Book Description

Wavelets: Theory, Algorithms, and Applications is the fifth volume in the highly respected series, WAVELET ANALYSIS AND ITS APPLICATIONS. This volume shows why wavelet analysis has become a tool of choice infields ranging from image compression, to signal detection and analysis in electrical engineering and geophysics, to analysis of turbulent or intermittent processes. The 28 papers comprising this volume are organized into seven subject areas: multiresolution analysis, wavelet transforms, tools for time-frequency analysis, wavelets and fractals, numerical methods and algorithms, and applications. More than 135 figures supplement the text.

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 ... Read more

Reviews (1)

High quality math papers from a conference, a bit outdated
This book consists of some well-written papers by some leading research scientists in the field of wavelets. It can be thought of as a very high quality proceedings book. The title is a bit misleading. The book is mostly about theory and algorithms. A high level of mathematical background is required to read the book. Some books which can be considered competitors to this one are the 2 older and even more outdated books edited by: (1) Combes et al. and (2) Ruskai et al. AND a book by Benedetto and Frazier, which was so successful that it was translated into Japanese (alhtough some of the weaker papers were omitted in the Japanese version). This Chui book and the Benedetto-Frazier book contain extensive references, though many are now out-of-date. The applications section is also bit out-of-date. Mathematics can live on as is, though tastes and fashions vary over time.

Chui is a very successful organizer of conferences and has written and edited many books, either alone or with others. This book is well put together and suffers from fewer faults as Chui's earlier book (Vol 2. of this series).

The theoretical papers are more interesting and cover topics not found in Daubechies' famous book: Ten Lectures on Wavelets. Some of the applications address genuine laboratory problems. Image and acoustical digital signal processing are barely touched upon, but it doesn't matter since the publication of Mallat's A Wavelet Tour of Signal Processing obviates the need. This book does not contain problems to be assigned for homework, but it can still be used in a course -- perhaps a graduate seminar or a special topics course. Be warned that the book is a bit pricey to use as a supplementary text in a course. ... Read more

Book Description

This book presents a mathematical introduction to the theory of orthogonal wavelets and their uses in analyzing functions and function spaces, both in one and in several variables. Starting with a detailed and self-contained discussion of the general construction of one dimensional wavelets from multiresolution analysis, the book presents in detail the most important wavelets: spline wavelets, Meyer's wavelets and wavelets with compact support. It then moves to the corresponding multivariable theory and gives genuine multivariable examples. The author discusses wavelet decompositions in Lp spaces, Hardy spaces and Besov spaces and provides wavelet characterizations of those spaces. Also included are periodic wavelets or wavelets not associated with a multiresolution analysis. This will be an invaluable book for those wishing to learn about the mathematical foundations of wavelets. ... Read more

Reviews (1)

The mathematics of wavelets
The literature on wavelets has been growing at an increasing pace, as testified by 129 titles listed in Amazon.com, the great majority which were first published in the last five years. Many of these books are either (i)popular accounts for the general reader, (ii) user's guides for the practitioner or (iii) advanced mathematical treatises for experts. Thus we find very few textbooks where the serious student can obtain an honest introduction to the subject from a recognized authority. This is even more surprising, given that the origins of the subject can be traced back to the mathematics literature in the early part of the 20th century under the names of Alfred Haar, Phillip Franklin and Sir Edmund Whittaker---and later to further work by Alberto Calderon in the 1960's. With the flurry of activity in the 1980's in the French school, the subject came alive and is now a permanent part of the literature in mathematical analysis, growing by leaps and bounds.

The book of Wojtaszczyk is a welcome addition to the literature on wavelets. Without the benefit of glossy pictures or computer output, the author has been extremely successful in presenting a clear and correct approach to the subject for readers who have a minimal acquaintance with mathematical analysis at the level of integration theory and elementary Fourier analysis. Going beyond other introductory works, the book contains a systematic sets of exercises at the end of each chapter, as a sort of "reality check" for the student, to test his/her understanding of the theory.

The first four chapters of the book deal with wavelets in one dimensions, continued in chapter 9. Following the examples of the classical Haar system and the Stromberg spline wavelet, done in great detail, we are introduced to MRA systems as a systematic method for generating wavelet bases of the space of square-integrable functions on the line. An MRA (multi-reolution analysis) is defined by a "scaling function", which satisfies an orthonormality condition, a scaling condition and a smoothness condition in the Fourier domain. Any such function generates an MRA, which in turn generates a wavelet basis. In particular one can generate in this framework the smooth wavelets of Meyer by this method, followed by the compactly supported wavelets of Daubechies. Wavelet theory is first formulated for square integrable functions, but can be extended to other Banach spaces, where it often provides an "unconditional basis", which is not true of the classical Fourier series.

