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chet mean [
19
]. It is equivalent
to the conventional mean for vector spaces with a Euclidian distance.
The computation of the Fr
This generalization of the mean is called the Fr
é
cult
because of the presence of a minimization operator. Fortunately, a simple gradient
descent procedure can be used to compute the mean [
31
] when the exponential and
logarithmic maps are known. This procedure is summarized by the following
recurrent equation:
é
chet mean directly from the de
nition is dif
N
X
N
1
l
n
þ
1
¼
Exp
l
n
ð
Log
l
n
ð
x
i
ÞÞ:
ð
7
Þ
i
¼
0
This equation is guaranteed to converge. Moreover, in practice it converges
rather quickly. We observed that it generally converged in less than
five iterations
for articulated models of the spine.
To use Eq. (
7
), it is necessary to initialize the mean to start the procedure. The
initial value can be one of the points of the set from which the mean is computed.
Furthermore, more than one starting point can be tried to test the uniqueness of the
mean and escape local minimums.
The Fr
chet mean is not unique in general, and the starting point of the iterative
procedure is theoretically important. However, in the case of an articulated model of
the spine, multiple strategies were tried, but generally produced the exact same results.
é
3.3 Covariance and Variability
In conventional statistics, variations would
first be studied with the covariance
matrix. However, because the covariance matrix is de
ned using the conventional
mean, we need to de
chet mean.
The exponential map can be intuitively understood as a local linearized vector
space around a given point on a manifold. Thus, as a simple generalization, it is
possible to simply compute a covariance matrix around the Fr
ne its generalization around the Fr
é
chet mean on the
exponential map. Mathematically, this can be expressed by the following:
é
T
Log
l
ð
R
¼
E Log
l
ð
x
Þ
x
Þ
N
X
N
ð
8
Þ
1
T
Log
l
ð
Log
l
ð
x
i
Þ
x
i
Þ:
¼
i¼0
This generalized covariance computed in the tangent space of the mean and
associated variance are connected because Tr
2
, which is also the case for
ðRÞ
¼
r
the usual vector space de
nitions.
Articulated models contain several rigid transformations and sub-models, which
can all be correlated to each other. To explore the covariance of a whole articulated
