2. Signal reconstruction from multiscale edges A wavelet based algorithm

3. Algorithm

4. Decomposition Discrete Wavelet Transform Save edges e.g. local extrema Input

5. Reconstruction Find approximation Inverse Wavelet Transform Output local extrema contained two bugs

6. What is Discrete Wavelet Transform?

7. Discrete Wavelet Transform (1 level) input Detail Coefficients Approximation Coefficients

9. input Detail Level 1 Detail Level 2 Detail Level 3 Approximation Level 3

11. Input + Details (3 Levels)

12. Input + Approximation (3 Levels)

13. DWT + IDWT input Detail Coefficients Approximation Coefficients + output

14. Perfect Reconstruction Property + XX =

15. + XX = Bug: Incorrect coefficient given in paper

16. Perfect Reconstruction Property + XX = 0.054685 0.0546875 Given Correct

17. input Detail Coefficients Approximation Coefficients + output + XX =

18. Detail (L3) Approx. (L3) + + Detail (L2) + Detail (L1) output

19. Detail (L3) Approx. (L3) + + Detail (L2) + Detail (L1) output Bug: cascade not implemented correctly

20. Validation

21. Interpolation Space  check for interpolation DWT Space  check for perfect reconstruction DWT -> IDWT = identity

22. Results

23. Step Edge (Length 32)

24. Details (5 levels)

25. Approximation (5 levels)

26. All coefficients discarded since last level zero

27. Detail (L3) Approx. (L3) + + Detail (L2) + Detail (L1) output

28. Extrema (20/32 points, 63% compression)

29. Find approximation (iterative) Alternate projections between two spaces

30. Reconstructed (Iteration 1, 11, 21, 31)

31. l2 error (31 iterations)

32. Previous Result with Bug (Iteration 1, 11, 21)

33. Previous Result with Bug

34. Gaussian (Length 32)

35. Details (5 Levels)

36. Approximation (5 Levels)

37. Extrema (19/32 points, 59% compression)

38. Reconstructed (Iteration 1, 11, 21, 31)

39. l2 error (31 iterations)

40. Previous Result with Bug (Iteration 1, 11, 21)

41. Previous Result with Bug

42. Dirac Delta (Length 32)

43. Details (5 Levels)

44. Approximation (5 Levels)

45. Extrema (20/32 points, 62% compression)

46. Reconstructed (Iteration 1, 11, 21, 31)

47. l2 error (31 iterations)

48. Sinusoid (Length 32)

49. Details (5 Levels)

50. Approximation (5 Levels)

51. Extrema (22/32 points, 69% compression)

52. Reconstructed (Iteration 1, 11, 21, 31)

53. l2 error (31 iterations)

54. Triangle (Length 32)

55. Details (5 Levels)

56. Approximation (5 Levels)

57. Extrema (33/32 points,- 3% compression)

58. Reconstructed (Iteration 1, 11, 21, 31)

59. l2 error (31 iterations)

60. Random Noise (Length 32)

61. Details (5 Levels)

62. Approximation (5 Levels)

63. Extrema (40/32 points,- 25% compression)

64. Reconstructed (Iteration 1, 11, 21, 31)

65. l2 error (31 iterations)

66. Audio Signals

67. Source: Zoolander (2001)

69. Reconstructed Signal

71. Zoom 1

72. Reconstructed Signal

73. Zoom 2

74. Reconstructed Signal

75. Zoom 3

76. WindowExtremaWindowExtremaWindowExtremaWindowExtrema 190099271786525756 28681082718108126781 3877118411988927841 4846128302091628947 5779138262199529920 6787148752287030293 776015770239553120 877616757248473220 Extrema Saved For Each Window

77. Extrema versus Level (Window 9)

78. Input

79. Reconstructed

80. 1)Do not do maximum DWT decomposition Lowest approximation coefficient will not be constant 2)Interpolate approximation coefficients instead Future Work

81. input Detail Level 1 Detail Level 2 Detail Level 3 Approximation Level 3

83. input Detail Level 1 Detail Level 2 Approximation Level 2

85. Questions?

86. Supplemental Slides

87. How to find approximation?

88. What do we know about saved points? 1) Local extrema 2) Come from wavelet transform

89. Find approximation (iterative) Alternate projections between two spaces

90. Find approximation (iterative) sequences of functionswhose H1 normis finite

91. Find approximation (iterative)

92. sequences of functions: interpolate input signal’s wavelet extrema

93. Sample Element from Γ

95. Find approximation (iterative) What happens when element is projected onto Γ? 1) Saved points are interpolated 2) H1 norm minimized

96. Q: Why minimize over H1 norm? A: Interpolation points act like local extrema

97. Find approximation (iterative) dyadic wavelet transforms of L^2 functions

98. Find approximation (iterative) How to project onto V space (DWT space)?

99. Project onto V space (DWT space) IDWT input (interpolated functions) DWT output (DWT of l2 function) l2 function not necessarily DWT of l2 function

100. Find approximation (iterative) start at zero element

101. What do we know about saved points? 1) Local extrema 2) Come from wavelet transform

102. Find approximation (iterative) intersection = space of solutions interpolation space wavelet transform space

103. Error Analysis (Step Edge)

104. Step Edge (Length 32)

105. Discrete Wavelet Transform (3 levels)

106. Save Points (Extrema + Ends)

107. Alternate Projection (Initialization) zero out all wavelet values

108. Alternate Projection (Iteration 1a) project onto Γ space (interpolate points)

109. DWT after Γ Projection (1 iteration) extrema points interpolated

110. DWT after Γ Projection (1 iteration) same concavity

111. Alternate Projection (Iteration 1b) project onto V space (DWT space)

112. false edge(periodic padding)

113. save end points(interpolate end intervals)

114. Discrete Wavelet Transform input edge detection (scale 1) edge detection (scale 2) edge detection (scale 4) = = =