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Wavelet Transform

Wavelet Transform. Definition of The Continuous Wavelet Transform CWT. The continuous-time wavelet transform (CWT) of f(x) with respect to a wavelet (x):. L 2 (R). Mother Wavelet Dilation / Translation.  Mother Wavelet a Dilation Scale b Translation.

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Wavelet Transform

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  1. Wavelet Transform

  2. Definition of The Continuous Wavelet Transform CWT The continuous-time wavelet transform (CWT) of f(x) with respect to a wavelet (x): L2(R)

  3. Mother WaveletDilation / Translation  Mother Wavelet a Dilation Scale b Translation

  4. Properties of a Basic Wavelet   L2(R) is called a Basic Wavelet if the following admissibility condition is satisfied: Admissibility condition. Necessary condition to obtain the inverse from the CWT by the basic Wavelet . Sufficient, but not a necessary condition to obtain the inverse by general Wavelet. Oscillation (Wave) 1. Finite energy (Let) fast decay 2. Oscillation + fast decay = Wave + let = Wavelet

  5. Haar Wavelet Dilation / Translation Haar 1 1 1 2-1/2 2-1/2 1 4 2 4 4 2 -1 -1 -1

  6. Morlet Wavelet Dilation / Translation Morlet

  7. Forward / Inverse Transform [1/5] Forward Inverse Admissibility condition.

  8. Forward / Inverse Transform [2/5] Theorem cwt_001 Proof

  9. Forward / Inverse Transform [3/5] Theorem cwt_002 Proof

  10. Forward / Inverse Transform [4/5] Theorem cwt_003 Proof

  11. Forward / Inverse Transform [5/5] Theorem cwt_004 Proof

  12. Wavelet TransformMorlet Wavelet - Stationary Signal

  13. Wavelet TransformMorlet Wavelet - Transient Signal

  14. Wavelet TransformMorlet Wavelet - Transient Signal

  15. Wavelet TransformMorlet Wavelet - Non-visible Oscillation [1/3]

  16. Wavelet TransformMorlet Wavelet - Non-visible Oscillation [2/3]

  17. Wavelet TransformMorlet Wavelet - Non-visible Oscillation [3/3]

  18. Wavelet TransformHaar Wavelet - Stationary Signal

  19. Wavelet TransformHaar Wavelet - Transient Signal

  20. Wavelet TransformMexican Hat - Stationary Signal

  21. Wavelet TransformMexican Hat - Transient Signal

  22. Wavelet TransformMorlet WaveletFourier/Wavelet Fourier Wavelet

  23. Wavelet TransformMorlet WaveletFourier/Wavelet Fourier Wavelet

  24. CWT - Correlation 1 Cross- correlation CWT CWT W(a,b) is the cross-correlation at lag (shift)  between f(x) and the wavelet dilated to scale factor a.

  25. CWT - Correlation 2 W(a,b) always exists The global maximum of |W(a,b)| occurs if there is a pair of values (a,b) for which ab(t) = f(t). Even if this equality does not exists, the global maximum of the real part of W2(a,b) provides a measure of the fit between f(t) and the corresponding ab(t) (se next page).

  26. CWT - Correlation 3 The global maximum of the real part of W2(a,b) provides a measure of the fit between f(x) and the corresponding ab(x) ab(x) closest to f(x) for that value of pair (a,b) for which Re[W(a,b)] is a maximum. -ab(x) closest to f(x) for that value of pair (a,b) for which Re[W(a,b)] is a minimum.

  27. CWT - Localization both in time and frequency The CWT offers position/time and frequency selectivity; that is, it is able to localize events both in position/time and in frequency. Time: The segment of f(x) that influences the value of W(a,b) for any (a,b) is that stretch of f(x) that coinsides with the interval over which ab(x) has the bulk of its energy. This windowing effect results in the position/time selectivity of the CWT. Frequency: The frequency selectivity of the CWT is explained using its interpretation as a collection of linear, time-invariant filters with impulse responses that are dilations of the mother wavelet reflected about the time axis (se next page).

  28. CWT - Frequency - Filter interpretation Convolution CWT CWT is the output of a filter with impulse response *ab(-b) and input f(b). We have a continuum of filters parameterized by the scale factor a.

  29. CWT - Time and frequency localization 1 Time Center of mother wavelet Frequency Center of the Fourier transform of mother wavelet

  30. CWT - Time and frequency localization 2 Time Frequency Time-bandwidth product is a constant

  31. CWT - Time and frequency localization 3 Time Frequency Small a: CWT resolve events closely spaced in time. Large a: CWT resolve events closely spaced in frequency. CWT provides better frequency resolution in the lower end of the frequency spectrum. Wavelet a natural tool in the analysis of signals in which rapidly varying high-frequency components are superimposed on slowly varying low-frequency components (seismic signals, music compositions, …).

  32. CWT - Time and frequency localization 4  a=1/2 a=1 a=2 t Time-frequency cells for a,b(t)

  33. Filtering / Compression Data compression Remove low W-values Highpass-filtering Lowpass-filtering Replace W-values by 0 for high a-values Replace W-values by 0 for low a-values

  34. CWT - DWT CWT DWT Binary dilation Dyadic translation Dyadic Wavelets

  35. Mexican Hat

  36. Rotation - Scaling2 dim Rotation Scaling

  37. Translation - Rotation - Scaling3 dim Translation Rotation Scaling

  38. Mexican Hat - 3 Dim

  39. End

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