Chapters 5-8 deal with some of the multi-dimensional theory, where several wavelets are necesary to generate the MRA, suitably defined. Chapter 6 contains a self-contained treatment of some important topics in analysis: the Hardy-Littlewood maximal inequality, the Banach spaces H^1 and BMO, the John-Nirenberg inequalty. These are used to develop the property of unconditional basis for wavelets in the spaces L^p and H^1.

The author suceceds admirably in carrying out his stated goal beyond any reasonable expectation: in the preface we read "This is a purely mathematical book, although I constantly try to make the calculations as explicit as possible and I concentrate on theoretical questions that should have relevance in appplications, but regrettably discuss no real applications". With the flurry of literature on the uses of wavelets, these applications are best left to other works.

One can expect that this book will be in print for many years to come. ... Read more

Book Description

Mathematical Properties of Sequences and Other Combinatorial Structures is an excellent reference for both professional and academic researchers working in telecommunications, cryptography, signal processing, discrete mathematics, and information theory.

The work represents a collection of contributions from leading experts in the field. Contributors have individually and collectively dedicated their work as a tribute to the outstanding work of Solomon W. Golomb.

Mathematical Properties of Sequences and Other Combinatorial Structures covers the latest advances in the widely used and rapidly developing field of information and communication technology. ... Read more

Book Description

The finite-infinite interplay is central in the human thinking, from ancient philosophers and mathematicians (Zenon, Pythagoras), to modern mathematics (Cantor, Hilbert) and computer science (Turing, Goedel). Recent developments in mathematics and computer science suggest radically new answers to classical questions such as: Does infinity exist? Where does infinity come from? How can we reconcile the finiteness of the human brain with the infinity of ideas it produces? Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics. ... Read more

Book Description

This volume provides a comprehensive, up-to-date survey of inequalities that involve a relationship between a function and its derivatives or integrals. The book is divided into 18 chapters, some of which are devoted to specific inequalities such as those of Kolmogorov-Landau, Wirtinger, Hardy, Carlson, Hilbert, Caplygin, Lyapunov, Gronwell and others. Over 800 references to the literature are cited; proofs are given when these provide insight into the general methods involved; and applications, especially to the theory of differential equations, are mentioned when appropriate. This volume will interest all those whose work involves differential and integral equations. It can also be recommended as a supplementary text. ... Read more

Book Description

"Wavelets Made Easy", written at the level of a first course in calculus and linear algebra, offers a lucid and concise explanation of mathematical wavelets. Evolving from ten years of classroom use, its accessible presentation is designed for undergraduates in a variety of disciplines -- computer science, engineering, mathematics, mathematical sciences -- as well as for practicing professionals in these areas.

This unique text starts the first chapter with a description of the key features and applications of wavelets, focusing on Haar's wavelets but using only high school mathematics. The next two chapters introduce one-, two-, and three-dimensional wavelets, with only the occasional use of matrix algebra.

The second part of this book provides the foundations of least squares approximation, the discrete Fourier transform, and Fourier series. The third part explains the Fourier transform and then demonstrates how to apply basic Fourier analysis to designing and analyzing mathematical wavelets. Particular attention is paid to Daubechies' wavelets.

Numerous exercises, a bibliography, and a comprehensive index combine to make this book an excellent text for the classroom as well as a valuable resource for self study.

Key features of this work:

Written at an undergraduate level, prerequisites being only a first course in calculus and linear algebra Early chapters focus on Haar wavelets; only high school math required Gradual progression to one-, two-, and three-dimensional wavelets (chapters 4,5,6) -- good background in Fourier analysis; provides the foundations of least squares approximation, the discrete Fourier transform, and Fourier series Demonstrates how to apply basic Fourier analysis to designing and analyzing mathematical wavelets; particular attention is paid to Daubechies' wavelets Numerous exercises, a bibliography, and comprehensive index An excellent text for the classroom as well as a valuable resource for self study ... Read more

Reviews (4)

Excellent.
The book is excellent in the sense that it has the right mixture of both theoretic and practical aspects. The portions on Fourier transforms also are apt for a newcomer to understand the theory. Of course, the first impression formed by seeing the title is that the matter inside would be plain english without any mathematics. However, the mathematics in the book has been kept elementary and in places where that wasn't possible, a thorough introduction has been provided.

The applications are explicitly mentioned and algorithms are written down clearly, instead of simply providing the theory and leaving the implementation for the student to figure out for himself. For the less mathematically oriented, the exercises at the end of each chapter will provide a good intuition to what will be encountered in the consequent sections. Once this book is read, other books that concentrate on specific aspects of wavelets will appear to be more understandable.

