Image Processing Reference
In-Depth Information
is simply the square of the length of the basis vectors that can be computed ahead
of the time, and in fact also be used to rescale
{
e
}
i
. For this reason, it is justified to
assume that the length of basis vectors are normalized to 1,
e
i
=
e
i
/
e
i
,
e
i
(3.34)
so that
e
i
,
e
i
=1and the equation (3.33) reduces to
c
i
=
x
(
i
)=
x
,
e
i
(3.35)
The hat on
e
i
is a way to tell that this vector has the length 1. Often, the hat is even
omitted, when this fact is clear from the context.
Definition 3.4.
The vectors
u
m
and
u
n
are orthonormal if
u
m
,
u
n
=
δ
(
m
−
n
)
(3.36)
where
δ
is the
Kronecker delta
and is defined as
δ
(
m
)=
1
,
if
m
=0
;
0
,
otherwise.
(3.37)
In analogy with
E
3
,evenin
E
N
(as well as in
C
N
) there must be exactly
N
vectors
in order that they will be just sufficient, neither too many nor too few, to represent
any point in
E
N
. Also in
E
N
, such a set of vectors is called a
basis
. If the basis
vectors are orthogonal the basis is called an orthogonal basis.
3.6 Tensors as Hilbert Spaces
Discussed in linear algebra textbooks (e.g., [147, 210]) matrices are arrays with two
indices having scalar (real or complex) elements. Because addition and scaling are
well defined for matrices, we already know from Sect. 3.2 that they constitute a
vector space. Furthermore, in Sect. 3.2 we saw that actually all arrays with multiple
indices are vector spaces, and many of them correspond to real images. We only
need a suitable scalar product to make such spaces Hilbert spaces. For simplicity we
assume four indices below, but the conclusions are readily generalizable.
Lemma 3.2.
Let
A
and
B
be two arrays that have the same size. Then, with the
definition
=
k,l,m,n
A
(
k, l, m, n
)
∗
B
(
k, l, m, n
)
A
,
B
(3.38)
the space of such arrays is a Hilbert space.
To show that this fulfills the conditions of a scalar product, one can proceed as in Eq.
(3.27). This scalar product prompts the following norm:
=
A
2
=
|
2
A
,
A
A
(
k, l, m, n
)
(3.39)
k,l,m,n
|
