Nonideal Sampling
Practical sampling devices use
rectangular pulses of finite width. That is, the
samplingfunction is an array of rectangular pulses rather than impulses.
Therefore, the sampling function can be written as
where p (x, y) is a rectangular
pulse of finite extent and of unit area. Then the sampling process is
equivalent to convolving the continuous image with the pulse p (x, y) and then
sampling with a Dirac delta function .The net effect of sampling with a finite-width
pulse array is equivalent to prefilteringthe image with a lowpass filter
corresponding to the pulse p (x, y) followed by ideal sampling. The lowpass
filter will blur the image. This is the additional distortion to the aliasing
distortion that we discussed above. To illustrate the
effect the nonideal sampling has on the sampled image.
Consider an image shown in Figure 1.a, where
the image is a rectangular array of size 64 × 64 pixels. It is sampled by a
rectangular pulse of size 32 × 32 pixels as shown in Figure 1.b. The sampled
image is shown in Figure 1.c. We can clearly see the smearing effect of
sampling with a rectangular pulse of finite width. The 2D Fourier transforms of
the input image, sampling pulse, and the sampled image are shown in Figures 1.d–f,
respectively. Since the smearing effect is equivalent to lowpass filtering, the
Fourier transform of the sampled image is a bit narrower than that of the input
image. The Fourier transform of an ideal impulse and the Fourier transform of
the corresponding sampled image are shown in Figures 1.g,h, respectively. In
contrast to the sampling with a pulse of finite width, we see that the Fourier
transform of the ideally sampled image is the same as that of the input image.
Finally, the reconstructed image using an ideal lowpass filter is shown in
Figure 1.i. It is identical to the sampled image because we employed sampling
with a pulse of finite width.
Figure 1Effect of nonideal
sampling: (a) a 64 × 64 BW image, (b) rectangular sampling
image of size 32 × 32
pixels, (c) sampled image, (d) 2D Fourier transform of (a), (e) 2D Fourier transform
of (b), (f) 2D Fourier transform of the sampled image in (b), (g) 2D Fourier
transform of an ideal impulse, (h) 2D Fourier transform of (a) sampled by
impulse, and (i) image in (c) obtained by filtering by an ideal lowpass filter.
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