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Overview For over a decade now, wavelets have been and continue to be an evolving subject of intense interest. Their allure in signal processing is due to many factors, not the least of which is that they offer an intuitively satisfying view of signals as being composed of little pieces of wa'ues. Making this concept mathematically precise has resulted in a deep and sophisticated wavelet theory that has seemingly limitless applications. This book and its supplementary hands-on electronic: component are meant to appeal to both students and professionals. Mathematics and en­ gineering students at the undergraduate and graduate levels will benefit greatly from the introductory treatment of the subject. Professionals and advanced students will find the overcomplete approach to signal represen­ tation and processing of great value. In all cases the electronic component of the proposed work greatly enhances its appeal by providing interactive numerical illustrations. A main goal is to provide a bridge between the theory and practice of wavelet-based signal processing. Intended to give the reader a balanced look at the subject, this book emphasizes both theoretical and practical issues of wavelet processing. A great deal of exposition is given in the beginning chapters and is meant to give the reader a firm understanding of the basics of the discrete and continuous wavelet transforms and their relationship. Later chapters promote the idea that overcomplete systems of wavelets are a rich and largely unexplored area that have demonstrable benefits to offer in many applications.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
Although the theory of wavelet analysis is a relatively new and still evolving discipline, there is a deep and sophisticated body of work currently available. Much of this work, however, requires a fairly in-depth knowledge of several areas of advanced mathematics and hence limits its accessibility. It is a main objective of this work to strike a balance between accessibility and rnathernatical rigor that sacrifices as little as possible of both. To help achieve this goal, the dissemination of the material is provided by a hybrid combination of traditional (text) and nontraditional (Internet and electronic) media.
Anthony Teolis

2. Mathematical Preliminaries

Abstract
This chapter lays the basic mathematical foundations of the book, establishes notation, and defines various spaces and operators necessary for the development of the material. It is meant rnore as a reference and index of notation than a comprehensive mathematical introduction. Most of the basic notation and concepts that are used may be found in standard texts on real analysis ([Roy68], [Ben76]), operator theory ([GG80]), and discrete signal processing ([OS75], [Tre76]). The reader is assumed to have some familiarity with complex analysis and linear algebra.
Anthony Teolis

3. Signal Representation and Frames

Abstract
A pervasive and useful idea in mathematics is that functions (signals) may be decomposed into elementary atomic functions. Suppose that {φ n } is such a collection of building blocks or atoms from which functions may be constructed. Any such constructed function f has the form
$$ f(t) = \sum {{{c}_{n}}{{\phi }_{n}}(t)} $$
(3.1)
for some constants c n that may be chosen. By specifying both an underlying atomic set {φ n } and an associated scalar sequence {c n }, one may provide a description of the function f that has both practical and analytic value. This is true with the caveat that the atomic set is chosen with some care. In particular, the interpretation of the equality in Equation (3.1) is not straightforward for arbitrary atomic sets. In addition, there are deep and interesting convergence issues concerning the right-hand side of (3.1) when the atomic set has an infinite number of members. In a practical sense, it may be argued that choosing atomic sets that lead to fundamental analytical problems such as these are bad choices and should be avoided. Such problems may be circumvented by placing some modest requirements on the atomic set; namely, that it form a frame for a large enough space of interest.
Anthony Teolis

4. Continuous Wavelet and Gabor Transforms

Abstract
The continuous wavelet transform (CWT) as well as the continuous Gabor transform (CGT) (also known as the short-time Fourier transform) and their inverses are presented in this chapter. Both the CGT and the CWT take a one-dimensional time signal to a two-dimensional function of time and frequency. As such, they both seek to extract the time-frequency characteristics of the one-dimensional signal.
Anthony Teolis

5. Discrete Wavelet Transform

Abstract
In general, discrete wavelet transforms are generated by samplings (in the time—scale plane) of a corresponding continuous wavelet transform. Such a discrete wavelet transform is specified by the choice of items:
1.
a time—scale sampling set (a countable set of points), and
 
2.
an analyzing wavelet.
 
Anthony Teolis

6. Overcomplete Wavelet Transform

Abstract
This chapter deals with discrete wavelet transforms that are formed from the general samples of a continuous wavelet transform. Conceptually, there are few constraints on the spacing between sample points throughout the time—scale plane; however, computational consideration is restricted here to an interesting subclass of sampling sets that allow for the fast computation of the forward and inverse transforms.1 In this case, the freedom of choice of analyzing wavelets remains nearly unrestricted by the implementation. In general, the resulting transform is one that has underlying atoms that are nonorthogonal, and even more important, may be overcomplete. Consequently, such atomic functions have associated redundant (inner product) representations. For these reasons, the term overcomplete wavelet transform (OCWT) is used to describe the transform.
Anthony Teolis

7. Wavelet Signal Processing

Abstract
A basic motivation behind transform methods is the idea that some sorts of processing are better (or perhaps only possibly) achieved in the transform domain rather than in the original signal domain, In this sense, the utility of a transform is measured by its ability to facilitate desired signal processing tasks in the transform domain via algorithms that are digitally tractable, computationally efficient, concise, and noise robust. The efficacy of general wavelet transforms comes from the fact that wavelet domain algorithms exhibit all of these benefits when dealing with signals that are characterized by their time—frequency behavior. This chapter explores applications of overcomplete wavelet transforms in problems of data compression, noise suppression, digital communication, and signal identification.
Anthony Teolis

8. Object-Oriented Wavelet Analysis with MATLAB 5

Abstract
All of the numerical processing and examples presented in this book have been computed using a suite of object-oriented tools developed in MATLAB 5. This set of tools has been called collectively the Wavelet Signal Processing Workstation (WSPW) and is described generally in Section 8.1. Up-to-date code and detailed documentation is available online at www.​birkhauser.​com/​book/​ISBN/​0-8176-3909-8.
Anthony Teolis

Backmatter

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