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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

    10

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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

    15

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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 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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    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

    -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

    34

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    GaussianLowPassFilter

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    Highpass FiltersHighpass Filters

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    Highpass FiltersHighpass Filters

    Hhp= 1 - Hlp

    (I