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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

  1. Reconstruction of the Data Using 33 Up to 0.05 times the Largest Polymath Research Inc. WLT Coefficient e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  2. Reconstruction of the Data 34 Using the First (of 10) Level Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 of the MRD m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  3. Reconstruction of the Data 35 Using the First Two (of 10) Polymath Research Inc. Levels of the MRD e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  4. Reconstruction of the Data Using 36 the First Three (of 10) Levels of Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 the MRD ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  5. Reconstruction of the Data 37 Using the First Four (of 10) Polymath Research Inc. Levels of the MRD e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  6. Reconstruction of the Data 38 Using the First Five (of 10) Polymath Research Inc. Levels of the MRD e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  7. Reconstruction of the Data 39 Using the First Six (of 10) Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 Levels of the MRD ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  8. Reconstruction of the Data 40 Using the First Seven (of 10) Polymath Research Inc. Levels of the MRD e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  9. 41 Conclusions on Raw RT Mix Polymath Research Inc. Data Analysis Using DWT e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e • 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 of 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! BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  10. 42 Polymath Low Pass Filtered RT Mix Data Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  11. 43 The Filtering Has This Form Polymath Research Inc. and Effect in k-Space e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e Filter was of the form : � � � � 2 α k � � ( ) = exp − � � S k � � � � � � k width � � Where α =5 and k width = 400 BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  12. MRDs of the LP Filtered RT Mix 44 Data in 6 Different Daubechies Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 WLT Bases m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  13. Decay Rate of Largest Coefficient vs 45 Number of Coeffs Kept in LPF RT Polymath Research Inc. Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  14. Energy Accumulation Rate in 46 Coefficient Space vs # of WLTs Polymath Research Inc. Kept for LPF RT Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  15. Scaleograms: Waveleters Preferred 47 Way of Judging Tiling in Scale- Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 Translation Space for LPF RT Mix ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  16. Least Square Error Incurred By 48 Truncating the WLT Series at N Polymath Research Inc. of its Largest Coeffs LPF RT Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  17. Least Square Error Incurred by 49 Level Thresholding the DWT of Polymath Research Inc. LPF RT Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  18. Daubechies 5 Does Much Better than 50 Haar: 5 Quantitative Measures Polymath Research Inc. for LPF RT Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  19. Level by Level Decomposition of 51 the LPF RT Mix Data Using Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 Daub5 WLTs ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  20. 52 Reconstruction of the LPF Data Polymath with 5 Largest WLT Coeffs Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  21. 53 Reconstruction of the LPF Data Polymath Research Inc. with 10 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  22. 54 Reconstruction of the LPF Data Polymath Research Inc. with 15 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  23. 55 Reconstruction of the LPF Data Polymath Research Inc. with 20 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  24. 56 Reconstruction of the LPF Data Polymath Research Inc. with 30 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  25. 57 Reconstruction of the LPF Data Polymath Research Inc. with 50 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  26. 58 Reconstruction of the LPF Data Polymath Research Inc. with 100 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  27. 59 Reconstruction of the LPF Data Polymath Research Inc. with 200 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  28. 60 Reconstruction of the LPF Data Polymath Research Inc. with 400 Largest WLT Coeffs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  29. 61 Recons. of the LPF Data Using Polymath Research Inc. Up to 0.75 x the Largest WLTs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  30. 62 Recons. of the LPF Data Using Polymath Research Inc. Up to 0.5 x the Largest WLTs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  31. 63 Recons. of the LPF Data Using Polymath Research Inc. Up to 0.25 x the Largest WLTs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  32. 64 Recons. of the LPF Data Using Polymath Research Inc. Up to 0.1 x the Largest WLTs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  33. 65 Recons. of the LPF Data Using Polymath Research Inc. Up to 0.05 x the Largest WLTs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  34. 