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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.

1-D, 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,,M-1

Inv. Fourier Tr.:

=1

/2)()(

MMuxj

euFxf

x= 0,,M-1=0u

F(u) can be written as)(

)()(uj

euFuF=or)()()( ujIuRuF +=

22uIuRuF +=

= )(

tan)(1 uI

uu

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Example of 1Example of 1--D 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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22--Dimensional Discrete Fourier TransformDimensional Discrete Fourier Transform

2-D DFT

=

=

+=0 0

)//(2),(1

),(x y

NvyMuxjeyxfMN

vuF

u= frequency in xdirection, u= 0 ,, M-1

v= frequency in ydirection, v= 0 ,, N-1

2-D IDFT

+=1 1

)//(2),(),(

M NNvyMuxj

evuFyxf

= =u vx= 0 ,, M-1

y= 0 ,, N-1

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22--Dimensional 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

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x={1289}

So ution:N=4, ence

12

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= ,

(t)isdefinedas

Animpulsehassiftingpropertywithrespecttointegration,

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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.

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Example of 2Example of 2--D 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 2--D 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 2--D DFTD DFT

2D DFT

Original image

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Example of 2Example of 2--D 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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(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

-20-20Abrupt change in the amplitude

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How ringing effect happens (cont.)How ringing effect happens (cont.)

Surface Plot

x 10-3

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

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