Image Processing Reference
In-Depth Information
Theorem 5.2 (FC I).
There exists a set of scalars
F
(
ω
m
)
which can synthesize a
function
f
(
t
)
having the finite extension
T
,
T
2
1
2
π
F
(
ω
m
)=
f
(
t
) exp(
−
iω
m
t
)
dt
(Analysis)
(5.13)
T
2
−
such that
f
(
t
)=
2
π
T
F
(
ω
m
) exp(
iω
m
t
)
(Synthesis)
(5.14)
m
using
ω
m
=
m
2
T
and
m
=0
,
±
1
,
±
2
,
±
3
···
.
Exercise 5.1.
Prove the theorem.
HINT: Expand the function
T
2
π
f
in the Fourier basis.
While this is nothing but a convenient way of computing projection coefficients
in a function space where the functions are interpreted as points, few formulas have
had as much practical impact on science as these two, including signal analysis,
physics, and chemistry. Equations (5.13-5.14) are the famous Fourier
1
series. They
will be generalized further below to yield the usual Fourier transform on [0
,
∞
].
These formulas can be seen as a transform pair. In that sense the coefficients:
c
m
=
2
π
T
F
(
ω
m
)
(5.15)
constitute a
unique
representation of
f
, and vice versa.
It is worth noting that, per construction, the complex exponentials also constitute
a “basis” for sine and cosine functions whose periods exactly fit the basic period
one or more (integer) times. We can sometimes find the
projection coefficients
even
without integration since FCs are unique. We can, for example, write the sine and
cosine as weighted sums of complex exponentials by using the Euler formulas:
cos(
ω
1
t
)=
1
2
exp(
iω
1
t
)+
1
iω
1
t
)=
1
2
ψ
1
+
1
2
exp(
−
2
ψ
−
1
(5.16)
sin(
ω
1
t
)=
1
1
2
i
exp(
iω
1
t
)=
1
1
2
i
ψ
−
1
2
i
exp(
iω
1
t
)
−
−
2
i
ψ
1
−
(5.17)
The scalars in front of each orthogonal function
ψ
·
are then the projection coef-
ficients
c
m
s, very much like the coordinates of the ordinary vectors in
E
3
. They are
also referred to as
coordinates
.
/T
=
2
1
J. B. J. Fourier, 1768-1830, French mathematician and physicist.
cos(
ω
1
t
)
,
e
iω
1
t
Exercise 5.2.
Show that
