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Signal Processing Course: Wavelet Processing

E34ded36efe4b7abb12510d4e525fee8?s=47 Gabriel Peyré
January 01, 2012

Signal Processing Course: Wavelet Processing

E34ded36efe4b7abb12510d4e525fee8?s=128

Gabriel Peyré

January 01, 2012
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  1. Wavelet Processing Gabriel Peyré www.numerical-tours.com

  2. Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

    •Filter Constraints •2-D Multiresolutions
  3. Infinite continuous domains: Periodic continuous domains: Infinite discrete domains: Periodic

    discrete domains: f0 (t), t R f0 (t), t ⇥ [0, 1] R/Z The Four Settings ˆ f[m] = N 1 n=0 f[n]e 2i N mn ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n Note: for Fourier, bounded periodic. .. . .. . .. . .. . f[n], n Z f[n], n ⇤ {0, . . . , N 1} ⇥ Z/NZ
  4. Sampling idealization: Poisson formula: f0 f ˆ f0 ˆ f

    sampling periodization cont. FT discr. FT Commutative diagram: f[n] ˆ f0 ( ) ˆ f( ) Sampling and Periodization f[n] = f0 (n/N) ˆ f(⇥) = k ˆ f0 (N(⇥ + 2k )) (a) (c) (b) 1 0 (a) (c) (b) 1 0 (a) (b) (a) f0 (t)
  5. Sampling and Periodization (a) (c) (d) (b) 1 0

  6. Sampling and Periodization: Aliasing (b) (c) (d) (a) 0 1

  7. Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

    •Filter Constraints •2-D Multiresolutions
  8. Multiresolutions: Approximation Spaces

  9. Multiresolutions: Approximation Spaces

  10. Multiresolutions: Approximation Spaces

  11. Haar Multiresolutions 0 0.2 0.4 0.6 0.8 1 0 0.2

    0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
  12. Multiresolutions: Detail Spaces

  13. Multiresolutions: Detail Spaces

  14. Multiresolutions: Detail Spaces

  15. Multiresolutions: Detail Spaces

  16. Multiresolutions: Detail Spaces j,n \ j j0, 0 n <

    2 j ⇥ ⇥j0,n \ 0 n < 2 j0 ⇥
  17. Haar Wavelets −0.2 −0.1 0 0.1 0.2 0 0.2 0.4

    0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
  18. Haar Wavelets −0.2 −0.1 0 0.1 0.2 0 0.2 0.4

    0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2
  19. Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

    •Filter Constraints •2-D Multiresolutions
  20. Computing the Wavelet Coefficients

  21. Computing the Wavelet Coefficients

  22. Computing the Wavelet Coefficients

  23. Computing the Wavelet Coefficients

  24. Computing the Wavelet Coefficients

  25. Discrete Wavelet Coefficients 0 0.2 0.4 0.6 0.8 1

  26. −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

    0 0.2 0.4 0.6 0.8 1
  27. −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

    0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2
  28. −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

    0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
  29. −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

    0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.5 0 0.5
  30. −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

    0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.5 0 0.5
  31. Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1

  32. Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

    0.5 1
  33. Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

    0.5 1 0 0.5 1 1.5
  34. Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

    0.5 1 0 0.5 1 1.5 0 0.5 1 1.5 2
  35. Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

    0.5 1 0 0.5 1 1.5 0 0.5 1 1.5 2
  36. Haar Refinement

  37. Haar Refinement

  38. Haar Transform

  39. Haar Transform

  40. Haar Transform

  41. Haar Transform

  42. Haar Transform

  43. Inverting the Transform

  44. Inverting the Transform

  45. Inverting the Transform

  46. Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

    •Filter Constraints •2-D Multiresolutions
  47. Approximation Filter Constraints

  48. Approximation Filter Constraints

  49. Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

    ⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
  50. Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

    ⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
  51. Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

    ⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
  52. Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

    ⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
  53. { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

    2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
  54. { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

    2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
  55. { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

    2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
  56. { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

    2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
  57. { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

    2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
  58. Vanishing Moment Constraint −0.2 −0.1 0 0.1 0.2 −0.2 −0.1

    0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1
  59. Vanishing Moment Constraint −0.2 −0.1 0 0.1 0.2 −0.2 −0.1

    0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1
  60. Vanishing Moment Constraint −0.2 −0.1 0 0.1 0.2 −0.2 −0.1

    0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1
  61. Daubechies Family

  62. Daubechies Family

  63. Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

    •Filter Constraints •2-D Multiresolutions
  64. Anisotropic Wavelet Transform

  65. Anisotropic Wavelet Transform

  66. Anisotropic Wavelet Transform

  67. Anisotropic Wavelet Transform

  68. Anisotropic Wavelet Transform

  69. 2D Multi-resolutions

  70. 2D Multi-resolutions

  71. 2D Multi-resolutions

  72. 2D Wavelet Basis

  73. Discrete 2D Wavelets Coefficients

  74. Discrete 2D Wavelets Coefficients

  75. Discrete 2D Wavelets Coefficients

  76. Discrete 2D Wavelets Coefficients

  77. Discrete 2D Wavelets Coefficients

  78. Examples of Decompositions

  79. Separable vs. Isotropic

  80. Fast 2D Wavelet Transform

  81. Fast 2D Wavelet Transform

  82. Fast 2D Wavelet Transform

  83. Fast 2D Wavelet Transform

  84. Inverse 2D Wavelet Transform

  85. Inverse 2D Wavelet Transform

  86. Conclusion

  87. Conclusion

  88. Conclusion