Transcript Chapter 4 cont.

Filters in the Frequency Domain
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Image Smoothing Using Frequency Domain Filters:
◦ Ideal Lowpass Filters
◦ Butterworth Lowpass Filters
◦ Gaussian Lowpass Filters.
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Image sharpening Using Frequency Domain Filters:
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Ideal Highpass Filters
Butterworth Highpass Filters
Gaussian Highpass Filters.
The Laplacian in the Frequency Domain
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We’ll begin with lowpass filters.
Edges and other sharp intensity transitions (such as noise) in
an image contribute significantly to the high-frequency
content of its fourier transform. Hence, smoothing (blurring)
is achieved in the frequency domain by high frequency
attenuation; that is, by lowpass filtering.
In this section we consider three tuype of lowpass filters:
ideal, Butterworth, and Gaussian.
These three categories cover from very sharp (ideal) to very
smooth (Gaussian) filtering.
The Butterworth has a parameter called the filter order filter.
For high order values it approaches the ideal filter. Foe lower
order values, it is more like the Gaussian filter.
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A 2-D lowpass filter that passes without
attenuation all frequencies within a circle of
radius D0 from the origin and “cuts off” all
frequencies outside this circle is called an
ideal lowpass filter (ILPF)
It is specified by the function:
Where D(u,v) is the Distance from point (u,v) from
the origin of the frequency rectangle.
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The name ideal indicates that all frequencies
on or inside a circle of radius D0 are passed
without attenuation, where as all frequencies
outside the circle are completely attenuated
(filtered out).
ILPFs have blurring and ringing properties as
shown in figure 4.42
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The transfer function of a Butterworth
lowpass filter (BLPF) of order n, and with
cutoff frequency at a distance D0 from the
origin is defined as:
Unlike the ILPF, the BLPF transfer function does not have a sharp
discontinuity that gives a clear cut off between passed and filtered
frequencies.
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Gaussian lowpass filters (GLPFs) if 2-D is
given by:
• The GLPF achieved slightly less
smoothing than the BLPF of order 2
for the same value of cut off
frequency.
• GLPF ensure that there is no
ringing at all.
•In cases where tight control of the
transition between low and high
frequencies about the cut off
frequency are needed, then the BLPF
presents a more suitable choice.
In figure 4.51 the objective here is to blur out as much details as
possible while leaving large features recognizable.
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In this section edges and other abrupt changes in intensities
are associated with high- frequency components.
image sharpening can be achieved in the frequency domain
by highpass filtering, which attenuates the low-frequency
components without disturbing high-frequency information
in the Fourier transform.
In this section, we will consider ideal, Butterworth, and
Gaussian highpass filters. As before we’ll see that Butterworth
filters represent a transition between the sharpness of ideal
the ideal filter and the broad smoothness of the Gaussian
filter.
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A 2-D ideal highpass filter (IHPF) is defined as:
• Where D0 is the cutoff frequency.
•The IHPF is the opposite of the ILPF .
•IHPF has the same ringing properties as the ILPF.
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A 2-D Butterworth highpass (BHPF) of order n and cutoff
frequency D0 is defined as:
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The transfer function of the (GHPF) is given by:
The results obtained in figure 4.56 are more gradual than with the
previous two filters. Even the filtering of the smaller objects and
thin bars is cleaner with the Gaussian filter.
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The Laplacian is used in spatial filtering for
image enhancement and it yields equivalent
results using frequency domain teqniques.