1. Wavelet Transforms ( WT ) -Introduction and Applications
Presenter : Pei - Jarn Chen
2010/12/ E.E. Department of STUT .

2. Outline ☆ Theory ☆ Applications ☆ Matlab approach ☆ Reference
 methodology
 develop history
 mathematic description ( CWT & DWT)
☆ Applications
☆ Matlab approach
☆ Reference

3. Introduction Wavelet theory  Scaling
 Multi-resolution analysis( MRA )
 Mathematics description
 Wavelet transform ( CWT & DWT )
 Wavelet family

4. Wavelet theory Time - frequency analysis Scaling Δt*Δf≧(1/4)*π
Heinsberg uncertainty principle
Δt*Δf≧(1/4)*π

5. Wavelet theory Multiresolution analysis (MRA)
Multi_ scale analysis ( superposition )
dilation
translation

6. Wavelet theory Multi_ space analysis + = decomposition reconstruction
Approximate space
Detail space
decomposition
reconstruction

7. Wavelet theory Wavelet packet tree S A1 D1 AD2 AA2 DA2 DD2 AAA3 DAA3
DDD3

8. Wavelet Transform ( WT )
* Bandpass filter algorithm

9. Wavelet Transform ( WT )

10. Wavelet theory  Develope history
 1910 Haar orthogonal system
 1982 Strömberg first continuous wavelet
 1984 Grossman & Morlet-----wavelet transform
 1986 Meyer & Mallat ----multiresolution analysis &
mathematics description
 1987 Tchamitchian biorthogonal wavelets
 1988 Daubechies …………..

11. Wavelet Transform ( WT )
Mathematics description
define
j, k: scaling & translation parameters
Φ: scaling function ( j=k=0, father function) Vj
j, k : scaling & translation parameters
: wavelet function (j=k=0, mother function) Oj

12. Wavelet Transform ( WT )
Refinement ( dilation ) equation
Wavelet
family

13. Wavelet Transform ( WT )
The properties of mother wavelet w t

14. Wavelet Transform ( WT )
Wavelets basis
compactly supported wavelets
Harr
Daubechies……….
not compactly supported wavelets
Mexican hat function
Littlewood-Paley
Morlet
Meyer’s
B-spline………...

15. Wavelet Transform ( WT )
Harr
(t)
|()|
|()|
(t)

16. Wavelet Transform ( WT )
Meyer
(t)
|()|
|()|
(t)

17. Wavelet Transform ( WT )
Daubechies
|()|
(t)
(t)
|()|

18. Wavelet Transform ( WT )
Wavelet family

19. Wavelet Transform ( WT )
The technique of WT
Continuous Wavelet Transform (CWT)
a: scaling
b: translation
C=0.2247

20. Wavelet Transform ( WT )
Discrete Wavelet Transform (DWT)
Scaling function :
Wavelet function：
a= 2 j
CWT
DWT

21. Applications A. 1-D 1. * db3, level 5, DWT # A sum of sines
1. Detection breakdown points
2. Identifying pure frequency
3.The effect of wavelet on a sine
4. The level at which characteristcs
* db3, level 5, DWT

22. Applications 2.. db5, level 5, DWT # Frequency breakdown
1. Suppressing signals
2. Detecting long_term evolution
db5, level 5, DWT

23. Applications 3. # Color AR(3) Noise Processing noise
2. The relative importance
of different detail
3. The comparative importance
D1 and A1
* db3, level 5, DWT

24. Applications 4. # Two Proximal Discontinuties
1. Detecting breakdown points
2. Move the discontinuities closer
together and further apart
* db2 and db7, level 5, DWT

25. Applications 5. # A Triangle + A Sine + noise
Detecting long-term evolution
2. Splitting signal components
3. Identifying the frequency of
a sine
* db5, level 6, DWT

26. Applications 6. # A Real three-day Electrical Consumption Signal
* db3, level 5, DWT

27. Applications--Velocity dispersion ( T. Onsay and A. G. Haddow, J
Applications--Velocity dispersion ( T.Onsay and A.G. Haddow, J. Acoust. Soc. Am. Vol. 95, no. 3, pp , 1994 )
Fig .Signal_1 and signal_2 following the input
of glass ball on the free end of the beam
Fig. The CWT of the acceleration signal_2

28. Applications
B. 2-D ( imaging data compression, JPEG 2000)

29. Applications

30. Applications 1.

31. Matlab Approach (1) Using Wavelet Packets (2) Using Matlab command and
Matlab Approach (1) Using Wavelet Packets (2) Using Matlab command and *.m

33. Conclusion
The self -adjusting windows structure for WT provides an enhanced resolution compared to the Short Time Fourier Transform (STFT).
WT technique is not a panacea. It should be used with caution, depended by the problem itself.

34. Reference
[1]. A. Abbate, J. Koay, et. al., ‘Signal detection and noise suppression using a wavelet transform signal processor: Application to ultrasoic flaw detection’, IEEE Trans. On Ultrason., Ferroelect., and Freq. Contr., vol. 44, no. 1, pp , 1997.
[2]. B. M. Sadler, T. Pham, and L. C. Sadler,’ Optimal and wavelet-based shock wave detection and estimation’, J. Acoust. Soc. Am., vol. 104, no.2, pp , 1998
[3]. T. Onsay and A. G. Haddow, ’Wavelet transform analysis of transient wave propagation a dispersive medium’, J. Acoust. Soc. Am. Vol. 95, no. 3, pp , 1994
[4]. E. Meyer and T. Tuthill, ‘ Bayesian classification of ultrasound signal using wavelet coefficients’, IEEE Aerospace and Electronics Conference, vol. 1, pp , 1995
[5]. R. Polikar, L. Udpa, S. S. Udpa, and T. Taylor, ‘ Frequency invariant classification of ultrasound welding inspection signals’, IEEE Trans. On Ultrason., Ferroelect., and Freq. Contr., vol. 45, no. 3, p.p , 1998
[6]. W. X. Robert, S. Siffert and J. J. Kaufman, ’ Application of wavelet analysis to ultrasound characterization of bone’, IEEE 26 Asilomar conference, vol. 12, pp , 1994

35. Reference
[7]. M. Unser and A. Aldroubi, ’A review of Wavelet in Biomedical Appliocations’, IEEE Proceedings, vol. 84, no. 4 , pp , 1996
[8]. S. Mallat, ’ Wavelet tour of signal processing’, Academic Press,1998
[9]. M. R. Rao and A. S. Boparadikor, ’ Wavelet Transforms introduction to Theory and Application’ , Addison-Wesley Press, London, U.K. 1998
[10]. Wavelet Toolbox : for Use with MATLAB, 1996
[11]. M. Akay, ’ Time frequency and wavelets in biomedical signal processing’, IEEE
Press, U.S.A., 1998

36. Thank You !