BAE 790I BMME 231 Fundamentals of Image Processing Class 18

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BAE 790I / BMME 231Fundamentals of Image
ProcessingClass 18

• Restoration Filtering
• Inverse Filters Properties
• Wiener Filters
• MSE
• Properties

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Image Restoration

• Objective To quantitatively estimate the true
  image from its degraded measurement.

System H
Restoration process

g
f
n
An estimate of f
3
Noise Effects

• Problem The inverse filter generally magnifies
  the noise (note Fourier-domain example).

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Inverse Filter with Noise

24.2 dB
1.7 dB
High-frequency noise is amplified by the inverse
filter.
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Noise Effects

• Consider the error image between the true image
  and the inverse-filter estimate
• The estimate is unbiased. (On average, it is
  correct.)

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Noise Effects

• Consider the autocorrelation of the error image
  for the inverse filter
• If H has some small eigenvalues, H-1 has some BIG
  eigenvalues.
• If Rnn has content in the corresponding
  eigenimages, those elements of the noise get BIG.

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Mean Square Error

• Consider the mean square error (MSE) between the
  true image and the inverse-filter estimate
• This is a scalar quantity that gives a measure of
  how close the estimate is, on average.
• It is not zero.

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Restoration Filtering

• Objective To quantitatively estimate the true
  image from its degraded measurement.

System H
Filter Q

g
f
n
An estimate of f
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Noise Effects

• Consider the error image between the true image
  and the restoration filter estimate
• This is the case for any linear system Q.

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Mean Square Error for Q

• First, we compute an expression for the square
  error when using any linear system

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Mean Square Error for Q

• Second, we apply the expectation operator to get
  the mean. Only the noise is random, so the
  operator only applies to terms with n

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Mean Square Error for Q

• Second, we apply the expectation operator to get
  the mean. Only the noise is random, so the
  operator only applies to terms with n

Terms with En go to zero since n is zero mean
and uncorrelated with f.
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Mean Square Error for Q

• Second, we apply the expectation operator to get
  the mean. Only the noise is random, so the
  operator only applies to terms with n

Terms with En go to zero since n is zero mean
Terms with En go to zero since n is zero mean
and uncorrelated with f.
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Mean Square Error for Q

• Third, we want to find Q to minimize the MSE.
• Take the derivative with respect to Q

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Mean Square Error for Q

• The derivative operator is a square matrix that
  takes the derivative with respect to every
  element of Q.

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Aside Matrix Derivative Relations
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Mean Square Error for Q

• The derivative operator is a square matrix that
  takes the derivative with respect to every
  element of Q.

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Mean Square Error for Q

• Fourth, set the derivative to zero and solve for
  Q.

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Linear Wiener Filter

• The filter Q gives the minimum MSE for the linear
  case.
• The only stipulation we made was that n is
  zero-mean and uncorrelated with f.
• This is the linear version of the Wiener filter.

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LSI Wiener Filter

• For LSI systems, in the Fourier domain, this
  becomes

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LSI Wiener Filter

• Note the similarity between the linear and LSI
  forms

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LSI Wiener Filter

• Note that the noise-to-signal ratio appears in
  the denominator
• What happens at frequencies where noise is low?
• Frequencies where noise is high?

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Wiener Filter

• The Wiener filter automatically cuts off the
  filter at frequencies where noise becomes
  significantly higher than signal.
• It also restores some of the blurring.

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Wiener Filter - Blur Sigma 2
Original
Blur noise

SNR24.2 dB NMSE .0027

Wiener - noise-free
Wiener - noisy
SNR27.3 dB NMSE .0015
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Wiener Filter - Blur Sigma 5
Original
Blur noise

SNR23.5 dB NMSE .0074
Wiener - noise-free
Wiener - noisy
SNR26.0 dB NMSE .0041
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Wiener Filter Characteristics
Constant SNR 200 (26 dB)

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Wiener Filter Characteristics
Constant Gaussian Blur sigma 2

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Wiener Filter Bias

• The bias of the Wiener Filter result
• This is generally not zero. The Wiener filter
  estimate will be biased.

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Linear Wiener Filter

• Normally, this would require inversion of a large
  matrix
• Some approximations may help.

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Generalized Wiener Filter

• Implement the Wiener filter in another transform
  domain
• Hopefully, the inversion is easier to accomplish
  this way.

Unitary Transform A
Wiener Filter M
Inverse Transform AT

g
f