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are known to be more prevalent than left thoracic curvatures among scoliotic
patients, this imbalance is re
ected in the mean spine shape. The normal healthy
kyphosis and lordosis are also present in the average model.
The variability of the relative rigid transformations between vertebral levels is
also interesting to observe. Its largest departure from the mean shape in terms of
orientation appears to be along the posterior
fl
anterior axis. This means that the
variability would be most noticeable in a posterior
-
anterior radiograph, which is
the one on which the Cobb angle is generally measured. Finally, one can note that
the most important translational variability appears to be in the axial direction,
which is explained by the fact that most patients are growing adolescents and,
consequently, the variations in the patients
-
heights are important.
As an alternative to relative rigid transformations between neighboring verte-
brae, one might instead consider the absolute transformations, where one vertebra
serves as a global reference. This may seem like an appealing option, because it
would reduce the number of rigid transformation compositions needed to use the
model in inference applications. However, as can be seen in Fig.
6
(where L5is
used as a reference), the variabilities become much larger and their values then
depend on the arbitrary choice of the reference vertebra. For these reasons, absolute
transformations should be used with caution.
The tools presented so far can also be used to study the effect of orthopedic
treatment on the spine shape of patients. The computation of the mean shape before
and after treatment remains identical to the procedure used in Figs.
5
and
6
.
However, the variability is computed on the rigid transformations that transform the
articulated model before treatment into the articulated model recorded after treat-
ment. Figure
7
illustrates this procedure on a cohort of patients who received
orthopedic braces to slow down the progression of scoliosis. It is also possible to
test for differences between the effect and a control group [
5
] to locate signi
'
cant
effects and help optimize treatments.
3.6 Component Analysis
Another useful way to visualize the variability in a large dataset is to
find unidi-
mensional axes along which the variability is particularly strong. Then, a series of
models can be reconstructed along these axes and viewed either as an animation or
side-by-side (which has obvious advantages for printed media).
The best-known method in this family is called principal component analysis
[
15
]. This method was developed for multi-dimensional vector spaces. However, it
was shown that it could also be applied to manifolds under certain conditions [
17
].
The general idea is that, unlike the manifold itself, the tangent plane around the
mean is a vector space, and its basis can be changed by applying a linear trans-
formation. Thus, we seek an orthonormal matrix AAA
T
ð
¼ I
Þ
to linearly transform
the tangent plane (Log
l
ð
g
Þ
¼ALog
l
ð
f
Þ
) so that the resulting components would
be uncorrelated to each other and have decreasing variances. In other words, the
