Slides for a course on signal and image processing.

1. Linear and Non Linear
Denoising
Gabriel Peyré
www.numerical-tours.com

2. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

3. Noise in Images

4. Denoising Problem

5. Denoising Problem

6. Additive Noise Model

7. Noise Distributions
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

8. Noise Distributions
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

9. Noise Distributions
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

10. Data-dependent Noise

11. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

12. Linear Denoising Estimator

13. Fourier and Denoising

14. Optimal Filter Choice

15. Oracle Estimation of Optimal Filter

16. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

17. Diagonal Thresholding

18. Wavelet Diagonal Hard Thresholding

19. Sparse Signal Estimation

20. Optimal Threshold Selection

21. Non-linear Approximation and Estimation
W unit variance white noise.

22. Hard vs. Soft Thresholding

23. Hard vs. Soft Thresholding

24. Optimal Threshold

25. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

26. Translation Invariant Denoising

27. Translation Invariant Wavelets

28. Translation Invariant Haar (1D)

29. Translation Invariant Transform (2D)

30. Translation Invariant Thresholding

31. Optimal Invariant Threshold

32. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

33. Between Hard and Soft Thresholding

34. Stein Quadratic-Soft Thresholder

35. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

36. Block Thresholding

37. Optimal Block Choice

38. Comparison

39. Overview
• Noise in Signals and Images
• Linear Denoising by Blurring
• Non-linear Wavelet Denoising
• Translation Invariant Thresholding
• Other Diagonal Thresholders
• Non-diagonal Block Thresholding
• Data-dependent Noise

40. Poisson Noise

41. Poisson Noise Variance Stabilization
1.05
1
0.95
0.9
0.85
0.8
0.75
1 2 3 4 5 6 7 8 9 10

42. Multiplicative Noise

43. Multiplicative Noise Stabilization
0 0.5 1 1.5 2 2.5
−1.5 −1 −0.5 0 0.5 1 1.5

44. Conclusion