Resmi N.G.Reference:
Digital Signal ProcessingRafael C. GonzalezRichard E. Woods
� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties
� Notch filter� Lowpass Filter� Highpass Filter
� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters
� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering
� Homomorphic Filtering
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Basics of Filtering in Frequency Domain
1. Multiply the input image by (-1)x+y to center the transform.
2. Compute the DFT, F(u,v) of the resulting image.
3. Multiply F(u,v) by a filter function H(u,v) to obtain G (u,v).3. Multiply F(u,v) by a filter function H(u,v) to obtain G (u,v).
4.Compute the inverse DFT of G(u,v) to obtain g*(x,y).
5. Obtain the real part of g*(x,y).
6. Multiply the result by (-1)x+y to obtain g (x,y).
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Basic Steps for Filtering in Frequency Domain
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� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties
� Notch filter� Lowpass Filter� Highpass Filter
� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters
� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering
� Homomorphic Filtering
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Basic Filters and Properties� Notch Filter
� It is a constant function with a hole at the origin.� Sets F(0,0) to zero.
� Lowpass Filter� It attenuates high frequencies and passes low frequencies.
� Highpass Filter� It attenuates low frequencies and passes high frequencies.
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� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties
� Notch filter� Lowpass Filter� Highpass Filter
� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters
� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering
� Homomorphic Filtering
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Smoothing Frequency Domain Filters
�Low Pass Filter (Smoothing Filter)� The result in the spatial domain is equivalent to that of
a smoothing filter as the blocked high frequenciesa smoothing filter as the blocked high frequenciescorrespond to sharp intensity changes, i.e. to the fine-scale details and noise in the spatial domain image.
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High Pass Filter(Sharpening Filter)� A highpass filter attenuates the low-frequency
components without disturbing the high frequencyinformation in the Fourier Transform.
� It yields edge enhancement or edge detection in thespatial domain, because edges contain many highfrequencies. Areas of constant gray level consistmainly of low frequencies and are thereforesuppressed.
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Band Pass Filter� A bandpass filter attenuates very low and very high
frequencies, but retains a middle range band offrequencies. Bandpass filtering can be used to enhanceedges (suppressing low frequencies) while reducing theedges (suppressing low frequencies) while reducing thenoise(attenuating high frequencies).
� Bandpass filter is a combination of both lowpass andhighpass filters. These filters attenuate all frequenciesbelow a specific frequency and above a specific frequency,while retaining the frequencies between the two cut-offs.
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� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties
� Notch filter� Lowpass Filter� Highpass Filter
� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters
� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering
� Homomorphic Filtering
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Ideal Low Pass Filters
0
0
1 ( , )( , )
0 ( , )
.
Transfer Function
if D u v DH u v
if D u v D
D is a specified non negativequantity
≤= >
−
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( ) ( )
0
122 2
.
( , ) 2 2
D is a specified non negativequantity
D(u,v)is thedistance from point (u,v)to theoriginof
the frequency rectangle.
NMD u v u v
−
= − + −
� Ideal – because all frequencies inside a circle of radius D0are passed without any attenuation, whereas allfrequencies outside the circle are completely attenuated.
� The point of transition between H(u,v) = 1 and H(u,v) = 0is called the cut-off frequency.
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Ideal Low pass Filter
� Produces “Ringing” effect.� Cannot be realized in electronic components.� Cannot be realized in electronic components.� Not very Practical
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Butterworth Low Pass Filters� The transfer function of a BLPF of order n, and with cut-
off frequency at a distance D0 from the origin, is definedas
2
1( , )
( , )nH u v
D u v=
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( ) ( )
2
0
22
20
( , )1
1
2 21
n
n
D u vD
NMu v
D
+
= − + − +
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� Provides a smooth transition between low and highfrequencies.
� Butterworth filter of order 1 has neither ringing nornegative values.
� BLPF of order 2 has mild ringing and small negativevalues.
� Reduced ringing effect than ILPF.
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Gaussian Low Pass Filters2
2( , )
2( , )D u v
H u v e
D(u,v)is thedistance fromtheoriginof the Fourier
Transform.
σ−
=
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2
20
0
( , )2
0
.
,
( , )
.
