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