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the spine. When the optimization reaches the termination criterion, the polynomial
degree is increased by 1 and the optimization is restarted on the next level, using the
polynomial parameters from the previous level for initialization. The maximal
polynomial degree of 5 is adequate for modeling the axial vertebral rotation in both
normal and scoliotic spines.
The method was evaluated [
83
,
85
] on 30 normal and one scoliotic CT image of
the thoracolumbar spine, and the reported mean difference between the obtained
spine curve in 3D and manually de
4 mm. The
performance of the determination of the axial vertebral rotation was, using the sum
of absolute differences instead of the correlation coef
ned ground truth points was 2
:
1
1
:
cient (Eq.
32
), evaluated [
18
]
on 68 vertebrae extracted from CT images, and the reported mean difference against
reference values was around
6
with the corresponding 95% con
0
:
dence interval
8
.
of around 4
:
Automated Determination of the Spine Curve and Axial Vertebral Rotation
in MR Images
Vrtovec et al. [
86
] proposed a method for automated determination of the spine
curve and axial vertebral rotation in MR images. The method is based on ana-
tomical properties that vertebral bodies and intervertebral discs are nearly circular in
shape and that vertebrae are nearly symmetrical over the lines that pass through the
centers of vertebral bodies. In each axial cross-section z ¼ z
i
of the MR image
(Fig.
11
a), an arbitrary in-plane line yi
i
¼ y
i
ð
x
Þ
that divides the observed cross-
section into two image parts, i.e. part A and part B, can be de
ned as:
p
2
c
i
y
i
ð
Þ
¼
ð
k
i
Þ
;
ð
Þ
x
x
tan
34
where tan
2
c
i
is the slope and
k
i
is the intersection of the line with axis x of the
image-based coordinate system (Fig.
11
b). The line that splits the observed cross-
section into two symmetrical parts can be obtained by maximizing the similarity
between image parts A and B:
fc
i
; k
i
g
¼
argmax
fc
i
;k
i
g
ð
S
ð
I
ð
p
A
;
z
i
Þ;
I
ð
p
B
;
z
i
ÞÞÞ;
ð
35
Þ
where p
A
¼
ð
x
A
;
y
A
Þ; 8
p
¼
ð
x
;
y
Þ
[
y
i
ð
x
Þ
and p
B
¼
ð
x
B
;
y
B
Þ; 8
p
B
¼
ð
x
;
y
Þ
\
y
i
ð
x
Þ
represent all points p
that lie in image parts A and B, respectively, and S is
the similarity measure that can be computed as the standard mutual information
between image parts A and B:
¼
ð
x
;
y
Þ
!
X
X
p
Q
ð
p
A
;
p
B
Þ
p
Q
ð
S
ð
I
ð
p
A
;
z
i
Þ;
I
ð
p
B
;
z
i
ÞÞ
¼
p
Q
ð
p
A
;
p
B
Þ
log
;
ð
36
Þ
p
A
Þ
p
Q
ð
p
B
Þ
p
A
2
A
p
B
2
B
