Mean Square Error and Wiener Filter

2. WIENER FILTER

3. PERSONALITIES
©Wikipedia ©Brittanica
Andrey Kolmogorov
Norbert Wiener

4. CHARECTERISTICS
1. Restoration technique for deconvolution (high pass filtering).
2. Compression operation (low pass filtering)
3. Inverse filtering is very sensitive to additive noise.
4. Optimal tradeoff between inverse filtering and noise smoothing.
5. Removes the additive noise and inverts the blurring
simultaneously.
6. Better than Inverse filter
7. Degradation function & Noise SC
8. Image & Noise are random

5. OBJECTIVE
Reduce the error

6. OBJECTIVE
Reduce the error

7. OBJECTIVE
Reduce the error

8. OBJECTIVE
Reduce the error

9. OBJECTIVE
Reduce the error

10. ASSUMPTIONS
1. Noise & Image are uncorrelated
2. Any one’s mean is 0.
3. Gray levels in the estimate are
linear functions of the degraded image

11. VARIABLES
• H(u, v) = degradation function
• H*(u, v) = complex conjugate of H(u, v)
• ||H(u, v)||^2 = H*(u, v)H(u, v)
• Sn (u, v) = ||N(u, v)||^2 = power spectrum of the noise
• Sf (u, v) = ||F(u, v)||^2 = power spectrum of the
undegraded image

12. DERIVATIONS
• G(u,v) = F(u,v) H(u,v) + N(u,v) 1
• W (u,v) = Fc(u,v) /G(u,v) 2
• R(u,v) = G(u,v) *W(u,v) 3
• E{F(u,v) N*(u,v)} = E{F*(u,v) N(u,v)} =0 4
• E{||F (u,v)||^2} = SF (u,v) 5
• E{||N (u,v)||^2} = SN (u,v) 6
• ||F (u,v)||^2 = F (u,v) F* (u,v) 7

20. MAIN EQUATION

21. SIGNAL-TO-NOISE RATIO

22. MEAN SQUARE ERROR

23. SIGNAL-TO-NOISE RATIO

24. PRACTICAL EQUATION

25. FULL INVERSE FILTERING
k = 0.0025

26. RADIALLY LIMITTED INVERSE
FILTERING
k = 0.0025

27. WIENER FILTERING
k = 0.0025

28. MOTION BLUR AND ADDITIVE NOISE
100% OF σ
Corrupted Inverse Filter Wiener Filter

29. MOTION BLUR AND ADDITIVE NOISE
0.1% OF σ
Corrupted Inverse Filter Wiener Filter

30. MOTION BLUR AND ADDITIVE NOISE
0.00001% OF σ
Corrupted Inverse Filter Wiener Filter

31. REFERENCE
• Digital Image Processing, Third Edition Rafael C. Gonzalez,
Richard E. Woods
• http://www.owlnet.rice.edu/~elec539/Projects99/BACH/proj2/
wiener.html
• https://www.youtube.com/watch?v=aKSiZ8lu-lM
• https://www.youtube.com/watch?v=unkMkRUHULg
• https://www.youtube.com/watch?v=l3LMVLbDCic

32. THANK YOU