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In-Depth Information
of M concentric rings of radii
f
r
m
;
m ¼
1
;
2
; ...;
M
g
with
8
m
:
r
m
\
r
m
þ
1
and radial
width of each ring equal to
D
r ¼ r
m
þ
1
r
m
(Fig.
11
c):
P
m
¼
1
w
m
H
m
1
2
m
M
L
C
¼
H
P
m¼1
w
m
;
w
m
¼
exp
;
ð
37
Þ
where H
m
¼
P
q¼1
p
q
;
m
log p
q
;
m
is the entropy defined by the probability distri-
bution p
q
;
m
of image intensities in the mth ring, H
¼
P
q¼1
p
q
log p
q
is the
entropy de
ned by the probability distribution p
q
of image intensities within the
entire operator (i.e. within all rings), and Q is the number of bins used for proba-
bility estimation. The ring weights w
m
are chosen to be within L standard deviations
of the Gaussian distribution, so that the inner rings have a relatively stronger impact
to the operator response in comparison to the outer rings. The number of rings can
be automatically adjusted from M
¼
15 rings in the cervical region, to M
¼
20
rings in the thoracic region, and to M
30 rings in the lumbar region of the spine.
The radial width of each ring can be set to
¼
r
¼
1 mm, the ring weights to be
D
within L
2 standard deviations of the Gaussian distribution, and the probability
distributions can be computed using Q
¼
16 bins. The variation in image intensities
in the tangential direction is estimated by the sum of entropies H
m
in individual
concentric rings, while the variation in image intensities in the radial direction is
estimated by the entropy H within the entire operator, which also serves to penalize
the regions that are homogeneous in image intensity. The center of the vertebral
body
¼
x
i
;
y
i
Þ
is found by minimizing the response of the entropy-based operator
C
along the in-plane line of symmetry y
i
ð
ð
x
Þ
:
x
i
¼
argmin
x
ð
I
ð
x
;
y
i
ð
x
Þ;
z
i
ÞjCð
x
;
y
i
ð
x
ÞÞÞ;
ð
38
Þ
tan
p
2
c
i
y
i
¼
x
i
Þ
¼
x
i
k
i
y
i
ð
:
ð
39
Þ
f
c
i
¼
ð
x
i
;
y
i
;
z
i
Þ;
i ¼
represent the detected
centers of vertebral bodies in each ith axial cross-section of the MR spine image
(Fig.
11
d). A continuous representation of the spine curve cðiÞ
ð
i
Þ
The resulting points
1
;
2
; ...;
N
g
is obtained by
fitting
polynomial functions to points
f
c
i
g
to determine the optimal polynomial parameters
b
c
is obtained independently and therefore
outliers may be present, it is recommended to apply a robust regression method, for
example, the non-linear least trimmed squares (LTS) regression [
69
]:
(Eq.
25
). However, as each point in
f
c
i
g
!
X
h
c
b
c
¼
r
c
;
½i
j
argmin
b
c
b
c
;
ð
40
Þ
i
¼
1
2
where r
c
;
½i
¼
ð
c
i
c
ð
i
ÞÞ
represent
the ordered squared residuals in increasing
order; r
c
;
½1
r
c
;
½2
r
c
;
½N
, and h
c
is the trimming constant that satis
es the
