1. Yosemite National Park, California
The Wigner Function Representation of Multi-Resolution Analysis as a Key to Optimal Wavelet Basis Selection
Bedros B. Afeyan
Polymath Research Inc.,
Pleasanton, CA,
M. R. Douglas and
R. B. Spielman, Sandia
National Laboratories,
Albuquerque, NM
Presented at:
YES 2000
Yosemite National Park, California
10/29-11/1, 2000

2. Outline
What are Wavelets, MRD, Wavelet Transforms and Wigner Functions in Phase Space?
DWT-SP (PRI) Mathematica NoteBooks: An Introduction
Analyzing a Double Gaussian with Two Carriers (generic oscillatory band limited 1D Signal).
Comparison to each Gaussian with its own carrier cases.
What are the advantages of a Wigner Representation of the MRD for Best Wavelet Basis Selection?
Comparisons with other techniques that are less illuminating and less reliable.

3. What Are Wavelets. Start @ (www. wavelets. org) surf (Mathsoft, amara,
Mallat, Meyer, Daubechies, Beylkin, Coifman, Strang, Sweldens, Jawerth...
Wavelets are localized kernels or atoms in PHASE SPACE.
You may think of them as basis functions with prescribed dilation and translation properties.
They may or may not be orthonormal or have compact support or be differentiable everywhere, or be fractal, or have many zero momemts.
Wavelets are like breathing wave packets which can home in on structures in phase space better than FT or WFT ever could.
When the scale is decreased
translation steps between
wavelets should likewise be
decreased

4. What is MRD or Multi-resolution Decomposition?
Multiresolution: Zoom in and out on a number of successively finer scales in a sequence of nested approximation subspaces {Vj}j in Z.
In general, get an overcomplete basis set in L2(R) Approximate (or truncate) by bounding the scales of interest.
Scaling functions and the scaling equation:
The Wavelets:
Low pass filter
High pass filter
These filters decompose a sampled signal into 2 sub-sampled channels:
the coarse approximation of the signal and the missing details at finer scales.
The original signal can be reconstructed from these channels by interpolation.

