Image Processing Reference
In-Depth Information
μ
(
t
)=sinc(
πt
)=
sin(
Ω
t
)
2
.
(6.25)
Ω
2
it
If
Ω
, the extension or the support of
F
(
ω
) where it is nonzero, is an integer share
of 2
π
, i.e.,
Ω
=2
π/κ
for some integer
κ
, then the interpolator
μ
is a sinc function
in the
t
-domain with zero crosses at every
κ
th grid point. If
κ
=1, the zero crosses
will occur at every integer (except at the origin). We illustrate a
χ
,
μ
pair with
κ
=8
in Fig. 6.3. As a consequence of the previous discussion, shifting the above sinc
function yields a series of functions
μ
m
,
t
m
)) =
sin(
Ω
t
m
)=sinc(
Ω
(
t
−
t
m
))
2
μ
m
(
t
)=
μ
(
t
−
2
(
t
−
(6.26)
Ω
2
(
t
−
t
m
)
which are capable of reconstructing any band-limited function
f
via
f
(
t
)=
m
f
(
t
m
)
μ
m
(
t
)
(6.27)
However, it can also be shown
2
μ
m
}
m
=
−∞
that these functions
{
are orthogonal to
each other:
μ
m
,μ
n
=
2
π
Ω
δ
(
m − n
)
(6.28)
under the scalar product,
=
∞
−∞
f
∗
g
f, g
(6.29)
Thus, the band-limited functions constitute a Hilbert space with the above scalar
product.
3
Accordingly, any band-limited signal
f
can be reconstructed via an or-
thogonal expansion and the scalar product from Eq. (6.29)
f
(
t
)=
m
F
(
ω
m
)
μ
m
(
t
)
(6.30)
where
(6.31)
In consequence, the coefficients
F
(
ω
m
) in Eq. (6.30) must equal to
f
(
t
m
)=
f
(
m
2
Ω
) appearing in Eq. (6.27)
F
(
ω
m
)=
F
(
ω
m
)=
f, μ
m
/
μ
m
,μ
m
μ
m
,μ
m
μ
m
,f
=
μ
m
,f
=
f
(
m
2
π
Ω
)
(6.32)
(
2
Ω
)
In other words, the
projection coefficients
of a band-limited signal
f
onto the sinc
functions basis are the samples of the signal on the grid:
2
See Exercise 7.2.
3
This is not too surprising because just another representation of the same functions, i.e.,
their FTs, constitutes the space of limited extension functions. We have already seen that
such functions constitute a Hilbert space with a scalar product enabling their reconstruction
via FCs and the Fourier series.
