Chapter 8: Image Restoration

1. Chapter 8: Image Restoration • Image enhancement: Overlook degradation • processes, deal with images intuitively • Image restoration: Known degradation • processes; model the processes and • reconstruct images based on the inverse • model • ○ Degradations • e.g., noise, error, distortion, blurring

2. ◎ Degradation Model • g(x,y): degraded image, f(x,y): image, • h(x,y): degradation process • n(x,y): additive noise • From the convolution theorem, • Difficulties: • (a) unknown N(u,v), (b) small H(u,v)

3. ◎ Noise (originating from image acquisition, • digitization, or transmission) • ○White noise: the noise whose Fourier • spectrum is constant • ○ Periodic noise: • Noisy image Original image • ○ Additive noise: Each pixel is added a • value (noise) chosen from a • probability distribution

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

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

7. 。Gaussian noise:

8. 。Rayleigh noise:

9. 。Erlang (gamma) noise:

10. 。Exponential noise:

11. ◎ 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

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

13. Given • (b) Rayleigh noise:

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

15. ◎ Noise removal • ○ Salt-and-pepper noise • – high frequency image component Low-pass filter median filter (Linear filter) (nonlinear filter)

16. 。 Mean filter – tend to blur image (i) Arithmetic mean: 4 × 3 5 × 5

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

18. Median filter • 3 × 3 3 × 3 (twice) 5 × 5

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

20. 3×3 5×5 7×7 9×9

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

22. Example:

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

24. In general case, Corresponding spatial noise Fourier spectrum of noise 8-24

25. ○Inverse filtering:

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

27. ○Wiener filtering • -- Considers both degradation process and noise • Idea:

28. ◎ Wiener filter (Least Mean Square Filter): Let : correlation matrices of image f and noise n The ijth element of is given by We hope noise-to-signal ratio to be small : real symmetric matrices ○ For images, the correlation between pixels (20 to 30 pixels) in the image is in general limited. A typical correlation matrix has a bound of nonzero elements about the main diagonal and zeros in the right upper and left lower corner regions 8-28

29. can be made to approximate block circulant matrices and can be diagonalized Let 8-29

30. Let Then where D: diagonal matrix with Let where A, B: diagonal matrices Let Substitute into 8-30

31. From and Multiply 8-31

32. assume M = N Note When r = 1 => (Wiener filter) r = variable => parametric Wiener filter 8-32

33. If noise is zero, Wiener filter = inverse filter If noise is white noise, is constant

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,

37. Suppose uniform linear motion: • Note H vanishes at u = n / a • (n: an integer) • Restore image by the inverse • or Wiener filter