i
i
Þ
is the
unit tangent vector to the spine curve, and cðiÞ
ð
i
Þ
¼
ð
c
x
ð
i
Þ;
c
y
ð
i
Þ;
c
z
ð
i
ÞÞ
is the center of
rotation (Eq.
57
) for every point i on the spine curve cðiÞ.
. Exactly one coronal
orthogonal curved-planar cross-section passes through the spine curve (i.e.
ð
i
Þ
0)
and therefore displays the spinal anatomy along its midline, which is represented by
a straight line (Fig.
22
).
However, in the resulting cross-sections, anatomical deformations are present
that result from the intersections of coronal profiles due to the rotation for angle
bð
i
Þ
about axis y of the image-based coordinate system. To avoid anatomical defor-
mations, the coronal orthogonal curved-planar cross-section C
v
¼
v
c
can be obtained
as:
D
y
¼
