Image Restoration

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Chapter 5

• Image Restoration

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Preview

• Goal improve an image in some predefined sense.
• Image enhancement subjective process
• Image restoration objective process
• Restoration attempts to reconstruct an image that
  has been degraded by using a priori knowledge of
  the degradation process.
• Modeling the degradation and applying the inverse
  process to recover the original image.
• When degradation model is unknown ? blind
  deconvolution (ICA)

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A Model of Degradation

• or
• Given g(x,y), some knowledge about H, and some
  knowledge about the noise term, obtain an
  estimate of the original image.

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Noise Models

• Gaussian noise electronic circuit sensor noise
• Rayleigh noise range imaging
• Erlang (Gamma noise) laser imaging
• Exponential noise laser imaging
• Uniform noise
• Impulse (salt-and-pepper noise) faulty switching
• Periodic noise

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Gaussian Noise

• The PDF of a Gaussian random variable, z, is
  given by

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Rayleigh Noise

• The PDF of Rayleigh noise is given by
• Mean and variance are given by
• Useful for approximating skewed histograms.

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Erlang (Gamma) Noise

• The PDF of Erlang noise is given by
• Mean and variance

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Exponential Noise

• The PDF of exponential noise is given by
• where a gt0
• Mean and variance

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Uniform Noise

• The PDF of uniform noise is given by
• Mean and variance

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Impulse (Salt-and-Pepper) Noise

• The PDF of (bipolar) impulse noise is given by

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Periodic Noise

• Arises typically from electrical or
  electromechanical interference during image
  acquisition.
• The only type of spatially dependent noise
  considered in this chapter.

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Illustration (I)
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Illustration (II)
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Estimation of Noise Parameters

• Periodic noises from Fourier spectrum
• Others try to compute the mean and variance of a
  subimage S (containing only constant gray levels).

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Restoration in the Presence of Noise Only
Spatial Filtering

• Mean filters
• Arithmetic mean filters
• Geometric mean filter
• Harmonic mean filter
• Contraharmonic mean filterQ the order of
  the filter. Qgt0 eliminates pepper noise, Q lt0
  eliminates salt noise.

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Illustration (I)
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Illustration (II)
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Illustration (III)
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Order-Statistics Filters

• Median filters
• Max and min filters
• Midpoint filter
• Alpha-trimmed mean filter delete the d/2 lowest
  and d/2 highest gray-level values of g(s,t) in
  the neighborhood of Sxy , the average

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Illustration (I)
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Illustration (II)
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Illustration (III)
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Adaptive Filters

• Filters behavior changes based on statistical
  characteristics of the image inside the filter
  region defined by the mxn window.
• Adaptive, local noise reduction filter
• Adaptive median filter

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Adaptive, local noise reduction filter

• (a) g(x,y) the value of the noisy image at (x,y)
• (b) The variance of the noise
• (c) The mean of the pixels in Sxy
• (d) Local variance of the pixels in Sxy
• If (b) is zero, return g(x,y)
• If (d) is high relative to (b), the filter should
  return a value close to g(x,y)
• If the two variances are equal, return the
  arithmetic mean of the pixels in Sxy

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Illustration
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Adaptive Median Filter

• Notation
• zmin minimum gray level value in Sxy
• zmax maximum gray level value in Sxy
• zmed median of gray levels in Sxy
• zxy gray level value at (x,y)
• Smax maximum allowed size of Sxy
• Level A A1 zmed zmin, A2 zmed zmaxif A1gt
  0 and A2 lt0, go to level BElse increase the
  window sizeIf window size lt Smax repeat level
  Aelse output zxy
• Level B B1 zxy zmin, B2 zxy zmax if B1gt 0
  and B2 lt0, output zxyElse output zmed

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Illustration
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Periodic Noise Reduction

• By Fourier domain filtering
• Bandreject filters
• Bandpass filters
• Notch filters

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Illustration
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Ideal Notch Reject Filter

• Ideal notch reject filter
• where

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Butterworth Notch Reject Filter
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Gaussian Notch Reject Filter
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Notch Filters
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Linear, Position-Invariant Degradations

• Estimating the degradation function
• By image observation
• By experimentation
• By modeling

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Estimation by Image Observation

• In the strong signal area, using sample gray
  levels of the object and background to construct
  an unblurred image
• Then,
• Use Hs(u,v) to estimate H(u,v)

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Estimation by Experimentation

• Simulate an impulse by a (very) bright dot of
  light, the response G(u,v) is related to H(u,v)
  by

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Figure 5.24
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Estimation by Modeling

• Modeling atmospheric turbulence

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Atmospheric Turbulence
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Estimation by Modeling (contd)

• Modeling effect of planar motion x0(t),y0(t)
• If T is the duration of the exposure, then
• It can be shown that

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Motion Blur

• If x0(t)at/T and y0(t)0, then

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Motion Blur Example
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Deconvolution

• Inverse filtering
• Minimum mean square error (Wiener) filtering
• Constrained least squares filtering
• Geometric mean filter
• http//vision.cs.nccu.edu.tw/publications/CVPRIP20
  03_A.pdf

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Results (Inverse Filter)
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Results (Inverse and Wiener)
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Results (Motion Blurs)
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Results (Constrained LS Filter)
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Geometric Transformations

• Image warping
• Spatial transformations
• Gray-level interpolation