Wavelets Made Easy

ISBN-13: 9781461268239 / Angielski / Miękka / 2014 / 297 str.

Yves Nievergelt

Opis

This book explains the nature and computation of mathematical wavelets, which provide a framework and methods for the analysis and the synthesis of signals, images, and other arrays of data. The material presented here addresses the au- dience of engineers, financiers, scientists, and students looking for explanations of wavelets at the undergraduate level. It requires only a working knowledge or memories of a first course in linear algebra and calculus. The first part of the book answers the following two questions: What are wavelets? Wavelets extend Fourier analysis. How are wavelets computed? Fast transforms compute them. To show the practical significance of wavelets, the book also provides transitions into several applications: analysis (detection of crashes, edges, or other events), compression (reduction of storage), smoothing (attenuation of noise), and syn- thesis (reconstruction after compression or other modification). Such applications include one-dimensional signals (sounds or other time-series), two-dimensional arrays (pictures or maps), and three-dimensional data (spatial diffusion). The ap- plications demonstrated here do not constitute recipes for real implementations, but aim only at clarifying and strengthening the understanding of the mathematics of wavelets.

Recenzje

"The book explains in a nice way the nature and computation of mathematical wavelets, which provide a framework and methods for the analysis and synthesis of signals, images, and other arrays of data. A useful text for engineers, financiers, scientists, and students looking for explanation of wavelets."

-Journal of Information and Optimization Sciences

"Giving practice first and theory later, the author avoids discouraging readers whose main subject is not mathematics. The book is written in a very comprehensible and lively style. The text is essentially self-contained since many of the facts employed from analysis, linear algebra and functional analysis are stated and partially proved in the book."

