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coordinate system. The axial orthogonal multi-planar cross-section M
z¼z
c
is there-
fore obtained by selecting a fixed coordinate z ¼ z
c
, and sampling the 3D image I
along coordinates x and y in the image-based coordinate system:
M
z¼z
c
ð
x
;
y
Þ
¼
I
ð
x
;
y
;
z
c
Þ:
ð
51
Þ
In the case of normal spines, vertebrae are usually sagittally inclined, while in
the case of scoliotic spines, vertebrae are usually coronally inclined against axis z.
As a result, axial orthogonal multi-planar cross-sections (Fig.
16
) in general do not
show a completely geometrically correct shape of the vertebral anatomy, because
sampling planes cut through vertebrae at different anatomical locations. For
example, similarly as an ellipse can be obtained by intersecting a circular cone with
an inclined plane, the shape of the vertebral body is observed as a more elliptical
structure than it may actually be.
3.1.2 Oblique Multi-planar Reformation
An established type of MPR is oblique (slanted), meaning that the orthogonal
sampling plane is rotated (inclined) for selected angles about the axes of the image-
based coordinate system. By applying oblique MPR to a 3D spine image, the
following oblique multi-planar cross-sections can be obtained:
sagittal oblique multi-planar cross-sections are obtained by sampling the 3D
image on selected rotated sagittal orthogonal planes, de
ned in the image-based
coordinate system (section
Sagittal Oblique Multi-planar Cross-Sections
),
coronal oblique multi-planar cross-sections are obtained by sampling the 3D
image on selected rotated coronal orthogonal planes, de
ned in the image-based
coordinate system (section
Coronal Oblique Multi-planar Cross-Sections
),
axial oblique multi-planar cross-sections are obtained by sampling the 3D
image on selected rotated axial orthogonal planes, de
ned in the image-based
coordinate system (section
Axial Oblique Multi-planar Cross-Sections
),
generalized oblique multi-planar cross-sections are obtained by sampling the
3D image on planes that are arbitrarily de
ned in the image-based coordinate
system (section
Generalized Oblique Multi-planar Cross-Sections
).
The rotation for angles
a
,
b
and
c
about axes x, y and z, respectively, of the
image-based coordinate system can be represented by rotation matrices R
x
ðaÞ
,
R
y
ðbÞ
and R
z
ðcÞ
, respectively:
2
3
10 0
0
4
5
;
R
x
ðaÞ
¼
cos
a
sin
a
ð
52
Þ
0
sin
a
cos
a
