Frequency Domain Processing

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Frequency Domain Processing

• Lecture 3

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0. Overview of Linear Systems

• In image processing, linear systems are at the
  heart of many filtering operations, and they
  provide the basis for analyzing complex problems
  in areas such as image restoration.
• In this section we give a short review of linear
  systems

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0.1 Definitions
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0.1 Definitions
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0.1 Definitions
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0.1 Definitions
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0.1 Definitions
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0.2 Convolutions
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0.2 Convolutions
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0.2 Convolutions
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0.2 Convolutions
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0.2 Convolutions
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Frequency Domain Processing
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1 The 2D Discrete Fourier Transform

• Let f(x,y) for x0,1,2, ..., M-1 and y1,2, ...,
  N-1 denote an MN image. The 2D DFT of f is
  given by
• The frequency domain is simply the coordinate
  system spanned by F(u,v) with u and v as
  frequency variables.

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1 The 2D Discrete Fourier Transform

• The inverse DFT is given by
• Thus, given F(u,v), we can obtain f(x,y) back by
  means of the inverse DFT.

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1 The 2D Discrete Fourier Transform

• The value of the transform at the origin of the
  frequency domain that is F(0,0) is called the dc
  component of the Fourier transform.
• Even if f(x,y) is real, its transform is complex.
  
• The principal method of visually analyzing a
  transform is to compute its spectrum that is the
  magnitude of F(u,v).

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1 The 2D Discrete Fourier Transform

• The Fourier spectrum is defined as
• The phase angle is defined as

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1 The 2D Discrete Fourier Transform

• The polar representation of F(u,v) is defined by
• The power spectrum is defined as

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1 The 2D Discrete Fourier Transform

• It can be shown that
• The DFT is infinitely periodic in both u and v
  directions, with
• the periodicity determined by M and N.

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1 The 2D Discrete Fourier Transform

• Periodicity is also a property of the inverse
  DFT.
• An image obtained by taking the inverse DFT is
  also infinitely
• periodic in both u and v directions, with the
  periodicity
• determined by M and N.

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1 The 2D Discrete FourierTransform

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1 The 2D Discrete FourierTransform

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2 Computing and Vizualizing the 2D DFT

• The FFT of an M N image array is obtained by
  the syntax
• Ffft2(f)
• and with padding by the syntax
• Ffft2(f, P,Q)
• The Fourier spectrum is obtained as
• Sabs(F)

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2 Computing and Vizualizing the 2D DFT

• The funcion Ffft2 (f) moves the origin of
  transform to the center of the frequency
  rectangle.
• Fcfftshift (F)
• Log transformation
• S2log(1abs(Fc))
• Function ifftshift reverses the centering
• Fifftshift(Fc)

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2 Computing and Vizualizing the 2D DFT

• To compute inverse FFT
• fifft2 (F)
• If the imput used to compute F is real then the
  inverse should also be real. However fft2 often
  has small imaginary components resulting from
  round-off errors. It is good practice to extract
  the real part of the result
• freal(ifft2(F))

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2 Computing and Vizualizing the 2D DFT
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3 Filtering in the frequency domain

• The convolution theorem
• Linear spatial convolution is by convolving
  f(x,y) and h(x,y). The same result is obtained in
  the frequency domain by multiplying F(u,v) and
  H(u,v).
• The basic idea in frequency domain is to select a
  filter transfer function that modifies F(u,v) in
  a specified manner. A transfer function is
  multiplied by a centered F(u,v).
• A filter is called low pass filter if it
  attenuates the high frequency components to
  F(u,v) while leaving the low frequencies
  relatively unchanged.

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3 Filtering in the frequency domain
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3 Filtering in the frequency domain

• Example Linear spatial filtering.
• f
• 0 0 0 0 0
• 0 0 0 0 0
• 0 0 1 0 0
• 0 0 0 0 0
• 0 0 0 0 0

w 1 2 3 4 5 6 7
8 9
gimfilter(f, w, filtering_mode,
boundary_options, size_options)
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3 Filtering in the frequency domain
gimfilter(f, w, filtering_mode,
boundary_options, size_options)
gtgt gimfilter(f,w,'corr',0,'full') g 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 9
8 7 0 0 0 0 6 5
4 0 0 0 0 3 2 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
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3 Filtering in the frequency domain

• With specified values of P and Q we use the
  following syntax to compute the FFT using zero
  padding.
• gtgt Ffft2(f, 7,7)
• F
• Columns 1 through 2
• 1.00000000000000
  -0.22252093395631 - 0.97492791218182i
• -0.22252093395631 - 0.97492791218182i
  -0.90096886790242 0.43388373911756i
• -0.90096886790242 0.43388373911756i
  0.62348980185873 0.78183148246803i
• 0.62348980185873 0.78183148246803i
  0.62348980185873 - 0.78183148246803i
• 0.62348980185873 - 0.78183148246803i
  -0.90096886790242 - 0.43388373911756i
• -0.90096886790242 - 0.43388373911756i
  -0.22252093395631 0.97492791218182i
• -0.22252093395631 0.97492791218182i
  1.00000000000000
• .
• .
• .

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3.2 Basic steps in DFT filtering

• Obtain the padding parameters P and Q.
• Obtain the Fourier transform with padding
  Ffft2(f, P,Q)
• Generate a filter function, H of size PQ. If the
  filter is centered then use Hfftshift(H)
  before using the filter.
• Multiply the transform by the filter GH.F
• Obtain the real part of G greal(ifft2(G))
• Crop the top,left rectangle to the original
  size gg(1size(f,1), 1size(f,2))

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4. Lowpass frequency domain filters

• Ideal lowpass filter (ILPF)
• where D(u,v) is the distance from point (u,v) to
  the center of the
• filter.

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4. Lowpass frequency domain filters

• Butterworth lowpass filter (ILPF)
• where D(u,v) is the distance from point (u,v) to
  the center of the
• filter.

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4. Lowpass frequency domain filters

• Gaussian lowpass filter (ILPF)
• where D(u,v) is the distance from point (u,v) to
  the center of the
• filter.

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1 The 2D Discrete FourierTransform