5. The Scaling Function and Wavelet for Haar or Daubechies 1 in X-Space

6. The Scaling Function and Wavelet for Haar or Daubechies 1 in K- Space

7. The Scaling Functions and Wavelets for Daubechies 2-6 in X-Space

8. The Scaling Functions and Wavelets for Daubechies 2-6 in k-Space

9. A Little Wavelet Analysis: Pick a Signal, Any Signal!

10. Here’s MRD in Action for 6 Cases (Multi-Resolution Decomposition)

11. Data Compression Does Depend on Wavelet Choice

12. Cumulative Energy In # of Wavelets Kept Helps Discrimination Too

13. How Does Daubechies 5 Do It?

14. DWT-SP with Daubechies 5 The Actual Wavelet Coefficients

15. How Much Better Do You Do With Daubechies 5 than Daubechies 3?

16. The Least Square Error vs Number of Wavelets Kept for Daubechies 1-6

17. Comparisons of the Least Square Errors Between Daubechies 3 & 5

18. Least Square Error Reduction as More Levels Are Added to the MRD for Daubechies 1-6

19. The Relative Least Squares Error Does Not Go to Zero Very Fast Between Daubechies 3 & 5

20. How Well Do Box Windowed Fourier Transforms Do?

21. How Well Do Tapered Windowed Fourier Transforms Do?

22. What Do Coiflet Scaling Functions Look Like in X-Space?

23. What Do Coiflet Scaling Functions Look Like in K-Space?

24. What Do Coiflet Wavelets Look Like in X-Space?

25. What Do Coiflet Wavelets Look Like in K-Space?

26. What Do Least Asymmetric Daubechies Filter Scaling Functions Look Like in X-Space?

27. What Do Least Asymmetric Daubechies Filter Scaling Functions Look Like in K-Space?

28. What Do Least Asymmetric Daubechies Filter Wavelets Look Like in X-Space?

29. What Do Least Asymmetric Daubechies Filter Wavelets Look Like in k-Space?

30. We Can Now Do the Same Analysis with Coiflets and Least Asymmetric Daub Filters As We Did With Daubchies 1-6

31. The MRD Using Coifflets and Least Asymmetric Daub Filters

32. Data Compression via Coiflets and Asymmetric Daub Filters

33. Cumulative Energy Build Up via Coiflets and Least Asymmetric Daub Filters

34. This Is the MRD Using Least Asymmetric Daub Filter 4

35. The Actual Wavelet Coefficients Used in the MRD Using Least Asymmetric Daub Filter 4

36. How Does LADF4 Do Better Than LADF6? It Doesn’t!

37. RMS Error Decrease as a Function of Number of Wavelets Kept

38. Reconstruction Using LADF4 Adding a Few Wavelets at a Time

39. The Cumulative Error During LADF4 MRD Reconstruction

40. K-Space Reconstruction Using LADF4 A Few Wavelets at a Time

41. The Cumulative Error in K-Space During LADF4 MRD Reconstruction

42. What Do Biorthogonal Spline Scaling Functions of Odd and Even Order Look Like in X-Space?

43. What Do Biorthogonal Spline Scaling Functions of Odd and Even Order Look Like in K-Space?

44. What Do Biorthogonal Spline Wavelets of Odd and Even Order Look Like in X-Space?

45. What Do Biorthogonal Spline Wavelets of Odd and Even Order Look Like in K-Space?

46. What Do Spline Scaling Functions Look Like in X-Space?

47. What Do Spline Scaling Functions Look Like in K-Space?

48. What Do Spline Wavelets Look Like in X-Space?

49. What Do Spline Wavelets Look Like in K-Space?

50. What Do Shannon Scaling Functions Look Like in X-Space?

51. What Do Shannon Scaling Functions Look Like in K-Space?

52. What Do Shannon Wavelets Look Like in X-Space?

53. What Do Shannon Wavelets Look Like in K-Space?

54. We Can Now Do the Same Analysis with Biorthogonal Spline, Spline and Shanon Filters, as We Did With Daub1-6, LADF and Coiflets

55. The MRD Using Biorthogonal Spline, Spline and Shanon Filters

56. Data Compression via Biorthogonal Spline, Spline and Shanon Filters

57. Cumulative Energy Build Up via Biorthogonal Spline, Spline and Shanon Filters

58. This Is the MRD Using a Shannon Filter

59. Wavelet Coefficients Used in the MRD Using a Shannon Filter

60. How Does a Shannon Filter Do Better Than Spline Filter[2,8]

61. RMS Error Decrease as a Function of Number of Wavelets Kept

62. Scale Dependent Accumulation of RMS Error for Biorthognal Spline, Spline and Shannon MRD

63. Reconstruction Using a Shannon Filter Adding a Few Wavelets at a Time

64. Cumulative Error During Shannon Filter MRD Reconstruction

65. K-Space Reconstruction Using Shannon A Few Wavelets at a Time

66. The Cumulative Error in K-Space During Shannon MRD Reconstruction

67. What is A Suitable Phase Space Distribution Function Corresponding to a Signal f(x)? (the Wigner Function)
The Wigner function is a bilinear functional of the signal f(x)
which represents the signal in phase space- which is to say-
simultaneously in position and wavenumber space.

