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2002 | Buch

Brief Historical Introduction Kapitel lesen Erstes Kapitel lesen

Wavelet Transforms and Their Applications

verfasst von: Lokenath Debnath

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Brief Historical Introduction
Abstract
Historically, Joseph Fourier (1770–1830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonardo Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions. Thus, the Fourier series of a function f(x) defined on the interval (-ℓ, ℓ) is given by
$$ f\left( x \right) = \sum\limits_{n = - \infty }^\infty {c_n } {\rm{exp}}\left( {\frac{{in\pi x}}{\ell }} \right), $$
(1.1.1)
where the Fourier coefficients are
$$ c_n = \frac{{\rm{1}}}{{{\rm{2}}\ell }}\int\limits_{ - \ell }^\ell {f\left( t \right)} {\rm{exp}}\left( { - \frac{{in\pi t}}{\ell }} \right)dt. $$
(1.1.2)
Lokenath Debnath
Chapter 2. Hilbert Spaces and Orthonormal Systems
Abstract
Historically, the theory of Hilbert spaces originated from David Hilbert’s (1862–1943) work on quadratic forms in infinitely many variables with their applications to integral equations. During the period of 1904–1910, Hilbert published a series of six papers, subsequently collected in his classic book Grundzüge einer allemeinen Theorie der linearen lntegralgleichungen published in 1912. It contained many general ideas including Hilbert spaces (ℓ2 and L2), the compact operators, and orthogonality, and had a tremendous influence on mathematical analysis and its applications. After many years, John von Neumann (1903–1957) first formulated an axiomatic approach to Hilbert space and developed the modern theory of operators on Hilbert spaces. His remarkable contribution to this area has provided the mathematical foundation of quantum mechanics. Von Neumann’s work has also provided an almost definite physical interpretation of quantum mechanics in terms of abstract relations in an infinite dimensional Hilbert space. It was shown that observables of a physical system can be represented by linear symmetric operators in a Hilbert space, and the eigenvalues and eigenfunctions of the particular operator that represents energy are energy levels of an electron in an atom and corresponding stationary states of the system. The differences in two eigenvalues represent the frequencies of the emitted quantum of light and thus define the radiation spectrum of the substance.
Lokenath Debnath
Chapter 3. Fourier Transforms and Their Applications
Abstract
This chapter deals with Fourier transforms in L1(ℝ) and in L2 (ℝ) and their basic properties. Special attention is given to the convolution theorem and summability kernels including Cesáro, Fejér, and Gaussian kernels. Several important results including the approximate identity theorem, general Parseval’s relation, and Plancherel theorem are proved. This is followed by the Poisson summation formula, Gibbs’ phenomenon, the Shannon sampling theorem, and Heisenberg’s uncertainty principle. Many examples of applications of the Fourier transforms to mathematical statistics, signal processing, ordinary differential equations, partial differential equations, and integral equations are discussed. Included are some examples of applications of multiple Fourier transforms to important partial differential equations and Green’s functions.
Lokenath Debnath
Chapter 4. The Gabor Transform and Time-Frequency Signal Analysis
Abstract
Signals are, in general, nonstationary. A complete representation of nonstationary signals requires frequency analysis that is local in time, resulting in the time-frequency analysis of signals. The Fourier transform analysis has long been recognized as the great tool for the study of stationary signals and processes where the properties are statistically invariant over time. However, it cannot be used for the frequency analysis thai is local in time. In recent years, several useful methods have been developed for the time-frequency signal analysis. They include the Gabor transform, the Zak transform, and the wavelet transform.
Lokenath Debnath
Chapter 5. The Wigner-Ville Distribution and Time-Frequency Signal Analysis
Abstract
Although time-frequency analysis of signals had its origin almost fifty years ago, there has been major development of the time-frequency distributions approach in the last two decades. The basic idea of the method is to develop a joint function of time and frequency, known as a time-frequency distribution, that can describe the energy density of a signal simultaneously in both time and frequency. In principle, the time-frequency distributions characterize phenomena in a two-dimensional time-frequency plane. Basically, there are two kinds of time-frequency representations. One is the quadratic method covering the time-frequency distributions, and the other is the linear approach including the Gabor transform, the Zak transform, and the wavelet transform analysis. So, the time-frequency signal analysis deals with time-frequency representations of signals and with problems related to their definition, estimation and interpretation, and it has evolved into a widely recognized applied discipline of signal processing. From theoretical and application points of view, the Wigner-Ville distribution (WVD) or the Wigner-Ville transform (WVT) plays a major role in the time-frequency signal analysis for the following reasons. First, it provides a high-resolution representation in both time and frequency for non-stationary signals. Second, it has the special properties of satisfying the time and frequency marginals in terms of the instantaneous power in time and energy spectrum in frequency and the total energy of the signal in the time and frequency plane.
