Graphics Reference
In-Depth Information
ˆ
(
=
v
y
)
y
=
S
(
x
)
v
1
1
-
2
2
1
300
2
ˆ
(
y
=
v
)
y
=
T
(
x
)
=
S
x
400
v
1
1
-
2
2
Figure 19.2: When the band-limited function S at left is stretched along the x-axis to
form T, the transform of S at right is compressed, resulting in a more tightly band-limited
transform.
To resampl e
S
at 400 points, we're going to imagine three idealized but imprac-
tical steps, which we'll later refine to a practical algorithm.
1. Reconstruct
S
from the 300 samples in the image.
2. Stretch
S
along the
x
-axis by a factor of
400
300
.
3. Resample
S
at the 400 sample points
i
=
0,
...
399.
1
When we reconstruct
S
in step 1, it's band-limited at
2
. When we take
400 samples, if we want to avoid aliasing the signal must again be band-limited
at
v
0
=
1
2
. Fortunately, when we
stretch
the signal
S
on the
x
-axis, the Fourier transform
compresses
along the
-axis; the resultant signal is therefore still band-limited
at
2
. You can see, however, that when we want to
shrink
an image there will be
additional challenges. Figure 19.2 shows this schematically.
There is a difficulty with the idea of reconstructing a signal from its samples
at integer points. To do so, we need to know the value at
every
integer point, not
just 300 of them. For now, we're going to hide this problem by treating the source
image outside the range 0 to 299 as being zero. We'll return to this assumption in
Section 19.4.
By the way, for a function
f
on
Z
or
R
,the
support
of
f
is the set of all places
where it's nonzero, that is,
v
support
(
f
)=
{
x
:
f
(
x
)
=
0
}
.
(19.4)
If this set is contained in some finite interval, we say
f
has
finite support;
on
the other hand, if the support is not contained in any finite interval, then
f
has
infinite support.
Thus, the box function has finite support, while sinc has infinite
support. We've just chosen to treat our image samples as a function with finite
support. With this assumption, we can recover the original signal
S
as shown in
Listing 19.1. Note that rather than reconstructing the entire original signal, which
would require an infinite amount of work, we've merely given ourselves the ability
to evaluate this reconstructed signal at any particular point, thus converting the
abstract first step into something practical.
