Image Processing Reference
In-Depth Information
the filter
g
1
1.Im-
plementing the convolution directly via a 2D filter, as in Eq. (7.34), will require
m
2
operations/pixel. Implementing it as in Eq. (7.38) demands first
m
operations/pixel
to compute
h
=
g
2
∗
will be of size 1
×
m
, whereas the filter
g
2
will be of size
m
×
f
, then another
m
operations/pixel to compute the final
h
, total-
ing to 2
m
operations/pixel. It should be stressed that both ways of implementing the
filtering will yield results that are identical. Assuming a typical filter with
m
=20,
the gain in terms of arithmetic operations is, however, a factor of 10. This is an appre-
ciable difference for most applications because it translates to a speed-up of the same
amount in conventional computation environments. For this reason, separable filters
are highly attractive filters, in image and other multidimensional signal processing
applications.
7.5 Poisson Summation Formula, the Comb
We elaborate in this section on how to sample a function formally. We will use dis-
placed dirac distributions to achieve this. For a quick initiation, the theorems and
lemmas can be directly read without proofs.
Assume that we have an integrable function
F
(
ω
), which is not necessarily a
finite frequency function. Then we define the periodized version of it
F
(
ω
) as
F
(
ω
)=
n
F
(
ω
+
nΩ
)
(7.39)
Evidently,
F
(
ω
) is periodic with the period
Ω
, and we can expand it in terms of FCs,
see Eq. (6.12)
Ω
m
F
(
ω
)=
1
f
(
t
m
) exp(
−
it
m
ω
)
(7.40)
f
(
t
m
) through Eq. (6.13), if we keep in mind that
t
m
=
m
2
Ω
,
f
(
t
m
)=
where we obtain
Ω
2
F
(
ω
) exp(
it
m
ω
)
dω
Ω
2
−
=
Ω
2
F
(
ω
+
nΩ
) exp(
it
m
ω
)
dω
Ω
2
−
n
=
n
Ω
2
F
(
ω
+
nΩ
) exp(
it
m
ω
)
dω
−
Ω
2
Here, we changed the order of summation and integration which is permitted for
physically realizable functions. Accordingly,
f
(
t
m
) is obtained as:
