Signal Processing Course : Denoising

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Slides for a course on signal and image processing.

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Signal Processing Course : Denoising

  1. 1. Linear and Non Linear Denoising Gabriel Peyré www.numerical-tours.com
  2. 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. 3. Noise in Images
  4. 4. Denoising Problem
  5. 5. Denoising Problem
  6. 6. Additive Noise Model
  7. 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. 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. 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. 10. Data-dependent Noise
  11. 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. 12. Linear Denoising Estimator
  13. 13. Fourier and Denoising
  14. 14. Optimal Filter Choice
  15. 15. Oracle Estimation of Optimal Filter
  16. 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. 17. Diagonal Thresholding
  18. 18. Wavelet Diagonal Hard Thresholding
  19. 19. Sparse Signal Estimation
  20. 20. Optimal Threshold Selection
  21. 21. Non-linear Approximation and EstimationW unit variance white noise.
  22. 22. Hard vs. Soft Thresholding
  23. 23. Hard vs. Soft Thresholding
  24. 24. Optimal Threshold
  25. 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. 26. Translation Invariant Denoising
  27. 27. Translation Invariant Wavelets
  28. 28. Translation Invariant Haar (1D)
  29. 29. Translation Invariant Transform (2D)
  30. 30. Translation Invariant Thresholding
  31. 31. Optimal Invariant Threshold
  32. 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. 33. Between Hard and Soft Thresholding
  34. 34. Stein Quadratic-Soft Thresholder
  35. 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. 36. Block Thresholding
  37. 37. Optimal Block Choice
  38. 38. Comparison
  39. 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. 40. Poisson Noise
  41. 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. 42. Multiplicative Noise
  43. 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. 44. Conclusion

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