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2
. If the Frenet (tangent, normal, binormal) frame F = {T, N, B}
and the curvature are known at
a
i
ð
by arc length in
<
0
Þ
, a local approximation is obtained as follows:
s
2
2
j
0
N
0
s
3
a
i
ð
u
Þ
¼
a
i
ð
0
Þþ
sT
0
þ
6
j
0
s
0
B
0
:
ð
1
Þ
ned as the curves representing the spine
centerline derived from the segmented silhouettes si
i
on the PA and LAT images
respectively. Therefore for each value of u such that u ¼½0
;
2
s;
3
s; ...;
1
and
s
is
the equidistant step-size, a node in the tangent space n ¼
ð
x
PA
;
y
PA
;
x
LAT
;
y
LAT
; h
PA
;
h
LAT
; j
PA
; j
LAT
Þ
Hence,
a
PA
ð
u
Þ
and
a
LAT
ð
u
Þ
are de
ned, where x
i
and y
i
are the image projection coordinates of
X,
h
i
the orientation of projected tangent in the image planes, and
j
i
the image
curvatures, with i representing the biplanar X-rays. Here, the 3D position X and
tangent T are computed using standard methods [
22
]. To determine the normal
N and the 3D curvature
j
at a 3D space curve point on C
k
ð
is de
from biplanar views,
the mathematical relationship proposed by Li and Zucker [
40
] can be used where
the 3D normal N, the curvature
j
and the parameters from the viewing geometry is
formulated as:
u
Þ
3
2
2
f
ð
1
ð
u
PA
T
Þ
Þ
ð
u
PA
T
Þ
N
j
¼
j
PA
ð
2
Þ
3
2
2
dð
1
ð
u
PA
t
PA
Þ
Þ
where u
PA
¼
, p is determined from the vector pointing to the projection
in the image plane and f is the vector pointing to the image center, both in the
camera coordinate system. The
d
parameter is the depth computed from calibration
while t is the local tangent vector on the 2D image. The curvature can then be
computed as
j
¼
k
p
PA
=k
p
PA
k
N
jk
, given the constraint
k
N
k
¼
1. Hence, the normal vector
can be determined by N ¼ N
j=j
.
3.3.3 Self-calibration by Means of Optimization of the Radiographic
Parameters
The proposed self-calibration algorithm involves explicit use of the description of
the calibration matrices Mi
i
in order to estimate the geometrical parameters of the
radiographic setup leading to an optimal 3D reconstruction of all the spine
s ver-
tebrae [
7
,
8
]. The projective matrices Mi
i
used for the stereo-reconstruction of 3D
landmarks are modeled as:
'
ð
3
Þ
where R
i
is the rotation matrix de
ned by angular
ða
i
; b
i
; c
i
Þ
, T
i
is the translation
vector
ð
X
S
i
;
Y
S
i
;
Z
S
i
Þ
. Intrinsic parameters are modeled by the x
p
i
, y
p
i
coordinates of
