Chap4 Frequency Domain

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    Chapter 4:

    Frequency Domain Processing

    Image Transformations

    1

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    Introduction

    Although we discuss other transforms in some

    detail in this chapter, we emphasize the Fourier

    transform because of its wide range of

    applications in image processing problems.

    3

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    Fourier Transform (1-D)

    uXuXsin

    AXuF

    euXsinu

    Adxux2jexpxfuF uXj

    4

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    Fourier Transform (2-D)

    vYvYsin

    uX

    uXsinAXYv,uFdydxvyux2jexpy,xf)v,u(F

    5

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    Discrete Fourier Transform

    In the two-variable case the discrete Fourier transform pair is

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    Discrete Fourier Transform

    When images are sampled in a squared array, i.e. M=N,

    we can write

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    Discrete

    Fourier

    Transform

    Examples

    At all of theseexamples, the Fourier

    spectrum is shifted

    from top left corner to

    the center of the

    frequency square.

    8

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    Discrete

    Fourier

    Transform

    Examples

    At all of theseexamples, the Fourier

    spectrum is shifted

    from top left corner to

    the center of the

    frequency square.

    9

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    Discrete

    Fourier

    Transform

    Display

    At all of theseexamples, the Fourier

    spectrum is shifted

    from top left corner to

    the center of the

    frequency square.

    10

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    Discrete Fourier Transform

    Example

    Main Image (Gray Level) DFT of Main image

    (Fourier spectrum)

    Logarithmic scaled

    of the Fourier spectrum

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    Discrete Fourier Transform

    (Properties)

    Separability

    1N

    0v

    1N

    0u

    1N

    0y

    1N

    0x

    N/vy2jexpv,uFN/ux2jexpN

    1y,xf

    N/vy2jexpy,xfN/ux2jexpN

    1v,uF

    The discrete Fourier transform pair can be expressed in the seperable forms:

    12

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    Discrete Fourier Transform

    (Properties)

    Translation

    N/vyux2jexpv,uFyy,xxfand

    vv,uuFN/yvxu2jexpy,xf

    0000

    0000

    The translation properties of the

    Fourier transform pair are :

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    Discrete Fourier Transform

    (Properties)

    Periodicity

    The discrete Fourier transform and its

    inverse areperiodic with period N; that is,

    F(u,v)=F(u+N,v)=F(u,v+N)=F(u+N,v+N)

    If f(x,y) is real, the Fourier transform also

    exhibits conjugate symmetry:

    F(u,v)=F*(-u,-v)

    Or, more interestingly:

    |F(u,v)|=|F(-u,-v)|

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    Discrete Fourier Transform

    (Properties)

    Rotation

    sinvcosu

    sinrycosrx

    If we introduce the polar coordinates

    Then we can write:

    00 ,F,rf

    In other words, rotating F(u,v)

    rotates f(x,y) by the same angle.

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    Discrete Fourier Transform

    (Properties)

    Convolution

    v,uG*v,uFy,xgy,xfand

    v,uGv,uF)y,x(g*y,xf

    The convolution theorem in

    two dimensions is expressed

    by the relations :

    Note :

    ddy,xg,fy,xg*y,xf

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    Discrete Fourier Transform

    (Properties)

    Correlation

    dxgfxgxf

    *

    The correlation of two continuous

    functions f(x) and g(x) is defined

    by the relation

    So we can write:

    v,uGv,uFy,xgy,xf

    and

    v,uGv,uFy,xgy,xf

    *

    *

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    Discrete

    Fourier

    Transform

    Sampling

    (Properties)

    1-D

    f(x) : a given function

    F(u): Fourier Transform of f(x)

    which is band-limited

    s(x) : sampling function

    S(u): Fourier Transform of s(x)

    G(u): window for recovery of the

    main function F(u) and f(x).

    Recovered f(x) from sampled data

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    Discrete

    Fourier

    Transform

    Sampling

    0000 y,xfdydxyy,xxy,xf

    (Properties)

    2-D

    The sampling process for 2-Dfunctions can be formulated

    mathematically by making use

    of the 2-D impulse function

    (x,y), which is defined as

    A 2-D sampling function is

    consisted of a train of impulses

    separated x units in the x

    direction and y units in the y

    direction as shown in the figure.

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    Discrete

    Fourier

    Transform

    Sampling

    0

    1

    v,uG

    (Properties)

    2-D

    If f(x,y) is band limited (that is, its

    Fourier transform vanishes outsidesome finite region R) the result of

    covolving S(u,v) and F(u,v) might

    look like the case in the case shown

    in the figure. The function shown is

    periodic in two dimensions.

    (u,v) inside one of the rectangles

    enclosing R

    elsewhere

    The inverse Fourier transform of

    G(u,v)[S(u,v)*F(u,v)] yields f(x,y).

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    The Fast Fourier Transform (FFT) Algorithm21

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    Other Seperable Image Transforms

    AFAT

    BTBF

    AB

    BAFABBTB

    1

    1N

    0x

    1N

    0y

    v,u,y,xgy,xfv,uT

    Where F is the NN image matrix,

    A is an NN symmetric transformation matrix

    T is the resulting N

    N transform.

    If the kernel g(x,y,u,v) is seperable and symmetric,

    also may be expressed in matrix form:

    And for inverse transform we have:

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    Discrete Cosine Transform23

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    Discrete Cosine Transform (DCT)

    Each block consists

    of 44 elements,

    corresponding to x

    and y varying from 0to 3. The highest

    value is shown in

    white. Other values

    are shown in grays,

    with darker meaning

    smaller.

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    Discrete Cosine Transform

    Example

    Main Image (Gray Level) DCT of Main image

    (Cosine spectrum)

    Logarithmic scaled

    of the Cosine spectrum

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    DCT and Fourier Transform26

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    DCT Example27

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    Blockwise DCT Example28

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    Comparison Of Various Transforms29

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    Comparison Of Various Transforms30

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    The DFT and Image Processing

    To filter an image in the frequency domain:1. Compute F(u,v) the DFT of the image

    2. Multiply F(u,v) by a filter functionH(u,v)

    3. Compute the inverse DFT of the result

    ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

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    Some Basic Frequency Domain Filters

    ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002) Low Pass Filter

    High Pass Filter

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    Smoothing Frequency Domain Filters

    Smoothing is achieved in the frequency domain by

    dropping out the high frequency components

    The basic model for filtering is:

    G(u,v) = H(u,v)F(u,v)where F(u,v) is the Fourier transform of the imagebeing filtered andH(u,v) is the filter transformfunction

    Low pass filters only pass the low frequencies,drop the high ones

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    Ideal Low Pass Filter

    Simply cut off all high frequency components thatare a specified distance D0 from the origin of the

    transform

    changing the distance changes the behaviour of

    the filter

    ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

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    Ideal Low Pass Filter (cont)

    The transfer function for the ideal low pass filter

    can be given as:

    whereD(u,v) is given as:

    0

    0

    ),(if0

    ),(if1

    ),( DvuD

    DvuD

    vuH

    2/122 ])2/()2/[(),( NvMuvuD

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    Ideal Low Pass Filter (cont)

    Above we show an image, its Fourier spectrum

    and a series of ideal low pass filters of radius 5,

    15, 30, 80 and 230 superimposed on top of it

    ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

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    Ideal Low Pass Filter (cont)

    ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

    Original