Information Technology Reference
In-Depth Information
The distance d actually comes from the fact that rigid transformations constitute
a Riemannian manifold equipped with a metric. Because of this Riemannian
structure, it is possible to locally de
ne tangent planes to the manifold and map
vectors from these tangent planes to the manifold itself in such a way that the
magnitude of the vectors are consistent with the distances on the manifold.
We can express this mapping (called the exponential map) and its inverse
(the logarithmic map) around the identity transformation as follows:
R
ð
r
Þ
r
ð
R
Þ
Exp
Id
ð
T
Þ
¼
and Log
Id
ð
T
Þ
¼
;
ð
4
Þ
t
k
t
where R
are the conversion from the rotation vector to rotation matrix
and vice versa, respectively.
The exponential and logarithmic map around any rigid transformation
l
can then
be related to the map around the identity transformation. Because of the left-
invariance property of the distance d, that relation can then be expressed as follows:
ð
r
Þ
and r
ð
R
Þ
!
Þ
¼
l
J
L
ðlÞ
ð
1
Þ
!
Þ
Exp
l
ð
Exp
Id
ð
ð
5
Þ
Log
Id
ðl
1
Log
l
ð
T
Þ
¼
J
L
ðlÞ
T
Þ;
T
!
j
T
!
¼Id
). Its
detailed derivation can be found in [
32
]. Even though these exponential and
logarithmic maps are advanced mathematical concepts, they will be precious tools
to generalize statistical concepts for rigid transformations and, by extension, for
articulated models of the spine.
T
!
Þ
T
!
Þ
¼
@
T
!
where J
L
ð
is the Jacobian of the composition (J
L
ð
@
T
!
3.2 Mean and Centrality
As mentioned earlier, rigid transforms cannot be added together. They can, how-
ever, be composed, inversed, and compared using a valid distance. We should
therefore use a generalization of the conventional mean that takes advantage of
these properties.
It can be observed that the conventional mean minimizes the Euclidian distance
of the measures with the mean. Thus, given a general distance, a generalization of
the conventional mean would be to de
ne the mean as the element
l
of a manifold
M
that minimizes the sum of the distances with a set of elements x
0
...
N
of the same
manifold
M
. Thus,
l
is given by the following:
X
N
2
l
¼
arg min
x
2
M
d
ð
x
;
x
i
Þ
:
ð
6
Þ
i
¼
0