Lokenath Debnath
Chapter 6. The Wavelet Transform and Its Basic Properties
Abstract
Morlet et al. (1982a,b) modified the Gabor wavelets to study the layering of sediments in a geophysical problem of oil exploration. He recognized certain difficulties of the Gabor wavelets in the sense that the Gabor analyzing function g t,ω (τ) = g(τ-t)e iωτ oscillates more rapidly as the frequency ω tends to infinity. This leads to significant numerical instability in the computation of the coefficients (f,g ω,t ). On the other hand, g t,ω oscillates very slowly at low frequencies. These difficulties led to a problem of finding a suitable reconstruction formula. In order to resolve these difficulties, Morlet first made an attempt to use analytic signals f(t) = a(t) exp{iϕ(t)} and then introduced the wavelet y defined by its Fourier transform
$$ \hat \psi \left( \omega \right) = \sqrt {{\rm{2}}\pi } \omega ^{\rm{2}} {\rm{exp}}\left( { - \frac{{\rm{1}}}{{\rm{2}}}\omega ^{\rm{2}} } \right), \omega > {\rm{0}}{\rm{.}} $$
(6.1.1)
Lokenath Debnath
Chapter 7. Multiresolution Analysis and Construction of Wavelets
Abstract
The concept of multiresolution is intuitively related to the study of signals or images at different levels of resolution almost like a pyramid. The resolution of a signal is a qualitative description associated with its frequency content. For a low-pass signal, the lower its frequency content (the narrower the bandwidth), the coarser is its resolution. In signal processing, a low-pass and subsampled version of a signal is usually a good coarse approximation for many real world signals. Multiresolution is especially evident in image processing and computer vision, where coarse versions of an image are often used as a first approximation in computational algorithms.
Lokenath Debnath
Chapter 8. Newland’s Harmonic Wavelets
Abstract
So far, all wavelets have been constructed from dilation equations with real coefficients. However, many wavelets cannot always be expressed in functional form. As the number of coefficients in the dilation equation increases, wavelets get increasingly longer and the Fourier transforms of wavelets become more tightly confined to an octave band of frequencies. It turns out that the spectrum of a wavelet with n coefficients becomes more boxlike as n increases. This fact led Newland (1993a,b) to introduce a new harmonic wavelet (x)>(x) whose spectrum is exactly like a box, so that the magnitude of its Fourier transform \( \hat \psi \left( \omega \right) \) (ω) is zero except for an octave band of frequencies. Furthermore, he generalized the concept of the harmonic wavelet to describe a family of mixed wavelets with the simple mathematical structure. It is also shown that this family provides a complete set of orthonormal basis functions for signal analysis.
Lokenath Debnath
Chapter 9. Wavelet Transform Analysis of Turbulence
Abstract
Considerable progress has been made over the last three decades in our understanding of turbulence through new developments of theory, experiment, and computation. More and more evidence has been accumulated for the physical description of turbulent motions in both two and three dimensions. Consequently, turbulence is now characterized by a remarkable degree of order even though turbulence is usually defined as disordered fluid flows. In spite of tremendous progress, there are still a number of open questions and unsolved problems. These include coherent structures and intermittency effects, singularities of the Navier-Stokes equations, non-Gaussian statistics of turbulent flows, perturbations to the small scale produced by nonisotropic, non-Gaussian, and inhomogeneous large-scale motions, and measurements and computations of small-scale turbulence. No complete theory is yet available for the problem of how the eddy structure of turbulence evolves both under the action of mean distortion and even during the mutual random interaction of eddies of different sizes or scales.
Lokenath Debnath
Backmatter
Metadaten
Titel
Wavelet Transforms and Their Applications
verfasst von
Lokenath Debnath
Copyright-Jahr
2002
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0097-0
Print ISBN
978-1-4612-6610-5
DOI
https://doi.org/10.1007/978-1-4612-0097-0