By Stephane Mallat

This ebook is meant to function a useful reference for somebody excited by the applying of wavelets to sign processing. It has developed from fabric used to educate "wavelet sign processing" classes in electric engineering departments at Massachusetts Institute of expertise and Tel Aviv college, in addition to utilized arithmetic departments on the Courant Institute of recent York collage and ?‰colePolytechnique in Paris. Key gains* presents a huge standpoint at the ideas and functions of temporary sign processing with wavelets* Emphasizes intuitive knowing, whereas supplying the mathematical foundations and outline of quickly algorithms* quite a few examples of genuine purposes to noise elimination, deconvolution, audio and photograph compression, singularity and part detection,multifractal research, and time-varying frequency measurements* Algorithms and numerical examples are carried out in Wavelab, that's a Matlab toolbox freely to be had over the web* content material is out there on a number of point of complexity, reckoning on the person reader's needsNew to the second one variation* Optical stream calculation and video compression algorithms* photograph versions with bounded version services* Bayes and Minimax theories for sign estimation* 2 hundred pages rewritten and such a lot illustrations redrawn* extra difficulties and subject matters for a graduate path in wavelet sign processing, in engineering and utilized arithmetic

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**Extra resources for A Wavelet Tour of Signal Processing, Second Edition (Wavelet Analysis & Its Applications)**

**Example text**

By working in the Hilbert space Lz(R), we also have access to all the facilities provided by the existence of an inner product. The inner product of f E L 2(R) and g E L 2 (R) is (f ,g) - f + ~ f ( t ) g*(t)dt, and the resulting norm in L 2(]~) is +~ [Ifll 2 - (f ,f) - f If(t)12dt. O(3 26 CHAPTERII FOURIERKINGDOM The following theorem proves that inner products and norms in L2(R) are conserved by the Fourier transform up to a factor of 27r. 26) are called respectively the Parseval and Plancherel formulas.

Hence, there exists )~ C L2(R) such that lim n-++~ IIf- JCnl[- 0. By definition, f is the Fourier transform of f. 24). 7. A Dirac 5 associates to a function its value at t = 0. Since e iwt = 1 at t = 0 it seems reasonable to define its Fourier transform by 6(~)-/+~(t)e-i~'dt-1. 27) This formula is justified mathematically by the extension of the Fourier transform to tempered distributions [66, 69]. 3 Examples The following examples often appear in Fourier calculations. They also illustrate important Fourier transform properties.

Linear versus Non-Linear A signal f[n] of size N is contaminated by the addition of a noise. 4 BASESFOR WHAT?. I5 probability distribution is known. The measured data are X[n] = f [n] + W[n] . The signal f is estimated by transforming the noisy data X with an operator D: F=DX. The risk of the estimator/3 of f is the average error, calculated with respect to the probability distribution of the noise W: r(D, f ) = E{IIf - DXll2}. It is tempting to restrict oneself to linear operators D, because of their simplicity.