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technique [
16
]. By randomly selecting K
þ
1
¼
4 points
f
p
k
;
k ¼
1
;
2
; ...;
K
þ
1
g
, a curve in the form of a polynomial function c0ðiÞ
0
ð
i
Þ
from set
f
p
j
g
, i.e.
f
p
k
gf
p
j
g
of
degree K
3 can be generated, resulting in a combination of polynomial param-
eters b
c
. To evaluate
¼
c
0
ð
i
Þ
the
agreement of
curve
against points
in
f
p
j
;
j ¼
1
;
2
; ...;
J
g
, the criterion C
ð
b
c
Þ
is computed as the number of points in
f
p
j
g
¼
3 mm distant from c0ðiÞ
0
ð
Þ
that are less than r
i
in each axial cross-section:
c
0
ð
C
ð
b
c
Þ
¼
count
ð
d
ð
i
Þj
b
c
;
p
j
Þ
\
r
Þ;
j
¼
1
;
2
; ...;
J
;
ð
47
Þ
c
0
ð
denotes the Euclidean distance between curve c0ðiÞ
0
ð
where d
ð
i
Þj
b
c
;
p
j
Þ
i
Þ
and point
p
j
. Among 1000 generated polynomial functions c
0
ð
(the number of iterations can
be even larger), the optimal polynomial parameters b
c
i
Þ
of the spine curve cðiÞ
ð
i
Þ
correspond to the maximal criterion C
ð
b
c
Þ
:
b
c
¼
argmax
b
c
ð
C
ð
b
c
ÞÞ:
ð
48
Þ
fit to the maxima of
the 3D accumulator, which are located in the middle of vertebral body walls
(Fig.
13
).
The method was evaluated [
78
] on 42 3D images of the lumbar spine (29 CT
images and 13 MR images), and the reported mean difference between the obtained
spine curves in 3D and manually de
The obtained spine curve cðiÞ
ð
i
Þ
therefore represents the best
ned ground truth points was 1
:
8
1
:
1mm
(1
:
7
1
:
0 mm for CT images and 2
:
3
1
:
5 mm for MR images).
can be
obtained by extracting planes that are orthogonal to the spine curve. By intersecting
each of these planes with a line passing through cðiÞ
Once the spine curve cðiÞ
ð
i
Þ
is determined, the axial vertebral rotation
uð
i
Þ
ð
i
Þ
and inclined for an angle that
corresponds to the polynomial function de
ned by parameters b
u
, the optimal
polynomial parameters b
u
representing the axial vertebral rotation
uð
(Eq.
26
) can
be computed by maximizing the in-plane similarity of the resulting image parts, e.g.
by
i
Þ
finding the maximal correlation of image intensities between mirror image
halves (Eqs.
30
-
33
) or the maximal mutual
information of image intensities
between plane parts (Eqs.
34
36
), or by using a different similarity measure.
-
3 Cross-Sectional Reformation of 3D Spine Images
Volumetric image visualization can be de
ned as the transformation of image
information from a 3D image space onto a 2D display device. The most straight-
forward 2D visualization of 3D images is based on original cross-sections that
display the primarily reconstructed images, composed of original pixels in image
reconstruction planes. A 2D cross-section of a 3D image is de
ned as the
