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Polytechnique in Paris.Provides a broad perspective on the principles and applications of transient signal processing with waveletsEmphasizes intuitive understanding, while providing the mathematical foundations and description of fast algorithmsNumerous examples of real applications to noise removal, deconvolution, audio and image compression, singularity and edge detection, multifractal analysis, and time-varying frequency measurementsAlgorithms and numerical examples are implemented in Wavelab, which is a Matlab toolbox freely available over the InternetContent is accessible on several level of complexity, depending on the individual reader's needs

New to the Second Edition

Optical flow calculation and video compression algorithmsImage models with bounded variation functionsBayes and Minimax theories for signal estimation200 pages rewritten and most illustrations redrawnMore problems and topics for a graduate course in wavelet signal processing, in engineering and applied mathematicsChapters 10 and 11 consist of original research and are written in a more advanced style. In Chapter 10 it is shown that the structure of Maxwell's equations implies the existence of a wavelet analysis specifically adopted to electromagnetic radiation. The associated "eletromagnetic wavelets" are pulses parameterized by their point of origin and their scale, and can be made arbitrarily short by choosing fine scales. Furthermore, it is shown that every electromagnetic wave can be composed of such localized wavelets. This is applied in Chapter 11 to give a new formulation of radar based on electromagnetic wavelets. Since this theory is fully relativistic, its description of the Doppler effect is exact. In particular, it is three-dimensional, and does not make the usual assumption that the outgoing signal has a narrow bandwidth. Thus it should be useful in the construction of ultra-wideband radar systems.

Wavelets represent an area that combines signal in image processing, mathematics, physics and electrical engineering.

As such, this title is intended for the wide audience that is interested in mastering the basic techniques in this subject area, such as decomposition and compression.

Features theory, techniques, and applications

Presents alternative theoretical approaches including multiresolution analysis, splines, minimum entropy, and fractal aspects

Contributors cover a broad range of approaches and applications

Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent an entire range of properties of wavelets and coherent states. Many concrete examples, such as coherent states from semisimple Lie groups, Gazeau-Klauder coherent states, coherent states for the relativity groups, and several kinds of wavelets, are discussed in detail. The book concludes with a palette of potential applications, from the quantum physically oriented, like the quantum-classical transition or the construction of adequate states in quantum information, to the most innovative techniques to be used in data processing.

Intended as an introduction to current research for graduate students and others entering the field, the mathematical discussion is self-contained. With its extensive references to the research literature, the first edition of the book is already a proven compendium for physicists and mathematicians active in the field, and with full coverage of the latest theory and results the revised second edition is even more valuable.

Emphasis is given on ideas and intuition, avoiding the heavy computations which are usually involved in the study of wavelets. Readers should have a basic knowledge of linear algebra, calculus, and some familiarity with complex analysis. Basic knowledge of signal and image processing is desirable.

This text originated from a set of notes in Portuguese that the authors wrote for a wavelet course on the Brazilian Mathematical Colloquium in 1997 at IMPA, Rio de Janeiro.

What are wavelets? What makes them increasingly indispensable in statistical nonparametrics? Why are they suitable for "time-scale" applications? How are they used to solve such problems as denoising, regression, or density estimation? Where can one find up-to-date information on these newly "discovered" mathematical objects? These are some of the questions Brani Vidakovic answers in Statistical Modeling by Wavelets. Providing a much-needed introduction to the latest tools afforded statisticians by wavelet theory, Vidakovic compiles, organizes, and explains in depth research data previously available only in disparate journal articles. He carefully balances both statistical and mathematical techniques, supplementing the material with a wealth of examples, more than 100 illustrations, and extensive references-with data sets and S-Plus wavelet overviews made available for downloading over the Internet. Both introductory and data-oriented modeling topics are featured, including:

* Continuous and discrete wavelet transformations.

* Statistical optimality properties of wavelet shrinkage.

* Theoretical aspects of wavelet density estimation.

* Bayesian modeling in the wavelet domain.

* Properties of wavelet-based random functions and densities.

* Several novel and important wavelet applications in statistics.

* Wavelet methods in time series.

Accessible to anyone with a background in advanced calculus and algebra, Statistical Modeling by Wavelets promises to become the standard reference for statisticians and engineers seeking a comprehensive introduction to an emerging field.

—Bulletin of the AMS

An Introduction to Wavelet Analysis provides a comprehensive presentation of the conceptual basis of wavelet analysis, including the construction and application of wavelet bases.