The complex cannot be made easy, but it can be made clear
When I first saw this book, which is heavy on equations, I thought "wow, if this is wavelets made easy, I'd hate to see the hard version". And indeed this book is not light reading. But it is clear and there are plenty of examples. This book is far and away the best book on wavelets that I've found.
Wavelets are a complex topic. It would be unfair to think that you can pick up a book like this an read it like a popular science book. Reading a book like this takes concentration and effort. Implementing wavelet software is even more work, since some of the "in-place" algorithms are difficult. "Wavelets Made Easy" is a great foundation for anyone who wants to apply Haar wavelets for data analysis and filtering.
...

Nievergelt's coverage of Daubechies wavelets is not as
good as his coverage of Haar. He clearly explains the matrix
(linear algebra) view of Daubechies wavelets. The
coverage of edge effect issues is also good. However, as the
chapter proceeds, it fell apart, at least for me. There
appear to be errors in some of the algorithms. For example,
Nievergelt uses an average of the input time series over the
Daubechies wavelet. As far as I can see, this average cannot
be inverted to yield the original time series. Nievergelt's
Daubechies algorithm also does not seem to match algorithms
like the one published in Numerical Recipies.

So in summary, I'd say that this is a book that started out
well in the first few chapters but became less useful as
I worked through it.

wavelet made easy
This is an excellent book. Wavelet IS a quite advanced concept in Math. The author did a good job to minimize the math backgrond needed to understand wavelet. DON'T BLAM author because you have way too little math training.

Disappointing! Not at all Wavelets made "easy".
This book was quite a disappointment. It was not, as titled, Wavelets Made Easy. Despite the author's claims to the contrary, the book was not easy to read, largely due to the strongly professional mathematician's approach with unnecessarily obscure and involved notation. Later in the book the author adopted the mathematical formalism of proposition, proof, and lemma which is out of place in an introductory text aimed at non-mathematicians. Though the book treats a variety of applications of wavelets, little attention is paid to the background for these applications. Again, mathematical formalism takes precedence. Not recommended except perhaps for mathematicians, and for these, better texts are available. ... Read more

Book Description

Almost a century ago, harmonic analysis entered a (still continuing) Golden Age, with the emergence of many great masters throughout Europe. They created a wealth of profound analytic methods, to be successfully exploited and further developed by succeeding generations. This flourishing of harmonic analysis is today as lively as ever, as the papers presented here demonstrate. In addition to its own ongoing internal development and its basic role in other areas of mathematics, physics and chemistry, financial analysis, medicine, and biological signal processing, harmonic analysis has made fundamental contributions to essentially all twentieth century technology-based human endeavours, including telephone, radio, television, radar, sonar, satellite communications, medical imaging, the Internet, and multimedia. This ubiquitous nature of the subject is amply illustrated.The book not only promotes the infusion of new mathematical tools into applied harmonic analysis, but also to fuel the development of applied mathematics by providing opportunities for young engineers, mathematicians and other scientists to learn more about problem areas in today's technology that might benefit from new mathematical insights. ... Read more

Book Description

Wavelets and wavelet packets provide a theory analogous to Fourier analysis and tools analogous to coherent state methods. Among their numerous applications, wavelets have been used to data compression in both image and sound processing. They are intimately related to splines, and wavelet applications in spline theory are significant. Wavelets have become a tool in analyzing fractals and iterative schemes associated with dynamical systems. Signal processing methods such as quadrature mirror filters go hand in hand with wavelet techniques in studying a host of communcations problems. The profound issues of classical turbulence are being studied using wavelet packets. Both wavelet packet software and wavelet transform microchips are now available. There are also applications of wavelet theory in theoretical physics, oil exploration, irregular sampling, and singular integral operators.Many of the world's experts in the field of wavelets were principal speakers at the ASI, and their papers appear in this volume. Furthermore, these renowned scientists addressed their talks to an audience which consisted of a broad spectrum of pure and applied mathematicians, as well as a diverse group of engineers and scientists. Thus, the reader has the opportunity to both learn or reinforce the fundamental concepts from the individuals who have created and developed the blossoming field of wavelets, and to see them discuss in accessible terms their profound contributions and ideas for future research. ... Read more

Reviews (1)

very readable
A clear and very readable presentation of key wavelet ideas, and some modern and substantial real life applications. Multiresolutions, noise reduction in data, the conjugate gradient method via wavelets, text-to-speech, and turbulence, are some of the topics. Each chapter can be read by beginners, and yet the reader is quickly led to the heart of the matter, progressing through each chapter. Done by different expert authors, the chapters retain some independence, and yet there is a sense of unity to the book. ... Read more

Book Description

Wavelets have recently been enjoying a period of popularity and rapid growth, and the influence of wavelet methods now extends well beyondmathematics into a number of practical fields, including statistics. The theory of hypergroups can be traced back to the turn of the century, andfollowing its formalization in the early 1970s, the area has now reached maturity.
Hypergroups provide a very general and flexible context in which many of the classical techniques of harmonic analysis can be fruitfully employed.It is, therefore, natural to seek to exploit the newer techniques of wavelet analysis in this area. This text addresses itself to this challenge in somedepth, providing a thorough and authoritative account of wavelet methods applied to hypergroups. ... Read more

Book Description

Sampling, wavelets, and tomography are three active areas of contemporary mathematics sharing common roots that lie at the heart of harmonic and Fourier analysis.The advent of new techniques in mathematical analysis has strengthened their interdependence and led to some new and interesting results in the field.

