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Signal Processing Course: Denoising

Signal Processing Course: Denoising

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Gabriel Peyré

January 01, 2012
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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. 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