1. ELE 488 F06 ELE 488 Fall 2006 Image Processing and Transmission (10-5-06) Wiener Filtering Derivation Comments Re-sampling and Re-sizing 1D  2D 10/5/06

2. ELE 488 F06 Linear Spatial-Invariant Distortion and Inverse Filtering Assumption – no noise, known distortion – often used to correct blur Consider 1D case first, then take up 2D v = u * h, w = v * g = u * (h * g) want w = u, so h * g = δ. Inverse Filter: G(ω) = 1 / H(ω). (divide by 0?) Pseudo Inverse Filter: G(ω) = 1 / H(ω) if | H(ω) | > ε G(ω) = 0 if | H(ω) | < ε 2D: inverse filter: G(ω, λ ) = 1 / H(ω, λ ). pseudo inverse filter: G(ω, λ ) = 1 / H(ω, λ ) if | H(ω, λ ) | > ε G(ω, λ ) = 0 if | H(ω, λ ) | < ε h, H uv  g, G w UMCP ENEE631 Slides (created by M.Wu © 2001) restoration distortion

3. ELE 488 F06 Inverse & Pseudo-inverse Filtering From Jain Fig.8.10 UMCP ENEE631 Slides (created by M.Wu © 2001)

4. ELE 488 F06 Another Solution – cut off high frequencies Limit restoration to low frequency components to avoid amplifying noise at high freq and at nulls ( 480x480 image through severe turbulence) UMCP ENEE631 Slides (created by M.Wu © 2004) Gonzalez/ Woods

5. ELE 488 F06 A Different Approach to Deconvolution Consider distortion correction and noise suppression: –Treat image and noise as random fields –Minimize MSE between the original u and restored w: E{ [ u – w ] 2 }, given v –Best estimate is conditional mean: E { u | v } needs info not usually available usually difficult to solve, non-linear problem –To find the best linear estimate  Wiener filtering linear estimate from observed image by minimizing MSE UMCP ENEE631 Slides (created by M.Wu © 2001/2004) h, H uv  g, G w

6. ELE 488 F06 Wiener Filtering Best linear estimate –minimize MSE E{ [ w – u ] 2 } –spatial invariant filter w = v * g –wide-sense stationary signal and noise –noise uncorrelated with original signal power spectral densities: S uu (ω) S nn (ω) Solution: G(ω) = H*(ω) S uu (ω) / { |H| 2 (ω) S uu (ω) + S nn (ω) } = 1 / { H + S nn / ( H* S uu ) } (Notation: Φ uu Φ nn for power spectral densities last time) UMCP ENEE631 Slides (created by M.Wu © 2001) h, H uv  g, G w

7. ELE 488 F06 Wiener Filtering (cont) G = H* S uu / { |H| 2 S uu + S nn } Consider deblurring noisy image, need to balance between: –HPF filter for de-blurring (undo H distortion) –LPF for suppressing noise Noiseless case ~ S nn = 0 (inverse filter) –Wiener filter becomes pseudo-inverse filter for S nn  0 –G(ω) = H* S uu / { |H| 2 S uu + S nn } G(ω)  1 / H(ω), if H(ω) ≠ 0 G(ω)  0, if H(ω) = 0 No blur, H = 1 (smoothing Filter) –Wiener filter attenuates noise according to SNR at each freq. G(ω) = S uu / {S uu + S nn } = (S uu / S nn )/ { 1 + S uu / S nn } UMCP ENEE631 Slides (created by M.Wu © 2001)

8. ELE 488 F06 Comparisons From Jain Fig.8.11 UMCP ENEE631 Slides (created by M.Wu © 2001)

9. ELE 488 F06 Wiener Filtering vs. Inverse Filtering UMCP ENEE631 Slides (created by M.Wu © 2004) Gonzalez - Woods (480x480 image through severe turbulence)

