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Discrete Wavelet Transform Techniques for Denoising, Pattern - - PowerPoint PPT Presentation

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Discrete Wavelet Transform Techniques for Denoising, Pattern Detection and Compression of Turbulent Rayleigh-Taylor Mix Data International Workshop on Bedros Afeyan, Polymath Research Inc. Praveen Ramaprabhu & Malcolm J. the Physics of

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SLIDE 1

Discrete Wavelet Transform Techniques for Denoising, Pattern Detection and Compression of Turbulent Rayleigh-Taylor Mix Data

Bedros Afeyan, Polymath Research Inc. Praveen Ramaprabhu & Malcolm J. Andrews, Texas A&M University

International Workshop on the Physics of Compressible Turbulent Mixing

Cal Tech

Pasadena, CA December 9-14, 2001

Polymath Research Inc.

ωpe

2 = 4π ne e2

me e2 c ≈ 1 137

slide-2
SLIDE 2

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

2

What Are Wavelets? Start @ (www.wavelets.org) surf (Mathsoft, amara, ...)

  • 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.

ψ j, k x

( ) = 2 j 2 Ψ 2j x − k

2

j

  • ; j,k ∈Ζ

Ψn x

( ) = −1 ( )

n dn

dxn exp −κ x − xc

( )

2 2

( )

[ ]

When the scale is decreased translation steps between wavelets should likewise be decreased

Mallat, Meyer, Daubechies, Beylkin, Coifman, Strang, Sweldens, Jawerth...

slide-3
SLIDE 3

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

3

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:

ϕ x

( ) = 2

hk ϕ 2x − k

( )

k =0 2 N −1

  • hk = 1

k

  • ψ x

( ) = 2

gk ϕ 2x − k

( )

k= 0 2N −1

  • gk = −1

( )

k h2 N−1− k

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.

ϕ x

( )

−∞ ∞

  • dx = 1

Low pass filter High pass filter

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SLIDE 4

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

4

What Are Discrete Wavelet Decompositions Good for?

  • Wavelet decompositions are very useful for the analysis of

intermittent or bursty data.

  • Spatial and scale localized information is efficiently represented.
  • Because the trends you want are captured efficiently (get large

coefficients in the expansion)very high quality denoising is possible.

  • Similarly, pattern recognition and detection capability is enhanced.
  • Compression is achieved where a few coefficients can represent what

is needed in the data.

  • All this depends on nonlinear (or largest coefficient)thresholding and

not scale or level thresholding . The latter is rather reminiscent of what is done with Fourier expansions.

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SLIDE 5

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

5

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

slide-6
SLIDE 6

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

6

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

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SLIDE 7

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

7

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

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SLIDE 8

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

8

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

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SLIDE 9

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

9

Raw Thermocouple RT Strong Mix Data (30 cm Downstream, theta ~ 0.71) from Texas A&M

T − TAVE TMAX − TAVE

Time, arb. units (Delta t=0.012 sec, Sampling Rate = 85 Hz)

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SLIDE 10

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

10

The Faded and Padded Version

  • f the Data 8192 Points Long

80 pts to fade 16 pts to pad per side

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SLIDE 11

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

11

The Fourier Transform of the RT Mix Data

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SLIDE 12

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

12

MRDs of the RT Mix Data in 6 Different Daubechies WLT Bases

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SLIDE 13

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

13

Decay Rate of Largest Coefficient vs Number of Coefficients Kept in 6 Different Daub WLT Decomps

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SLIDE 14

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

14

Energy Accumulation Rate in Coefficient Space vs # of WLTs Kept in 6 Different Daub Decomps

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SLIDE 15

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

15

Scaleograms: Waveleters Preferred Way of Judging Tiling in Scale-Translation Space

