By Stephane Mallat
ISBN-10: 0080520839
ISBN-13: 9780080520834
ISBN-10: 012466606X
ISBN-13: 9780124666061
This ebook is meant to function a useful reference for a person desirous about the appliance 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 collage, in addition to utilized arithmetic departments on the Courant Institute of recent York collage and ?‰colePolytechnique in Paris. Key positive factors* presents a huge viewpoint at the ideas and functions of temporary sign processing with wavelets* Emphasizes intuitive realizing, whereas delivering the mathematical foundations and outline of speedy algorithms* a number of examples of genuine functions to noise elimination, deconvolution, audio and picture compression, singularity and aspect detection,multifractal research, and time-varying frequency measurements* Algorithms and numerical examples are applied in Wavelab, that is a Matlab toolbox freely to be had over the web* content material is available on a number of point of complexity, reckoning on the person reader's needsNew to the second one version* Optical move calculation and video compression algorithms* photo versions with bounded edition features* Bayes and Minimax theories for sign estimation* two hundred pages rewritten and so much illustrations redrawn* extra difficulties and subject matters for a graduate direction in wavelet sign processing, in engineering and utilized arithmetic
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Additional info for A Wavelet Tour of Signal Processing, Second Edition (Wavelet Analysis & Its Applications)
Sample text
Linear approximations of uniformly smooth signals in wavelet and Fourier bases have similar properties and characterize nearly the same function spaces. Suppose that we want to approximate a class of discrete signals of size N, modeled by a random vector F[n]. The average approximation error when projecting F over the first M basis vectors of an orthonormal basis/3 {gm}O<_m
42) O(3 Proof 1. The Fourier transform of the k th order derivative f(k)(t) is (i~)k~(~). 42) implies that f + ~ I~(~)11~1 ~ d~ < + ~ for any k < p, so f(k)(t) is continuous and bounded. II This result proves that if there exist a constant K and e > 0 such that - 1 + ]&l p-4-1-k-e' then f E C p. 42) implies that f C C ~. The decay of If @) I depends on the worst singular behavior of f . For example, f -- l[-r,r] is discontinuous at t - i T , so If( )I decays like In this case, it could also be important to know that f(t) is regular for t # +T.
D~(t). ~(t)= f(t-r). 5) Stability and Causality A filter is said to be causal if L f (t) does not depend on the values f(u) for u > t. Since f +e~ L f (t) -- h(u) f (t - u)du, OO 22 CHAPTERII FOURIERKINGDOM this means that h(u) = 0 for u < 0. Such impulse responses are said to be causal. The stability property guarantees that Lf(t) is bounded if f(t) is bounded. Since [Lf(t)l <_ f+oo [ h ( u ) l l f ( t - u ) l d u < _ s u p l f ( u ) l cx) u E1R f+~ Ih(u)ldu, oo it is sufficient that f_+~ Ih(u)ldu < + ~ .
A Wavelet Tour of Signal Processing, Second Edition (Wavelet Analysis & Its Applications) by Stephane Mallat
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