First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book.

Wavelets and Subband Coding

1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

Suppose a discrete-time signal S(t), 0/spl les/t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1/sub {t=/spl tau/}/ and sinusoids exp{2/spl pi/iwt/N}//spl radic/N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l/sup 1/ norm of the coefficients among all decompositions. Here "highly sparse" means that N/sub t/+N/sub w/</spl radic/N/2 where N/sub t/ is the number of time atoms, N/sub w/ is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l/sup 1/ uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l/sup 1/ norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2/sup j/2/.

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Uncertainty principles and ideal atomic decomposition

We present the lifting scheme, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included.

The lifting scheme: A construction of second generation wavelets

The lifting scheme: a construction of second generation wavelets

We prove the Kato conjecture for elliptic operators on Jfin. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L =-div (AV) with bounded measurable coefficients in IEtn iS the

The solution of the Kato square root problem for second order elliptic operators on Rn

Beyond Calderon-Zygmund operators Basic $L^2$ theory for elliptic operators $L^p$ theory for the semigroup $L^p$ theory for square roots Riesz transforms and functional calculi Square function estimates Miscellani Appendix A. Calderon-Zygmund decomposition for Sobolev functions Appendix. Bibliography.

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On Necessary and Sufficient Conditions for LP-Estimates of Riesz Transforms Associated to Elliptic Operators on RN and Related Estimates

— We present in this work recent progress on the square root problem of Kato for differential operators in divergence form on R n . We discuss topics on functional calculus, heat and resolvent kernel estimates, square function estimates and Carleson measure estimates for square roots. In the first chapter, we show in a quantitative way how the theorems of Aronson-Nash and of De Giorgi are equivalent. In the central chapters, we take advantage of recent development in functional calculus and in harmonic analysis to propose a new point of view on Kato's problem which allows us to unify previous results and extend them. In the last chapter we study the associated Riesz transforms, their relation to Calderón-Zygmund operators and their behavior on L p -spaces. Résumé (Problème de la racine carrée pour les opérateurs sous forme divergence et sujets connexes). — Ce travail a pour thème principal le problème de Kato concernant la racine carrée des opérateurs différentiels elliptiques sous forme divergence dans R n . Pour mener à bien cette étude, nous nous intéressons à des questions relatives au calcul fonctionnel, aux estimations de noyaux, aux fonctionnelles quadratiques et aux mesures de Carleson associées aux racines carrées. Dans le premier chapitre, nous montrons en un sens précis comment les théorèmes d'Aronson-Nash et de De Giorgi sont équivalents. Dans les deux chapitres centraux, nous tirons parti de développements récents sur le calcul fonctionnel et en analyse harmonique pour proposer un nouveau point de vue sur le problème de Kato qui permet d'unifier les résultats antérieurs et de les généraliser. Enfin, dans le dernier chapitre, nous étudions les transformées de Riesz associées, leur relation aux opérateurs de Calderón-Zygmund et leur comportement sur les espaces Lp. © Astérisque 249, S M F 1998

Square root problem for divergence operators and related topics

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.

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Riesz transform on manifolds and heat kernel regularity

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is $L^p$ bounded on such a manifold, for $p$ ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain $L^p$ estimate in the same interval of $p$'s.

Pascal Auscher

Papers

The solution of the Kato square root problem for second order elliptic operators on Rn

On Necessary and Sufficient Conditions for LP-Estimates of Riesz Transforms Associated to Elliptic Operators on RN and Related Estimates

Square root problem for divergence operators and related topics

Riesz transform on manifolds and heat kernel regularity

Riesz transform on manifolds and heat kernel regularity