-ZAA

Spis treści

A Algorithms for Wavelet Transforms.- 1 Haar’s Simple Wavelets.- 1.0 Introduction.- 1.1 Simple Approximation.- 1.2 Approximation with Simple Wavelets.- 1.2.1 The Basic Haar Wavelet Transform.- 1.2.2 Significance of the Basic Haar Wavelet Transform.- 1.2.3 Shifts and Dilations of the Basic Haar Transform.- 1.3 The Ordered Fast Haar Wavelet Transform.- 1.3.1 Initialization.- 1.3.2 The Ordered Fast Haar Wavelet Transform.- 1.4 The In-Place Fast Haar Wavelet Transform.- 1.4.1 In-Place Basic Sweep.- 1.4.2 The In-Place Fast Haar Wavelet Transform.- 1.5 The In-Place Fast Inverse Haar Wavelet Transform.- 1.6 Examples.- 1.6.1 Creek Water Temperature Analysis.- 1.6.2 Financial Stock Index Event Detection.- 2 Multidimensional Wavelets and Applications.- 2.0 Introduction.- 2.1 Two-Dimensional Haar Wavelets.- 2.1.1 Two-Dimensional Approximation with Step Functions.- 2.1.2 Tensor Products of Functions.- 2.1.3 The Basic Two-Dimensional Haar Wavelet Transform.- 2.1.4 Two-Dimensional Fast Haar Wavelet Transform.- 2.2 Applications of Wavelets.- 2.2.1 Noise Reduction.- 2.2.2 Data Compression.- 2.2.3 Edge Detection.- 2.3 Computational Notes.- 2.3.1 Fast Reconstruction of Single Values.- 2.3.2 Operation Count.- 2.4 Examples.- 2.4.1 Creek Water Temperature Compression.- 2.4.2 Financial Stock Index Image Compression.- 2.4.3 Two-Dimensional Diffusion Analysis.- 2.4.4 Three-Dimensional Diffusion Analysis.- 3 Algorithms for Daubechies Wavelets.- 3.0 Introduction.- 3.1 Calculation of Daubechies Wavelets.- 3.2 Approximation of Samples with Daubechies Wavelets.- 3.2.1 Approximate Interpolation.- 3.2.2 Approximate Averages.- 3.3 Extensions to Alleviate Edge Effects.- 3.3.1 Zigzag Edge Effects from Extensions by Zeros.- 3.3.2 Medium Edge Effects from Mirror Reflections.- 3.3.3 Small Edge Effects from Smooth Periodic Extensions.- 3.4 The Fast Daubechies Wavelet Transform.- 3.5 The Fast Inverse Daubechies Wavelet Transform.- 3.6 Multidimensional Daubechies Wavelet Transforms.- 3.7 Examples.- 3.7.1 Hangman Creek Water Temperature Analysis.- 3.7.2 Financial Stock Index Image Compression.- B Basic Fourier Analysis.- 4 Inner Products and Orthogonal Projections.- 4.0 Introduction.- 4.1 Linear Spaces.- 4.1.1 Number Fields.- 4.1.2 Linear Spaces.- 4.1.3 Linear Maps.- 4.2 Projections.- 4.2.1 Inner Products.- 4.2.2 Gram—Schmidt Orthogonalization.- 4.2.3 Orthogonal Projections.- 4.3 Applications of Orthogonal Projections.- 4.3.1 Application to Three-Dimensional Computer Graphics.- 4.3.2 Application to Ordinary Least-Squares Regression.- 4.3.3 Application to the Computation of Functions.- 4.3.4 Applications to Wavelets.- 5 Discrete and Fast Fourier Transforms.- 5.0 Introduction.- 5.1 The Discrete Fourier Transform (DFT).- 5.1.1 Definition and Inversion.- 5.1.2 Unitary Operators.- 5.2 The Fast Fourier Transform (FFT).- 5.2.1 The Forward Fast Fourier Transform.- 5.2.2 The Inverse Fast Fourier Transform.- 5.2.3 Interpolation by the Inverse Fast Fourier Transform.- 5.2.4 Bit Reversal.- 5.3 Applications of the Fast Fourier Transform.- 5.3.1 Noise Reduction Through the Fast Fourier Transform.- 5.3.2 Convolution and Fast Multiplication.- 5.4 Multidimensional Discrete and Fast Fourier Transforms.- 6 Fourier Series for Periodic Functions.- 6.0 Introduction.- 6.1 Fourier Series.- 6.1.1 Orthonormal Complex Trigonometric Functions.- 6.1.2 Definition and Examples of Fourier Series.- 6.1.3 Relation Between Series and Discrete Transforms.- 6.1.4 Multidimensional Fourier Series.- 6.2 Convergence and Inversion of Fourier Series.- 6.2.1 The Gibbs—Wilbraham Phenomenon.- 6.2.2 Piecewise Continuous Functions.- 6.2.3 Convergence and Inversion of Fourier Series.- 6.2.4 Convolutions and Dirac’s “Function” ?.- 6.2.5 Uniform Convergence of Fourier Series.- 6.3 Periodic Functions.- C Computation and Design of Wavelets.- 7 Fourier Transforms on the Line and in Space.- 7.0 Introduction.- 7.1 The Fourier Transform.- 7.1.1 Definition and Examples of the Fourier Transform.- 7.2 Convolutions and Inversion of the Fourier Transform.- 7.3 Approximate Identities.- 7.3.1 Weight Functions.- 7.3.2 Approximate Identities.- 7.3.3 Dirac Delta (?) Function.- 7.4 Further Features of the Fourier Transform.- 7.4.1 Algebraic Features of the Fourier Transform.- 7.4.2 Metric Features of the Fourier Transform.- 7.4.3 Uniform Continuity of Fourier Transforms.- 7.5 The Fourier Transform with Several Variables.- 7.6 Applications of Fourier Analysis.- 7.6.1 Shannon’s Sampling Theorem.- 7.6.2 Heisenberg’s Uncertainty Principle.- 8 Daubechies Wavelets Design.- 8.0 Introduction.- 8.1 Existence, Uniqueness, and Construction.- 8.1.1 The Recursion Operator and Its Adjoint.- 8.1.2 The Fourier Transform of the Recursion Operator.- 8.1.3 Convergence of Iterations of the Recursion Operator.- 8.2 Orthogonality of Daubechies Wavelets.- 8.3 Mallat’s Fast Wavelet Algorithm.- 9 Signal Representations with Wavelets.- 9.0 Introduction.- 9.1 Computational Features of Daubechies Wavelets.- 9.1.1 Initial Values of Daubechies’ Scaling Function.- 9.1.2 Computational Features of Daubechies’ Function.- 9.1.3 Exact Representation of Polynomials by Wavelets.- 9.2 Accuracy of Signal Approximation by Wavelets.- 9.2.1 Accuracy of Taylor Polynomials.- 9.2.2 Accuracy of Signal Representations by Wavelets.- 9.2.3 Approximate Interpolation by Daubechies’ Function.- D Directories.- Acknowledgments.- Collection of Symbols.

Blurb

This book, 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 practising 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.