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Wavelets ?

Wavelets ?. Raghu Machiraju Contributions: Robert Moorhead, James Fowler, David Thompson, Mississippi State University Ioannis Kakadaris, U of Houston. Simulations, scanners. State-Of-Affairs. Concurrent. Presentation. Retrospective. Analysis. Representation. Why Wavelets?

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Wavelets ?

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  1. Wavelets ? Raghu Machiraju Contributions: Robert Moorhead, James Fowler, David Thompson, Mississippi State University Ioannis Kakadaris, U of Houston

  2. Simulations, scanners State-Of-Affairs • Concurrent • Presentation • Retrospective • Analysis • Representation

  3. Why Wavelets? • We are generating and measuring larger datasets every year • We can not store all the data we create (too much, too fast) • We can not look at all the data (too busy, too hard) • We need to develop techniques to store the data in better formats

  4. Data Analysis • Frequency spectrum correctly shows a spike at 10 Hz • Spike not narrow - significant component at between 5 and 15 Hz. • Leakage - discrete data acquisition does not stop at exactly the same phase in the sine wave as it started.

  5. QuickFix

  6. Windowing &Filtering

  7. Image Example • 8x8 Blocked Window (Cosine) Transform • Each DCT basis waveform represents a fixed frequency in two orthogonal directions • frequency spacing in each direction is an integer multiple of a base frequency

  8. Windowing & Filtering Windows – fixed in space and frequencies Cannot resolve all features at all instants

  9. Linear Scale Space input s= 1 s= 16 s= 24 s= 32

  10. Successive Smoothing

  11. Sub-sampled Images • Keep 1 of 4 values from 2x2 blocks • This naive approach and introduces aliasing • Sub-samples are bad representatives of area • Little spatial correlation

  12. Image Pyramid

  13. Image Pyramid – MIP MAP • Average over a 2x2 block • This is a rather straight forward approach • This reduces aliasing and is a better representation • However, this produces 11% expansion in the data

  14. Image Pyramid – Another Twist

  15. Time Frequency Diagram

  16. Ideally ! Create new signal G such that ||F-G|| = e

  17. Wavelet Analysis • A1 D1 D2 D3 • D3 • D2 • D1 • A1

  18. Why Wavelets? Because … • We need to develop techniques to analyze data better through noise discrimination • Wavelets can be used to detect features and to compare features • Wavelets can provide compressed representations • Wavelet Theory provides a unified framework for data processing

  19. Scale-Coherent Structures • Coherent structure - frequencies at all scales • Examples - edges, peaks, ridges • Locate extent and assign saliency

  20. Wavelets – Analysis

  21. Wavelets – DeNoising

  22. Wavelets – Compression Original 50:1

  23. Wavelets – Compression Original 50:1

  24. Wavelets – Compression

  25. Yet Another Example 50% 7%

  26. 2% Final Example 50% 100% 1%

  27. 1.0 0.8 ) E ( n o i t a 0.6 m r o f density n u momentum i v momentum d e 0.4 w momentum z energy i l a m r o n 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 normalized rate Information Rate Curve • Energy Compaction – Few coefficients can efficiently represent functions • The Curve should be as vertical as possible near 0 rate

  28. Filter Bank Implementation • G: High Pass Filter • H: Low Pass Filter

  29. Synthesis Bank

  30. Successive Approximations

  31. Successive Details

  32. Wavelet Representation

  33. Coefficients

  34. Lossey Compression

  35. Lossey Compression

  36. Image Example A Frame Another Frame

  37. Image Example Average Difference

  38. Wavelet Transform

  39. Frequency Support

  40. Image Example LvLh LvHh HvHh HvLh

  41. Image Example LvLh LvHh HvHh HvLh

  42. How Does One Do This ?

  43. Dilations • Rescaling Operation t --> 2t • Down Sampling, n --> 2n • Halve function support • Double frequency content • Octave division of spectrum- Gives rise to different scales and resolutions • Mother wavelet! - basic function gives rise to differing versions

  44. Dilations

  45. Successive Approximations

  46. Translations • Covers space-frequency diagram • Versions are

  47. Wavelet Decomposition • Induced functional Space - Wj. • Related to Vjs • Space Wj+1 is orthogonal to Vj+1 • Also • J-level wavelet decomposition -

  48. Successive Differences

  49. Wavelet Expansion • Wavelet expansion (Tiling- j: scale, k: translates), Synthesis • Orthogonal transformation, Coarsest level of resolution - J • Smoothing function - f, Detail function - y • Analysis: • Commonly used wavelets are Haar, Daubechies and Coiflets

  50. Scaling Functions • Compact support • Bandlimited - cut-off frequency • Cannot achieve both • DC value (or the average) is defined • Translates of f are orthogonal

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