Image Processing Reference
In-Depth Information
Under the passage to the limit, the discrete sequence of
F
(
ω
m
) will approach a func-
tion
F
(
ω
) that is continuous in
ω
almost everywhere
1
. This is because at any fixed
ω
,
F
(
ω
) can be approximated by
F
(
ω
m
) as (i) for some
m
, the difference between
ω
ω
m
will be negligible, and (ii) the analysis formula, Eq. (5.13), exists almost
everywhere when
ω
m
changes. Thus, the synthesis formula, Eq. (6.4), approximates
the “measure/area” (integral) of the limit function
F
(
ω
) exp(
iωt
) better and better,
−
to the effect that
will be replaced by
while
Δω
will be replaced by d
ω
. The anal-
ysis formula remains as an integral, but the integration domain approaches the entire
real axis, [
]. In consequence,
f
T
will approximate
f
, the function that we
started with, ever better. Because
f
is integrable, the analysis formula will converge
as
T
increases. As mentioned, the value
ω
m
is a constant in the analysis integral and
it can therefore be replaced by the fixed point
ω
, yielding
F
(
ω
). We formulate these
results in the following theorem:
−∞
,
∞
Theorem 6.1 (FT).
Provided that the integrals exist, the integral transform pair
∞
1
2
π
F
(
f
)(
ω
)=
F
(
ω
)=
f
(
t
) exp(
−
iωt
)
dt
(Forward) FT
(6.8)
−∞
F
−
1
(
F
)(
t
)=
f
(
t
)=
∞
−∞
F
(
ω
) exp(
iωt
)
dω
(Inverse) FT
(6.9)
defines the
Fourier transform
(FT) and the
Inverse Fourier transform
, which relate a
pair of complex-valued functions
f
(
t
)
and
F
(
ω
)
, both defined on the domain of the
real axis. The function
F
is called the Fourier Transform of the function
f
.
As their counterparts in the FC transform, the FT and inverse FT equations are
also referred to as
analysis formula
and
synthesis formula
, respectively. The pair
(
f, F
) is commonly referred to as the FT pair. Because the forward FT is the same
as the inverse FT except for a reflection in the origin, i.e.,
F
−
1
(
f
)(
ω
),
the forward FT can be used to implement the inverse FT in practice. Reflection can
be implemented by reordering the result. Notwithstanding its simplicity, this obser-
vation is useful when practicing the FT and results in the following convenience.
F
(
f
)(
ω
)=
−
Lemma 6.1 (Symmetry of FT).
If
(
f
(
t
)
,F
(
ω
))
(6.10)
is an FT pair, so is
(
F
(
ω
)
,f
(
−
t
))
(6.11)
The conclusions and principles established assuming the forward direction are also
valid in the inverse direction.
1
Almost everywhere means everywhere except for a set of points with zero measure. See
[226] for further reading on measure and integral.
