Wavelets Made Easy

by Nievergelt, Yves

• ISBN13: 9780817640613
• ISBN10: 0817640614
• Format: Hardcover
• Copyright: 1999-04-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	ix
Outline	xi
A Algorithms for Wavelet Transforms	1	(114)
Haar's Simple Wavelets	3	(33)
Introduction	3	(1)
Simple Approximation	4	(4)
Approximation with Simple Wavelets	8	(6)
The Basic Haar Wavelet Transform	8	(2)
Significance of the Basic Haar Wavelet Transform	10	(1)
Shifts and Dilations of the Basic Haar Transform	11	(3)
The Ordered Fast Haar Wavelet Transform	14	(7)
Initialization	14	(1)
The Ordered Fast Haar Wavelet Transform	15	(6)
The In-Place Fast Haar Wavelet Transform	21	(7)
In-Place Basic Sweep	22	(1)
The In-Place Fast Haar Wavelet Transform	23	(5)
The In-Place Fast Inverse Haar Wavelet Transform	28	(3)
Examples	31	(5)
Creek Water Temperature Analysis	31	(2)
Financial Stock Index Event Detection	33	(3)
Multidimensional Wavelets and Applications	36	(37)
Introduction	36	(1)
Two-Dimensional Haar Wavelets	37	(12)
Two-Dimensional Approximation with Step Functions	37	(2)
Tensor Products of Functions	39	(3)
The Basic Two-Dimensional Haar Wavelet Transform	42	(4)
Two-Dimensional Fast Haar Wavelet Transform	46	(3)
Applications of Wavelets	49	(11)
Noise Reduction	49	(3)
Data Compression	52	(6)
Edge Detection	58	(2)
Computational Notes	60	(5)
Fast Reconstruction of Single Values	60	(3)
Operation Count	63	(2)
Examples	65	(8)
Creek Water Temperature Compression	65	(2)
Financial Stock Index Image Compression	67	(1)
Two-Dimensional Diffusion Analysis	68	(1)
Three-Dimensional Diffusion Analysis	69	(4)
Algorithms for Daubechies Wavelets	73	(42)
Introduction	73	(1)
Calculation of Daubechies Wavelets	73	(9)
Approximation of Samples with Daubechies Wavelets	82	(3)
Approximate Interpolation	83	(1)
Approximate Averages	84	(1)
Extensions to Alleviate Edge Effects	85	(10)
Zigzag Edge Effects from Extensions by Zeros	85	(3)
Medium Edge Effects from Mirror Reflections	88	(2)
Small Edge Effects from Smooth Periodic Extensions	90	(5)
The Fast Daubechies Wavelet Transform	95	(6)
The Fast Inverse Daubechies Wavelet Transform	101	(6)
Multidimensional Daubechies Wavelet Transforms	107	(3)
Examples	110	(5)
Hangman Creek Water Temperature Analysis	110	(2)
Financial Stock Index Image Compression	112	(3)
B Basic Fourier Analysis	115	(88)
Inner Products and Orthogonal Projections	117	(30)
Introduction	117	(1)
Linear Spaces	117	(6)
Number Fields	117	(3)
Linear Spaces	120	(2)
Linear Maps	122	(1)
Projections	123	(11)
Inner Products	124	(5)
Gram--Schmidt Orthogonalization	129	(2)
Orthogonal Projections	131	(3)
Applications of Orthogonal Projections	134	(13)
Application to Three-Dimensional Computer Graphics	134	(2)
Application to Ordinary Least-Squares Regression	136	(2)
Application to the Computation of Functions	138	(4)
Applications to Wavelets	142	(5)
Discrete and Fast Fourier Transforms	147	(28)
Introduction	147	(1)
The Discrete Fourier Transform (DFT)	147	(10)
Definition and Inversion	148	(7)
Unitary Operators	155	(2)
The Fast Fourier Transform(FFT)	157	(8)
The Forward Fast Fourier Transform	157	(4)
The Inverse Fast Fourier Transform	161	(1)
Interpolation by the Inverse Fast Fourier Transform	161	(2)
Bit Reversal	163	(2)
Applications of the Fast Fourier Transform	165	(6)
Noise Reduction Through the Fast Fourier Transform	165	(2)
Convolution and Fast Multiplication	167	(4)
Multidimensional Discrete and Fast Fourier Transforms	171	(4)
Fourier Series for Periodic Functions	175	(28)
Introduction	175	(1)
Fourier Series	176	(9)
Orthonormal Complex Trigonometric Functions	176	(1)
Definition and Examples of Fourier Series	177	(5)
Relation Between Series and Discrete Transforms	182	(1)
Multidimensional Fourier Series	183	(2)
Convergence and Inversion of Fourier Series	185	(15)
The Gibbs-Wilbraham Phenomenon	185	(2)
Piecewise Continuous Functions	187	(4)
Convergence and Inversion of Fourier Series	191	(1)
Convolutions and Dirac's ``Function'' +	192	(2)
Uniform Convergence of Fourier Series	194	(6)
Periodic Functions	200	(3)
C Computation and Design of Wavelets	203	(82)
Fourier Transforms on the Line and in Space	205	(33)
Introduction	205	(1)
The Fourier Transform	205	(4)
Definition and Examples of the Fourier Transform	205	(4)
Convolutions and Inversion of the Fourier Transform	209	(4)
Approximate Identities	213	(7)
Weight Functions	214	(1)
Approximate Identities	215	(4)
Dirac Delta (δ) Function	219	(1)
Further features of the Fourier Transform	220	(9)
Algebraic Features of the Fourier Transform	221	(2)
Metric Features of the Fourier Transform	223	(4)
Uniform Continuity of Fourier Transforms	227	(2)
The Fourier Transform with Several Variables	229	(5)
Applications of Fourier Analysis	234	(4)
Shannon's Sampling Theorem	234	(2)
Heisenberg's Uncertainty Principle	236	(2)
Daubechies Wavelets Design	238	(24)
Introduction	238	(1)
Existence, Uniqueness, and Construction	238	(15)
The Recursion Operator and its Adjoint	239	(4)
The Fourier Transform of the Recursion Operator	243	(2)
Convergence of Iterations of the Recursion Operator	245	(8)
Orthogonality of Daubechies Wavelets	253	(5)
Mallat's Fast Wavelet Algorithm	258	(4)
Signal Representations with Wavelets	262	(23)
Introduction	262	(1)
Computational Features of Daubechies Wavelets	262	(12)
Initial Values of Daubechies' Scaling Function	263	(3)
Computational Features of Daubechies Function	266	(7)
Exact Representation of Polynomials by Wavelets	273	(1)
Accuracy of Signal approximation by Wavelets	274	(11)
Accuracy of Taylor Polynomials	274	(4)
Accuracy of Signal Representations by Wavelets	278	(3)
Approximate Interpolation by Daubechies' Function	281	(4)
D Directories	285	(2)
Acknowledgments	287	(2)
Collection of Symbols	289	(2)
Bibliography	291	(4)
Index	295