Image Processing Reference
In-Depth Information
represents periodic functions by a basis defined as a set of infinite complex exponentials.
That is,
jk
t
ct
() =
ce
(7.2)
k
k
=-
Here, ω defines the fundamental frequency and it is equal to
T
/2π where
T
is the period of
the function. The main feature of the Fourier expansion is that it defines an orthogonal
basis. This simply means that
T
ftftdt
()
() = 0
(7.3)
k
j
0
for
k
j
. This property is important for two main reasons. First, it ensures that the expansion
does not contain redundant information (each coefficient is
unique
and contains no information
about the other components). Secondly, it simplifies the computation of the coefficients.
That is, in order to solve for
c
k
in Equation 7.1, we can simply multiply both sides by
f
k
(
t
)
and perform integration. Thus, the coefficients are given by
T
T
2
c
=
ctf tdt
()
()
f tdt
()
(7.4)
k
k
k
0
0
By considering the definition in Equation 7.2 we have that
T
=
1
-
jk
t
c
cte
()
dt
(7.5)
k
T
0
In addition to the exponential form given in Equation 7.2, the Fourier expansion can also
be expressed in trigonometric form. This form shows that the Fourier expansion corresponds
to the summation of trigonometric functions that increase in frequency. It can be obtained
by considering that
jk
t
-jk t
ct
( ) =
c
+
(
ce
+
c e
)
(7.6)
0
k
-
k
k
=1
In this equation the values of
e
jk
ω
t
and
e
-
jk
ω
t
define pairs of complex conjugate vectors.
Thus
c
k
and
c
-
k
describe a complex number and its conjugate. Let us define these numbers
as
c
k
=
c
k
,1
-
jc
k
,2
and
c
-
k
=
c
k
,1
+
jc
k
,2
(7.7)
By substitution of this definition in Equation 7.6 we obtain
jk
t
-
jk
t
jk
t
-
jk
t
e
+
2
e
-
e
+
2
e
ct
( ) =
c
+ 2
c
+
jc
(7.8)
0
k
,
1
k
,2
k
=1
That is,
ct
( ) =
c
+ 2
(
c
cos(
k t
) +
c
sin(
k t
))
(7.9)
0
k
,1
k
,2
k
=1
If we define
a
k
= 2
c
k
,1
and
b
k
= 2
c
k
,2
(7.10)
