Information Technology Reference
In-Depth Information
This example illustrates that powerful prior models such as a statistical model of
the spine based on articulated modeling make it possible to mitigate the effect of
noise. In the particular case of the reconstruction of the spine from two radiographs,
it appears that using an articulated model of the spine may be interesting only when
there is a high level of noise.
However, a more important problem for 3D spine reconstruction using radio-
graphs is the amount of user interaction required and the impossibility of
finding
meaningful correspondences for certain anatomical landmarks.
4.2 Completing Partial Models
Thus, another interesting problem would be to complete a three-dimensional spine
model that has been partially reconstructed. Incomplete models can be caused by
several factors. Surgical instrumentation could have occulted parts of the spine.
Certain vertebrae may be outside the radiograph
s borders, or a user may just want
to save time by reconstructing only a few vertebrae.
The problem then involves reconstructing an articulated model of the whole
spine based on a few vertebrae. The idea is to use the statistical model to
'
ll the
gaps by computing the most likely model given a set of constraints. This can be
done, in the case of a Gaussian distribution, by minimizing the Mahalanobis dis-
tance while preserving certain constraints. This idea may be formalized by the
following equation:
S
S
R
1
S
T
¼
arg min
S
ð
15
Þ
T
absolute
i
T
i
T
absolute
i
1
:
¼
8
2
Subject to
T
i
i
K
p
i
;
j
¼ p
i
;
j
8
i
;
j
Þ2
L
;
where T
absolute
i
~
p
i
;
j
are known anatomical
landmarks, K is the set of all known vertebrae, and L is the set of known landmarks.
In summary, Eq. (
15
) states that we seek the closest model to the mean with
respect to the Mahalanobis distance. This model should, however, match the poses
and shapes of the already available vertebrae.
The optimization method used to minimize Eq. (
15
) is sequential quadratic
programming [
40
]. It was selected because the cost function is quadratic and the
constraints are usually close to linear constraints when an initial solution suf
are the absolute poses of known vertebrae,
ciently
close to the optimum is provided. Sequential quadratic programming is a gener-
alization of Newton
s method for unconstrained optimization, which iteratively
solves a quadratic model of the problem using linear approximations of the con-
straints. Like Newton
'
is method, it is a local optimization method, and it is subject
to entrapments in local minima.
'
