This document provides an overview of wavelet processing and wavelet transforms. It begins by reviewing Fourier transforms and introducing 1D multiresolutions and wavelet transforms. It describes the filter constraints for approximation and detail filters. It then discusses 2D multiresolutions and wavelet transforms, including anisotropic, separable, and isotropic transforms. It also covers fast wavelet transforms, discrete wavelet coefficients, and inverting the transform. The document concludes with examples of wavelet decompositions.

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. The Four Settings
Note: for Fourier, bounded periodic.
Infinite continuous domains: f0 (t), t R ... ...
+⇥
ˆ
f0 ( ) = f0 (t)e i t
dt
⇥
Periodic continuous domains: f0 (t), t ⇥ [0, 1] R/Z
1
ˆ
f0 [m] = f0 (t)e 2i mt
dt
0
Infinite discrete domains: f [n], n Z ... ...
ˆ
f( ) = f [n]ei n
n Z
Periodic discrete domains: f [n], n ⇤ {0, . . . , N 1} ⇥ Z/N Z
N 1
ˆ
f [m] = f [n]e
2i
N mn
n=0

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

5. Sampling and Periodization
(a)
(b)
1
(c) 0
(d)

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

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
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0

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
1
0.8
0.6
0.4
0.2
0
1 0.2
0.8
0.1
0.6
0
0.4
−0.1
0.2
0 −0.2

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

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

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

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

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

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

31. Fast Wavelet Transform
1
0.8
0.6
0.4
0.2
0

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

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

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

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

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] ⇥⇤
¯ |⌅(⇤ + 2k⇥)|2 = 1
ˆ
k

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

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

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

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

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

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

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

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

58. Vanishing Moment Constraint
1
0.8
0.6
0.4
0.2
0
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

59. Vanishing Moment Constraint
1
0.8
0.6
0.4
0.2
0
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

60. Vanishing Moment Constraint
1
0.8
0.6
0.4
0.2
0
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

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