Chap 4 Image Enhancement in the Frequency Domain

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Chap 4 Image Enhancement in the Frequency Domain
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Background

• Any periodic function can be expressed as the sum
  of sines and cosines of different frequencies,
  each multiplied by a different coefficient.
• We called this sum a Fourier series.
• Even function that are not periodic can be
  expressed as the integral of sines and cosines
  multiplied by a weighting function.
• This formation is the Fourier transform.

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Periodic Function

• A function f is periodic with period P greater
  than zero if
• Af(x P) Af(x), where A denotes amplitude.
• f(x) sinx, P 2p, frequency1/ 2p, A1.
• f(x) Asinnx, P 2p/n, frequencyn/ 2 p.
• n?, frequency?.

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Fourier Series

• Suppose f(x) is a function defined on the
  interval -p,p. The Fourier series expansion of
  f(x) is
• where an and bn are constants called the Fourier
  coefficients, and

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Coefficients of Any Period T 2L

• Replace v by px/L to obtain the Fourier series of
  the function (x) of period 2L

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Complex Fourier Series

• Complex exponentials
• According to Eulers formula
• and so,
• Using these two equations we can find the
  complex exponential form of the trigonometric
  functions as

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Complex Fourier Series
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Continuous Spectra

• Consider the following function
• Only a single pulse remains and the resulting
  function is no longer periodic. A function which
  is not periodic can be considered as a function
  with very large period.

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Continuous Spectra
These two integrals form the conclusion of
Fouriers integral theorem.
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Alternative Forms

• Note that there are a number of alternative forms
  for Fourier transform, such as
• The third form is popular in the field of signal
  processing and communications systems.

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4.2 Fourier Transform in theFrequency Domain

• Fourier transform F(u) of f(x) is defined as
• The inverse Fourier Transform is
• DFT for Discrete function f(x), x0,1,..M-1
• for u0,1,..M-1
• Inverse DFT

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• Eulers formula
• Each term of the Fourier transform is composed of
  the sum of all values of the function f(x).
• M2 summations and multiplications
• The values of f(x) are multiplied by sines and
  cosines of various frequencies.
• The domain (values of u) over which the values of
  F(u) range is appropriately called the frequency
  domain, because u determines the frequency of the
  components of the transform.
• Each of the M terms of F(u) is called a frequency
  component of the transform.

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Complex Spectra

• In general, the components of Fourier transform
  are complex quantities in the following form
• F(u) R(u) jI(u)
• and can be written as
• F(u) F(u)ej?(u)
• The spectra is usually represented by the
  amplitude of a specific frequency
• Amplitude or spectrum of Fourier transform
• F(u) (R2(u)I2(u))1/2

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Complex Spectra

• These complex coefficients couples
• Amplitude spectrum value
• Magnitude of each of the harmonic components.
• Phase spectrum value
• The phase of each harmonic relative to the
  fundamental harmonic frequency ?0.

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The frequency spectrum is centered at 0. To
visual easily, we sometimes multiply f(x) by
(-1)x before applying the transform.
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Why (-1)x?
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4.2.2 The Two-dimensional Discrete Fourier
Transform (DFT)

• 2D-DFT of f(x, y) of size M?N
• Inverse 2-D DFT

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• Modulation in the space domain
• F(-1)xyf(x, y) F(u-M/2,v-N/2)
• Shift the origin of F(u,v) to frequency
  coordinates (M/2, N/2),
• the center of (u, v), u0,M-1, v0,N-1.
• frequency rectangle
• Average of f(x,y)
• For real f(x,y)
• F(u, v) F(-u, -v)
• F(u, v) F(-u, -v)
• The spectrum of the Fourier transform is
  symmetric.

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Implementation
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4.2.3 Filtering in the Frequency Domain

• What is the frequency of an image?
• Since frequency is directly related rate of
  change, it is not difficult intuitively to
  associate frequencies with pattern of intensity
  variations in an image.
• The low frequencies correspond to the slowly
  varying components of an image.
• The higher frequencies begin to correspond to
  faster and faster gray level changes in the
  image.
• such as edges.
• F(0, 0) the average gray level of an image.

