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 ,
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 )=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.
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