Information Technology Reference
In-Depth Information
2
X
i
1
r
p
¼
k
i
:
ð
11
Þ
¼
1
...
n
A shape model can be recreated from the coordinates of the transformed tangent
space simply by going back to the original tangent space and projecting the model
on the manifold using the exponential map. Thus, if
a
i
is the coordinate associated
with the ith principal component, the following equation can be used to re-create a
shape model:
X
k
S ¼
Exp
l
ð
a
i
a
i
Þ:
ð
12
Þ
i¼1
The visualization of the components allows an analysis of not only the variations
of individual inter-vertebral transformations, but also of the variations of the spine
shape as a whole. In a sense, the method presented in Sect.
3.4
allowed us to
analyze the variations of individual translations and rotations in the articulated
models of the spine, and component analysis can be used for the major modes of
variation of the whole spine. These visualization methods are thus complementary.
Component analysis helps identifying and analyzing deformation trends that
changes the shape of the whole spine. This can often be used to better understand
different concurrent factors that contribute to the variation in the spine
'
s shape. For
instance, the
first principal deformation mode of our database of scoliotic patients
illustrated in Fig.
8
shows an elongation of the spine combined with the develop-
ment of a thoracic curve. This component explains the largest amount of variance in
the spine shapes. It could also be analyzed more generally as being a combination
Fig. 8 First principal deformation modes for a dataset of adolescent idiopathic scoliosis patients.
Spine models were rendered (from left to right) for
3, 0, and 3 times the standard deviation
explained by the corresponding deformation mode. a Front view. b Lateral view
−
