Information Technology Reference
In-Depth Information
describes the longitudinal axis of the spine). The maximal polynomial degree can
be determined from the
fl
flexion points in normal and scoliotic spinal curvatures, as
the number of
fl
flexion points of a polynomial function is equal
to its degree
decreased by one (e.g. a straight line is of the
flexion
points). For example, when observing the thoracolumbar section of the spine, three
distinctive
rst degree and has zero
fl
flexion points exist in a normal spinal curvature, i.e. the maximal tho-
racic kyphosis, the thoracolumbar junction, and the maximal lumbar lordosis, and
therefore polynomial functions of the fourth degree represent an adequate choice.
Accordingly, a scoliotic spinal curvature can be adequately described by polyno-
mial functions of the
fl
fifth degree.
As the axial vertebral rotation is de
ned as the rotation of vertebrae about the
spine curve, it can be determined automatically in planes that are orthogonal to the
spine curve cðiÞ,
ned by polynomial parameters b
c
(Eq.
29
). In each such ith
plane, a circular domain of radius d and centered at point cðiÞ
ð
i
Þ
,de
ð
i
Þ
¼
ð
c
x
ð
i
Þ;
c
y
ð
i
Þ;
c
z
ð
i
ÞÞ
is de
ned (Fig.
10
a). The circular domain is then bisected by a line that passes
through the center of the domain and is inclined for angle
u
against the projection
e
Iy
ð
i
Þ
I
to the plane (Eq.
14
). In the obtained halves A
and B of the ith circular domain, s
A
and s
B
represent image intensities at mirror
pixels according to the line of bisection:
of the unit vector
e
Iy
¼½
0
;
1
;
0
s
A
ð
i
;
j
; uÞ
¼
I
ð
R
z
ðuÞ
½
u
;
v
;
c
z
ð
i
ÞÞ
ð
30
Þ
¼
I
ð
R
t
ð
i
Þ
ðuÞ
½
x
;
y
;
0
þ
½
c
x
ð
i
Þ;
c
y
ð
i
Þ;
c
z
ð
i
ÞÞ;
s
B
ð
i
;
j
; uÞ
¼
I
ð
R
z
ðuÞ
½
þ
u
;
v
;
c
z
ð
i
ÞÞ
ð
31
Þ
¼
I
ð
R
t
ð
i
Þ
ðuÞ
½
þ
x
;
y
;
0
þ
½
c
x
ð
i
Þ;
c
y
ð
i
Þ;
c
z
ð
i
ÞÞ;
where j is the index of the mirror point pair (a total of J mirror point pairs exist),
consecutively assigned on the basis of each
ð
u
;
v
Þ
with u
[
0 and u
2
þ
v
2
d
2
,or
d
2
(Fig.
10
b). Matrices R
z
(Eq.
17
) and R
t
ð
i
Þ
(Eq.
19
) represent, respectively, the rotation
1
about axis w of the spine-based
coordinate system and the rotation about the unit tangent vector t^ðiÞ
ð
i
Þ
0 and x
2
y
2
each
ð
x
;
y
Þ
with x
þ
[
to the spine
curve c
ð
at point i in the image-based coordinate system.
Radius d is de
i
Þ
ned so that the anatomy of the whole vertebra (and not only of
the vertebral body) is captured within the circular domain, and can be de
ned from
the quantitative morphometrical vertebral analysis [
52
54
]. From the obtained
mirror image intensity pairs, the in-plane similarity between the two mirror halves
of the ith circular domain at inclination
u
can be quantitatively evaluated by the
correlation coef
-
cient R
AB
ð
i
; uÞ
:
1
It is assumed that point
p
¼½
x
;
y
;
z
is a column vector. If p is a row vector, vector transpose
operation is required,
therefore the equation p
0
¼
ð
x
0
;
y
0
;
z
0
Þ
¼R½x
;
y
;
z
¼Rp turns into
T
T
.
