Understanding Image Restoration Techniques in the Digital World

1. Chapter 8: Image Restoration ○ Types of image degradations: Noise, error, distortion, blurring, etc. Degradation model: where g(x,y): degraded image, f(x,y): image, h(x,y): degradation process n(x,y): additive noise 8-1

2. ○ Two ways to recover image degradations: • 1) Image enhancement: Overlook degradation • processes, deal with images intuitively • 2) Image restoration: Known degradation • processes; model the processes and • reconstruct images based on the inverse • model

3. From the convolution theorem, ○ Recall the degradation model Fourier transform Difficulties: (a) Unknown N(u,v), (b) Small H(u,v) 8-3

4. ◎ Types of Noises: • ○White noise: the noise whose Fourier spectrum • is constant • ○ Periodic noise: • Original image Noisy image • ○ Additive noise: each pixel is added a value • (noise) chosen from a probability distribution

5. (a, b can be + or -) • 。Salt-and-pepper (impulse) noise • Let x : noise value e.g.,

6. 。Uniform noise: (a, b can be + or -)

8. Given histogram h

9. 。Gaussian noise:

10. Method 1: 2-10

11. Method 2: 2-11

12. Homework : (1) Create an image g(x,y) whose pixels all have the same gray value of 100. Show the image g(x,y). (2) Generate Gaussian noise n(x,y), with , using methods 1 and 2. Show the noisy image f(x,y) = g(x,y) + n(x,y). (3) Display the histogram h(i) of f(x,y). (4) Comment on your results. Example: Input image g(x,y) of gray values of 100 Noisy image f(x,y) Histogram of g(x,y) Histogram of f(x,y) 12

13. 。Rayleigh noise:

14. 。Erlang (gamma) noise:

15. 。Exponential noise:

16. ◎ Estimation of noise parameters • Steps: 1. Choose a uniform image region • 2. Compute histogram • 3. Compute mean and variance • 4. Determine the probability distribution • from the shape of • 5. Estimate the parameters of the probability • distribution using

17. Given • Examples: • (a) Uniform noise:

18. Given • (b) Rayleigh noise:

19. ○ Multiplicative noise: Each pixel is multiplied • with a value (noise) chosen from a probability • distribution, e.g., speckle noise

20. ◎ Noise removal • ○ Salt-and-pepper noise • – high frequency image component low-pass filter median filter

21. 。 Mean filter (i) Arithmetic mean: 4 × 3 5 × 5

22. (ii) Geometric mean: • (iii) Harmonic mean: • (iv) Contraharmonic mean:

23. 3 × 3 median filter • 3 × 3 (twice) 5 × 5

24. 。Adaptive filter -- change characteristics according • to the pixels under the window

25. 3×3 5×5 7×7 9×9

26. ○ Gaussian noise • Assume Gaussian noise n(x,y) is uncorrelated • and has zero mean Image averaging:

27. Example:

28. ○ Periodic noise Notch filter • Band reject • filter

29. In general case, Fourier spectrum noise Corresponding spatial noise

30. ○Inverse filtering

31. Low-pass Filtering:  Constrained Division d = 40 60 80 100

32. ○Wiener filtering • -- Considers both degradation process and noise • Idea: (Parametric Wiener filter)

33. When r = 1, (Wiener filter) If noise is zero, , (Inverse filter) If noise is white noise, is constant 8-33

34. Input image k = 0.001 k = 0.0001 k = 0.00001

35. ○ Motion debluring • Image f(x,y) undergoes planar motion • : the components of motion • T: the duration of exposure • Fourier transform,

36. Shifting property:

37. Suppose uniform linear motion: Restore image by the inverse or Wiener filter