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Introduction to Wavelets -part 2

Introduction to Wavelets -part 2. By Barak Hurwitz. Wavelets seminar with Dr ’ Hagit Hal-or. List of topics. Reminder 1D signals Wavelet Transform CWT,DWT Wavelet Decomposition Wavelet Analysis 2D signals Wavelet Pyramid some Examples. Reminder – from last week.

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Introduction to Wavelets -part 2

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  1. Introduction toWavelets -part 2 By Barak Hurwitz Wavelets seminar with Dr’ Hagit Hal-or

  2. List of topics • Reminder • 1D signals • Wavelet Transform • CWT,DWT • Wavelet Decomposition • Wavelet Analysis • 2D signals • Wavelet Pyramid • some Examples

  3. Reminder – from last week • Why transform? • Why wavelets? • Wavelets like basis components. • Wavelets examples. • Wavelets advantages. • Continuous Wavelet Transform.

  4. Reminder -Why transform?

  5. Reminder –Noise in Fourier spectrum Noise

  6. 1D SIGNAL Coefficient * sinusoid of appropriate frequency The original signal

  7. Wavelet Properties • Short time localized waves • 0 integral value. • Possibility of time shifting. • Flexibility.

  8. Wavelets families

  9. Wavelet Transform Coefficient * appropriatelyscaled and shiftedwavelet The original signal

  10. CWT Step 1 Step 2 Step 3 Step 4 Step 5 Repeat steps 1-4 for all scales

  11. Example –A simulated lunar landscape

  12. CWT of the “Lunar landscape” 1/46 scale time mother

  13. Scale and Frequency • Higher scale correspond to the most “stretched” wavelet. • The more stretched the wavelet– the coarser the signal features being measured by the wavelet coefficient. Low scale High scale

  14. Scale and Frequency (Cont’d) • Low scale a : Compressed wavelet :Fine details (rapidly changing) : High frequency • High scale a : Stretched wavelet: Coarse details (Slowly changing): Low frequency

  15. Shift Smoothly over the analyzed function

  16. The DWT • Calculating the wavelets coefficients at every possible scale is too much work • It also generates a very large amount of data Solution: choose only a subset of scales and positions, based on power of two (dyadic choice) Discrete Wavelet Transform

  17. LPF Input Signal HPF Approximations and Details: • Approximations: High-scale, low-frequency components of the signal • Details: low-scale, high-frequency components

  18. Decimation • The former process produces twice the data • To correct this, we Down sample(or: Decimate) the filter output by two. A complete one stage block : A* LPF Input Signal D* HPF

  19. Multi-level Decomposition • Iterating the decomposition process, breaks the input signal into many lower-resolution components: Wavelet decomposition tree: high pass filter Low pass filter

  20. Wavelet reconstruction • Reconstruction (or synthesis) is the process in which we assemble all components back Up sampling (or interpolation) is done by zero inserting between every two coefficients

  21. Example*: * Wavelet used: db2

  22. What was wrong with Fourier? • We loose the time information

  23. Short Time Fourier Analysis • STFT - Based on the FT and using windowing :

  24. STFT • between time-based and frequency-based. • limited precision. • Precision <= size of the window. • Time window - same for all frequencies. What’s wrong with Gabor?

  25. Wavelet Analysis • Windowing technique with variable size window: • Long time intervals - Low frequency • Shorter intervals - High frequency

  26. The main advantage:Local Analysis • To analyze a localized area of a larger signal. • For example:

  27. Local Analysis (Cont’d) low frequency • Fourier analysis Vs. Wavelet analysis: scale Discontinuity effect time High frequency NOTHING! exact location in time of the discontinuity. more

  28. ( ) ) ( Y = Y - x b 1 a , b x a a 2D SIGNAL Wavelet function • b– shift coefficient • a – scale coefficient • 2D function 1D function

  29. Time and Space definition 1D • Time– for one dimension waves we start point shifting from source to end in time scale . • Space– for image point shifting is two dimensional . 2D

  30. Image Pyramids

  31. Wavelet Decomposition

  32. Wavelet Decomposition- Another Example LENNA

  33. high pass high pass high pass more

  34. Coding Example Original @ 8bpp DWT @0.5bpp DCT @0.5 bpp

  35. Zoom on Details DWTDCT

  36. Another Example 0.15bpp 0.18bpp 0.2bpp DCT DWT

  37. Where do we use Wavelets? • Everywhere around us are signals that can be analyzed • For example: • seismic tremors • human speech • engine vibrations • medical images • financial data • Music Wavelet analysis is a new and promising set of tools for analyzing these signals

  38. THE END

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