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image intensities in different image modalities (i.e. CT, T
1
T1-weighted MR and
T
2
T2-weighted MR images). As a result, two opposite edge points p
m
and p
m
are
respectively determined, and by repeating the procedure for all edge points
f
p
m
;
m
¼
1
;
2
; ...;
M
g
extracted from the 3D spine image, two sets of pairs of
p
m
Þ;
M
þ
;
M
þ
p
m
Þ;
opposite edge points
fð
p
m
;
m
¼
1
;
2
; ...;
M
g
and
fð
p
m
;
m
¼
M
;
M
1
;
2
; ...;
M
g
are respectively formed. The
final set of pairs of opposite
fð
p
m
;
p
m
Þ
¼
ð
p
m
;
p
m
Þ[ð
p
m
;
p
m
Þ;
M
0
;
M
0
edge points
is
obtained by joining the results of the search in the positive and negative gradient
vector directions.
As the spine curve passes through the center of each vertebral body, it is located
where the lines connecting pairs
m
¼
1
;
2
; ...;
2M
g
fð
p
m
;
p
m
Þ;
m ¼
; ...;
M
0
g
of opposite edge
points most often intersect. For this purpose, a 3D accumulator A
I
, which is of the
same size as the observed 3D spine image I, is generated in the image-based
coordinate system and initialized with zero values, i.e. A
I
ð
1
;
2
Þ
¼
; 8
¼
ð
;
;
Þ2
:
p
0
p
x
y
z
I
p
m
Þ
Each line connecting a pair
of opposite edge points is assigned a weighting
function, normally distributed according to the distance d
ð
p
m
;
p
m
Þ
ð
p
m
;
and scaled
p
m
Þ
between opposite edge points p
m
and p
m
. The
according to the congruence S
ð
p
m
;
3D accumulator value A
I
ð
at point pl
l
is then increased by the value of the
weighting function at each point
p
l
Þ
f
p
l
¼
ð
x
l
;
y
l
;
z
l
Þ;
l
¼
1
;
2
; ...;
L
g
along the con-
necting line:
!
;
2
p
m
Þ=
ð
d
ð
p
m
;
p
l
Þ
d
ð
p
m
;
2
Þ
A
I
ð
p
l
Þ
¼A
I
ð
p
l
Þþ
S
ð
p
m
;
p
m
Þ
exp
ð
46
Þ
2
p
m
Þ=
2
ð
d
ð
p
m
;
6
Þ
where d
ð
p
m
;
p
l
Þ
is the Euclidean distance between edge point pl
m
and each point pl
l
on
p
m
Þ=
the line, and d
6 represents the standard deviation of the normally distrib-
uted weighting values. By accumulating the lines connecting all pairs
ð
p
m
;
p
m
Þ;
fð
p
m
;
m
¼
M
0
g
1
of opposite edge points in the 3D spine image, the values in the 3D
accumulator A
I
increase most along the longitudinal axes of vertebral bodies, as
vertebral body walls contribute to most pairs of opposite edge points. The resulting
normalized accumulator values,
;
2
; ...;
1, therefore
represent the probability that the spine curve passes through the corresponding
locations (Fig.
12
d).
Although maximal accumulator values point to the location of the spine curve,
they may not always represent its exact location, as the generation of the accu-
mulator is obstructed by edges that do not represent vertebral body walls. To
determine the exact location of the spine curve, the coordinates of H
8
p ¼
ð
x
;
y
;
z
Þ2
I
)
0
A
I
ð
p
Þ
j
j
5 largest
maxima are extracted from each axial cross-section of the 3D accumulator A
I
and
connected into line segments. The line segments that are shorter than one half of the
average size of the human vertebral body d
vb
¼
¼
30 mm [
47
,
52
,
53
] are discarded
(i.e. segments shorter than d
vb
=
2
¼
15 mm). The remaining line segments, i.e. the
set of points
, represent the candidate locations for the spine
curve. A robust estimation of the polynomial parameters b
c
of the spine curve cðiÞ
f
p
j
;
j
¼
1
;
2
; ...;
J
g
Þ
(Eq.
25
) can be obtained by applying the random sample consensus (RANSAC)
ð
i
