Image Processing Reference
In-Depth Information
We will up-sample
f
(
t
) with the finer step size of
T
, i.e.,
T
=
T
κ
. There is ideally
no risk of destroying information with a finer step size, which should be contrasted
to down-sampling, cancelling the highest frequency contents that must be void for
the best fidelity. As before, sampling is achieved by replacing
t
with
nT
:
f
(
nT
)=
m
mT
0
)=
m
f
(
mT
0
)
μ
((
n
f
(
mT
0
)
μ
(
nT
−
κ
−
m
)
T
0
)
(9.8)
Here too we can identify the sum as a scalar product between the two discrete
sequences: the values of
f
on the original coarse grid
mT
0
and the continuous in-
terpolation function
μ
sampled with the original step size
T
0
. This is because the
summation index
m
generates the interpolator samples, whereas the quantity
nT
is
an offset that remains unchanged as
m
changes. We note that the values
f
will be de-
livered at a finer grid when changing
n
, i.e., at
κ
fractions of the original, coarser grid.
This will require as many arithmetic operations as the size of the filter
μ
, sampled at
the step of
T
0
. This size, which we denote by
M
, can be determined by truncating
the filter when it reaches a sufficiently low value at the boundary as compared to its
maximum value, typically at 1%. The filter size of
μ
is proportional to the sampling
step
T
0
. In up-sampling we make
T
smaller than
T
0
, but we neither change
μ
nor
T
0
so that the number of arithmetic operations
per point in the up-sampled signal
is
M
and remains constant for any
κ
. We note in the last expression of Eq. (9.8) that a
change of
n
results in a shift of
μ
, but only as much as an integer multiple of
T
0
/κ
,
where
κ
itself is a positive integer. After
κ
consecutive changes of
n
, the original
grid is obtained cyclically.
The function
f
(
nT
) can be generated as
κ
convolutions, each using its own frac-
tionally shifted up-sampling filter,
μ
. This is because in Eq. (9.8) the filter coef-
ficients used to generate
f
(
nT
) are the same as those generating
f
(
n
T
) only if
Mod (
n − n
,κ
)=0. An alternative is to view
μ
(
nT
mκT
) as a continuous
function
μ
sampled with the step
T
, but every
κ
th point is retained when forming the
scalar product. This is equivalent to filling
κ
−
1 zeros between the available values
of the discrete signal to be up-sampled, i.e.,
f
(
nT
0
), while retaining all points of
μ
:
f
(
nT
)=
m
−
mκT
)=
l
f
(
nT
)
μ
((
n
f
(
mκT
)
μ
(
nT
−
−
l
)
T
)
(9.9)
where
f
(
mT
)=
f
(
mT
)
,
if
m
Mod
κ
=0;
0
,
(9.10)
else.
Effectively, Eq. 9.9 implements up-sampling by first generating a sparse se-
quence
f
and then smoothing this by the filter
μ
,
sampled with the step size T
. This
process is simpler to remember and to implement, but one might argue that it is less
efficient because it involves
κ
1 multiplications with zeros. This is indeed the case,
if these are truly computed. On the other hand, multiplications with zeros occur cycli-
cally, meaning that they can be eliminated with some care in the implementation.
Example 9.1. We illustrate, in Fig. 9.1, the down-sampling process in1Dfor
κ
=2,
i.e., we wish to reduce the number of samples of a discrete signal by half keeping as
much descriptive power in the resulting samples as possible.
−