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Wavelets

Wavelets. History, Family, and Functionality Max Reichlin, ME 535. Introduction to Wavelets. Definition: A wavelet is a finite, zero average oscillation. They are used in a number of circumstances where many complex signals interact.

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Wavelets

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  1. Wavelets History, Family, and Functionality Max Reichlin, ME 535

  2. Introduction to Wavelets • Definition: • A wavelet is a finite, zero average oscillation. They are used in a number of circumstances where many complex signals interact. • Wavelets are used for a number of different purposes including: • Compression • Noise Reduction • Signal Decomposition • Edge Detection, Computer Vision • (6, Zorin)

  3. History of Wavelets • The first wavelet was the Haar Wavelet • Developed in the early 1900s • Discontinuous • Came before the general class of objects was recognized. • The Gabor Wavelet was developed in 1946 by Dennis Gabor as an alternative to the Fourier Transform • In the 1980’s, Jean Morlet developed what is now referred to as the Morlet Wavelet, although it is very similar to the Gabor Wavelet. • Since then, many wavelets have been developed including the Mexican Hat (or Ricker) Wavelet and the Daubechies families of wavelets. • (6, Zorin)

  4. Why use a Wavelet? • Example: The surface sea temperatures over the 20th Century • Contains many different signals of interest • Long waves vary in space and time throughout the century • Measuring merely variance from the mean: • No information about the frequency of oscillations • Must choose finite window for variance • (4, Torrence)

  5. The Wavelet Transform Windowed Fourier Transform: Wavelet Transform • Fourier Transform vs. Wavelet Transform • A Fourier Transform solves the frequency problem – giving a great deal of information about the frequencies present. • However, even with a windowed Fourier Transform, wide differences in frequency are lost • The Wavelet Transform can capture detail from a very short pulse while still providing high quality information about the surrounding noise. • Crazy fact: The human ear uses the equivalent of a physical wavelet transform when changing pressure waves into neural signals. • (3, Daubechies, 4, Torrence)

  6. Mother and Daughter Wavelets Each wavelet has parent formula, which can then be scaled to the frequency range of interest. Mother, Morlet: Child, Morlet: (1, Arobone, 4, Torrence)

  7. The Results By integrating the wavelet formula over the entire time in question and using a series of daughter wavelets, scaled according to the period in question, changes in the period of powerful elements become evident. But there is no free lunch: Many methods that are available to the Fourier Transform, such as the convolution, are not currently available for Wavelet analysis. (4, Torrence)

  8. Tools for Wavelet Analysis • FAWAVE • http://www.uwec.edu/walkerjs/FAWAVE/ • We saw in class • Free Program • Can enter functions • Simple audio and image editors • MathWorks’ Matlab Wavelet Toolbox • http://www.mathworks.com/products/wavelet/ • Costs money • Much of the same functionality and more • I didn’t use it • Wavelab 850 • http://statweb.stanford.edu/~wavelab/Wavelab_850/index_wavelab850.html • Free toolbox • Connects to Matlab • Fairly easy to use • Wide range of de-noising, compression, and analysis functions • Many examples and tutorials • Some bugs

  9. References 1) Arobone, E. (2009, May 07). Introduction to Wavelets. Retrieved from CSE 262 Lectures: http://cseweb.ucsd.edu/classes/sp09/cse262/Lectures/WAVELET.pdf 2) Buckheit, J., Chen, S., Donoho, D., Johnstone, I., & Scargle, J. (2005). Wavelab 850. Retrieved from Wavelab 850: http://statweb.stanford.edu/~wavelab/Wavelab_850/index_wavelab850.html 3) Daubechies, I. (1992). 10 Lecutres on Wavelets.Philadephia: Society of Industrial and Applied Mathematics. 4) Torrence, C., & Compo, G. P. (n.d.). Wavelet Analysis & Monte Carlo. Retrieved from A Practical Guide to Wavlets: http://paos.colorado.edu/research/wavelets/ 5) Various. (n.d.). Morlet Wavelet. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Morlet_wavelet 6) Zorin, D., & Owens, J. (2000, September 29). Introduction to Wavelets. Retrieved from EE 482 Lectures: http://cva.stanford.edu/classes/ee482a/docs/lect01_sample.pdf

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