Image Processing Reference
In-Depth Information
The way in is to look at the Fourier transform. This is a highly theoretical topic, but do not
let that put you off. The Fourier transform has found many uses in image processing and
understanding; it might appear to be a complex topic (that's actually a horrible pun!) but
it is a very rewarding one to study. The particular concern is the appropriate sampling
frequency of (essentially, the value for
N
), or the rate at which pixel values are taken from,
a camera's video signal.
2.3
The Fourier transform
The
Fourier transform
is a way of mapping a signal into its component frequencies.
Frequency
measures in hertz (Hz) the rate of repetition with
time
, measured in seconds (s);
time is the
reciprocal
of frequency and vice versa (hertz = 1/second; s = 1/Hz).
Consider a music centre: the sound comes from a CD player (or a tape) and is played on
the speakers after it has been processed by the amplifier. On the amplifier, you can change
the bass or the treble (or the loudness which is a combination of bass and treble).
Bass
covers the
low
frequency components and
treble
covers the
high
frequency ones. The
Fourier transform is a way of mapping the signal from the CD player, which is a signal
varying continuously with time, into its frequency components. When we have transformed
the signal, we know which frequencies made up the original sound.
So why do we do this? We have not changed the signal, only its representation. We can
now visualise it in terms of its frequencies, rather than as a voltage which changes with
time. But we can now change the frequencies (because we can see them clearly) and this
will change the sound. If, say, there is hiss on the original signal then since hiss is a high
frequency component, it will show up as a high frequency component in the Fourier
transform. So we can see how to remove it by looking at the Fourier transform. If you have
ever used a graphic equaliser, then you have done this before. The graphic equaliser is a
way of changing a signal by interpreting its frequency domain representation; you can
selectively control the frequency content by changing the positions of the controls of the
graphic equaliser. The equation which defines the
Fourier transform
,
Fp
, of a signal
p
, is
given by a complex integral:
-
jt
Fp
( =
)
p t e
()
dt
(2.1)
-
where:
Fp
(ω ) is the Fourier transform;
ω is the
angular
frequency, ω = 2π
f
measured in
radians/s
(where the frequency
f
is the reciprocal of time
t
,
f
= (1/
t
);
j
is the complex variable (electronic engineers prefer
j
to
i
since they cannot
confuse it with the symbol for current - perhaps they don't want to be mistaken for
mathematicians!)
p
(
t
) is a
continuous
signal (varying continuously with time); and
e
-
j
ω
t
= cos(ω
t
) -
j
sin(ω
t
) gives the frequency components in
x
(
t
).
We can derive the Fourier transform by applying Equation 2.1 to the signal of interest.
We can see how it works by constraining our analysis to simple signals. (We can then say
that complicated signals are just made up by adding up lots of simple signals.) If we take
a pulse which is of amplitude (size)
A
between when it starts at time
t
= -
T
/2 and when
it ends at
t
=
T
/2, and is zero elsewhere, the pulse is:
