1. Chapter 5
Image Restoration

2. 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)

3. 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.

4. 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

5. Gaussian Noise
The PDF of a Gaussian random variable, z, is given by:

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

7. Erlang (Gamma) Noise The PDF of Erlang noise is given by:
Mean and variance:

8. Exponential Noise The PDF of exponential noise is given by:
where a >0
Mean and variance:

9. Uniform Noise
The PDF of uniform noise is given by:
Mean and variance:

10. Impulse (Salt-and-Pepper) Noise
The PDF of (bipolar) impulse noise is given by:

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

12. Illustration (I)

13. Illustration (II)

14. 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).

15. Restoration in the Presence of Noise Only – Spatial Filtering
Mean filters:
Arithmetic mean filters
Geometric mean filter
Harmonic mean filter:
Contraharmonic mean filter: Q: the order of the filter. Q>0 eliminates pepper noise, Q <0 eliminates salt noise.

16. Illustration (I)

17. Illustration (II)

18. Illustration (III)

19. 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

20. Illustration (I)

21. Illustration (II)

22. Illustration (III)

23. Adaptive Filters
Filter’s 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

24. 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

25. Illustration

26. 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 – zmax if A1> 0 and A2 <0, go to level B Else increase the window size If window size <= Smax repeat level A else output zxy
Level B: B1= zxy – zmin, B2= zxy – zmax if B1> 0 and B2 <0, output zxy Else output zmed

27. Illustration

28. Periodic Noise Reduction
By Fourier domain filtering:
Bandreject filters
Bandpass filters
Notch filters

29. Illustration

30. Ideal Notch Reject Filter
where

31. Butterworth Notch Reject Filter

32. Gaussian Notch Reject Filter

33. Notch Filters

34. Linear, Position-Invariant Degradations
Estimating the degradation function
By image observation
By experimentation
By modeling

35. 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)

36. 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:

37. Figure 5.24

38. Estimation by Modeling
Modeling atmospheric turbulence

39. Atmospheric Turbulence

40. Estimation by Modeling (cont’d)
Modeling effect of planar motion x0(t),y0(t):
If T is the duration of the exposure, then
It can be shown that:

41. Motion Blur
If x0(t)=at/T and y0(t)=0, then

42. Motion Blur Example

43. Deconvolution Inverse filtering
Minimum mean square error (Wiener) filtering
Constrained least squares filtering
Geometric mean filter

44. Results (Inverse Filter)

45. Results (Inverse and Wiener)

46. Results (Motion Blurs)

47. Results (Constrained LS Filter)

48. Recent Developments Blind deconvolution
Removing camera shake from a single photograph