Image Processing Reference
In-Depth Information
7.3 Convolution, FT, and the
δ
We define first the convolution operation, which is bilinear.
Definition 7.2.
Given two signals
f
and
g
,the
convolution
operation, denoted by
∗
,
is defined as
g
)(
t
)=
∞
−∞
h
(
t
)=(
f
∗
f
(
t
−
τ
)
g
(
τ
)
dτ
(7.16)
It takes an ordinary variable substitution to show that the convolution is commutative:
g
)(
t
)=
∞
−∞
τ
)
g
(
τ
)
dτ
=
∞
−∞
h
(
t
)=(
f
∗
f
(
t
−
f
(
τ
)
g
(
t
−
τ
)
dτ
=(
g
∗
f
)(
t
)
(7.17)
Now we study the FT of the function
h
=
f
∗
g
.
iωt
)[
1
2
π
h
(
t
)=
F
(
f
∗
g
)(
ω
)=
exp(
−
f
(
t
−
τ
)
g
(
τ
)
dτ
]
dt
g
(
τ
)[
1
2
π
=
f
(
t
−
τ
) exp(
−
iωt
)
dt
]
dτ
g
(
τ
)[
1
2
π
=
f
(
t
) exp(
−
iω
(
t
+
τ
))
dt
]
dτ
1
2
π
iωτ
)
dτ
[
2
π
2
π
=
g
(
τ
) exp(
−
f
(
t
) exp(
−
iωt
)
dt
]
=2
πF
(
ω
)
G
(
ω
)
(7.18)
Consequently we have the following result, which establishes the behavior of
convo-
lution under the Fourier transform
.
Theorem 7.3.
The bilinear operator * transforms as
·
under the FT
h
=
f
∗
g
⇔
H
=2
πF
·
G
(7.19)
This is a useful property of FT in theory and applications. Below we bring further
precision as to how it relates to sampled functions. Before doing that, we present
the
Fourier transform of
δ
(
x
), the
Dirac distribution
, as a theorem for its practical
importance.
Theorem 7.4.
The Dirac-
δ
acts as the element of unity (one) for the convolution
operation:
δ
∗
f
=
f
∗
δ
=
f
(7.20)
