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Signal reconstruction from multiscale edges

Signal reconstruction from multiscale edges. A wavelet based algorithm. Algorithm. Decomposition. Input. Discrete Wavelet Transform. Save edges e.g. local extrema. Reconstruction. local extrema. Find approximation. Inverse Wavelet Transform. Output. contained two bugs.

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Signal reconstruction from multiscale edges

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  1. Signal reconstruction from multiscale edges A wavelet based algorithm

  2. Algorithm

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

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

  5. What is Discrete Wavelet Transform?

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

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

  8. Input + Details (3 Levels)

  9. Input + Approximation (3 Levels)

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

  11. Perfect Reconstruction Property = + X X

  12. Perfect Reconstruction Property Bug: Incorrect coefficient given in paper = + X X

  13. Perfect Reconstruction Property = + X X Given 0.054685 Correct 0.0546875

  14. input Detail Coefficients = + + X X Approximation Coefficients output

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

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

  17. Validation

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

  19. Results

  20. Step Edge (Length 32)

  21. Details (5 levels)

  22. Approximation (5 levels)

  23. Approximation (5 levels) All coefficients discarded since last level zero

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

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

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

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

  28. l2 error (31 iterations)

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

  30. Previous Result with Bug

  31. Gaussian (Length 32)

  32. Details (5 Levels)

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

  34. Previous Result with Bug

  35. Dirac Delta (Length 32)

  36. Details (5 Levels)

  37. Sinusoid (Length 32)

  38. Details (5 Levels)

  39. Triangle (Length 32)

  40. Details (5 Levels)

  41. Random Noise (Length 32)

  42. Details (5 Levels)

  43. Audio Signals

  44. Source: Zoolander (2001)

  45. Reconstructed Signal

  46. Reconstructed Signal

  47. Zoom 1

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