This state-of-the-art book not only presents new results in these research areas, but it also demonstrates the role of sampling in both wavelet theory and tomography.Specific topics covered include:

* Robustness of Regular Sampling in Sobolev Algebras * Irregular and Semi-Irregular Weyl-Heisenberg Frames * Adaptive Irregular Sampling in Meshfree Flow Simulation* Sampling Theorems for Non-Bandlimited Signals* Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets * Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces * Sampling Theory and Parallel-Beam Tomography* Thin-Plate Spline Interpolation in Medical Imaging * Filtered Back-Projection Algorithms for Spiral Cone Computed Tomography

Aimed at mathematicians, scientists, and engineers working in signal and image processing and medical imaging, the work is designed to be accessible to an audience with diverse mathematical backgrounds.Although the volume reflects the contributions of renowned mathematicians and engineers, each chapter has an expository introduction written for the non-specialist.One of the key features of the book is an introductory chapter stressing the interdependence of the three main areas covered.A comprehensive index completes the work.

Contributors:

J.J. Benedetto N.K. Bose P.G. Casazza Y.C. Eldar H.G. Feichtinger A. Faridani A. Iske S. Jaffard A. Katsevich S. Lertrattanapanich G. Lauritsch B. Mair M. Papadakis P.P. Vaidyanathan T. Werther D.C. Wilson A.I. Zayed ... Read more

Book Description

In 1909 Alfred Haar introduced into analysis a remarkable system which bears his name. The Haar system is a complete orthonormal system on [0,1] and the Fourier--Haar series for arbitrary continuous function converges uniformly to this function.This volume is devoted to the investigation of the Haar system from the operator theory point of view. The main subjects treated are: classical results on unconditional convergence of the Haar series in modern presentation; Fourier--Haar coefficients; reproducibility; martingales; monotone bases in rearrangement invariant spaces; rearrangements and multipliers with respect to the Haar system; subspaces generated by subsequences of the Haar system; the criterion of equivalence of the Haar and Franklin systems.Audience: This book will be of interest to graduate students and researchers whose work involves functional analysis and operator theory. ... Read more

Book Description

This is the first complete treatment of the metrical theory of regular continued fractions and related representations of real numbers. The authors have attempted to give the best possible results now known, with proofs that are the simplest and most direct. The book is essentially based on recent work of the authors. In order to unify and generalize the results obtained so far, new concepts have been introduced, e.g., an infinite order chain representation of the continued fraction expansion of irrationals, the conditional measures associated with, and the extended random variables corresponding to that representation. Also, such procedures as singularization and insertion allow to obtain most of the continued fraction expansions related to the regular continued fraction expansion. The authors present and prove with full details for the first time in book form, the most recent developments in solving the celebrated 1812 Gauss' problem which originated the metrical theory of continued fractions. At the same time, they study exhaustively the Perron-Frobenius operator, which is of basic importance in this theory, on various Banach spaces including that of functions of bounded variation on the unit interval. The book is of interest to research workers and advanced Ph.D. students in probability theory, stochastic processes and number theory. ... Read more

Book Description

Topics include periodic functions, Fourier series, vector space concepts, Fourier transforms, generalized functions, special functions, and applications. ... Read more

Book Description

First appearing in the late 1980s, wavelets have developed rapidly, and their applications now reach from signal processing to numerical analysis, operator theory, and beyond. Among the generalizations of wavelets that have emerged, multiwavelets stand out because of their potential for short support and for their capability of being both symmetric and orthogonal. Some existing wavelet books have a short section covering multiwavelets, but to date, none offer a comprehensive treatment. Pulling together information scattered throughout the literature of mathematics and engineering, Wavelets and Multiwavelets offers a full exposition of wavelet and multiwavelet theory. Maintaining the delicate balance between mathematical rigor and readability, the author presents the theory of classical wavelets and general m-band multiwavelets in parallel. This provides a more coherent approach and yields alternative proofs and new insights even for standard wavelets. MATLAB routines that allow you to implement and experiment with multiwavelet algorithms are available from the CRC Press Web site. ... Read more