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Fig. 11 a The search for the line of in-plane symmetry is performed in each ith axial orthogonal
multi-planar cross-section of the MR spine image. b The in-plane line y
i
ð
x
Þ
is defined by
parameters
c
i
and
k
i
. c The operator
C
is centered in
ð
x
i
;
y
i
Þ
and consists of M concentric rings;
x
i
;
y
i
Þ
m ¼ 1
;
2
; ...;
M, each with radial width of
r. d The resulting center of the vertebral body
ð
D
and in-plane rotation
c
i
where p
Q
ð
are the probability distributions of image intensities in
image parts A and B, respectively, p
Q
ð
p
A
;
p
B
Þ
p
A
Þ
and p
Q
ð
p
B
Þ
is the joint probability distribution of
image intensities, and Q is the number of bins used for probability estimation. The
resulting parameters
c
i
and
k
i
(Eq.
34
)de
line the in-plane line of symmetry, which
passes through the center of the vertebral body in the observed ith axial cross-
section.
To determine the exact location of the center of the vertebral body, the shape
properties of the vertebral anatomy are combined with the appearance of the vertebral
body in MR images. When observed in axial cross-sections, the shape of the vertebral
body is relatively circular. For a circular structure displayed in a MR image, a certain
amount of variation in image intensities is always present along any radial direction,
however, in the tangential direction, the variation in image intensities is relatively
small. To obtain a quantitative estimation of these properties, the entropy-based
operator
C
is introduced that is centered at an arbitrary point p ¼
ð
x
;
y
Þ
and consists
