Digital Signal Processing Reference
In-Depth Information
Obviously, any sup/inf input-preserving operator involves necessarily supre-
mum/infimum matrices belonging to the set PDS
.
We have now this classical result which can be easily proven.
(
n
)
Proposition 1
Any total ordering in
PDS
(
n
)
leads to
sup
/
inf
input-preserving oper-
ators.
We can now introduce two families of total orderings.
1.2.2 Lexicographic Total Orderings Based
on Tensor Invariants
N
i
Let us consider that the
N
matrices of
A ={
A
i
}
have been factorized in their
=
1
spectral form, i.e.,
V
i
i
V
i
A
i
=
where
i
is the diagonal matrix of ordered eigenvalues
i
=
diag
(λ
1
(
A
i
),
···
, λ
n
(
A
i
)) ,
with
λ
1
(
A
i
)
≥ ··· ≥
λ
n
(
A
i
)
, and
V
i
∈
SO
(
n
)
is the orthogonal matrix of eigenvec-
tor basis
v
1
(
A
i
),
···
,
v
n
(
V
i
=
A
i
)
,
v
1
(
v
A
i
),
v
k
(
such that
A
i
)
=
1 and
(
A
i
)
=
0,
∀
j
=
k
. This representation
j
is frequently used in this study.
We introduce the lexicographic spectral partial ordering
0
lex
≤
as follows.
0
Definition 2
lex
B
if
the ordered sequence of the eigenvalues of
A
is lexicographically smaller or equal
to the corresponding sequence of eigenvalues of
B
, i.e., if there exists an index
j
,
1
Let
A
and
B
be two PDS
(
n
)
matrices. We define that
A
≤
≤
j
≤
n
such that
λ
i
(
A
)
=
λ
i
(
B
)
for all
i
<
j
, and
λ
j
(
A
)<λ
j
(
B
)
if
j
≤
n
To be precise, it is a total ordering for the space of eigenvalues however is only an
antisymmetric preorder for PDS
. In fact, using their interpretation as ellipsoids,
two unequal matrices
A
and
B
can have the same shape, given by their eigenvalues
but different orientation in the space given by the orthogonal matrix basis. The most
natural way to complete the spectral ordering in order to have a total spectral ordering
involves fixing a reference orthogonal basis
R
0
, in such a way that for
A
and
B
having
the same eigenvalues the biggest is the matrix having an orthogonal basis closer to
R
0
;
this distance should be of course measured in
SO
(
n
)
. An additional question should
be taking into account concerning the choice of
R
0
. If the value of the reference
is independent of the image to be morphologically processed involves that a global
(
n
)
