DISCRETE WAVELET TRANSFORMATIONS An Elementary Approach with Applications PATRICK J. VAN FLEET University of St. Thomas BICENTENNIAL ; WILEY- INTERSCIENCE BICENTENNIAL A JOHN WILEY & SONS, INC., PUBLICATION This page intentionally left blank DISCRETE WAVELET TRANSFORMATIONS 1 β Ο 7 ®WILEY 2 Ο Ο 7 « •ι THE WILEY BICENTENNIAL-KNOWLEDGE FOR GENERATIONS £^ach generation has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how. For 200 years, Wiley has been an integral part of each generation's journey, enabling the flow of information and understanding necessary to meet their needs and fulfill their aspirations. Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities. Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it! WILLIAM J. PESCE PETER BOOTH WILEY PRESIDENT AND CHIEF EXECUTIVE OFFICER CHAIRMAN QF THE BOARD DISCRETE WAVELET TRANSFORMATIONS An Elementary Approach with Applications PATRICK J. VAN FLEET University of St. Thomas BICENTENNIAL ; WILEY- INTERSCIENCE BICENTENNIAL A JOHN WILEY & SONS, INC., PUBLICATION All photographs in the text and on the front cover appear courtesy of Radka Tezaur. Copyright © 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our web site at www.wiley.com. Wiley Bicentennial Logo: Richard J. Pacifico Library of Congress Cataloging-in-Publication Data: Van Fleet, Patrick J., 1962- Discrete wavelet transformations : an elementary approach with applications Patrick J. Van Fleet p. cm. Includes bibliographical references and index. ISBN 978-0-470-18311-3 (cloth : acid-free paper) 1. Wavelets (Mathematics) 2. Transformations (Mathematics) 3. Digital images—Mathematics. I. Title. QA403.3.V36 2008 515'.2433—dc22 2007027174 Printed in the United States of America. 10 9 8 7 6 5 4 3 21 For Andy This page intentionally left blank CONTENTS Preface xiii Acknowledgments xxjji 1 Introduction: Why Wavelets? 1 2 Vectors and Matrices 15 2.1 Vectors, Inner Products, and Norms 16 Problems 20 Computer Lab 22 2.2 Basic Matrix Theory 22 Problems 36 Computer Lab 38 2.3 Block Matrix Arithmetic 38 Problems 45 Computer Lab 48 vii