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silhouettes of the morphed 3D models would therefore match the 2D information on
the biplanar X-rays in the image domain ui,
i
, replicating the speci
cs of a particular
scoliotic deformity.
4.4.3 Bundle Adjustment of the 3D Vertebral Landmarks
The crude statistical 3D model of the personalized spine is subsequently re
ned by
adjusting the 3D coordinates of the vertebrae. For a bundle adjustment of the 3D
landmark coordinates, a non-linear optimization method minimizes the cost function
E
and updates the 4 pedicle extremities and 2 endplate centers (6 anatomical
landmarks) in sl
l
at each vertebral level l (starting from L5 and progressing to T1),
based on the measures taken on the biplanar images.
ð
s
l
Þ
Cost Function
The Powell-Brent optimization method minimizes a cost function combining image
edge alignment from the 3D surface model, epipolar geometry correspondence and
morphological constraints formulated by Eqs. (
11
), (
13
) and (
14
) which are
described below:
E
ð
s
l
Þ
¼
x
1
D
edges
þ x
2
D
epipolar
x
3
D
morphology
ð
10
Þ
where
s
l
¼
R
ð
s
l
1
Þ
½
s
l
þ
T
ð
s
l
1
Þ
takes into account the previous updated vertebra
model and
is the rigid displacement of landmarks pi
i
at the previous vertebra
level s
l
1
before/after optimization. The weights
x
are dynamically assigned on a
vertebral level basis with
x
1
representing the image-based criterion of the cost
function regulated by a pixel coherence factor [
30
],
x
2
represents the epipolar
geometry constraint regulated by the calibration accuracy obtained in Sect.
3
, while
x
3
enforces the criterion such that
x
1
þ x
2
þ x
3
¼
ð
R
;
T
Þ
1. The set of 3D landmarks pi
i
for each vertebra si
i
are globally adjusted based on the following measures.
Image Gradient Edge Alignment
In order to integrate image-based information in the optimization process, a simi-
larity measure estimates the distance of the projection of a 3D deformed model to
the computed gradient of the X-rays. The approach would: (1) deform the prior
generic high resolution 3D vertebra model obtained from CT acquisitions, using the
level set surface evolution technique with the set of landmarks pi
i
evolving with the
same deformation Eq. (
9
); (2) project
the triangulated mesh and the distance
measure
uð
of the 3D model using the projection parameters of the 3D
radiographic scene to create a silhouette onto the images; (3) compute a 2D distance
map for these edges and; (4) sum over the distance map values at the locations
x
;
y
;
z
;
t
Þ
