EE 7730

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EE 7730

• Image Enhancement (Frequency Domain)

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Frequency-Domain Filtering

• Compute the Fourier Transform of the image
• Multiply the result by filter transfer function
• Take the inverse transform

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Frequency-Domain Filtering
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Frequency-Domain Filtering

• Ideal Lowpass Filters

Non-separable
gtgt f1,f2 freqspace(256,'meshgrid') gtgt H
zeros(256,256) d sqrt(f1.2 f2.2) lt 0.5 gtgt
H(d) 1 gtgt figure imshow(H)
Separable
gtgt f1,f2 freqspace(256,'meshgrid') gtgt H
zeros(256,256) d abs(f1)lt0.5 abs(f2)lt0.5 gtgt
H(d) 1 gtgt figure imshow(H)
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Frequency-Domain Filtering

• Butterworth Lowpass Filter

As order increases the frequency response
approaches ideal LPF
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Frequency-Domain Filtering

• Butterworth Lowpass Filter

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Frequency-Domain Filtering

• Gaussian Lowpass Filter

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Frequency-Domain Filtering
Ideal LPF
Butterworth LPF
Gaussian LPF
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Example
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Highpass Filters
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Example
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Homomorphic Filtering

• Consider the illumination and reflectance
  components of an image

Illumination
Reflectance

• Take the ln of the image

• In the frequency domain

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Homomorphic Filtering

• The illumination component of an image shows slow
  spatial variations.
• The reflectance component varies abruptly.
• Therefore, we can treat these components somewhat
  separately in the frequency domain.

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With this filter, low-frequency components are
attenuated, high-frequency components are
emphasized.
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Homomorphic Filtering
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Summary

• Digital Image Fundamentals Pixel, resolution,
  bit depth, ...
• Linear Systems Shift invariance, causality,
  convolution, impulse response, ...
• Fourier Transform 2D Fourier Transform of
  continuous and discrete signals, 2D FT properties
  (linearity, shifting, modulation, convolution,
  multiplication, energy conservation, etc.), Dirac
  delta function, Kronecker delta function,
• 2D Sampling Aliasing, anti-aliasing filter,
  downsampling, interpolation,
• Discrete Fourier Transform Periodicity, other
  properties, frequency-domain filtering,
• Discrete Cosine Transform Properties (real basis
  functions, good energy compaction), relationship
  with DFT, matrix representation of DCT,
• Image Enhancement Image enhancement by point
  processing (intensity transformation, histogram
  equalization, histogram specification, etc.),
  Image enhancement by spatial-domain filtering
  (lowpass filtering, highpass filtering, median
  filtering, high-boost filtering, gradient and
  laplacian operators, etc.), Image enhancement by
  frequency-domain filtering (lowpass/highpass
  filters, homomorphic filtering, etc.)