Image Processing Reference
In-Depth Information
Infinitesimal
Signal amplitude by
infinitesimal width sampling
Averaged sampled point
by wide sampling width
Finite
sampling width
Signal amplitude by
finite width sampling
Input signal
FIGURE 6.3
Sampling width and the sampled signal amplitude.
points are in the middle of the peaks and troughs (corresponding to gray in the images).
The signal curve that is obtained by tracing the sampled points is flat with no amplitude,
as shown in Figure 6.2b. In this case, neither the amplitude nor the frequency is retained.
Thus, this phase is also important, especially around the Nyquist frequency, as will be seen.
An input signal whose frequency is higher than the Nyquist frequency is shown in
Figure 6.2a. The reproduced signal obtained by tracing the sampled points is very differ-
ent from the original input signal, as shown in the figure. This false signal is called alias-
ing or folding noise. Thus, a higher sampling frequency is necessary to obtain accurate
signal information for a higher Nyquist frequency.
In the above description, the sampling width is assumed to be infinitesimal. However,
the actual sampling operation cannot be realized with a zero sampling width, but requires
some finite width. Figure 6.3 shows the impact of the sampling width on the sampled result.
In the case of an infinitesimal sampling width, the maximum and minimum values of
the input signal are reflected in the sampled result. However, in the case of sampling with
a finite width, the sampling is carried out during the sampling period by integration or
averaging. Thus, the maximum and minimum values cannot be directly reflected as sam-
pled points. Accordingly, a larger amplitude of the sampled signal is obtained by sampling
with a narrower sampling width.
6.2 Sampling in Space Domain
A schematic diagram of spatial sampling is shown in Figure 6.4a. Pixels are periodically
arrayed with sampling pitch
p
and aperture
a
in real space. Since only light that passes
through an aperture can reach the sensor parts, the sampling operation is only carried
out in the aperture area; that is, the aperture width is the same as the sampling width. As
the sampling pitch is
p
, the sampling frequency
f
s
equals 1/
p
and the sampling width is
a
.
Under this condition, the frequency dependency of the sampled signal amplitude of sine
wave input signals is shown in Figure 6.4b. The frequency is normalized by the sampling
