Digital Image Processing

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Digital Image Processing

• 5 Image Restoration

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Preview

• The ultimate goal of image restoration techniques
  is to improve an image in some predefined sense.
• Restoration attempts to reconstruct or recover an
  image that has been degraded by using a priori
  knowledge of the degradation phenomenon.
• Restoration techniques are oriented toward
  modeling the degradation and applying the inverse
  process in order to recover the original image.

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5.1 A model of degradation/Restoration

• A model of the image degradation/restoration
  process

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5.2 Noise models

• Sources of noise
• Image acquisition
• Image transmission
• Spatial and frequency properties of noise
• Noise parameters
• Correlation of the noise with the image.
• Frequency distribution of noise
• Assumptions
• Noise is independent of spatial coordinates.
• Noise is uncorrelated with respect to the image
  itself.

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5.2 Noise models

• Some important noise probability density
  functions
• Gaussian noise
• Reyleigh noise

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5.2 Noise models

• Some important noise probability density
  functions
• Erlang (Gamma) noise
• Exponential noise

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5.2 Noise models

• Some important noise probability density
  functions
• Uniform noise
• Impulse (salt-and-pepper) noise

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5.2 Noise models
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5.2 Noise models

• Test pattern

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5.2 Noise models
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5.2 Noise models
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5.2 Noise models

• Periodic noise

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5.2 Noise models

• Estimation of noise parameters
• Mean
• Variance

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5.2 Noise models
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5.3 Restoration spatial filtering

• Mean filters
• Arithmetic mean filter
• Geometric mean filter
• Harmonic mean filter
• Contraharmonic mean filter
• Order-statistics filters
• Median filter
• Max and min filters
• Midpoint filter
• Alpha-trimmed mean filter
• Adaptive filter
• Adaptive, local noise reduction filter
• Adaptive median filter

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5.3 Restoration spatial filtering
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5.3 Restoration spatial filtering
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5.3 Restoration spatial filtering
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5.3 Restoration spatial filtering
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5.3 Restoration spatial filtering
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5.3 Restoration spatial filtering
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5.4 Periodic noise reduction

• Bandreject filters
• Ideal bandreject filter
• Butterworth bandreject filter
• Gaussian bandreject filter

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5.4 Periodic noise reduction
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5.4 Periodic noise reduction

• Bandpass filters

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5.4 Periodic noise reduction

• Notch filters
• Ideal notch filter
• Butterworth notch filter
• Gaussian notch filter

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5.4 Periodic noise reduction
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5.4 Periodic noise reduction
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5.5 Linear position-invariant systems

• Linearity
• Definition assume that
• H is linear if and only if
• Additivity
• Homogeneity

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5.5 Linear position-invariant systems

• Position invariant
• Definition assume that
• H is position invariant if
• Image expression in terms of impulses

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5.5 Linear position-invariant systems

• Convolution integral
• Output image
• If the system is linear
• If the system is position invariant

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5.5 Linear position-invariant systems

• Expression of convolution integral
• Degradation model
• Degradation model in frequency domain

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5.6 Estimating the degradation function

• Three principal ways
• Observation
• Experimentation
• Mathematical modeling

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5.6.1 Estimation by image observation

• Principle
• Look at a small section of the image containing
  simple structures as the observed subimage,
• Construct an unblurred subimage of the same size
  and characteristics
• degradation function of subimage
• degradation function

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5.6.2 Estimation by experimentation

• Principle
• Use an equipment to imitate the equipment used to
  acquire the degraded image
• Obtain the impulse response of the degradation
  equipment
• Due to the fact that the Fourier transform of an
  impulse is a constant, we have

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5.6.2 Estimation by experimentation
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5.6.3 Estimation by modeling

• An instance of modeling atmospheric turbulence

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5.6.3 Estimation by modeling

• Another instance uniform linear motion

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5.6.3 Estimation by modeling
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5.7 Inverse filtering

• Inverse filtering
• In the presence of noise

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5.7 Inverse filtering
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5.8 Wiener filtering

• Mean square error
• Conditions
• The noise and image are uncorrelated
• Zero mean
• The gray levels in the estimate are a linear
  function of the gray levels in the degraded image

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5.8 Wiener filtering

• Wiener filtering

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5.8 Wiener filtering
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5.8 Wiener filtering
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5.9 Constrained least square filtering

• Degradation expressed in vector-matrix form
• Suppose that g(x,y) is of size M ? N.
• So, g, f and ? all have dimension M N ?1, and
  the matrix H has dimension M N ? M N.

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5.9 Constrained least square filtering

• Principle

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5.9 Constrained least square filtering

• Solution in the frequency domain

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5.9 Constrained least square filtering
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5.10 Geometric mean filter

• Geometric mean filter

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5.11 Geometric transformation

• Geometric transformations modify the spatial
  relationships between pixels in an image.
• Two steps
• Spatial transformation
• Gray level interpolation

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5.11.1 Spatial transformation

• Principle
• Suppose that an image f (x, y) undergoes
  geometric distortion to produce an image g (x,
  y).
• Transformation may be expressed as
• x r (x, y) and y s (x, y)

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5.11.2 Gray level interpolation

• Approaches
• Nearest neighbor, or zero-order interpolation
• Cubic convolution interpolation
• Bilinear interpolation

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5.11.2 Gray level interpolation
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