Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 32

Anti-aliasing
filter

f a ( x , y ) Sampler f ( n , m )

Bandlimited analog signal

FIGURE 2.10 Anti-aliasing filter.

the original signal should be low-pass filtered prior to sampling. In particular, the

low-pass filter should ideally eliminate all the frequencies that are higher than half

the sampling rate. The low-pass filter used to reduce or eliminate aliasing is referred

to as the anti-aliasing filtering has to be

applied prior to the sampling. Once the image has been inadequately sampled and the
high frequencies have been aliased into the low frequencies, there is no way to
eliminate the aliasing effect. The block diagram of a sampler with an anti-aliasing

filter. It should be noted that the anti-aliasing

filter is shown in Figure 2.10.
Aliasing is a common problem in digital cameras that use CCD sensor arrays.
This is usually due to the fact that the camera lens is of high quality and is capable of
accurately reproducing scenes containing frequencies higher than half the Nyquist
frequency (determined by the sensor resolution). The anti-aliasing

filter should be

placed somewhere between the lens and the sensor.

Example 2.1

In the first example, we consider the 1-D analog signal

f a (x) ¼ 2 cos (400 px)

The signal has one frequency component f 0 ¼ 200 cycles per unit length. The

Nyquist rate for this signal is f Nyquist ¼ 2f 0 ¼ 2 200 ¼ 400. If we undersample

this signal at the rate of f s ¼ 300 samples per unit length, we obtain

¼

400 p

300 n

4 3 n 4 3 n 2 3 n

f (n) ¼ 2 cos ¼ 2 cos 2 cos 2 pn ¼ 2 cos

Reconstructing the signal from these samples will result in a signal of lower
frequency. This lower frequency is given by

v 1 ¼ f s 2

3 ¼

200 p

Figure 2.11 shows the original analog signal and the reconstructed signal. As is
obvious from this figure, the signal is aliased due to undersampling.

Example 2.2

Consider the analog signal

f a (x, y) ¼ 2 cos (400 px þ 600 py)