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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
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
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