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Signal Processing Course : Denoising
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Signal Processing Course : Denoising

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Slides for a course on signal and image processing.

Slides for a course on signal and image processing.


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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. Non-linear Approximation and EstimationW 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