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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. Reference. “Characterization of Signals from Multiscale Edges” Stephane Mallat and Sifen Zhong

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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. Reference • “Characterization of Signals from Multiscale Edges” • Stephane Mallat and Sifen Zhong • IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, pp 710-732, July 1992

  4. Motivation

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

  6. Compressed Signal (37 points) What else for reconstruction?

  7. Compressed Signal (37 points) sharp one-sided edge

  8. Compressed Signal (37 points) sharp two-sided edge

  9. Compressed Signal (37 points) “noisy” edges

  10. Calculation Original: (256 points) • Reconstruction: • edges • edge type information (37 points) (x points)

  11. Compression 37 + x < 256 edges edge type

  12. Summary Save edges

  13. Summary sharp two-sided edge sharp one-sided edge “noisy” edges Save edge type

  14. Summary + = edges edge type reconstruct

  15. Algorithm Decomposition + Reconstruction

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

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

  18. What is Discrete Wavelet Transform? Input Discrete Wavelet Transform

  19. What is DWT? • Choose mother wavelet • Dilate mother wavelet • Convolve family with input DWT

  20. 1) Choose mother wavelet

  21. 2) Dilate mother wavelet mother wavelet dilate

  22. 2) Dilate mother wavelet

  23. Convolve family with input wavelet scale 1 = input = input wavelet scale 2 input = wavelet scale 4

  24. Convolve “family” wavelet scale 1 = input DWT = input wavelet scale 2 multiscale input = wavelet scale 4

  25. What is DWT? (mathematically)

  26. How to dilate? mother wavelet

  27. How to dilate? dyadic (powers of two)

  28. How to dilate? scale

  29. How to dilate? z halve amplitude double support

  30. Mother Wavelet (Haar) scale 1, j=0

  31. Mother Wavelet (Haar) scale 2, j=1

  32. Mother Wavelet (Haar) scale 4, j=2

  33. What is DWT? Convolution of dilates of mother wavelets against original signal.

  34. What is DWT? Convolution of dilates of mother wavelets against original signal. convolution

  35. What is DWT? Convolution of dilates of mother wavelets against original signal. dilates

  36. What is DWT? Convolution of dilates of mother wavelets against original signal. original signal

  37. What is convolution? (best match operation) • mother wavelet Input 2)dilation Discrete Wavelet Transform 3)convolution

  38. Convolution (best match operator) dummy variable

  39. Convolution (best match operator) flip g around y axis

  40. Convolution (best match operator) shifts g by t

  41. Convolution (best match operator) do nothing to f

  42. Convolution (best match operator) pointwise multiplication

  43. Convolution (best match operator) integrate over R

  44. Convolution (one point) flip g and shift by 7.7

  45. Convolution (one point) do nothing to f

  46. Convolution (one point) multiply f and g pointwise

  47. Convolution (one point) integrate over R

  48. Convolution (one point) scalar

  49. Convolution of two boxes

  50. Convolution of two boxes

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