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SLIDE 16

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

16

Least Square Error Incurred By Truncating the WLT Series at N of its Largest Coefficients

slide-17
SLIDE 17

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

17

Least Square Error Incurred by Level Thresholding the DWT

slide-18
SLIDE 18

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

18

Daubechies 5 Does Much Better than Haar: 5 Quantitative Measures

slide-19
SLIDE 19

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

19

Level by Level Decomposition of the RT Mix Data Using Daub5 WLTs

slide-20
SLIDE 20

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

20

Reconstruction of the Data Using the 5 Largest WLT Coefficients

slide-21
SLIDE 21

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

21

Reconstruction of the Data Using the 10 Largest WLT Coefficients

slide-22
SLIDE 22

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

22

Reconstruction of the Data Using the 15 Largest WLT Coefficients

slide-23
SLIDE 23

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

23

Reconstruction of the Data Using the 20 Largest WLT Coefficients

slide-24
SLIDE 24

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

24

Reconstruction of the Data Using the 30 Largest WLT Coefficients

slide-25
SLIDE 25

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

25

Reconstruction of the Data Using the 50 Largest WLT Coefficients

slide-26
SLIDE 26

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

26

Reconstruction of the Data Using 100 Largest WLT Coefficients

slide-27
SLIDE 27

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

27

Reconstruction of the Data Using 200 Largest WLT Coefficients

slide-28
SLIDE 28

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

28

Reconstruction of the Data Using 400 Largest WLT Coefficients

slide-29
SLIDE 29

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

29

Reconstruction of the Data Using Up to 0.75 times the Largest WLT Coefficient

slide-30
SLIDE 30

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

30

Reconstruction of the Data Using Up to 0.5 times the Largest WLT Coefficient

slide-31
SLIDE 31

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

31

Reconstruction of the Data Using Up to 0.25 times the Largest WLT Coefficient

slide-32
SLIDE 32

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

32

Reconstruction of the Data Using Up to 0.1 times the Largest WLT Coefficient

slide-33
SLIDE 33

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

33

Reconstruction of the Data Using Up to 0.05 times the Largest WLT Coefficient

slide-34
SLIDE 34

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

34

Reconstruction of the Data Using the First (of 10) Level

  • f the MRD
slide-35
SLIDE 35

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

35

Reconstruction of the Data Using the First Two (of 10) Levels of the MRD

slide-36
SLIDE 36

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

36

Reconstruction of the Data Using the First Three (of 10) Levels of the MRD

slide-37
SLIDE 37

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

37

Reconstruction of the Data Using the First Four (of 10) Levels of the MRD

slide-38
SLIDE 38

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

38

Reconstruction of the Data Using the First Five (of 10) Levels of the MRD

slide-39
SLIDE 39

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

39

Reconstruction of the Data Using the First Six (of 10) Levels of the MRD

slide-40
SLIDE 40

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

40

Reconstruction of the Data Using the First Seven (of 10) Levels of the MRD

slide-41
SLIDE 41

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

41

Conclusions on Raw RT Mix Data Analysis Using DWT

  • Compression of around a factor of 20 seems likely with full data set.
  • Will see what low pass filtering will do to initial data and its

subsequent WLT analysis.

  • Looks like 25% of the largest coefficients are enough to reconstruct

the clean parts of the data.

  • We should compare different stages of evolution of RT Mix in terms
  • f their optimum WLT representations.
  • Significant dynamical degrees of freedom vs insignificant ones which

vary more slowly or not at all or randomly might be obtainable if we keep at it!

slide-42
SLIDE 42

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

42

Low Pass Filtered RT Mix Data

slide-43
SLIDE 43

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

43

The Filtering Has This Form and Effect in k-Space

Filter was of the form:

S k

( ) = exp −

k kwidth

  • 2 α
  • Where α=5 and kwidth = 400
slide-44
SLIDE 44

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

44

MRDs of the LP Filtered RT Mix Data in 6 Different Daubechies WLT Bases

slide-45
SLIDE 45

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

45

Decay Rate of Largest Coefficient vs Number of Coeffs Kept in LPF RT Mix Data

slide-46
SLIDE 46

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

46

Energy Accumulation Rate in Coefficient Space vs # of WLTs Kept for LPF RT Mix Data

slide-47
SLIDE 47

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

47

Scaleograms: Waveleters Preferred Way of Judging Tiling in Scale- Translation Space for LPF RT Mix

slide-48
SLIDE 48

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

48

Least Square Error Incurred By Truncating the WLT Series at N

  • f its Largest Coeffs LPF RT Mix Data
slide-49
SLIDE 49

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

49

Least Square Error Incurred by Level Thresholding the DWT of LPF RT Mix Data

slide-50
SLIDE 50

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

50

Daubechies 5 Does Much Better than Haar: 5 Quantitative Measures for LPF RT Mix Data

