Introduction and Overview

1. Introduction and Overview Dr Mohamed A. El-Gebeily Department of Mathematical Sciences KFUPM mgebeily@kfupm.edu.sa

2. Why Wavelets? • Comparison With Fourier Analysis • What is Wavelet Analysis? • The Continuous Wavelet Transform • The Discrete Wavelet Transform • Introduction to Wavelet Families • Applications

3. Why Wavelets? • Wavelets have scale and time zooming aspects • Scale zooming • Used to analyze the regularity of a signal. • Biology for cell membrane recognition. • Metallurgy for the characterization of rough surfaces • Finance (which is more surprising), for detecting the properties of quick variation of values. • In Internet traffic description, for designing the services size.

4. Time zooming • Used to detect ruptures and short-time phenomena such as transient processes • Applications: • Industrial supervision of gear-wheel faults • Non destructive control quality processes • Detection of short pathological events as epileptic crises or normal ones as evoked potentials in EEG (medicine) • SAR imagery • Automatic target recognition • Intermittence in physics

5. Wavelet Decomposition as a Whole • Many applications use the wavelet decomposition taken as a whole. • The common goals are de-noising or compression. • Compression: FBI fingerprints • It is almost impossible to sum up several thousand papers written within the last 15 years. • It is difficult to get information on real-world industrial applications from companies.

6. Fourier Analysis • Breaks down a signal into constituent sinusoids of different frequencies. • Useful when the signal's frequency content is of great importance. • Has a serious drawback: time information is lost.

7. Short-Time Fourier Analysis (STFT) • Dennis Gabor (1946) adapted the Fourier transform to analyze only a small section of the signal at a time -- a technique called windowing the signal • Limited precision, because of the fixed size of the window for all frequencies.

8. Wavelet Analysis • Wavelet: a windowing technique with variable-sized regions. • Long time intervals where we want more precise low-frequency information, and shorter regions where we want high-frequency information.

9. What Can Wavelet Analysis Do? • The ability to perform local analysis -- that is, to analyze a localized area of a larger signal. • Example: a sinusoidal signal with a small discontinuity

10. What Is Wavelet Analysis? • A wavelet is a waveform of effectively limited duration that has an average value of zero. • Compare wavelets with sine waves

11. The Continuous Wavelet Transform • The continuous wavelet transform (CWT) is defined as:

12. Scaling Shifting

13. Five Easy Steps to a Continuous Wavelet Transform • Take a wavelet and compare it to a section at the start of the original signal. • Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal.

14. 3.Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal. C=.767 4.Scale (stretch) the wavelet and repeat steps 1 through 3. 5.Repeat steps 1 through 4 for all scales.

15. A scalogram is a 3-D plot of the wavelet coefficients against time and scale.

16. The Discrete Wavelet Transform (DWT) • The wavelet coefficients are computed at the dyadic points • It turns out that this is more efficient and enough to recover the original function from the wavelet coefficients.

17. One-Stage Filtering: Approximations and Details • In wavelet analysis, we often speak of approximations and details. The approximations are the high-scale, low-frequency components of the signal. The details are the low-scale, high-frequency components.

18. Downsampling The WT produces two sequences called cA and cD.

19. Example A pure sinusoid with high-frequency noise added to it. s = sin(20.*linspace(0,pi,1000)) + 0.5.*rand(1,1000); [cA,cD] = dwt(s,'db2');

20. Multiple-Level Decomposition wavelet decomposition tree.

21. Reconstruction Filters • The low- and highpass decomposition filters (L and H), together with their associated reconstruction filters (L' and H'), form a system of what is called quadrature mirror filters

22. Reconstructing Approximations and Details • It is possible to reconstruct the approximations and details themselves from their coefficient vectors.

23. Extending this technique to the components of a multilevel analysis, we find that similar relationships hold for all the reconstructed signal constituents. That is, there are several ways to reassemble the original signal:

24. An Introduction to the Wavelet Families Several families of wavelets have proven to be especially useful: Haar Haar wavelet is the first and simplest. Haar wavelet is discontinuous, and resembles a step function. Same as Daubechies db1.

25. Daubechies Ingrid Daubechies invented what are called compactly supported orthonormal wavelets -- thus making discrete wavelet analysis practicable.

26. Biorthogonal

27. Coiflets

28. Symlets

29. Morlet

30. Mexican Hat

31. Meyer

32. Example: Signal Processing The purpose of this example is to show how analysis by wavelets can detect the exact instant when a signal changes. The discontinuous signal consists of a slow sine wave abruptly followed by a medium sine wave.

33. Example: Image processing The image below is compressed to a ratio of 25:1 of its original size by using the two dimensional wavelet transform