Information Technology Reference
In-Depth Information
model, one only needs to consider the logarithmic map of the Cartesian product of
its components. In other words, it is possible to concatenate the logarithmic map of
individual inter-vertebral transformations and sub-models to obtain the logarithmic
map of the whole articulated model and use it to compute the generalized covari-
ance matrix.
3.4 Graphical Visualization
It is crucial to be able to visualize a statistical shape model in an intuitive way in
order to ef
filled with numbers are far
from being ideal. Fortunately, there is an easier and more intuitive method to do so
using an articulated statistical model.
Visualizing the mean model is rather simple; one can simply render the mean
model, as it would be done for any of the individual models that were used to build
the statistical models. For instance, Fig.
5
was produced by rendering a template
model that was deformed based on the positions and orientations of the Fr
ciently communicate results. Large tables
chet
mean computed over a large set of individual spine models from scoliotic patients.
However, it is more challenging to properly illustrate the variabilities. One
relatively easy way to do so is to focus on the individual covariance matrices
associated with the rotations and translations. Each three-by-three matrix can then
be visualized as a 3D ellipsoid. The ellipsoid geometric description is obtained by
performing an eigenvalue decomposition on the covariance matrix. The eigenvec-
tors become the principal axes of the ellipsoid, and the eigenvalues are used to scale
the length of the ellipsoid along the principal axes.
In the case of translations, the ellipsoid can be interpreted by thinking of the
lengths along the principal axes as the standard deviations of the translations
measured along these three-dimensional directions. For rotations, the interpretation
relies on the fact that the exponential map is given by the rotation vector. Therefore,
the lengths of the ellipsoid along the principal axes may be interpreted as the
standard deviations of the rotations measured around these axes (see Fig.
4
).
é
ʻ
1
v
1
ʻ
2
v
2
z
y
ʻ
0
v
0
x
Fig. 4 Three-dimensional ellipsoids can be used to visualize covariance matrices associated with
translations or rotations. The generalized covariance matrices of these transformations are
decomposed into their eigenvectors v and eigenvalues
k
, which are used to determine the directions
and lengths of the ellipsoid