68. The Wigner Function Representation of our Gaussian Enveloped Two Separate Carriers Signal

69. Contour Plots of the Wigner Function Representation of our Two Gaussians with Two Carriers Signal

70. Mean Square Error in Daubechies 1 or Haar MRD’s Performance in Wigner Land (8/57%, 15/36%, 24/12%, 30/7%, 38/11%)

71. Mean Square Error in Daubechies 3 MRD’s Performance in Wigner Land (8/60%, 15/33%, 23/12%, 30/7%, 38/4%)

72. Mean Square Error in Daubechies 5 MRD Using the Wigner Function Representation (8/27%, 15/13%, 23/6%, 30/4%, 38/2%)

73. Mean Square Error in Phase Space for Daubechies 6 MRD Using the Wigner Function (6/58%, 13/23%, 19/9.2%, 25/5%, 32/3%, 38/1.7%)

74. Wigner Function Representation of the MRD Using LADF2 (6/77%,13/50%,19/30%,26/18%,32/11%,38/7%)

75. Wigner Function Representation of the MRD Using LADF4 (6/67%,13/30%,19/17%,26/11%,32/6.7%,38/4%)

76. Wigner Function Representation of the MRD Using LADF6 (6/66%,13/33%,19/14%,26/6%,32/3%,38/2%)

77. Spline Filters Do a Great Job Also Even Competes with Daubechies 5

78. Wigner Function Representation of the MRD Using Shannon (6/49%,13/37%,19/30%,25/24%,32/21%,38/21%)

79. A Single Gaussian with a LF Carrier Is (Less than) Half the Story

80. How Does Daubechies 5 Break Down the LHS Gaussian?

81. The WLT Coefficients Corresponding to the Daub 5 MRA of LHS Wavepacket Are

82. The Wigner Function of One of the Gaussians with a LF Carrier

83. Haar Has Lots of Trouble Even with Just a Single Scale Gaussian (8/40%,18/16%,23/8%,40/3.3%, 38/0.7%)

84. Daubechies 3 Is Not Ideal But Does Consistently Beat Haar (8/30%, 18/10%, 23/3%, 30/1%, 38/.5%)

85. Daubechies 5 Starts with Least Amount of Pollution at Lowest Levels of MRA (8/12%, 18/5%, 23/1%,30/0.3%,38/0.1%)

86. How About the High Frequency Carrier RHS Wavepacket Acting Alone?

87. How Does Daubechies 5 Break Down the RHS Gaussian?

88. The WLT Coefficients Corresponding to the Daub 5 MRA of RHS Wavepacket Are

89. The Wigner Function of the RHS Wavepacket with a High frequency Carrier

90. Haar/Daub 1 Has Trouble with High Frequency Carrier Wave Packets (8/39%, 18/31%, 23/15%,30/9.5%,38/4.4%)

91. Daub 3 Reconstruction is Initially Even Worse But Quickly Recovers (8/52%, 18/19%, 23/10%,30/4%,38/2.5%)

92. Daub 5 Does Not Even Blink an Eye to Catch the HF Signal (8/21%, 18/10%, 23/6%,30/4%,38/2%)

93. Haar (or Daub 1)Wavelet’s Wigner Representation In Phase Space

94. Daubechies 3 Wavelet’s Wigner Representation In Phase Space

95. Daubechies 5 Wavelet’s Wigner Representation In Phase Space

96. Daubechies 6 Wavelet’s Wigner Representation In Phase Space

97. Least Asymmetric Daub Filter 2 Wavelet’s Wigner Representation

98. Least Asymmetric Daub Filter 4 Wavelet’s Wigner Representation

99. Least Asymmetric Daub Filter 6 Wavelet’s Wigner Representation

100. Quadratic Spline Filter Wavelet’s Wigner Representation

101. Shannon Wavelet’s Wigner Representation

102. A PRI Prescription
The Combined application of Wigner function phase space techniques and those of Multi Resolution Decomposition with Wavelets allows us to detect patterns across scales which would be lost otherwise.
By analysing the time evolution of a signal on different scales in space, we can detect changes in patterns that are random from those that are systematic, coherent or resonant.
Specially constructed phase space functionals beyond Wigner (bilinear) ones can capture even more pertinent information such as signaling the onset of 3 wave interactions, inverse cascades, etc. These, combined with MRA with the optimum wavelet family for that signal type lead to optimally useful representations.