Image Processing Reference
In-Depth Information
1. Confine
F
to the basic period around the origin. This operation has the purpose
to hinder
F
from being periodic because if
F
is not periodic but limited, then
f
is continuous. This is achieved by multiplying
F
with a suitable characteristic
function,
χ
D
(
ω
). Such a multiplication is equivalent to a convolution in the spa-
tial domain, leading to a reconstruction of the continuous signal from its sampled
signals, Eq. (8.6):
f
(
t
)=
m
f
(
t
m
)
μ
0
(
t
−
t
m
)
(9.1)
F
−
1
(
χ
D
).
2. Resample the continuous signal at the desired rate, which at all circumstances
should be done with a finer step size than what the highest frequency of the
signal content allows (Nyquist theorem). Effectively, this sampling of Eq. (9.1)
amounts to, sampling the interpolation function at the desired points. The equa-
tion itself becomes an ordinary discrete scalar product between the filter samples
and the old function samples,
f
(
t
m
):
f
(
t
n
)=
m
where
μ
0
=
f
(
t
m
)
μ
0
(
t
n
−
t
m
)
(9.2)
In Sects. 9.2 and 9.3 we will discuss further the choice of
μ
0
in practice. To establish
the principles of how the interpolation functions are used when performing up and
down-sampling of discrete signals, which we do next, it is sufficient to imagine them
as functions that look like tents for now.
Down-sampling
Here we will
down-sample
a given discrete signal
f
(
mT
0
) with an integer factor of
κ
by destroying as little information as possible. In that, we follow the procedure
outlined in items and 1 and 2, above.
We reconstruct the continuous signal
f
(
t
) and obtain
f
(
t
)=
m
f
(
mT
0
)
μ
0
(
t
−
mT
0
)
(9.3)
where
μ
0
is the interpolation function, the effective width of which is inversely pro-
portional to
Ω
0
. The highest frequency content of the signal
f
(
t
) is
Ω
2
and the grid
is a regular grid given by
t
m
=
mT
0
where
1
T
0
=
2
Ω
0
.
Before sampling
f
(
t
) with the new step size
T>T
0
, we must make sure that the
continuous
f
does not contain frequencies outside the new interval [
T
,
T
]. Some
of the high frequencies that were possible to represent with the smaller step size
T
0
are no longer possible to represent with the coarser step,
T
. This can be achieved first
in the mind, by convolving the continuous
f
(
t
) with a filter,
μ
1
having a frequency
extension that is the same as the new frequency interval, yielding:
π
−
1
For the sake of simplicity we have made this choice, although the argumentation still holds
if we choose
T
0
as
T
0
≤
2
Ω
0
.
