Image Processing Reference
In-Depth Information
f
(
t
)=
m
f
(
mT
0
)
μ
(
t
−
mT
0
)
(9.4)
where
μ
is the new interpolation function obtained as a continuous convolution be-
tween two lowpass filters,
μ
=
μ
1
∗
μ
0
. The interpolation function
μ
can be assumed
to be equal to
μ
1
for the following reasons. In the frequency domain this convolution
will be realized as a multiplication between two characteristic functions. Ideal char-
acteristic functions
2
assume, by construction, only values 1 and 0 to define intervals,
regions, volumes, and so on. In the case of down-sampling,
μ
will be equal to
μ
1
because a large step size implies a characteristic function with a pass region that is
narrower than that of a smaller step size. After the substitution
t
=
nT
we obtain:
f
(
nT
)=
m
f
(
mT
0
)
μ
(
nT
−
mT
0
)
(9.5)
We can now restate this result by the substitution
T
=
κT
0
, where
κ
is a positive
integer:
f
(
nκT
0
)=
m
mT
0
)=
m
f
(
mT
0
)
μ
(
nκT
0
−
f
(
mT
0
)
μ
((
nκ
−
m
)
T
0
)
(9.6)
Here we used the values of
f
on the original fine
grid
mT
0
to form the scalar prod-
uct with the filter
μ
sampled on the same fine grid. We note, however, that the values
f
(
nκT
0
) are to be computed at a coarser grid, i.e., at every
κ
th point of the fine grid.
As compared to a full convolution, a reduction of the number of arithmetic operations
by the factor
κ
is possible by building the scalar products only at the new grid points,
i.e., at every
κ
th point of the original grid. In other words, the number of arithmetic
operations per new grid point is
M/κ
, where
M
is the size of the filter
μ
sampled at
the original grid positions. However, the size of the discrete filter is directly propor-
tional to the step size
T
=
κT
0
. Consequently, the number of arithmetic operations
per grid point does not change when changing
κ
.
Up-sampling
Up-sampling
with a positive integer factor of
κ
works in nearly the same way as
down-sampling.
First, the continuous signal
f
(
t
) is obtained as in Eq. (9.3):
f
(
t
)=
m
f
(
mT
0
)
μ
(
t
−
mT
0
)
(9.7)
where
μ
is the interpolation function associated with the original
grid
having the
period
T
0
.
2
In practice where computational efficiency, numerical, and perceptional trade-offs must be
made simultaneously, the interpolation filters will be inverse FTs of smoothly decreasing
functions.
