Image Processing Reference
In-Depth Information
Together, Equation 2.1 and Equation 2.9 form a relationship known as a
transform pair
that
allows us to transform into the frequency domain, and back again. By this process, we can
perform operations in the frequency domain or in the time domain, since we have a way of
changing between them. One important process is known as
convolution
. The convolution
of one signal
p
1
(
t
) with another signal
p
2
(
t
), where the convolution process denoted by *,
is given by the integral
pt p t
()
() =
p
( )
p t
( -
)
d
(2.10)
1
2
1
2
-
This is actually the basis of systems theory where the output of a system is the convolution
of a stimulus, say
p
1
, and a system's
response
,
p
2
. By inverting the time axis of the system
response, to give
p
2
(
t
-
) we obtain a
memory
function. The convolution process then
sums the effect of a stimulus multiplied by the memory function: the current output of the
system is the cumulative response to a stimulus. By taking the Fourier transform of Equation
2.10, where the Fourier transformation is denoted by
F
,
the Fourier transform of the
convolution of two signals is
-
jt
Fp t
[
( )
p t
( )] =
p
(
)
p t
( -
)
d
e
dt
1
2
1
2
-
-
(2.11)
=
pt
( -
)
e
-
jt
dtp
( )
d
2
1
-
-
)] =
e
-
j
ωτ
Now since
F
[
p
2
(
t
- τ
Fp
2
(ω ) (to be considered later in Section 2.6.1), then
-
j
Fp t
[
( ) ( )] =
p t
Fp
(
)
p
(
)
e
d
1
2
2
1
-
-
j
=
Fp
(
)
p
( )
e
d
(2.12)
2
1
-
=
Fp
2
(ω
) ×
Fp
1
(ω )
As such, the frequency domain dual of convolution is multiplication; the
convolution
integral
can be performed by
inverse
Fourier transformation of the
product
of the transforms
of the two signals. A frequency domain representation essentially presents signals in a
different way but it also provides a different way of processing signals. Later we shall use
the duality of convolution to speed up the computation of vision algorithms considerably.
Further,
correlation
is defined to be
pt
() () =
pt
p
( ) ( + )
p t
d
(2.13)
1
2
1
2
-
where
is another symbol which is used sometimes, but there is
not much consensus on this symbol). Correlation gives a measure of the
match
between the
two signals
p
2
(
denotes correlation (
) we are correlating a signal with itself and
the process is known as
autocorrelation
. We shall be using correlation later, to
find
things
in images.
) and
p
1
(
). When
p
2
(
) =
p
1
(
