Image Processing Reference
In-Depth Information
trigonometric circle that represents the angles from 0 to 2
π
. This can always be
achieved by stretching or shrinking the argument of the periodic function
f
(
t
) to
yield
t
=
2
T
t
. In terms of physics,
t
becomes a “dimensionless” quantity since it is
an angle. The quantity
ω
1
=
2
π
T
(5.2)
serves the goal that
t
obtains values in [0
,
2
π
] when
t
is assigned values in [0
,T
].
Theorem 5.1.
The space of periodic functions that share the same period is a Hilbert
space with the scalar product:
=
T
2
f
∗
g
f, g
(5.3)
T
2
−
This is the scalar product that we have seen before, except that it has been given a
precision with respect to function type and the integration domain. It is straightfor-
ward to extend it to higher dimensions where the integration domains will be squares,
cubes, and hypercubes.
5.2 Fourier Coefficients (FC)
Since there are many types of orthogonal functions that are periodic, one speaks of
orthogonal function families
. A very useful orthogonal function family is the
com-
plex exponentials
:
ψ
m
(
t
)=exp(
iω
m
t
)
(5.4)
where
ω
m
=
m
2
π
T
and,
m
=0
,
±
1
,
±
2
,
±
3
···
(5.5)
In other words for each
ω
m
, we have a different member of the same family. The
function family is then
,
e
−i
2
ω
1
t
,
e
−iω
1
t
,
1
,
e
iω
1
t
,
e
i
2
ω
1
t
,
{
ψ
m
}
m
=
{···
···}
(5.6)
This is also called the
Fourier basis
, because the set acts as a basis for function spaces
and defines the Fourier transform, Sect. 6.1. Here, we discuss the Fourier series as
an intermediary step.
We can verify that the members of the Fourier basis are orthogonal via their
mutual scalar products. Assuming that
m
=
n
, we can find the primitive function to
the exponential easily and obtain
