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8/3/2019 Chap4IEFD [Compatibility Mode]
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ImageEnhancement
inFrequencyDomainProf. D. S. Pandya
Computer Dept.U. V. Patel College of Engg.Mahesana
1
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ImageEnhancementin
requency oma n
Ima eEnhancementinS atialDomainvs.
ImageEnhancementinFrequencyDomain
our er er es
FourierTransform
2
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Background: Fourier SeriesBackground: Fourier Series
An eriodic si nals can beFourier series:
viewed as weighted sum
of sinusoidal signals withdifferent frequencies
Frequency Domain:view frequency as anindependent variable
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
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Fourier Tr. and Frequency DomainFourier Tr. and Frequency Domain
Time, spatial FrequencyFourier Tr.
DomainSignals DomainSignalsInv Fourier Tr.
1D, Continuous case
= dxexfuF uxj 2)()(Fourier Tr.:
= dueuFxf uxj 2)()(Inv. Fourier Tr.:
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Fourier Tr. and Frequency Domain (cont.)Fourier Tr. and Frequency Domain (cont.)

=
=1
/2)(
1)(
MMuxj
exfuFFourier Tr.:
u= 0,,M1
Inv. Fourier Tr.:
=1
/2)()(
MMuxj
euFxf
x= 0,,M1=0u
F(u) can be written as)(
)()(uj
euFuF=or)()()( ujIuRuF +=
22uIuRuF +=
= )(
tan)(1 uI
uu
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Example of 1Example of 1D Fourier TransformsD Fourier Transforms
Notice that the longer
the time domain signal,The shorter its Fouriertransform
(Images from Rafael C. Gonzalez and Richard
Wood, Digital Image Processing, 2nd Edition.
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Relation BetweenRelation Betweenx andx anduu
between samples in f(x) and let frequency resolution ube spacebetween frequencies components in F(u), we have
u = 1
Example: for a signal f(x) with sampling period 0.5 sec, 100 poin
Hz02.01
==u5.0100
apart by 0.02 Hertz or more.
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22Dimensional Discrete Fourier TransformDimensional Discrete Fourier Transform
2D DFT
=
=
+=0 0
)//(2),(1
),(x y
NvyMuxjeyxfMN
vuF
u= frequency in xdirection, u= 0 ,, M1
v= frequency in ydirection, v= 0 ,, N1
2D IDFT
+=1 1
)//(2),(),(
M NNvyMuxj
evuFyxf
= =u vx= 0 ,, M1
y= 0 ,, N1
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22Dimensional Discrete Fourier Transform (cont.)Dimensional Discrete Fourier Transform (cont.)
F(u,v) can be written as
),( vujevuFvuF
=orvuIvuRvuF +=
whereuI
,,, vuvuvu +=
= ),(
tan,vuR
vu
For the purpose of viewing, we usually display only the,
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forconvenience.Itcanbevisualizedas
WhereWiskernel,WN istheNth
root
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TheinverseDFTcanbewrittenas
11
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x={1289}
So ution:N=4, ence
12
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= ,
(t)isdefinedas
Animpulsehassiftingpropertywithrespecttointegration,
13
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14
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ConvolutionConvolution
Convolutionoftwofunctions
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o econvo u ono wo unc ons n espatialdomainisequaltotheproductinthe
requency oma no o e wo unc ons.
Conversely,convolutioninfrequencydomainisanalogoustomultiplicationinthespatial
domain.
16
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Example of 2Example of 2D DFTD DFT
o ce a e onger e me oma n s gna ,
The shorter its Fourier transform (Images from Rafael C. Gonzalez and RichardWood, Digital Image Processing, 2nd Edition.
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Example of 2Example of 2D DFTD DFT
Notice that direction of an
object in spatial image andIts Fourier transform are
.
(Images from Rafael C. Gonzalez and RichardWood, Digital Image Processing, 2nd Edition.
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Example of 2Example of 2D DFTD DFT
2D DFT
Original image
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Example of 2Example of 2D DFTD DFT
2D DFT
Original image
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Basic Concept of Filtering in the Frequency DomainBasic Concept of Filtering in the Frequency Domain
),(),(),(),(),(),( vuGvuHvuFyxhyxfyxg ==
We cam perform filtering process by using
Multiplication in the frequency domainis easier than convolution in the spatial
Domain.
(Images from Rafael C. Gonzalez and Richard E.Wood, Digital Image Processing, 2nd Edition.
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Filtering in the Frequency Domain with FFT shiftFiltering in the Frequency Domain with FFT shift
F(u,v)H(u,v)
(User defined)g(x,y)
FFT shift2D IFFTX
2D FFT FFT shift
f(x,y) G(u,v)
In this case, F(u,v) and H(u,v) must have the same size anhave the zero frequency at the center.
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Filtering in the Frequency Domain : ExampleFiltering in the Frequency Domain : Example
, ,which means that the zero frequencycomponent is removed.
Note: Zero frequency = average
intensity of an image (Images from Rafael C. Gonzalez and Richard E.Wood, Digital Image Processing, 2nd Edition.
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Filtering in the Frequency Domain : ExampleFiltering in the Frequency Domain : Example
Lowpass Filter
Highpass Filter
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
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Filter Masks and Their Fourier TransformsFilter Masks and Their Fourier Transforms
(Images from Rafael C. Gonzalez and RichardWood, Digital Image Processing, 2nd Edition.
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Ideal Lowpass FilterIdeal Lowpass Filter
Ideal LPF Filter Transfer function
= 0),(1
),(DvuD
vuH0
,
where D(u,v) = Distance from (u,v) to the center of the mask.
Ima es from Rafael C. Gonzalez and Richard E. . .
Wood, Digital Image Processing, 2nd Edition.
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Examples of Ideal Lowpass FiltersExamples of Ideal Lowpass Filters
Ima es from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
The smaller D0, the more high frequency components are remo
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Results of Ideal Lowpass FiltersResults of Ideal Lowpass Filters
obviously seen!
(Images from Rafael C. Gonzalez and Richard E.Wood, Digital Image Processing, 2nd Edition.
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How ringing effect happensHow ringing effect happens
>=
0
0
),(0
,),(
DvuD
vuvuH
Surface Plot
0.8
1
Ideal Lowpass Filterwith D0= 5 0.2
0.4
0.6
0
200
20
0
2020Abrupt change in the amplitude
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How ringing effect happens (cont.)How ringing effect happens (cont.)
Surface Plot
x 103
5
10
15
Spatial Response of IdealLow ass Filter with D = 5
20
0
20
0
20
20
0
Ripples that cause ringing effect
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How ringing effect happens (cont.)How ringing effect happens (cont.)
(Images from Rafael C. Gonzalez and RichardWood, Digital Image Processing, 2nd Edition.
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Butterworth Lowpass FilterButterworth Lowpass Filter
Transfer function
NDvuD
vuH2
/,1
1),(
+=
Where D0= Cut off frequency, N= filter order.
(Images from Rafael C. Gonzalez and Richard E.Wood, Digital Image Processing, 2n Edition.
R l f B h L FilR l f B h L Fil
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Results of Butterworth Lowpass FiltersResults of Butterworth Lowpass Filters
There is less ringing
effect compared tothose of ideal lowpass
ers
(Images from Rafael C. Gonzalez and Richard
Wood, Digital Image Processing, 2nd Edition.
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GaussianLowPassFilter
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GaussianLowPassFilter
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Highpass FiltersHighpass Filters
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Highpass FiltersHighpass Filters
Hhp= 1  Hlp
(I