Image Processing Reference
In-Depth Information
Any rule that operates on two vectors and produces a number out of them is not good
enough to be qualified to be a scalar product. To be called a scalar product, such an
operator must obey the
scalar product rules
:
∗
1.
u
,
v
=
v
,
u
α
∗
2.
α
u
,
v
=
u
,
v
3.
u
+
v
,
z
=
u
,
z
+
v
,
z
=0 iff
u
=0
Here the star is the complex conjugate, and “iff” is a short way of writing “if and
only if”. Remembering the first requirement, we note that the second is equivalent to
4.
u
,
u
>
0
if
u
=0
,
and
u
,
u
u
,α
v
=
α
u
,
v
(3.24)
As a byproduct, the last relationship offers a natural way of producing a norm for
a vector space having a scalar product:
u
=
u
,
u
(3.25)
Since the scalar product can be used to express the norm, the distance between
two vectors is easily expressed by scalar products as well:
u
−
v
2
=
u
−
v
,
u
−
v
(3.26)
Definition 3.2.
A vector space which has a scalar product defined in itself is called
a Hilbert Space.
We use the term “scalar product” and not the term “real” since this allows us to
use the same concept of Hilbert Space for vector spaces having complex-valued ele-
ments. The scalar products can thus be complex-valued in general, but never the norm
associated with it. Such vector spaces are represented by the symbol
C
N
. Hence a
scalar product for
C
N
must be defined in such a way that the auto-scalar product of
a vector represents the square of the length. The norm must be strictly positive or be
zero if and only if the vector is null. As a scalar product for
E
N
would be a special
case of the scalar product for
C
N
, it should be inherited from that of
C
N
. For this
reason the definition of a scalar product on
C
N
must be done with some finesse.
Lemma 3.1.
A scalar product for vectors
u
,
v
∈
C
N
, i.e., vectors with complex
elements having the same finite dimension, yields
=
i
u
(
i
)
∗
v
(
i
)
u
,
v
(3.27)
where
∗
is the complex conjugate and
u
(
i
)
,
v
(
i
)
are the elements of
u
,
v
.
We show the lemma by first observing that conditions 1-3 on scalar products are
fulfilled because these properties follow from the definition of
u
and
v
, and be-
cause
u
and
v
belong to the vector space. For any complex number
u
(
i
), the product