D u vD
Transform.
is ameasureof the spread of theGaussiancurve
When D
H u v e
whereD is thecut off frequency
σσ
−
=
=
−
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20
20
12 20
( , ) 0, ( , ) 1
( , ) , ( , ) 0.607D
D
WhenD u v H u v
WhenD u v D H u v e e−
−
= =
= = = =
Gaussian Low Pass Filters
� Very smooth filter function.� Inverse DFT of the Gaussian lowpass filter is Gaussian.� No “Ringing” effect.� No “Ringing” effect.
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Applications of Low Pass Filters� In the field of machine perception
� Character Recognition
� In printing and publishing industry.� In printing and publishing industry.� Cosmetic processing prior to printing
� For processing satellite and aerial images.
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� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties
� Notch filter� Lowpass Filter� Highpass Filter
� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters
� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering
� Homomorphic Filtering
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Sharpening Frequency Domain Filters
� Ideal High Pass Filters� Transfer Function of high pass filter is given by
( , ) 1 ( , )hp lpH u v H u v= −
� That is, when low pass filter attenuates frequencies, high pass filter passes them and vice versa.
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( , )
.
hp lp
lpH u v is thetransfer functionof corresponding
low pass filter
� Opposite of ideal lowpass filter.
� Sets to zero all frequencies inside a circle of radius D0while all frequencies outside the circle are passed without
0
0
0 ( , )( , )
1 ( , )
if D u v DH u v
if D u v D
≤= >
while all frequencies outside the circle are passed withoutattenuation.
� Not physically realizable with electronic components.
� Produces ringing effect.
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D0 = 15,30,80
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Butterworth High Pass Filter
2
0
1( , )
1( , )
nH u vD
D u v
=
+
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� Represents a transition between the sharpness of IHPF andthe total smoothness of Gaussian filter.
D0 = 15,30,80
Gaussian High Pass Filter2
20
( , )2( , ) 1
D u vDH u v e
−
= −
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D0 = 15,30,80
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� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties
� Notch filter� Lowpass Filter� Highpass Filter
� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters
� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering
� Homomorphic Filtering
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Enhancement using The Laplacian
2 2
( )( ) ( )
( , ) ( , )
nn
n
d f xju F u
dx
d f x y d f x y
ℑ =
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2 22 2
2 2
2 2
2 2
2 2
( , ) ( , )( ) ( , ) ( ) ( , )
( ) ( , )
( , ) ( , )( , ).
d f x y d f x yju F u v jv F u v
dx dy
u v F u v
d f x y d f x yis the Laplacianof f x y
dx dy
ℑ + = +
= − +
+
( ) ( )
2 2 2
2 2
22
( , ) ( ) ( , )
, ( , ) ( ).
( , ) ( )2 2
f x y u v F u v
Laplaciancanbeimplemented in the frequency domain
using the filter H u v u v
NMH u v u v shifted
∴ℑ ∇ = − +
= − +
= − − + −
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The Laplacian filtered imageinthe spatial domain is
obtained by comput
( ) ( )222 1
( , ) ( , ) :
( , ) ( , )2 2
ing theinverse FourierTransform
of H u v F u v
NMf x y u v F u v− ∇ = ℑ − − + −
Unsharp Masking and High Boost Filtering� High pass filters eliminate the zero frequency component
of their Fourier transforms and hence average backgroundintensity reduces to near black.
� Solution: Add a portion of the image back to the filteredresult.
� Enhancement using Laplacian adds the entire image backto the filtered result.
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� Unsharp masking consists of generating a sharp image by subtracting a blurred version of an image from itself.
� That is, obtaining a highpass-filtered image by subtracting from the image a lowpass-filtered version of itself.
( , ) ( , ) ( , )f x y f x y f x y= −
� High-boost filtering generalizes this by multiplying f(x,y) by a constant A≥1.
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( , ) ( , ) ( , )hp lpf x y f x y f x y= −
( , ) ( , ) ( , )hp lpf x y Af x y f x y= −
� High-boost filtering thus increases the contribution made by the image to the overall enhanced result.
� When A=1, high-boost filtering reduces to regular highpass filtering.
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High Frequency Emphasis Filtering� To increase the contribution made by high-frequency
components of an image.
� Multiply a highpass filter function by a constant and addan offset so that the zero frequency term is not eliminatedby the filter.by the filter.
� Filter transfer function is given by
� Where a ≥0 and b>a.
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( , ) ( , )hfe hpH u v a bH u v= +
Module 2 Assignment� Explain the following point operations:
� Contrast Stretching� Range Compression� Image Clipping� Image Clipping
� Explain Homomorphic Filtering.� Explain Convolution and Correlation Theorems.
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Thank YouThank You
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