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Fig. 6 Local articulation
model
are the subsequent bundle
vertebrae. As shown in Fig.
6
, the spatial relations between anchor and bundle
vertebrae are modeled as
Assume
m
i
is an anchor vertebra and
fm
i
þ
1
; ...; m
i
þ
M
g
½
T
i
;
T
i
T
i
þ
1
; ...;
T
i
;
T
i
þ
1
...
T
i
þ
M
1
, where T
i
de
nes a local similarity transformation between vi
i
and v
i
þ
1
. S
1
ð
V
B
j
V
A
Þ
is de
ned as:
X
T
2
e
ðwð
T
i
Þl
T
i
Þ
N
T
i
ðwð
T
i
Þl
T
i
Þ
e
ckwð
T
i
Þwð
T
i
þ
1
Þk
S
1
ð
V
B
j
V
A
Þ
¼
þ
2
=ð
1
þ
Þ
ð
8
Þ
i
Here,
wð:Þ
is an operator that converts Ti
i
to a vector space, i.e., the rotation part of
T
i
is converted to its quaternion.
l
T
i
and
N
T
i
are the Frechet mean and generalized
covariance of local transformation Ti,
i
, calculated as [
27
]. The
first term contains the
prior information of local transformations across population. The second term
evaluates the difference between local Ti
i
across the same spine. These two terms
complement each other, such that a scoliotic spine still get a high value of S
1
, due to
the continuity of its local transformations.
Spatial con
, is modeled
with two assumptions: (1) A vertebral disc is roughly perpendicular to the line
connecting its neighboring vertebrae centers; and (2) Center of a vertebral disc is
gurations between vertebrae and discs, S
2
ð
D
j
V
A
;
V
B
Þ