Graphics Reference
In-Depth Information
To scale up a source image in two dimensions, we simply scale each row first,
then scale each column (see Figure 19.3).
Scale up
rows
Inline Exercise 19.2:
Convince yourself that scaling rows-then-columns
results in the same image as scaling columns-then-rows.
Scale up
columns
We now turn to the more complicated problem of scaling
down
an image
I
, that is,
making it smaller. We'll assume that the source image has
N
pixels and the target
image has only
K
N
pixels.
Once again we can reconstruct from integer samples by convolving with sinc
to get the original signal
S
. But in the next step, when we build
T
(
x
)=
S
K
N
<
x
,
(19.6)
y
=
F
(
I
)(
v
)
1
the resultant function is no longer band-limited at
2
; instead, it's band-limited at
N
2
K
1
2
. Before we could safely sample
T
, we would have to band-limit it to
frequency
2
by convolving with sinc.
>
y
=
b
(
v
)
Let's look at the process in the frequency domain, as shown schematically in
Figure 19.4. In reconstructing
S
, we convolved with sinc, that is, we multiplied
by a width-one box in the frequency domain. When we squashed
S
to produce
T
,
we stretched the Fourier transform of
S
correspondingly, and then needed to once
again multiply by a unit-width box.
1
1
-
2
2
Reconstruct
F
(
S
)
=
F
(
I
)
?
b
N before
stretching, as
shown in Figure 19.5? Then when we stretch the result by
N
What if we instead multiplied by a box of width
K
/
F
(
T
)
/
Stretch
K
, we'll already be
v
0
=
2
. So, in the frequency domain, the sequence of operations
band-limited at
is as follows.
F
(
T
)
?
b
Filter
1. Multiply by
v
→
b
(
)
to reconstruct.
v
K
2. Multiply by
v
→
b
((
N
/
K
)
)
to band-limit to
2
N
.
v
1
3. Stretch by a factor of
N
/
K
; the result is band-limited at
2
.
Figure
19.4:
Band-limiting
S
after scaling.
4. Convolve with
ψ
as a result of sampling at integer points.
Notice that the first two steps can be combined: Multiplying by a wide box and
then a narrow box gives the same result as multiplying by just the narrow box! This
means that instead of reconstructing and then band-limiting, we can reconstruct
and band-limit in a single step, by using a wider sinc-like function in the recon-
struction process. Instead of
x
y
=
F
(
I
)(
v
)
1
N
2
v
y
=
b
K
1
1
-
Reconstruct
and band-limit
N
sinc
(
N
x
)
.After
that, all the remaining steps are the same. The program is shown in Listing 19.3.
K
2
→
sinc
(
x
)
, we need to use
x
→
2
N
1
1
22
F
(
S
)
?
v
A
b
v
K
Listing 19.3: Scaling down a one-dimensional source image.
Stretch
1
2
3
4
5
6
void
scaledown(source, target, N, K)
{
F
(
T
)
?
b
assert(K <= N);
for
(j = 0; j < K; j++) {
target[j] = SL((N/K)
*
(j + 0.5) - 0.5, source, N, K);
Figure
19.5:
Band-limiting
S
}
before scaling.