66 Reconstruction of the LPF Data Polymath Using the First MRD Level Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  35. 67 Reconstruction of the LPF Data Polymath Research Inc. Using the First 2 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  36. 68 Reconstruction of the LPF Data Polymath Research Inc. Using the First 3 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  37. 69 Reconstruction of the LPF Data Polymath Research Inc. Using the First 4 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  38. 70 Reconstruction of the LPF Data Polymath Research Inc. Using the First 5 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  39. 71 Conclusions Regarding the WLT Polymath Research Inc. Analysis of the LPF RT Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e • 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 or 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 of the correlation properties of the data. • More to come! BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  40. Raw RT Weak Mix Data (2 cm 72 Downstream, Theta = 0.7) Polymath Research Inc. � c ≈ 1 e 2 2 = 4 π n e e 2 from Texas A&M ω pe 137 m e T − T AVE T MAX − T AVE BBA WLTs and RT Mix Time, arb. units Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  41. 73 The Faded and Padded Version of Polymath Research Inc. the RT Weak Mix Data: 8192 Points e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  42. 74 The Fourier Transform of the Polymath Research Inc. RT Weak Mix Data e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  43. MRD Coefficients of the RT Weak 75 Mix Data in 6 Different Daubechies Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 WLT Bases ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  44. Actual MRDs of the RT Weak 76 Mix Data in 6 Different Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 Daubechies WLT Bases ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  45. Decay Rate of Largest Coefficient vs 77 Number of Coefficients Kept in 6 Polymath Research Inc. Different Daub WLT Decomps e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  46. Energy Accumulation Rate in 78 Coefficient Space vs # of WLTs Polymath Research Inc. Kept in 6 Different Daub Decomps e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  47. Scaleograms: Waveleters 79 Preferred Way of Judging Tiling Polymath Research Inc. in Scale-Translation Space e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  48. Least Square Error Incurred By 80 Truncating the WLT Series at N Polymath Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 of its Largest Coefficients ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  49. 81 Least Square Error Incurred by Polymath Research Inc. Level Thresholding the DWT e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  50. 82 Daubechies 5 Does Much Better Polymath Research Inc. than Haar: 5 Quantitative Measures e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  51. 83 Level by Level Decomposition of the RT Polymath Research Inc. Weak Mix Data Using Daub5 WLTs e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  52. 84 Reconstruction of the Data Using Polymath Research Inc. the 5 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  53. 85 Reconstruction of the Data Using Polymath Research Inc. the 10 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  54. 86 Reconstruction of the Data Using Polymath the 15 Largest WLT Coefficients Research Inc. e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  55. 87 Reconstruction of the Data Using Polymath Research Inc. the 20 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  56. 88 Reconstruction of the Data Using Polymath Research Inc. the 25 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  57. 89 Reconstruction of the Data Using Polymath Research Inc. the 30 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  58. 90 Reconstruction of the Data Using Polymath Research Inc. the 35 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  59. 91 Reconstruction of the Data Using Polymath Research Inc. the 40 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  60. 92 Reconstruction of the Data Using Polymath Research Inc. the 45 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  61. 93 Reconstruction of the Data Using Polymath Research Inc. the 50 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  62. 94 Reconstruction of the Data Using Polymath Research Inc. the 100 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  63. 95 Reconstruction of the Data Using Polymath Research Inc. the 200 Largest WLT Coefficients e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  64. 96 Reconstruction of the LPF Data Polymath Research Inc. Using the First MRD Level e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  65. 97 Reconstruction of the Weak Mix Polymath Research Inc. Data Using First 2 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  66. 98 Reconstruction of the Weak Mix Polymath Research Inc. Data Using First 3 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  67. 99 Reconstruction of the Weak Mix Polymath Research Inc. Data Using First 4 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

  68. 100 Reconstruction of the Weak Mix Polymath Research Inc. Data Using First 5 MRD Levels e 2 � c ≈ 1 2 = 4 π n e e 2 ω pe 137 m e BBA WLTs and RT Mix Cal Tech Pasadena CA 8th IW PCTM 12-11-01

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