Image Processing Reference
In-Depth Information
The effects of sampling can often be seen in films, especially in the rotating wheels of
cars, as illustrated in Figure 2.10 . This shows a wheel with a single spoke, for simplicity.
The film is a sequence of frames starting on the left. The sequence of frames plotted in
Figure 2.10 (a) is for a wheel which rotates by 20° between frames, as illustrated in Figure
2.10 (b). If the wheel is rotating much faster, by 340°
between frames, as in Figure 2.10 (c)
and Figure 2.10 (d) then the wheel will appear to rotate the other way. If the wheel rotates
by 360°
between frames, then it will appear to be stationary. In order to perceive the wheel
as rotating forwards, then the rotation between frames must be 180° at most. This is
consistent with sampling at at least twice the maximum frequency. Our eye can resolve this
in films (when watching a film, I bet you haven't thrown a wobbly because the car's going
forwards whereas the wheels say it's going the other way) since we know that the direction
of the car must be consistent with the motion of its wheels, and we expect to see the wheels
appear to go the wrong way, sometimes.
20 °
(a) Oversampled rotating wheel
(b) Slow rotation
340 °
(c) Undersampled rotating wheel
(d) Fast rotation
Figure 2.10
Correct and incorrect apparent wheel motion
2.5
The discrete Fourier transform (DFT)
2.5.1
One-dimensional transform
Given that image processing concerns sampled data, we require a version of the Fourier
transform which handles this. This is known as the discrete Fourier transform (DFT). The
DFT of a set of N points p x (sampled at a frequency which at least equals the Nyquist
sampling rate) into sampled frequencies Fp u is:
2
1
N
-1
-
j
xu
N
Fp
=
=0
p
e
(2.15)
u
x
N
x
This is a discrete analogue of the continuous Fourier transform: the continuous signal is
replaced by a set of samples, the continuous frequencies by sampled ones, and the integral
is replaced by a summation. If the DFT is applied to samples of a pulse in a window from
sample 0 to sample N /2 - 1 (when the pulse ceases), then the equation becomes:
 
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