Image Processing Reference
In-Depth Information
=
T
2
ψ
n
,ψ
m
exp(
−
inω
1
t
) exp(
imω
1
t
)
dt
T
2
−
=
n
)
ω
1
t
)
t
=
T
1
2
exp(
i
(
m
−
i
(
m
−
n
)
ω
1
T
2
t
=
−
1
i
(
m − n
)
ω
1
=
[exp(
i
(
m
−
n
)
ω
1
T
)
−
exp(
i
0)]=0
(5.7)
where
m
=0. Now we also need to find the norms of these basis vectors, and
therefore we assume that
m
=
n
.
−
n
Ψ
m
2
=
T
2
ψ
m
,ψ
m
=
exp(
−
imω
1
t
) exp(
imω
1
t
)
dt
T
2
−
=
T
2
1
dt
=
T
(5.8)
T
2
−
The norm of the complex exponential is consequently independent of
m
. That is, any
member of this orthogonal family has the same norm as the others. This result is not
obvious, but thanks to the scalar product we could reveal it conveniently. Accord-
ingly, we obtain:
ψ
n
,ψ
m
=
Tδ
(
m − n
) (5.9)
where
δ
is the Kronecker delta, see Eq. (3.37). Since complex exponentials are or-
thogonal and we know their norms, any function
f
that has a limited extension
T
can
be reconstructed by means of them as:
f
(
t
)=
m
c
m
exp(
iω
m
t
)
(5.10)
where
c
m
s are complex-valued scalars. An arbitrary coefficient
F
(
n
) can be deter-
mined by taking the scalar product of both sides of Eq. (5.10) with exp(
iω
n
t
):
exp(
iω
n
t
)
,
m
exp(
iω
n
t
)
,f
=
c
m
exp(
iω
m
t
)
=
m
c
m
exp(
iω
n
t
)
,
exp(
iω
m
t
)
=
m
c
m
Tδ
(
m
−
n
)=
c
n
T
(5.11)
yielding,
T
2
1
T
1
T
c
n
=
exp(
iω
n
t
)
,f
=
f
(
t
) exp(
−
iω
n
t
)
dt
(5.12)
T
2
−
Consequently,
c
n
s are the
projection coefficients
of
f
on the Fourier basis. They are
also known as the
Fourier coefficients
(FCs). We summarize this result in a slightly
different way in the next theorem, to facilitate the derivation of the Fourier transform
in Sect. 6.1.
