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BAE 790I / BMME 231 Fundamentals of Image Processing Class 18

BAE 790I / BMME 231 Fundamentals of Image Processing Class 18. Restoration Filtering Inverse Filters: Properties Wiener Filters MSE Properties. Image Restoration. Objective: To quantitatively estimate the true image from its degraded measurement. System H. Restoration process. +. g.

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BAE 790I / BMME 231 Fundamentals of Image Processing Class 18

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  1. BAE 790I / BMME 231Fundamentals of Image ProcessingClass 18 • Restoration Filtering • Inverse Filters: Properties • Wiener Filters • MSE • Properties

  2. 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).

  4. Inverse Filter with Noise 24.2 dB 1.7 dB High-frequency noise is amplified by the inverse filter.

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

  6. 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.

  7. 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.

  8. Restoration Filtering • Objective: To quantitatively estimate the true image from its degraded measurement. System H Filter Q + g f n An estimate of f

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

  10. Mean Square Error for Q • First, we compute an expression for the square error when using any linear system:

  11. 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:

  12. 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 E[n] go to zero since n is zero mean and uncorrelated with f.

  13. 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 E[n] go to zero since n is zero mean Terms with E[n] go to zero since n is zero mean and uncorrelated with f.

  14. Mean Square Error for Q • Third, we want to find Q to minimize the MSE. • Take the derivative with respect to Q:

  15. Mean Square Error for Q • The derivative operator is a square matrix that takes the derivative with respect to every element of Q.

  16. Aside: Matrix Derivative Relations

  17. Mean Square Error for Q • The derivative operator is a square matrix that takes the derivative with respect to every element of Q.

  18. Mean Square Error for Q • Fourth, set the derivative to zero and solve for Q.

  19. 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.

  20. LSI Wiener Filter • For LSI systems, in the Fourier domain, this becomes:

  21. LSI Wiener Filter • Note the similarity between the linear and LSI forms:

  22. 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?

  23. 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.

  24. Wiener Filter - Blur Sigma = 2 Original Blur + noise SNR=24.2 dB NMSE = .0027 Wiener - noise-free Wiener - noisy SNR=27.3 dB NMSE = .0015

  25. Wiener Filter - Blur Sigma = 5 Original Blur + noise SNR=23.5 dB NMSE = .0074 Wiener - noise-free Wiener - noisy SNR=26.0 dB NMSE = .0041

  26. Wiener Filter Characteristics Constant SNR = 200 (26 dB)

  27. Wiener Filter Characteristics Constant Gaussian Blur sigma = 2

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

  29. Linear Wiener Filter • Normally, this would require inversion of a large matrix: • Some approximations may help.

  30. 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

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