10. ELE 488 F06 enlarged

11. ELE 488 F06 Wiener Filtering vs. Inverse Filtering UMCP ENEE631 Slides (created by M.Wu © 2004) Gonzalez - Woods

12. ELE 488 F06

14. Real discrete-time, wide sense stationary random signals

15. ELE 488 F06 Derivation of Wiener Filter h, H uv  g, G w

16. ELE 488 F06 Derivation of Wiener Filter

17. ELE 488 F06 Derivation of Wiener Filter

18. ELE 488 F06 Wiener Filter: From Theory to Practice Have assumed knowing power spectral densities of image & noise and frequency response Assumptions valid? Implementation issues UMCP ENEE631 Slides (created by M.Wu © 2004)

19. ELE 488 F06 Wiener Filter: Issues to Be Addressed Filter size –Theoretically infinite impulse response  large-size DFTs –Impose filter size constraint: find the best FIR that minimizes MSE Need to estimate power spectrum density (psd) S nn,S uu –Estimate psd of blurred image and compensate for effect of noise –Estimate pds from representative images similar to image to be restored –Use statistical modeling for the image and estimate parameters –Constrained least square filter ~ (Jain Sec.8.8, Gonzalez Sec.5.9) Optimize smoothness in restored image (least-square of the rough transitions) Constrain differences between blurred image and blurred version of reconstructed image Estimate the restoration filter w/o the need of estimating p.s.d. Unknown distortion H ~ Blind Deconvolution UMCP ENEE631 Slides (created by M.Wu © 2001)

20. ELE 488 F06 Basic Ideas of Blind Deconvolution Three ways to estimate H: observation, experimentation, assume model Estimate H via spectrum’s zero patterns –Two major classes of blur (motion blur and out-of-focus) –H has nulls related to the type and the parameters of the blur Maximum-Likelihood blur estimation –Each set of image model and blur parameters gives a “typical” blurred output. –Given the observation of blurred image, try to find the set of parameters that is most likely to produce that blurred output Iteration ~ Expectation-Maximization approach (EM) Given estimated parameters, restore image via Wiener filtering Examine restored image and refine parameter estimation Get local optimum “Blind Image Deconvolution” by Kundur et al, IEEE Sig. Proc. Magazine, vol.13, 1996 UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

21. ELE 488 F06 Image Resizing: Scaling and Interpolation 512x512 image resample to 216x216 and to 128x128

22. ELE 488 F06

24. Image Resampling 512x512 image resample to 216x216 and to 128x128

25. ELE 488 F06 Image Resizing: Scaling and Interpolation 512x512 256x256 zoom in

26. ELE 488 F06 Image Resizing: Scaling and Interpolation 512x512 128x128 zoom in

27. ELE 488 F06

29. Image Resizing: Scaling and Interpolation 512x512 800x600

30. ELE 488 F06 Image Resizing: Scaling and Interpolation 512x512 320x200

31. ELE 488 F06 Integer Subsampling by L (=2) Beware of Aliasing! Review of 1 D sampling and re-sampling

32. ELE 488 F06 1 D Sampling and Reconstruction signal x(t) samples x(nT) h(t) = 1, 0<t<T h(t) = 0, other t piecewise constant x r (t) Reconstruction from samples : x r (t) =x(nT) h(t - nT)

33. ELE 488 F06 h(t) = 1 - | t |/T, 0<t<T h(t) = 0, other t piecewise linear x r (t) h(t) = sinc(t/T) = sin(πt /T) / (π t /T) bandlimited x r (t) Sampling Theorem (Shannon, Nyquist, Whittaker, Komogorov,...)

34. ELE 488 F06 Sampling, Reconstruction, Aliasing

35. ELE 488 F06

36. Decrease Sampling Rate by M

37. ELE 488 F06 Example

38. ELE 488 F06 Increasing Sampling Rate by L Use LPF (stopband edge ωc = π/M) to remove “image”.

39. ELE 488 F06 Change Sampling Rate by L/M

40. ELE 488 F06 Integer Subsampling by L L=2 Beware of Aliasing!

41. ELE 488 F06 Sampling Subsampled by 5 Noticable aliasing! Gaussian LPF (2) then Subsampled by 5 Gaussian LPF (4) then Subsampled by 5

42. ELE 488 F06 Resizing from M x N down to J x K M x N J x K Sinc interpolation at new grid points – not practical Use approximate reconstruction with reduced computation sample reconstruct sample