Wavelet Transforms ( WT ) -Introduction and Applications Presenter : Pei - Jarn Chen 2010/12/08 E.E. Department of STUT .

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

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

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

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

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

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

Wavelet Transform ( WT ) * Bandpass filter algorithm

Wavelet Transform ( WT )

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 …………..

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

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

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

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

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

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

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

Wavelet Transform ( WT ) Wavelet family

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

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

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

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

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

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

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

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

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. 1441-1449, 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

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

Applications

Applications 1.

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

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.

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. 14-26, 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. 955- 963, 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. 1441-1449, 1994 [4]. E. Meyer and T. Tuthill, ‘ Bayesian classification of ultrasound signal using wavelet coefficients’, IEEE Aerospace and Electronics Conference, vol. 1, pp. 240-243, 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. 614-625, 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. 1090-1094, 1994

Reference [7]. M. Unser and A. Aldroubi, ’A review of Wavelet in Biomedical Appliocations’, IEEE Proceedings, vol. 84, no. 4 , pp.626-638, 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

Thank You !