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Signal reconstruction from multiscale edges

Signal reconstruction from multiscale edges. A wavelet based algorithm. Author Yen-Ming Mark Lai ( ylai@amsc.umd.edu ) Advisor Dr. Radu Balan rvbalan@math.umd.edu CSCAMM, MATH. Motivation. Save edges. Motivation. sharp two-sided edge. sharp one-sided edge. “noisy” edges.

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Signal reconstruction from multiscale edges

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  1. Signal reconstruction from multiscale edges A wavelet based algorithm

  2. Author Yen-Ming Mark Lai (ylai@amsc.umd.edu) Advisor Dr. Radu Balan rvbalan@math.umd.edu CSCAMM, MATH

  3. Motivation Save edges

  4. Motivation sharp two-sided edge sharp one-sided edge “noisy” edges Save edge type

  5. Motivation + = edges edge type reconstruct

  6. Algorithm Decomposition + Reconstruction

  7. Decomposition Input Discrete Wavelet Transform Save edges e.g. local extrema “edges+edge type”

  8. Decomposition = input edge detection (scale 1) = input edge detection (scale 2) input = edge detection (scale 4)

  9. Reconstruction “edges+edge type” local extrema Find approximation Inverse Wavelet Transform Output

  10. How to find approximation? “edges+edge type” local extrema Find approximation

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

  12. Find approximation (iterative) sequences of functions whose H1 norm is finite

  13. Find approximation (iterative)

  14. Find approximation (iterative) sequences of functions: 1) interpolate input signal’s wavelet extrema 2) have minimal H1 norm

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

  16. Numerical Example algorithm interpolates between points unclear what to do outside interpolation points

  17. Find approximation (iterative)

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

  19. Find approximation (iterative) intersection = space of solutions

  20. Find approximation (iterative) Start at zero element to minimize solution’s norm

  21. Preliminary Results

  22. Step Edge (length 8)

  23. Quadratic Spline Wavelet

  24. Take DWT

  25. Take DWT

  26. Convolution in Matlab * conv ( [1,-1] [0,0,0,0,1,1,1,1] ) , current next + =next-current

  27. Convolution in Matlab * next-current =-1 next-current =0

  28. Convolution in Matlab * next-current next-current next-current next-current next-current next-current next-current next-current next-current = -1 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 1 =

  29. Save Local Extrema

  30. Save Local Extrema

  31. Interpolate DWT (Level 1) unclear what to do outside interpolation points interpolation to minimize H1 norm

  32. Original DWT – Level 1 Interpolated DWT – Level 1 error

  33. Original DWT – Level 2 Interpolated DWT – Level 2 error

  34. Original DWT – Level 3 Interpolated DWT – Level 3 error

  35. Original DWT – Level 4 Interpolated DWT – Level 4 matrix inversion failed

  36. Interpolated DWT

  37. Take IDWT to Recover Signal

  38. Recovered Signal (Red) and Original Step Edge (Blue)

  39. Summary

  40. Choose Input

  41. Take DWT

  42. Save Local Extrema of DWT

  43. Interpolate Local Extrema of DWT

  44. Take IDWT

  45. Issues • Convolution detects false edges • What to do with values outside interpolations points? • What to do when matrix inversion fails?

  46. Timeline Oct/Nov – code Alternate Projections (90%) Dec – write up mid-year report (85%) Jan– code local extrema search (100%)

  47. Timeline • February/March – test and debug entire system (8 weeks) • April – run code against database (4 weeks) • May – write up final report (2 weeks)

  48. Questions?

  49. Supplemental Slides

  50. Input Signal (256 points) Which points to save?

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