Digital Signal Processing Reference
In-Depth Information
6.1.1 Mean as a Variational Optimization
M
of a finite set of SPD matrices
To define a mean
, we model it using
the following optimization framework for a distance function
D
:
{
M
1
,...,
M
n
}
n
1
n
M
=
arg
min
D
(
M
i
,
M
).
(6.1)
Sym
+
(
M
∈
d
)
i
=
1
Taking
d
=
1, and choosing the squared Euclidean distance
D
(
P
,
Q
)
=
(
P
−
n
i
=
1
M
i
,
the arithmetic mean (minimizing the variance). The squared Euclidean distance is
derived from the inner product
2
M
1
Q
)
for positive numbers
P
,
Q
>
0, we get the center of mass
=
P
T
Q
of the underlying vector space:
P
,
Q
=
2
(
,
)
=
−
=
−
,
−
.
D
P
Q
P
Q
P
Q
P
Q
(6.2)
Thus to define the mean of square
matrices
P
,
Q
∈
M
(
d
,
d
)
, we can choose the
tr
F
=
(
MM
T
)
Fröbenius matrix norm
M
, and find the arithmetic matrix mean
n
i
=
1
M
i
as the minimizer of (
6.1
)for
D
M
1
2
F
. Although
trivial to compute, this arithmetic matrix mean has several drawbacks in practice.
For example, in DT-MRI [
9
], the Euclidean matrix mean may have a determinant
bigger than the input which is physically not plausible as matrices denote water flow
properties.
=
(
P
,
Q
)
=
P
−
Q
6.1.2 Log-Euclidean Mean
,
where log
M
is the
principal logarithm
of matrix
M
. The logarithm of a SPD matrix
is defined as the reciprocal operator of the exponentiation exp
M
The Log-Euclidean distance [
9
] is defined as
D
(
P
,
Q
)
=
log
Q
−
log
P
=
i
=
0
1
i
M
i
.For
!
SPD matrices
M
, we compute the eigendecomposition
R
T
M
=
R
diag
(λ
1
,...,λ
d
)
(6.3)
and deduce the log/exp matrices as
R
T
log
M
=
R
diag
(
log
λ
1
,...,
log
λ
d
)
(6.4)
and
R
T
=
(
λ
1
,...,
λ
d
)
.
exp
M
R
diag
exp
exp
(6.5)
Note that in general log
MN
=
log
M
+
log
N
and exp
(
M
+
N
)
=
exp
M
exp
N
.
This is only true when matrix commutes, that is
MN
0. Symmetric matrices
commute if and only if they share the same eigen spaces. The
Log-Euclidean mean
[
9
] inherits a vector space structure, and has a closed-form solution:
−
NM
=