slide-51
SLIDE 51

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

51

Level by Level Decomposition of the LPF RT Mix Data Using Daub5 WLTs

slide-52
SLIDE 52

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

52

Reconstruction of the LPF Data with 5 Largest WLT Coeffs

slide-53
SLIDE 53

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

53

Reconstruction of the LPF Data with 10 Largest WLT Coeffs

slide-54
SLIDE 54

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

54

Reconstruction of the LPF Data with 15 Largest WLT Coeffs

slide-55
SLIDE 55

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

55

Reconstruction of the LPF Data with 20 Largest WLT Coeffs

slide-56
SLIDE 56

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

56

Reconstruction of the LPF Data with 30 Largest WLT Coeffs

slide-57
SLIDE 57

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

57

Reconstruction of the LPF Data with 50 Largest WLT Coeffs

slide-58
SLIDE 58

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

58

Reconstruction of the LPF Data with 100 Largest WLT Coeffs

slide-59
SLIDE 59

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

59

Reconstruction of the LPF Data with 200 Largest WLT Coeffs

slide-60
SLIDE 60

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

60

Reconstruction of the LPF Data with 400 Largest WLT Coeffs

slide-61
SLIDE 61

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

61

  • Recons. of the LPF Data Using

Up to 0.75 x the Largest WLTs

slide-62
SLIDE 62

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

62

  • Recons. of the LPF Data Using

Up to 0.5 x the Largest WLTs

slide-63
SLIDE 63

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

63

  • Recons. of the LPF Data Using

Up to 0.25 x the Largest WLTs

slide-64
SLIDE 64

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

64

  • Recons. of the LPF Data Using

Up to 0.1 x the Largest WLTs

slide-65
SLIDE 65

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

65

  • Recons. of the LPF Data Using

Up to 0.05 x the Largest WLTs

slide-66
SLIDE 66

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

66

Reconstruction of the LPF Data Using the First MRD Level

slide-67
SLIDE 67

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

67

Reconstruction of the LPF Data Using the First 2 MRD Levels

slide-68
SLIDE 68

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

68

Reconstruction of the LPF Data Using the First 3 MRD Levels

slide-69
SLIDE 69

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

69

Reconstruction of the LPF Data Using the First 4 MRD Levels

slide-70
SLIDE 70

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

70

Reconstruction of the LPF Data Using the First 5 MRD Levels

slide-71
SLIDE 71

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

71

Conclusions Regarding the WLT Analysis of the LPF RT Mix Data

  • Far better compression and denoising is achieved once a modest

amount of initial low pass filtering is done on the data.

  • Note the extremely small contributions levels 5 and above make to the

MRD while with the unfiltered data that contribution was of order 1

  • r 0.1
  • Far cleaner structures are observable in levels 1, 2 and 3, periodic

correlations in time, or so it seems to the eye!

  • The reconstruction with largest wavelets kept shows long patches of

flatness surrounded by localized structures which could be indicative

  • f the correlation properties of the data.
  • More to come!
slide-72
SLIDE 72

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

72

Raw RT Weak Mix Data (2 cm Downstream, Theta = 0.7) from Texas A&M

T − TAVE TMAX − TAVE

Time, arb. units

slide-73
SLIDE 73

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

73

The Faded and Padded Version of the RT Weak Mix Data: 8192 Points

slide-74
SLIDE 74

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

74

The Fourier Transform of the RT Weak Mix Data

slide-75
SLIDE 75

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

75

MRD Coefficients of the RT Weak Mix Data in 6 Different Daubechies WLT Bases

slide-76
SLIDE 76

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

76

Actual MRDs of the RT Weak Mix Data in 6 Different Daubechies WLT Bases

slide-77
SLIDE 77

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

77

Decay Rate of Largest Coefficient vs Number of Coefficients Kept in 6 Different Daub WLT Decomps

slide-78
SLIDE 78

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

78

Energy Accumulation Rate in Coefficient Space vs # of WLTs Kept in 6 Different Daub Decomps

slide-79
SLIDE 79

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

79

Scaleograms: Waveleters Preferred Way of Judging Tiling in Scale-Translation Space

slide-80
SLIDE 80

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

80

Least Square Error Incurred By Truncating the WLT Series at N

  • f its Largest Coefficients
slide-81
SLIDE 81

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

81

Least Square Error Incurred by Level Thresholding the DWT

slide-82
SLIDE 82

BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

82

Daubechies 5 Does Much Better than Haar: 5 Quantitative Measures

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

83

Level by Level Decomposition of the RT Weak Mix Data Using Daub5 WLTs

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

84

Reconstruction of the Data Using the 5 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