The book develops the basic theory of wavelet bases and transforms without assuming any knowledge of Lebesgue integration or the theory of abstract Hilbert spaces. The book elucidates the central ideas of wavelet theory by offering a detailed exposition of the Haar series, and then shows how a more abstract approach allows one to generalize and improve upon the Haar series. Once these ideas have been established and explored, variations and extensions of Haar construction are presented. The mathematical prerequisites for the book are a course in advanced calculus, familiarity with the language of formal mathematical proofs, and basic linear algebra concepts.

Features:

* Rigorous proofs with consistent assumptions about the mathematical background of the reader (does not assume familiarity with Hilbert spaces or Lebesgue measure).

* Complete background material on is offered on Fourier analysis topics.

* Wavelets are presented first on the continuous domain and later restricted to the discrete domain for improved motivation and understanding of discrete wavelet transforms and applications.

* Special appendix, "Excursions in Wavelet Theory, " provides a guide to current literature on the topic.

* Over 170 exercises guide the reader through the text.

An Introduction to Wavelet Analysis is an ideal text/reference for a broad audience of advanced students and researchers in applied mathematics, electrical engineering, computational science, and physical sciences. It is also suitable as a self-study reference guide for professionals.

This book offers a self-contained treatment of wavelets, which includes this theoretical pillar and it applications to the numerical treatment of partial differential equations. Its key features are:

1. Self-contained introduction to wavelet bases and related numerical algorithms, from the simplest examples to the most numerically useful general constructions.

2. Full treatment of the theoretical foundations that are crucial for the analysis

of wavelets and other related multiscale methods : function spaces, linear and nonlinear approximation, interpolation theory.

3. Applications of these concepts to the numerical treatment of partial differential equations : multilevel preconditioning, sparse approximations of differential and integral operators, adaptive discretization strategies.

Following a technical overview and Kotelnikov's article, the book includes a wide and coherent range of mathematical ideas essential for modern sampling techniques. These ideas involve wavelets and frames, complex and abstract harmonic analysis, the Fast Fourier Transform (FFT), and special functions and eigenfunction expansions. Some of the applications addressed are tomography and medical imaging.

Topics and features: • Relations between wavelet theory, the uncertainty principle, and sampling • Multidimensional non-uniform sampling theory and algorithms • The analysis of oscillatory behavior through sampling • Sampling techniques in deconvolution • The FFT for non-uniformly distributed data • Filter design and sampling • Sampling of noisy data for signal reconstruction • Finite dimensional models for oversampled filter banks • Sampling problems in MRI.

Engineers and mathematicians working in wavelets, signal processing, and harmonic analysis, as well as scientists and engineers working on applications as varied as medical imaging and synthetic aperture radar, will find the book to be a modern and authoritative guide to sampling theory.

The field of this book (Electrical Engineering/Computer Science) is currently booming, which is, of course, evident from the sales of the previous edition. Since the first edition came out there has been much development, especially as far as the applications. Thus, the second edition addresses new developments in applications-related chapters, especially in chapter 4 "Filterbrook Families: Design and Performance," which is greatly expanded.

Unified and coherent treatment of orthogonal transforms, subbands, and waveletsCoverage of emerging applications of orthogonal transforms in digital communications and multimediaDuality between analysis and synthesis filter banks for spectral decomposition and synthesis and analysis transmultiplexer structuresTime-frequency focus on orthogonal decomposition techniques with applications to FDMA, TDMA, and CDMAThrough expansive coverage and easy-to-follow explanations, A First Course in Wavelets with Fourier Analysis, Second Edition provides a self-contained mathematical treatment of Fourier analysis and wavelets, while uniquely presenting signal analysis applications and problems. Essential and fundamental ideas are presented in an effort to make the book accessible to a broad audience, and, in addition, their applications to signal processing are kept at an elementary level.

The book begins with an introduction to vector spaces, inner product spaces, and other preliminary topics in analysis. Subsequent chapters feature:

The development of a Fourier series, Fourier transform, and discrete Fourier analysis

Improved sections devoted to continuous wavelets and two-dimensional wavelets

The analysis of Haar, Shannon, and linear spline wavelets

The general theory of multi-resolution analysis

Updated MATLAB code and expanded applications to signal processing

The construction, smoothness, and computation of Daubechies' wavelets

Advanced topics such as wavelets in higher dimensions, decomposition and reconstruction, and wavelet transform

Applications to signal processing are provided throughout the book, most involving the filtering and compression of signals from audio or video. Some of these applications are presented first in the context of Fourier analysis and are later explored in the chapters on wavelets. New exercises introduce additional applications, and complete proofs accompany the discussion of each presented theory. Extensive appendices outline more advanced proofs and partial solutions to exercises as well as updated MATLAB routines that supplement the presented examples.

A First Course in Wavelets with Fourier Analysis, Second Edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level.