Digital Image Processing Week VI

1. Image Restoration Digital Image Processing Week VI Thurdsak LEAUHATONG

2. The Image Enhancement vs Restoration • The Concept Enhancement • Manipulate the image so that it is suitable for a specific application. Original Image Enhanced Image

3. The Image Enhancement vs Restoration • The Concept Restoration • Restore a degraded image back to the original image. Degraded Image Original Image

4. The Image Degradation • Model Degradation Fiilter + Original Image Blurred Image Degraded Image Additive Noise

5. The Image Degradation • Model Degraded Image

6. The Image Restoration Process Restoration Fiilter Degraded Image Degraded Image

7. Additive White Noise • Frequency Property • The Fourier transform of the additive white noise is constant and equal to. Additive White Noise

8. Additive White Noise • Frequency Property • The ideal white noise contains all frequencies. • The practical white noise is a band-limited signal.

9. Additive White Noise Property of Spatial Domain • Autocorrelation function of white noise Additive White Noise

10. Additive White Noise Property of Spatial Domain • If drops quickly with x and y, then changes quickly with position. • of additive white noise is impulse function, then the additive noise at a pixel is independent from the other pixels.

11. Statistic Model of Additive White Noise • The additive white noise cannot be predicted but can be approximately described in statistical way using the probability density function (PDF). Uniform Gaussian Salt& Pepper Lognormal

12. Statistic Model of Additive White Noise Rayleigh Exponential Erlang (Gamma)

13. Statistic Model of Additive White Noise

14. Statistic Model of Additive White Noise Degraded images Original image Histogram

15. Statistic Model of Additive White Noise Degraded images Original image Histogram

16. Periodic Noise Periodic noise looks like dots In the frequency domain

17. Estimation of Noise We cannot use the image histogram to estimate noise PDF. It is better to use the histogram of one area of an image that has constant intensity to estimate noise PDF.

18. Periodic Noise Reduction by Freq. Domain Filtering Degraded image DFT Periodic noise can be reduced by setting frequency components corresponding to noise to zero. Band reject filter Restored image

19. Band Reject Filters Use to eliminate frequency components in some bands Periodic noise from the previous slide that is Filtered out.

20. Notch Reject Filters A notch reject filter is used to eliminate some frequency components.

21. Notch Reject Filter: Notch filter (freq. Domain) Degraded image DFT Noise Restored image

22. Example: Image Degraded by Periodic Noise Degraded image DFT (no shift) DFT of noise Noise Restored image

23. Chapter 5 Image Restoration

24. Mean Filters Degradation model: To remove this part Arithmetic mean filter or moving average filter (from Chapter 3) Geometric mean filter mn = size of moving window

25. Geometric Mean Filter: Example Image corrupted by AWGN Original image Image obtained using a 3x3 geometric mean filter Image obtained using a 3x3 arithmetic mean filter AWGN: Additive White Gaussian Noise

26. Harmonic and Contraharmonic Filters Harmonic mean filter Works well for salt noise but fails for pepper noise Contraharmonic mean filter mn = size of moving window Positive Q is suitable for eliminating pepper noise. Negative Q is suitable for eliminating salt noise. Q = the filter order

27. Contraharmonic Filters: Example Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 contra- harmonic mean filter With Q = 1.5 Image obtained using a 3x3 contra- harmonic mean filter With Q=-1.5

28. Contraharmonic Filters: Incorrect Use Example Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 contra- harmonic mean filter With Q=-1.5 Image obtained using a 3x3 contra- harmonic mean filter With Q=1.5

29. Order-Statistic Filters: Revisit Original image subimage Statistic parameters Mean, Median, Mode, Min, Max, Etc. Moving window Output image

30. Order-Statistics Filters Median filter Max filter Reduce “dark” noise (pepper noise) Min filter Reduce “bright” noise (salt noise) Midpoint filter

31. Median Filter : How it works A median filter is good for removing impulse, isolated noise Salt noise Pepper noise Median Sorted array Moving window Degraded image Salt noise Filter output Pepper noise Normally, impulse noise has high magnitude and is isolated. When we sort pixels in the moving window, noise pixels are usually at the ends of the array.

32. Median Filter : Example 1 2 Image corrupted by salt-and-pepper noise with pa=pb= 0.1 3 4 Images obtained using a 3x3 median filter

33. Max and Min Filters: Example Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 max filter Image obtained using a 3x3 min filter

34. Adaptive Filter General concept: • Filter behavior depends on statistical characteristics of local areas • inside mxn moving window • More complex but superior performance compared with “fixed” • filters Statistical characteristics: Noise variance: Local mean: Local variance:

35. Adaptive, Local Noise Reduction Filter Purpose: want to preserve edges Concept: • 1. If sh2 is zero,  No noise • the filter should return g(x,y) because g(x,y) = f(x,y) • 2. If sL2 is high relative to sh2,  Edges (should be preserved), • the filter should return the value close to g(x,y) • 3. If sL2 = sh2,  Areas inside objects • the filter should return the arithmetic mean value mL Formula:

36. Adaptive Noise Reduction Filter: Example Image corrupted by additive Gaussian noise with zero mean and s2=1000 Image obtained using a 7x7 arithmetic mean filter Image obtained using a 7x7 adaptive noise reduction filter Image obtained using a 7x7 geometric mean filter

37. Adaptive Median Filter Purpose: want to remove impulse noise while preserving edges Algorithm: Level A: A1= zmedian – zmin A2= zmedian – zmax If A1 > 0 and A2 < 0, goto level B Else increase window size If window size <= Smax repeat level A Else return zxy Level B: B1= zxy – zmin B2= zxy – zmax If B1 > 0 and B2 < 0, return zxy Else return zmedian where zmin = minimum gray level value in Sxy zmax = maximum gray level value in Sxy zmedian = median of gray levels in Sxy zxy = gray level value at pixel (x,y) Smax = maximum allowed size of Sxy

38. Adaptive Median Filter: How it works Level A: A1= zmedian – zmin A2= zmedian – zmax Else  Window is not big enough increase window size If window size <= Smax repeat level A Else return zxy  zmedian is not an impulse B1= zxy – zmin B2= zxy – zmax If B1 > 0 and B2 < 0,  zxy is not an impulse return zxy to preserve original details Else return zmedian  to remove impulse Determine whether zmedian is an impulse or not If A1 > 0 and A2 < 0, goto level B Level B: Determine whether zxy is an impulse or not

39. Adaptive Median Filter: Example Image corrupted by salt-and-pepper noise with pa=pb= 0.25 Image obtained using a 7x7 median filter Image obtained using an adaptive median filter with Smax = 7 More small details are preserved