Digital Signal Processing Reference
In-Depth Information
A
1
/
2
A
−
1
/
2
BA
−
1
/
2
1
/
2
A
1
/
2
A
◦
B
=
γ(
1
/
2
)
=
,
(1.16)
which corresponds to the geometric mean of the matrices, i.e., a symmetrized version
of
2
.
The extension of the geometric mean computation of more than two matrices is
solved using the notion of Riemannian center, known as Karcher-Frechet barycen-
ter [
20
,
24
]. A fast and efficient algorithm proposed by F. Barbaresco [
8
,
9
]is
summarized as follows.
1
/
(
AB
)
N
i
Definition 5
For a set of matrices
A ={
A
i
}
1
, the Karcher-Frechet barycenter is
=
computed as
A
μ
(
A
)
=
X
k
+
1
such that
e
i
=
1
log
X
−
1
/
2
A
i
X
−
1
/
2
k
X
1
/
2
k
X
1
/
2
k
X
k
+
1
=
,
(1.17)
k
where
>
0 is the step parameter of the gradient descent.
For robustness purposes, it is probably more appropriate to consider the notion of
Riemannian median [
46
,
5
].
In summary, the algorithm for supremum matrix
A
sp
∨
:
diag
i
λ
1
(
A
i
),
···
,
i
1. Compute marginal supremum of eigenvalues:
∨
=
A
i
))
2. Compute Karcher-Frechet barycenter:
A
μ
=
λ
n
(
V
μ
μ
V
T
μ
3. Compute inverse spectral matrix:
A
sp
V
μ
∨
V
T
∨
=
μ
, a similar algorithm is defined for the matrix infimum
A
sp
∧
Mutatis mutandis
.
In Fig.
1.3
is given an example of the supremum/infimum obtained for a set of 10
PDS
∨
by
∧
matrices: the geometric mean, the supremum and the infimum are ellipsoids
with same orientation.
A
sp
(
2
)
and
A
sp
inherit the properties of the Karcher-Frechet barycenter. This ques-
tion will be considered in ongoing work. In any case, we insist again that
A
sp
∨
∧
and
∨
A
sp
do not produce dilation/erosion operators since they do not commute with
supremum/infimum, i.e., given two sets of PDS
∧
N
i
(
n
)
matrices
A ={
A
i
}
and
=
1
N
i
=
M
j
=
M
j
=
B ={
B
j
}
1
and let
C = A ∪ B ={
A
i
}
1
∪{
B
j
}
1
,wehave
sp
B
sp
A
sp
C
sp
∨
=
∨
∨
This is due to the fact that Karcher-Frechet barycenter is not associative, i.e.,
A
μ
◦
B
μ
=
C
μ
.
1.3.2 Spectral Sup/Inf on Optimized Basis
To complete this section, let us to mention briefly an alternative to tackle the problem
of defining the orthogonal basis of the supremum/infimum.
