Information Technology Reference
In-Depth Information
X
N
j¼0
½
k
2
x
a
i
ð
j
=
N
Þ
P
i
E
torsion
ðn
i
Þ
¼
ð
X
C
k
ð
j
=
N
Þ
Þk
ð
6
Þ
t
a
i
ð
j
=
N
Þ
P
i
2
þk
ð
T
C
k
ð
j
=
N
Þ
Þk
þkj
a
i
ð
j
=
N
Þ
P
i
2
ð
K
C
k
ð
j
=
N
Þ
Þk
where N is the number of Frenet frames along the spinal curve. The
rst two terms
evaluate the Euclidean distance between the analytical projection of X and T from
standard perspective transformation formulae using the current estimate of the
geometrical parameters and the image measures x and t respectively. The third term
measures the difference in curvature values
j
using (
14
). The method uses a bundle
adjustment approach based on an iterative nonlinear optimization process. The
Levenberg-Marquardt algorithm is used for optimization, iterating until the correc-
tion to the geometric parameters becomes negligible [
8
]. The set of parameters and
projection matrices is therefore regenerated and this procedure is repeated until the
system reaches a steady state, where the distance between the observed and computed
projection falls to a minimum. To avoid local minima, a directional optimization
approach is used to obtain a
first, coarse solution which is accurate enough to be used
as an initial guess. Moreover, higher reliability of the torsion parameters is ensured in
the optimization scheme by enforcing regularity in the Tikhonov sense with the term
b
. This helps to compensate the instability of the curvature parameters for patients
with strong deformations which can affect the convergence of the algorithm. The
regularization term acts as a dampening factor, controlling the quality of these
parameters by penalizing terms exhibiting very high tangential and torsional values.
4 3D Reconstruction of the Spine
In this section, we present the different methods for generating a 3D model of a
vertebra or spine from radiographic images. These methods are categorized in the
following classes: point-based, contour-based and statistical methods. We then
present a hybrid statistical and image-based approach to generate personalized 3D
reconstructions based on geometrical properties in Sect.
4.4
.
4.1 Point-Based Methods
The 3D reconstruction of point-based models is usually performed manually by an
expert operator and consists of locating six corresponding anatomical landmarks
(2 endplate midpoints + 4 pedicle extremities) on each vertebra from T1 (
rst
thoracic vertebra) to L5 (last lumbar vertebra) on a coronal and sagittal X-ray
(Fig.
5
). Additional non-stereo corresponding point (NSCP) landmarks on the
