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): L 2 (R)

3 Mother Wavelet Dilation / Translation  Mother Wavelet aDilationScale bTranslation

4 Properties of a Basic Wavelet Finite energy (Let) fast decay Oscillation (Wave) 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.   L 2 (R) is called a Basic Wavelet if the following admissibility condition is satisfied: Oscillation + fast decay = Wave + let = Wavelet

5 Haar Wavelet Dilation / Translation Haar /2

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 Transform Morlet Wavelet - Stationary Signal

13 Wavelet Transform Morlet Wavelet - Transient Signal

14 Wavelet Transform Morlet Wavelet - Transient Signal

15 Wavelet Transform Morlet Wavelet - Non-visible Oscillation [1/3]

16 Wavelet Transform Morlet Wavelet - Non-visible Oscillation [2/3]

17 Wavelet Transform Morlet Wavelet - Non-visible Oscillation [3/3]

18 Wavelet Transform Haar Wavelet - Stationary Signal

19 Wavelet Transform Haar Wavelet - Transient Signal

20 Wavelet Transform Mexican Hat - Stationary Signal

21 Wavelet Transform Mexican Hat - Transient Signal

22 Wavelet Transform Morlet Wavelet Fourier/Wavelet Fourier Wavelet

23 Wavelet Transform Morlet Wavelet Fourier/Wavelet Fourier Wavelet

24 CWT - Correlation 1 CWT Cross- correlation 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 W 2 (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 W 2 (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 t  Time-frequency cells for  a,b (t) a=1/2 a=1 a=2

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

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

35 Mexican Hat

36 Rotation - Scaling 2 dim Rotation Scaling

37 Translation - Rotation - Scaling 3 dim Rotation Scaling Translation

38 Mexican Hat - 3 Dim

39 End