Image Processing Reference
In-Depth Information
5.3 (Parseval-Plancherel) Conservation of the Scalar Product
We consider a scalar product between two members of our Hilbert space of periodic
functions,
f
=
2
T
m
F
(
ω
m
)
ψ
m
and
g
=
2
T
m
G
(
ω
m
)
ψ
m
:
=
2
π
T
2
F
(
ω
m
)
ψ
m
,
n
f, g
G
(
ω
n
)
ψ
n
m
=
2
π
T
2
F
∗
(
ω
m
)
G
(
ω
n
)
ψ
m
,ψ
n
mn
=
2
π
T
2
F
∗
(
ω
m
)
G
(
ω
n
)
Tδ
(
m
−
n
)
mn
(2
π
)
2
T
F
∗
(
ω
m
)
G
(
ω
m
)
=
(5.18)
m
To obtain Eq. (5.18), we changed the order of the summation and the integration (of
the scalar product). This is allowed for all functions
f
and
g
that are physically real-
izable. The
δ
(
m
n
) is the Kronecker delta whereby we obtained the last equality.
Notice that the Kronecker delta reduced the double sum to a single sum by replac-
ing
n
with
m
everywhere before it disappeared, a much appreciated behavior of
δ
under summation. We explain the reason for this “sum-annihilating” property of the
Kronecker delta. Assume that we have written down all the terms one after the other,
−
F
∗
(
ω
1
)
G
(
ω
1
)
δ
(1
−
1)
, F
∗
(
ω
1
)
G
(
ω
2
)
δ
(1
−
2)
, F
∗
(
ω
1
)
G
(
ω
3
)
δ
(1
−
3)
···
(5.19)
The only terms that are nonzero are those when both indices are equal,
n
=
m
.
Accordingly, we can reach all terms that are nonzero in the sum by using a single
index in a single sum. Behaviorally, this is the same as saying that
δ
“replaces” one
of its indices in every other terms such that the argument of the
δ
becomes zero and
it “erases” the sum corresponding to the disappeared index, before vanishing itself.
The set of the discrete Fourier coefficients
{
F
(
ω
m
)
}
m
=(
···
F
(
ω
−
2
)
,F
(
ω
−
1
)
,F
(
ω
0
)
,F
(
ω
1
)
,F
(
ω
2
)
···
)
(5.20)
can be viewed as a vector with an infinite number of elements. Members constitute a
faithful and unique representation of
f
because of the synthesis formula, Eq. (5.14).
Can such infinite dimensional vectors be considered a Hilbert space on their own ac-
count? Yes, indeed. Scaling and addition are extended versions of their counterparts
in finite dimensional vector spaces:
(
···
αF
(
ω
−
1
)
,αF
(
ω
0
)
,αF
(
ω
1
)
,
···
)=
α
(
···
F
(
ω
−
1
)
,F
(
ω
0
)
,F
(
ω
1
)
,
···
)
(5.21)
and
(
···F
(
ω
−
1
)+
G
(
ω
−
1
)
,F
(
ω
0
)+
G
(
ω
0
)
,F
(
ω
1
)+
G
(
ω
1
)
, ···
)
=(
···
F
(
ω
−
1
)
,F
(
ω
0
)
,F
(
ω
1
)
,
···
)+(
···
G
(
ω
−
1
)
,G
(
ω
0
)
,G
(
ω
1
)
,
···
)
