Image Processing Reference
In-Depth Information
T/
2
μ
(
ω
)=
1
T
exp(
−
iωt
)
dt
(6.16)
−T/
2
Clearly
F
(
ω
) is reconstructed from its samples
F
(
ω
m
) by using these as weight
functions for a series of a displaced versions of
μ
,the
interpolation function
. We can
identify
μ
in Eq. (6.16) as the Fourier transform of a piecewise constant function
χ
T
(
t
), which is constant inside the interval where
f
is nonzero, and zero elsewhere.
χ
T
(
t
)=
1
,
T
2
,
T
2
if
t
∈
[
−
];
(6.17)
0
,
otherwise
.
T
2
,
T
2
We will refer to
χ
T
as the
characteristic function
of the interval [
−
]. The rela-
tionship in Eq. (6.16) along with the definition in Eq. (6.17) reveals that
∞
μ
(
ω
)=
1
T
iωt
)
dt
=
2
π
χ
T
(
t
) exp(
−
T
F
(
χ
T
)(
ω
)
(6.18)
−∞
Computing the integral in Eq. (6.16) in a straightforward manner yields
exp(
t
=
T/
2
μ
(
ω
)=
1
T
−
iωt
)
(6.19)
−
iω
t
=
−T/
2
1
T
ω
2 sin(
ω
T
1
=
2
)
(6.20)
In consequence, defining the
sinc function
as
sinc(
ω
)=
sin
ω
ω
(6.21)
yields the interpolator function
μ
(
ω
)=sinc
T
2
ω
(6.22)
which, using Eq. (6.18), results in
2
π
sinc
T
2
ω
(
χ
T
)(
ω
)=
T
F
(6.23)
A characteristic function and its corresponding interpolation function is illustrated in
Fig. 6.3. Before we discuss the properties of the sinc functions further, we present a
few observations and elucidate the concept of functions with finite frequency exten-
sion, also known as band-limited functions.
ω
m−
1
=
2
T
, so that if the distance
between the samples defines the unit measurement,
ω
m
−
•
The sampling interval for
F
is given by
ω
m
−
ω
m−
1
=1, then
T
's
value is locked to
T
=2
π
. If another measurement unit is used to quantify the
length of the interval that samples
F
, alternatively the length of the period of
f
,
an ordinary scaling of the arguments will restore the correspondence. Because of
this, it is customary to assume that a finite extension 1D function is nonzero in
[
−
π, π
], and the sampling interval of its FT is equal to unit length.