85

Reconstruction of the Data Using the 10 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

86

Reconstruction of the Data Using the 15 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

87

Reconstruction of the Data Using the 20 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

88

Reconstruction of the Data Using the 25 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the Data Using the 30 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the Data Using the 35 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

91

Reconstruction of the Data Using the 40 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

92

Reconstruction of the Data Using the 45 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

93

Reconstruction of the Data Using the 50 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

94

Reconstruction of the Data Using the 100 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

95

Reconstruction of the Data Using the 200 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

96

Reconstruction of the LPF Data Using the First MRD Level

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

97

Reconstruction of the Weak Mix Data Using First 2 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

98

Reconstruction of the Weak Mix Data Using First 3 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

99

Reconstruction of the Weak Mix Data Using First 4 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

100

Reconstruction of the Weak Mix Data Using First 5 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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101

Conclusions Regarding the WLT Analysis of the RT Weak Mix Data

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

102

Low Pass Filtered (LPF) Padded and Faded RT Weak Mix Data

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

103

The Filtering Has This Form and Effect on the Data in k-Space

Filter was of the form:

S k

( ) = exp −

k kwidth

  • 2α
  • Where α=5 and kwidth = 400
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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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MRD Coefficients of the LPF RT Weak Mix Data in 6 Different Daubechies WLT Bases

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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Actual MRDs of the LPF RT Weak Mix Data in 6 Different Daubechies WLT Bases

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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106

Decay Rate of Largest Coefficient vs Number of Coefficients Kept in 6 Different Daub WLT Decomps

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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107

Energy Accumulation Rate in Coefficient Space vs # of WLTs Kept in 6 Different Daub Decomps.

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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108

Scaleograms: Waveleters Preferred Way of Judging Tiling in Scale- Translation Space

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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109

Least Square Error Incurred By Truncating the WLT Series at N

  • f its Largest Coefficients
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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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Least Square Error Incurred by Level Thresholding the DWT

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Daubechies 5 Does Much Better than Haar: 5 Quantitative Measures

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

112

Level by Level Decomposition of the LPF RT Weak Mix Data Using Daub5 WLTs

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

113

Reconstruction of the LPF RT Weak Mix Data Using the 5 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the LPF RT Weak Mix Data Using the 10 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the LPF RT Weak Mix Data Using the 15 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

116

Reconstruction of the LPF RT Weak Mix Data Using the 20 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the LPF RT Weak Mix Data Using the 25 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

118

Reconstruction of the LPF RT Weak Mix Data Using the 30 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

119

Reconstruction of the LPF RT Weak Mix Data Using the 35 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

120

Reconstruction of the LPF RT Weak Mix Data Using the 40 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

121

Reconstruction of the LPF RT Weak Mix Data Using the 45 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

122

Reconstruction of the LPF RT Weak Mix Data Using the 50 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

123

Reconstruction of the LPF RT Weak Mix Data Using the 100 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

124

Reconstruction of the LPF RT Weak Mix Data Using the 200 Largest WLT Coefficients

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the LPF RT Weak Mix Data Using the First MRD Level

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the LPF RT Weak Mix Data Using the First 2 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

127

Reconstruction of the LPF RT Weak Mix Data Using the First 3 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

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Reconstruction of the LPF RT Weak Mix Data Using the First 4 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

ωpe 2 = 4π ne e2 me e2 c ≈ 1 137

129

Reconstruction of the LPF RT Weak Mix Data Using the First 5 MRD Levels

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BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

Polymath Research Inc.

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Conclusions Based on the LPF RT Weak Mix Data’s WLT Analyses