Graphics Reference
In-Depth Information
• The transform of the unit-width box-function
b
is the unit-spacing sinc
function, sinc
(
)=
sin(
π
v
)
π
v
, while the transform of sinc is the box
b
.
• The scaling property. If
g
(
x
)=
f
(
ax
)
, then
v
(
f
)
a
and
)=
1
F
(
g
)(
a
F
(19.1)
v
F
(
f
)(
)=
a
F
(
g
)(
a
)
.
(19.2)
v
v
• Band limiting and reconstruction. If
-
f
is a function in
L
2
(
R
)
, and
-
1
2
, and
-
y
i
is the sample of
f
at
i
for
i
F
(
f
)(
)=
0for
|
v
|≥
v
Z
then
f
can be reconstructed from the
y
i
's by convolution with sinc, that is,
∈
∞
f
(
x
)=
y
i
sinc
(
x
−
i
)
.
(19.3)
i
=
−∞
1
2
, then sampling of
f
will produce
aliases, in the sense that there is a band-limited function
g
whose samples
are the same as those of
f
, and reconstruction using Equation 19.3 will
produce
g
rather than
f
.
• A sequence of
L
2
functions that approach a comb with unit tooth-spacing
have Fourier transforms that approach a comb with unit tooth-spacing.
Rather than talking about sequences that approach a comb, we'll use the
symbol
Furthermore, if
f
is
not
band-limited at
as if there really were such a thing as a “comb function,” and
you'll understand that in any such use, there's an implicit argument about
limits being suppressed.
ψ
In this and the next section, we'll consider the problem of enlarging, or scaling up,
and shrinking, or scaling down, an image. You might think that such operations
would be straightforward, at least in some cases. If we have a 300
×
300 image, for
example, and want a 150
150 version, it seems as if simply throwing away alter-
nate rows and alternate columns would provide the desired result. Exercise 19.7
shows that this simple solution leads to very bad results, so we'll need a different
approach. Fortunately, the different approach we describe will solve the problem
of scaling up and down not only by small integer factors, but by any factor at all.
The scaling-up operation, which we address in this section, is relatively easy. The
scaling-down operation, discussed in the next section, has additional subtleties.
We'll work in one dimension as usual, so we'll start with a 300-sample discrete
signal (which we'll call the source) that we want to turn into a 400-sample dis-
crete signal (which we'll call the target). We'll assume that the 300-sample image
was generated by sampling a function
S
×
L
2
(
R
)
at 300 consecutive integer
∈
points.
We'll also assume that the signal
S
:
R
R
was strictly band-limited at
v
0
=
2
so that there was no aliasing when the samples were taken.
→
