Image Processing Reference
In-Depth Information
vectors cannot be multiplied or divided with each other to yield new vectors. As a
result, an explicit computation of the matrix eigenvalues will become superfluous,
and the structure tensor will be automatically decomposed into a directional and a
nondirectional part.
Central to the structure tensor theory is the maximization of the scatter
k
T
Sk
.
Because
S
is a positive semidefinite matrix with real elements, i.e.,
S
=
A
T
A
for
some
A
having real coefficients, the scatter is (
Ak
)
T
(
Ak
) and is either zero or pos-
itive for
any
real vector
k
. By incorporating the complex conjugation into transposi-
tion, i.e.,
B
H
=(
B
∗
)
T
, the Hermitian transposition, the expression (
Ak
)
H
(
Ak
)=
k
H
Sk
will be either zero or positive even if the vector
k
has been expressed in a ba-
sis that has complex elements. Here, we will maximize
k
H
Sk
, assuming
k
may have
complex elements. First, we introduce a new basis that has complex vector elements,
using the unitary matrix:
U
H
1
i
i
1
,
1
.
(10.33)
1
√
2
1
√
2
−
i
k
=
U
H
k
,
U
H
=
where
and
U
=
−
i
1
Unitary matrices generalize orthogonal matrices in that a unitary matrix has in gen-
eral complex elements, and it obeys the relationship
U
H
U
=
UU
H
=
I
. Conse-
quently,
k
H
Sk
=(
Uk
)
H
SUk
=
k
H
Zk
(10.34)
where
Z
=
U
H
SU
(10.35)
1
2
S
(1
,
1) +
S
(2
,
2)
−
i
(
S
(1
,
1)
−
S
(2
,
2) +
i
2
S
(1
,
2))
=
S
(2
,
2) +
i
2
S
(1
,
2))
∗
i
(
S
(1
,
1)
−
S
(1
,
1) +
S
(2
,
2)
We call
Z
the
complex structure tensor
, and we can conclude that it represents the
same tensor as
S
, except for a basis change, and that both matrices have common
eigenvalues. They share also eigenvectors, but only up to the unitary transformation,
so that
k
Z
representing an eigenvector of
Z
is given by
k
Z
=
U
H
k
S
, with
k
S
being
an eigenvector of
S
. We define the elements of
Z
via the complex quantities
I
20
and
I
11
as follows.
Definition 10.3.
The matrix
I
11
−iI
20
iI
20
,
where
Z
=
1
2
I
20
=
S
(1
,
1)
−
S
(2
,
2) +
i
2
S
(1
,
2)
I
11
=
S
(1
,
1) +
S
(2
,
2)
(10.36)
I
11
is the complex representation of the structure tensor.
A matrix
Z
is called Hermitian if
Z
H
=
Z
, and if additionally
k
H
Zk
0, which is
the case by definition for the complex structure tensor, is called Hermitian positive
semidefinite. With the above representation, the elements of
Z
encode the
λ
max
,
λ
min
as well as
k
max
more explicitly than
S
. This is summarized in the following
theorem [28], the proof of which is found in Sect. 10.17.
≥
