THE WAVELET TUTORIAL PART I by ROBI POLIKAR FUNDAMENTAL CONCEPTS & AN OVERVIEW OF THE WAVELET THEORY Second Edition NEW! - Thanks to Noël K. MAMALET, this tutorial is now available in French Welcome to this introductory tutorial on wavelet transforms. The wavelet transform is a relatively new He has been a visiting professor in the ElectricalEngineering Department at Massachusetts Institute of Technology and in the Applied Mathematics Department at the University of Tel Aviv. Dr. Mallat received the 1990 IEEE Signal Processing Society's paper award, the 1993 Alfred Sloan fellowship in Mathematics, the 1997Outstanding Achievement Award from the SPIE Optical Engineering Society, and the 1997 Blaise Pascal Prize in applied mathematics, from theFrench Academy of Sciences. A base wavelet is needed in order to realize the wavelet transform, a mathematical tool that converts a signal into a different form and reveals the characteristics or " features" hidden within the original signal. Contributi importanti alla teoria delle wavelet possono essere attribuiti alla formulazione, da parte di Goupillaud, Grossmann e Morlet, di quella che ora è nota come CWT , ai lavori preliminari di Strömberg sulle wavelet discrete , alle wavelet ortogonali a supporto compatto di Daubechies , alla struttura a multirisoluzione di Mallat , all'interpretazione in tempo-frequenza della CWT da parte di Delprat , alla trasformata wavelet armonica di Newland e molti altri ancora. Step 1: The wavelet is placed at the beginning of the signal, and set s=1 (the most compressed wavelet); Step 2: The wavelet function at scale "1" is multiplied by the signal, and integrated over all times; then multiplied by ; Step 3: Shift the wavelet to t= , and get the transform value at t= and s=1; "Mallat: 05-ch01-p374370" — 2009/3/7 — 17:59 — page4—#4 4 CHAPTER 1 Sparse Representations The systematic theory for constructing orthonormal wavelet bases was estab-lished by Meyer and Mallat through the elaboration of multiresolution signal approximations [362],as presented in Chapter 7. It was inspired by original ideas Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. Irène Waldspurger. Stéphane Georges Mallat (born 24 October 1962) is a French applied mathematician, concurrently appointed as Professor at Collège de France and École normale supérieure. He made fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s. He has additionally done work in applied mathematics, Scribd è il più grande sito di social reading e publishing al mondo. Mallat shows that the Lipschitz continuity is related to the wavelet transform, and that if the wavelet transform is Lipschitz α, the function is also Lipschitz α: ( ) (2)α 2 W f x K j j ≤ The conclusions are summarized in the following table. ααα constraint Meaning Impact on Wavelet transform 0 < α <= 1 f(x) is differentiable at The theory of wavelets as presented in the previous chapters gives a harmonic analysis representation of an infinite-dimensional function space (like L 2 (R) for instance) in terms of an infinite orthonormal basis (or tight frame in the general case). For applications of this theory to real-world situations, it is necessary to deal