33
Fourier Transform
image(y_magnitude);
colormap(gray(256) );
xlabel(‘u’);
ylabel(‘v’);
The center in the DFT image shows the zero frequency, and, as a result, lowfrequency components of the signals are the dots close to the origin, while the
points far away from the center (origin) represent the high-frequency components
of the image.
The properties of 2-D DFT are very similar to continuous signals and 1-D DFT.
Some of these properties that play important roles in image processing and are
further explored in Problems section.
2.6 FILTER DESIGN
In this section, we discuss using frequency domain or DFT to design practical filters.
In preprocessing a signal, it is often the case that some details of the signals must
be altered. For example, high frequencies in a signal are often considered to have
been corrupted by high-frequency noise (which is the case for many applications). In
such cases, one needs to somehow be able to filter the high frequency in the signal.
The schematic diagram of Figure 2.14 shows how a filter H(f) can be used to filter a
signal (or image) P(f) to generate a processed signal (or image) Q(f).
Often the users of a system have a reasonably reliable idea on the range of
frequency for noise. For instance, for high-frequency noise, we may know that
majority of the noise energy is at a given frequency and above. In such cases, the
ideal scenario is to design a low-pass filter to eliminate the noise. The shape of
the ideal low-pass filter is shown in Figure 2.15.
Knowing that Q(f) = H(f), any frequency higher than the cutoff frequency D 0 in
the original signal P(f) is eliminated in the filtered signal Q(f). This is ideal filtering of high frequencies using an ideal low-pass filter. Similarly, one can imagine an
ideal high-pass filter (Figure 2.16) as well as an ideal band-pass filter (Figure 2.17).
In each case, the desired frequencies are preserved and the unwanted frequencies
are eliminated.
P( f )
Q( f )
H( f )
FIGURE 2.14 Filtering signals and images using filter H(f).
1
|H|
D 0
f
FIGURE 2.15 Ideal low-pass filter.
Fourier Transform
image(y_magnitude);
colormap(gray(256) );
xlabel(‘u’);
ylabel(‘v’);
The center in the DFT image shows the zero frequency, and, as a result, lowfrequency components of the signals are the dots close to the origin, while the
points far away from the center (origin) represent the high-frequency components
of the image.
The properties of 2-D DFT are very similar to continuous signals and 1-D DFT.
Some of these properties that play important roles in image processing and are
further explored in Problems section.
2.6 FILTER DESIGN
In this section, we discuss using frequency domain or DFT to design practical filters.
In preprocessing a signal, it is often the case that some details of the signals must
be altered. For example, high frequencies in a signal are often considered to have
been corrupted by high-frequency noise (which is the case for many applications). In
such cases, one needs to somehow be able to filter the high frequency in the signal.
The schematic diagram of Figure 2.14 shows how a filter H(f) can be used to filter a
signal (or image) P(f) to generate a processed signal (or image) Q(f).
Often the users of a system have a reasonably reliable idea on the range of
frequency for noise. For instance, for high-frequency noise, we may know that
majority of the noise energy is at a given frequency and above. In such cases, the
ideal scenario is to design a low-pass filter to eliminate the noise. The shape of
the ideal low-pass filter is shown in Figure 2.15.
Knowing that Q(f) = H(f), any frequency higher than the cutoff frequency D 0 in
the original signal P(f) is eliminated in the filtered signal Q(f). This is ideal filtering of high frequencies using an ideal low-pass filter. Similarly, one can imagine an
ideal high-pass filter (Figure 2.16) as well as an ideal band-pass filter (Figure 2.17).
In each case, the desired frequencies are preserved and the unwanted frequencies
are eliminated.
P( f )
Q( f )
H( f )
FIGURE 2.14 Filtering signals and images using filter H(f).
1
|H|
D 0
f
FIGURE 2.15 Ideal low-pass filter.