Image Processing Reference
In-Depth Information
(a)
(b)
FIGURE 6.1 (See color insert)
Comparison of sampling frequency dependence of spatial information quality: (a) fine sampling with 1318 × 1106
pixels; (b) course sampling with 64 × 54 pixels (without smoothing).
: Sampled point
: Reproduced signal
Aliasing
(a)
f > f s /2
(b)
f = f s /2
Phase 2
(c)
f = f s /2
Phase 1
(d)
f << f s
Sampling
points
p : Sampling pitch
f s : Sampling frequency
FIGURE 6.2
Sampling and sampling theorem.
points, it is easily understood that any input signal with a higher frequency than f s /2 can-
not be reproduced accurately by the sampling frequency f s . Therefore, the maximum fre-
quency that can be reproduced is just half the sampling frequency. This frequency is called
the Nyquist frequency, and the relation is called the Nyquist theorem or the sampling
theorem.
Denoting the sampling pitch and the Nyquist frequency as p and f N , respectively, we
obtain the following relations:
1
2
1
2
f
==
f
(6.1)
N
p
However, an input signal whose frequency is f s /2 is not always reproduced accurately.
Figure 6.2b shows an input signal whose frequency is f s /2, which is the same as Figure 6.2c,
the only difference being that the phase is a quarter cycle late. The positions of the sampling
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