Signal Processing Course: Denoising

Transcript

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

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 2 3 4 5 6

    7 8 9 10 0.75 0.8 0.85 0.9 0.95 1 1.05

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