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Filtering steps

• Multiply the input image by (-1)xy to center the
  transform.
• Compute DFT F(u, v)
• Multiply F(u,v) by a filter function H(u,v)
• G(u,v) F(u,v)H(u,v)
• Computer the inverse DFT of G(u,v)
• Obtain the real part of g(x,y)
• Multiply g(x,y) with (-1)xy

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Notch filter H(u,v) 0 if (u,v) (M/2, N/2),
H(u,v) 1 otherwise
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Lowpass filter Highpass filter
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4.2.4 Filtering in spatial and frequency domains

• The discrete convolution f(x,y)h(x,y)
• f(x,y)h(x,y) ? F(u,v)H(u,v)
• f(x,y)h(x,y) ? F(u,v)H(u,v)

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4.3 Smoothing Frequency-Domain Filters

• Frequency-Domain Filtering
• G(u,v) H(u,v)F(u,v)
• Filter H(u,v)
• Ideal filter
• Butterworth filter
• Gaussian Filter

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4.3.1 Ideal Low pass filter

• H(u,v) 1 if D(u,v) ? D0
• 0 if D(u,v) gt D0
• The center is at (u,v)(M/2, N/2)
• D(u,v)(u-M/2)2 (v-N/2)21/2
• Cutoff frequency is D0
• Power estimate
• The percentage a of power enclosed in the circle
  is

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• The blurring in this image is a clear indication
  that most of the sharp detail information in the
  picture is contained in the 8 power removed by
  the filter.
• The result of a 99.5 is quite close to the
  original, indicating little edge information is
  contained in the upper 0.5 of the spectrum power.

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4.3.2 Butterworth Lowpass Filter

• Butterworth lowpass filter (BLPF) of order n
• At the frequency as an half of the cutoff
  frequency D0, H(u, v)0.5.

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4.3.2 Butterworth Lowpass Filter
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4.3.2 Butterworth Lowpass Filter
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4.3.2 Butterworth Lowpass Filter
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4.3.3 Gaussian Lowpass Filter

• Gaussian filter
• Let ?D0
• When D(u, v)D0 , H(u, v)0.667

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4.3.3 Gaussian Lowpass Filter
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4.3.3 Gaussian Lowpass Filter
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4.3.4 Other Lowpass filtering examples
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4.3.4 Other Lowpass filtering examples
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4.4 Sharpening Frequency-Domain Filter

• Highpass filtering
• Hhp(u,v)1-Hlp(u,v)
• Given a lowpass filter Hlp(u,v), find the spatial
  representation of the highpass filter
• Compute the inverse DFT of Hlp(u,v)
• Multiply the real part of the result with (-1)xy

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4.4 Sharpening Frequency-Domain Filter
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4.4 Sharpening Frequency-Domain Filter
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4.4.1 Ideal Highpass Filter

• H(u,v)0 if D(u,v)?D0
• 1 if D(u,v)gtD0
• The center is at (u,v)(M/2, N/2)
• D(u,v)(u-M/2)2(v-N/2)21/2
• Cutoff frequency is D0

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4.4.1 Ideal Highpass Filter
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4.4.2 Butterworth Highpass Filter

• Butterworth filter has no sharp cutoff
• At cutoff frequency D0 H(u, v)0.5

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4.4.2 Butterworth Highpass Filter
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4.4.3 Gaussian Highpass Filter

• Gaussian highpass filter (GHPF)
• Let ?D0

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4.4.3 Gaussian Highpass Filter
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5.4 Periodic Noise Reduction by Frequency Domain
Filtering

• Periodic noise is due to the electrical or
  electromechanical interference during image
  acquisition.
• Can be estimated through the inspection of the
  Fourier spectrum of the image.

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Periodic Noise Reduction by Frequency Domain
Filtering
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5.4 Periodic Noise Reduction by Frequency Domain
Filtering

• Bandreject filters
• Remove or attenuate a band of frequencies.
• D0 is the radius.
• D(u, v) is the distance from the origin, and
• W is the width of the frequency band.

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Butterworth and Gaussian Bandreject Filters

• Butterworth bandreject filter (order n)
• Gaussian band reject filter

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Bandreject Filters
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Bandpass filter

• Obtained form bandreject filter
• Hbp(u,v)1-Hbr(u,v)
• The goal of the bandpass filter is to isolate the
  noise pattern from the original image, which can
  help simplify the analysis of noise, reasonably
  independent of image content.

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Result of The BandPass Filter
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5.4.3 Notch filters

• Notch filter rejects (passes) frequencies in
  predefined neighborhoods about a center
  frequency.
• where

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5.4.3 Notch filters

• Butterworth notch filter
• Gaussian notch filter
• Note that these notch filters will become
  highpass when u0v00

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Notch filters
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Example 5.8
Use 1-D Notch pass filter to find the horizontal
ripple noise