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Wavelet Transforms and Their Applications

  • Textbook
  • © 2002

Overview

Authors:
  1. Lokenath Debnath

    1. Department of Mathematics, University of Texas—Pan American, Edinburg, USA

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xv
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  2. Brief Historical Introduction

    • Lokenath Debnath
    Pages 1-21

  3. Hilbert Spaces and Orthonormal Systems

    • Lokenath Debnath
    Pages 23-142

  4. Fourier Transforms and Their Applications

    • Lokenath Debnath
    Pages 143-256

  5. The Gabor Transform and Time-Frequency Signal Analysis

    • Lokenath Debnath
    Pages 257-306

  6. The Wigner-Ville Distribution and Time-Frequency Signal Analysis

    • Lokenath Debnath
    Pages 307-360

  7. The Wavelet Transform and Its Basic Properties

    • Lokenath Debnath
    Pages 361-402

  8. Multiresolution Analysis and Construction of Wavelets

    • Lokenath Debnath
    Pages 403-474

  9. Newland’s Harmonic Wavelets

    • Lokenath Debnath
    Pages 475-489

  10. Wavelet Transform Analysis of Turbulence

    • Lokenath Debnath
    Pages 491-524

  11. Back Matter

    Pages 525-565
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Keywords

About this book

Overview Historically, the concept of "ondelettes" or "wavelets" originated from the study of time-frequency signal analysis, wave propagation, and sampling theory. One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean MorIet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary signals. However, this new concept can be viewed as the synthesis of various ideas originating from different disciplines including mathematics (Calder6n-Zygmund operators and Littlewood-Paley theory), physics (coherent states in quantum mechanics and the renormalization group), and engineering (quadratic mirror filters, sideband coding in signal processing, and pyramidal algorithms in image processing). Wavelet analysis is an exciting new method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, image processing, pattern recognition, computer graphics, the detection of aircraft and submarines, and improvement in CAT scans and other medical image technology. Wavelets allow complex information such as music, speech, images, and patterns to be decomposed into elementary forms, called the fundamental building blocks, at different positions and scales and subsequently reconstructed with high precision.

Reviews

"It contains a wealth of information that should make it useful in signal processing and perhaps some other areas of engineering . . . I like the book as a possible text for a beginning graduate course in, say, mathematical methods in engineering. It covers a number of topics that are quite useful but are rarely covered in mainstream mathematics courses . . . a lot of the proofs are short and computational, which is necessary in such a book that covers a large number of topics . . . it would serve as a good text, provided that the aim of the course is to present a variety of transforms useful in signal processing, as well as the wavelet transforms."

—Mathematical Reviews

"The last two decades have produced tremendous developments in the mathematical theory of wavelets and their great variety of applications. Since wavelet analysis is a relatively new subject, this monograph is intended to be self-contained. The book is designed as a modern and authoritative guide to wavelets, wavelet transform, time-frequency signal analysis and related topics.

It is known that some research workers look upon wavelets as a new basis for representing functions, others consider them as a technique for time-frequency analysis and some others think of them as a new mathematical subject. All these approaches are gathered in this book, which presents an accessible, introductory survey of new wavelet analysis tools and the way they can be applied to fundamental analysis problems. We point out the clear, intuitive style of [the] presentation, and the numerous examples demonstrated through[out] the book illustrate how methods work in a step-by-step manner.

This way, the book becomes ideal for a broad audience including advanced undergraduate students, graduate[s] and professionals in signal processing. Also, the book provides the reader with a thorough mathematical background, and the wide variety of applications coverthe interdisciplinary collaborative research in applied mathematics."

—Revue D’Analyse Numérique et de Théorie de L’Approximation

Authors and Affiliations

  • Department of Mathematics, University of Texas—Pan American, Edinburg, USA

    Lokenath Debnath

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