Graphics Reference
In-Depth Information
periodic function. The limit
is
a periodic function, but that's not in
L
2
(
R
)
, because
it doesn't go to zero as
. On the other hand, it turns out that this periodic
limit
is
the Fourier transform of
f
. The proof of these claims is quite subtle;
Dym and McKean [DM85] provide the necessary details for those who have stud-
ied real analysis.
To summarize the preceding paragraph briefly, if we take an
L
2
(
R
)
function
f
and sample it at integer points to get
f
ω →±∞
2
(
Z
)
, then
∈
(
f
)=
F
F
(
f
)
ψ
,
(18.76)
where
is the comb function.
Suppose that
f
is strictly band-limited, that is,
ψ
1
F
(
f
)(
ω
)=
0for
|ω|≥
2
.
(
f
)
consists of disjoint replicates of
Then
(
f
)
. To recover the Fourier trans-
form of
f
from this, we need only multiply it by a box of width 1, which is the
function
b
. Multiplication by
b
in the frequency domain corresponds to convolu-
tion with
F
F
F
−
1
(
b
)
in the value domain. The inverse Fourier transform of
b
is the
function
x
sinc
(
x
)
. We conclude that
to reconstruct a band-limited function
from its samples, it suffices to convolve the samples with sinc.
This is a pretty big result. It says, for instance, that if you have an image
created by sampling a band-limited function
f
, you can recover
f
from the image
by convolving with a sinc. In one dimension, that means that if the samples are
called
f
j
, you can compute
→
f
(
x
)=
j
f
j
sinc
(
j
−
x
)
.
(18.77)
If
x
happens to be an integer—say,
x
=
3—then this sum becomes
f
(
3
)=
j
f
j
sinc
(
j
−
3
)
.
(18.78)
The arguments to sinc in this sum are all integers, and sinc is 0 at every integer
point, except that sinc
(
0
)=
1. So the sum simplifies to say
f
(
3
)=
f
3
sinc
(
0
)=
f
3
.
(18.79)
That's good: It says that to reconstruct the value of
f
at an integer point, you just
need to look at the sample there. What if
x
is
not
an integer? Then the sinc is
nonzero at every argument, and the value of
f
at
x
involves a sum of infinitely
many terms. This is clearly not practical to implement.
We'll soon discuss other approaches to reconstruction, but the central idea—
that we can reconstruct a function from its samples by convolving with sinc—
remains important. We'll use it repeatedly in the next chapter when we discuss
shrinking and enlarging images. Typically we'll apply this theoretical result mul-
tiple times to determine what computation we should be doing, and then, with the
ideal computation at hand, we'll determine a good approximation.
We've now seen two applications of convolving with sinc. The first is that
for any function
f
sinc is the band-limited function closest to
f
.
As you'll recall, that's because in the frequency domain, convolution with sinc
becomes “multiplication by a box,” which removes all high frequencies from
f
,
while leaving the low frequencies untouched. That's the Fourier transform of the
band-limited function closest to
f
, hence its inverse transform is the band-limited
function in the value domain closest to
f
. The second application is in reconstruc-
tion: To reconstruct a band-limited function from its samples, we convolve the
samples with a sinc.
∈
L
2
(
R
)
,
f
