Wavelet Transforms And Their Applications

This textbook is an introduction to wavelet transforms and accessible to a larger audience with diverse backgrounds and interests in mathematics, science, and engineering. Emphasis is placed on the logical development of fundamental ideas and systematic treatment of wavelet analysis and its applications to a wide variety of problems as encountered in various interdisciplinary areas. Topics and Features: * This second edition heavily reworks the chapters on Extensions of Multiresolution Analysis and Newlands’s Harmonic Wavelets and introduces a new chapter containing new applications of wavelet transforms * Uses knowledge of Fourier transforms, some elementary ideas of Hilbert spaces, and orthonormal systems to develop the theory and applications of wavelet analysis * Offers detailed and clear explanations of every concept and method, accompanied by carefully selected worked examples, with special emphasis given to those topics in which students typically experience difficulty * Includes carefully chosen end-of-chapter exercises directly associated with applications or formulated in terms of the mathematical, physical, and engineering context and provides answers to selected exercises for additional help Mathematicians, physicists, computer engineers, and electrical and mechanical engineers will find Wavelet Transforms and Their Applications an exceptionally complete and accessible text and reference. It is also suitable as a self-study or reference guide for practitioners and professionals.

It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet. Hence, if they actually meet, that is, if they are at least so much related to each other that a real interaction can take place, then one may hope that new and interesting developments may follow. Werner Heisenberg This volume contains papers presented at the August 1992 NATO Advanced Study Institute on Wavelets and Their Applications. The conference was held at the beautiful Il Ciocco resort near Lucca, in the glorious Tuscany region of northern Italy. Once again we gathered at this idyllic spot to explore and extend the reciprocity between mathematics and engineering. The dynamic interaction between world-renowned scientists from the usu ally disparate communities of pure mathematicians and applied scientists, which occurred at our 1989 and 1991 ASI's, continued at this meeting. Wavelet theory and technology is in an important growth stage at which theoretical and practical results are being compared with existing methods. There have been spectacular wavelet successes and sobering comparisons with traditional ideas-but still there is a wide expanse of scientific problems to explore. Since these problems lie at the forefront of both pure mathematics and applied science, our NATO ASI was especially pertinent at this time.

An "applications first" approach to discrete wavelettransformations Discrete Wavelet Transformations provides readers with a broadelementary introduction to discrete wavelet transformations andtheir applications. With extensive graphical displays, thisself-contained book integrates concepts from calculus and linearalgebra into the construction of wavelet transformations and theirvarious applications, including data compression, edge detection inimages, and signal and image denoising. The book begins with a cursory look at wavelet transformationdevelopment and illustrates its allure in digital signal and imageapplications. Next, a chapter on digital image basics, quantitativeand qualitative measures, and Huffman coding equips readers withthe tools necessary to develop a comprehensive understanding of theapplications. Subsequent chapters discuss the Fourier series,convolution, and filtering, as well as the Haar wavelet transformto introduce image compression and image edge detection. Thedevelopment of Daubechies filtersis presented in addition tocoverage of wavelet shrinkage in the area of image and signaldenoising. The book concludes with the construction of biorthogonalfilters and also describes their incorporation in the JPEG2000image compression standard. The author's "applications first" approach promotes a hands-ontreatment of wavelet transforma-tion construction, and over 400exercises are presented in a multi-part format that guide readersthrough the solution to each problem. Over sixty computer labs andsoftware development projects provide opportunities for readers towrite modules and experiment with the ideas discussed throughoutthe text. The author's software package, DiscreteWavelets, is usedto perform various imaging and audio tasks, compute wavelettransformations and inverses, and visualize the output of thecomputations. Supplementary material is also available via thebook's related Web site, which includes an audio and videorepository, final project modules, and softwarefor reproducingexamples from the book. All software, including theDiscreteWavelets package, is available for use withMathematica®, MATLAB®, and Maple. Discrete Wavelet Transformations strongly reinforces the use ofmathematics in digital data applications, sharpens programmingskills, and provides a foundation for further study of moreadvanced topics, such as real analysis. This book is ideal forcourses on discrete wavelet transforms and their applications atthe undergraduate level and also serves as an excellent referencefor mathematicians, engineers, and scientists who wish to learnabout discrete wavelet transforms at an elementary level.

This book explores the use of wavelet transforms in signal processing, including image, finance, and communication systems. It covers five contemporary applications, including the interaction between inertial sensors and wavelet filtering techniques, geophysical prospecting, volatility patterns in asset returns, computerized tomography (CT), and fault detection techniques. The book provides a foundation for further exploration, focusing on wavelet transformations' basic principles, their application in geophysical prospecting, and their use in identifying volatility patterns in asset returns. The book is intended for students, researchers, and professionals interested in understanding wavelet transforms and their practical implementations.

The use of the wavelet transform to analyze the behaviour of the complex systems from various fields started to be widely recognized and applied successfully during the last few decades. In this book some advances in wavelet theory and their applications in engineering, physics and technology are presented. The applications were carefully selected and grouped in five main sections - Signal Processing, Electrical Systems, Fault Diagnosis and Monitoring, Image Processing and Applications in Engineering. One of the key features of this book is that the wavelet concepts have been described from a point of view that is familiar to researchers from various branches of science and engineering. The content of the book is accessible to a large number of readers.

The continuous wavelet transform has deep mathematical roots in the work of Alberto P. Calderon. His seminal paper on complex method of interpolation and intermediate spaces provided the main tool for describing function spaces and their approximation properties. The Calderon identities allow one to give integral representations of many natural operators by using simple pieces of such operators, which are more suited for analysis. These pieces, which are essentially spectral projections, can be chosen in clever ways and have proved to be of tremendous utility in various problems of numerical analysis, multidimensional signal processing, video data compression, and reconstruction of high resolution images and high quality speech. A proliferation of research papers and a couple of books, written in English (there is an earlier book written in French), have emerged on the subject. These books, so far, are written by specialists for specialists, with a heavy mathematical flavor, which is characteristic of the Calderon-Zygmund theory and related research of Duffin-Schaeffer, Daubechies, Grossman, Meyer, Morlet, Chui, and others. Randy Young's monograph is geared more towards practitioners and even non-specialists, who want and, probably, should be cognizant of the exciting proven as well as potential benefits which have either already emerged or are likely to emerge from wavelet theory.