Image Processing Reference
In-Depth Information
extension
T
that is finite. This technique relies heavily on the fact that
T
is finite,
and therefore we can make copies of
f
to obtain a periodic function. This function
can in turn be expanded in Fourier series by use of a scalar product defined on a
period, e.g., [0
,T
]. In Eq. (5.13), we obtained the discrete set of values used in the
reconstruction. We viewed these values as an equidistant
sampling
of a continuous
function, although we did not speculate how this function of “imagination” behaved
between the grid points. As long as the function values turn out to be the correct
values on the grid, we can always take this view, by letting the values of
F
between
the discrete points
ω
m
=
m
2
T
vary according to some rule of our choice.
However, theorem 6.1 (FT) states that we can reconstruct
f
even if it is not a fi-
nite extension signal, albeit that the reconstruction now cannot be achieved by using
a discrete sequence of values
F
(
ω
m
), but instead a continuous function
F
(
ω
).By
using the FT theorem we are now, in fact, capable of reconstructing
f
even
with-
out
periodizing it. Assuming that the signal
f
is nonzero in the symmetric interval
[
−
T/
2
,T/
2], the FT of
f
is:
∞
T/
2
1
2
π
1
2
π
F
(
ω
)=
f
(
t
) exp(
−
iωt
)
dt
=
f
(
t
) exp(
−
iωt
)
dt
(6.14)
−∞
−T/
2
Eq. (6.14) tells us that not only the values of
F
are unique on the grid
ω
m
, but also
in the continuum between the grid points! The conclusion must be that there exists a
continuous function
F
(
ω
) that can be represented via its discrete values, also referred
to as
discrete samples
or
samples
, to such an extent that even the function values
between the discrete grid points are uniquely determined by the simple knowledge
of its discrete values on the grid
ω
m
. In other words, the discrete samples
F
(
ω
m
)
constitute not only an exact representation of the periodized original signal
f
,but
also an exact representation of
F
(
ω
), which is in turn an exact representation of the
original (unperiodized) function
f
. Notice that
F
(
ω
) is the Fourier transform of the
limited extension function
f
(without periodization).
How do we estimate (interpolate)
F
(
ω
) when we only know its values at discrete
points? The answer to this is obtained by substituting Eq. (5.14) in Eq. (6.14).
T/
2
1
2
π
F
(
ω
)=
f
(
t
) exp(
−
iωt
)
dt
−T/
2
T/
2
(
m
1
2
π
F
(
ω
m
)
2
π
T
=
exp(
iω
m
t
)) exp(
−
iωt
)
dt
−T/
2
T/
2
=
m
F
(
ω
m
)
2
π
T
1
2
π
exp(
−
i
(
ω
−
ω
m
)
t
)
dt
−T/
2
T/
2
=
m
F
(
ω
m
)
1
T
exp(
−
i
(
ω
−
ω
m
)
t
)
dt
−T/
2
=
m
F
(
ω
m
)
μ
(
ω
−
ω
m
)
(6.15)
where
