Information Technology Reference
In-Depth Information
M
p
1
;
p
2
;
p
3
(Eq.
74
)withp
1
as the spine curve start point, p
2
as the spine curve end
point, and p
3
¼ p
c
¼ c
ð
i
p
Þ
as a point on the spine curve at i
¼
i
p
.
3.2 Curved-Planar Reformation
Curved-planar reformation is an ef
cient technique for cross-sectional visualization
of curved 3D anatomical structures, where the goal is to visualize the structure
along its entire length within a single cross-section. The volume of the 3D structure
is cut by a plane, and image intensities are sampled on that plane. According to
the orientation of the sampling plane in the spine-based coordinate system, the
following types of CPR can be applied to 3D images of the spine:
orthogonal CPR, where the sampling plane is orthogonal to one of the axes of
the spine-based coordinate system (Sect.
3.2.1
),
oblique CPR, where the orthogonal sampling plane, de
ned in the spine-based
coordinate system,
is rotated about
the axes of the spine-based coordinate
system (Sect.
3.2.2
).
The common characteristic of all types of CPR is that sampling planes are de
ned
3
on the basis of the spine-based coordinate system
S
. However, image intensities
can be accessed only in the image-based coordinate system, therefore a transfor-
mation from the spine-based to the image-based coordinate system is required
(Sect.
2.2.3
) and achieved through a continuous representation of the spine curve
c
ð
R
i
Þ
(Sect.
2.3.1
) and axial vertebral rotation
uð
i
Þ
(Sect.
2.3.2
). As the axial vertebral
rotation
uð
i
Þ
represents axes u and v in the spine-based coordinate system, it must be
de
ned in planes orthogonal to axis w. The axial vertebral rotation can be therefore
represented by matrix R
w
ðuð
of rotation about axis w of the spine-based coor-
dinate system, which has the same form as R
z
(Eq.
54
), i.e. R
w
ðuð
i
ÞÞ
.
However, the rotation in the image-based coordinate system has to be performed
about the axis de
i
ÞÞ
¼
R
z
ðuð
i
ÞÞ
ned by the unit tangent vector t^ðiÞ
ð
i
Þ
¼
ð
t
x
ð
i
Þ;
t
y
ð
i
Þ;
t
z
ð
i
ÞÞ
to the spine
curve c
, which represents axis w in the spine-based coordinate system. The
rotation in the form of axis-angle representation can be achieved by matrix
R
t
ð
i
Þ
ðuð
ð
i
Þ
ÞÞ
i
(Eq.
19
).
3.2.1 Orthogonal Curved-Planar Reformation
The most straightforward approach to CPR is orthogonal, meaning that the sam-
pling plane is orthogonal to one of the axes of the spine-based coordinate system.
By applying orthogonal CPR to a 3D spine image, the following orthogonal curved-
planar cross-sections can be obtained:
