Signal Processing Course: Wavelet Processing

Transcript

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