Discovering Wavelets

by Aboufadel, Edward; Schlicker, Steven

• ISBN13: 9780471331933
• ISBN10: 0471331937
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-10-05
• Publisher: Wiley-Interscience

Summary

An accessible and practical introduction to wavelets With applications in image processing, audio restoration, seismology, and elsewhere, wavelets have been the subject of growing excitement and interest over the past several years. Unfortunately, most books on wavelets are accessible primarily to research mathematicians. Discovering Wavelets presents basic and advanced concepts of wavelets in a way that is accessible to anyone with only a fundamental knowledge of linear algebra. The basic concepts of wavelet theory are introduced in the context of an explanation of how the FBI uses wavelets to compress fingerprint images. Wavelet theory is further developed in the setting of function spaces. The book then moves on to present more advanced topics such as filters, multiresolution analysis, Daubechies' wavelets, and further applications. The book concludes with a series of projects and problems that introduce advanced topics and offer starting points for research. Sample projects that demonstrate real wavelet applications include image compression, a wavelet-based search engine, processing with Daubechies' wavelets, and more. Among the special features of Discovering Wavelets are: * Real-life, hands-on examples that involve actual wavelet applications * A companion Web site containing Pixel Images software and Maple files to be used with the projects in the book * Challenging problems that reinforce and expand on the ideas being developed * An appendix containing the linear algebra needed to understand wavelets as presented in the book

Author Biography

EDWARD ABOUFADEL, PhD, and STEVEN SCHLICKER, PhD, are Associate Professors of Mathematics at Grand Valley State University in Allendale, Michigan.

Table of Contents

Preface	v
Acknowledgments	ix
Wavelets, Fingerprints, and Image Processing	1	(22)
Problems of the Digital Age	1	(1)
Digitizing Fingerprint Images	2	(2)
Signals	4	(1)
The Haar Wavelet Family	5	(4)
Processing Signals	9	(3)
Thresholding and Compression of Data	12	(6)
The FBI Wavelet/Scalar Quantization Standard	18	(5)
Wavelets and Orthogonal Decompositions	23	(20)
A Lego World	23	(3)
The Wavelet Sons	26	(2)
Sibling Rivalry: Two Bases for Vn	28	(4)
Averaging and Differencing	32	(3)
Projecting Functions Onto Wavelet Spaces	35	(2)
Function Processing and Image Boxes	37	(4)
A Summary of Two Approaches to Wavelets	41	(2)
Multiresolutions, Cascades, and Filters	43	(40)
Extending the Haar Wavelets to the Real Line	43	(2)
Other Elementary Wavelet Families	45	(4)
Multiresolution Analysis	49	(6)
The Haar Scaling Function Rediscovered	55	(7)
Relationships Between the Mother and Father Wavelets	62	(4)
Daubechies Wavelets	66	(4)
High and Low Pass Filters	70	(7)
More Problems of the Digital Age: Compact Discs	77	(6)
Sample Projects	83	(10)
Introduction: Overview of Projects	83	(1)
Linear Algebra Project: Image Processing and Compression	84	(2)
A Wavelet-Based Search Engine	86	(2)
B-Splines	88	(1)
Processing with the D4 Wavelets	89	(2)
Daubechies Wavelets with Six Refinement Coefficients	91	(2)
Appendix A Vector Spaces and Inner Product Spaces	93	(14)
A.1 Vector Spaces	93	(4)
A.2 Subspaces	97	(1)
A.3 Inner Product Spaces	98	(3)
A.4 The Orthogonal Decomposition Theorem	101	(6)
Appendix B Maple Routines	107	(6)
B.1 Matrix Generator	107	(1)
B.2 Processing Sampled Data	108	(1)
B.3 Projections onto Wavelet Spaces	109	(1)
B.4 The Cascade Algorithm	110	(1)
B.5 Processing an Image from Pixel Images	111	(2)
Appendix C Answers to Selected Problems	113	(4)
Appendix D Glossary of Symbols	117	(2)
References	119	(4)
Index	123