Wavelet Analysis with Applications to Image Processing PDF
Preview Wavelet Analysis with Applications to Image Processing
ANALYSIS with Applications to IMAGE PROCESSING L. Prasad S. S. Iyengar <%j @ CRC Press Taylor&FrancisGroup BocaRaton London NewYork CRCPressisanimprintofthe Taylor&FrancisGroup,aninformabusiness CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 - BocaRaton,FL334872742 ©1997byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks - - ISBN13:978-0-849331695(hbk) Thisbookcontainsinformationobtainedfromauthenticandhighly regardedsources.Reasonable effortshavebeenmadetopublishreliabledataandinformation,buttheauthorandpublishercannot assumeresponsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse.Theauthors and publishers have attempted to trace the copyright holders of all material reproduced in this publicationandapologizetocopyrightholders ifpermissiontopublishinthisformhasnotbeen obtained.Ifanycopyrightmaterialhasnotbeenacknowledgedpleasewriteandletusknowsowe mayrectifyinanyfuturereprint. 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VisittheTaylor&FrancisWebsiteat http://www.taylorandfrancis.com andtheCRCPressWebsiteat http://www.crcpress.com LibraryofCongressCard Number97-11042 LibraryofCongressCataloging-in-PublicationData Prasad,L.(Lakshman) Waveletanalysiswithapplicationstoimageprocessing/L.Prasad,S.S.Iyengar p. cm. Includesbibliographicalreferencesandindex. ISBN0-8493-3169-2— (alk.paper) 1.Imageprocessing Mathematics.2.Wavelets(Mathematics)I.Iyengar,S.S. (SundararajaS.)II.Title. TA1637.P71997 — 621.36’7 dc21 97-11042 CIP Coverdesign:DawnBoyd Ill Preface Wavelet transforms havecome to be an important tool of mathematical analysis, with a wide and ever increasing range of applications, in recent years. Amongtheareaswhere wavelet analysisfinds application arediffer- entialequations,numericalanalysis,andsignalprocessing, tomentiononly afew. Waveletanalysisisamongthenewestand most powerfuladditionsto thearsenalsof mathematicians,scientistsandengineers,and hassucceeded in bringing together a vast pool of researchers by providing a novel and elegant “world-view”. The versatility of wavelet analytic techniques has forged new interdisciplinary bonds byofferingcommon solutions toappar- ently diverseproblemsand providinganew unifying perspectiveonseveral existingproblemsinvariedfields. Tobesure,waveletanalysisisaculmina- tionof theattemptsof workersinseveralfieldstodesign new toolstosolve problems in their areas. It has its origins in Calderon-Zygmund operator theoryof harmonicanalysis,in thetheoryof coherent statesand renormal- ization grouptheoryof quantum physics,and in subbandfilteringof signal processing. To some extent, this hybrid origin explains and endorses the flexibility and universal applicability of wavelet analytic methods. Fourier analysis has been the dominant principal analytical tool in the mathematicalsciencesandengineering,offeringarichframeworkwith pow- erful techniques tostudy propertiesoffunctions andoperators,andsignals via their spectra. In some problems, however, Fourier analytic techniques areinadequateorlead toextremelyonerouscomputations. A casein point is the time-frequency analysisof signals. The wavelet transform of afunc- tion (signal) capturesthelocalized time-frequencyinformationof thefunc- tion, unlike the Fourier transform, which sacrifices localization in one do- main in order tosecureit in the complementary domain. The property of time-frequencylocalizationgreatlyenhancestheabilitytostudythebehav- iors (e.g., smoothness, singularities) of signals as well as to change these featureslocally,withoutsignificantlyaffectingthestateofthesignals’char- acteristics in other regions of frequency or time. Also, unlike the Fourier transform, the continuous wavelet transform is closely linked to the dis- crete wavelet transform. This relation allows one to speak of a “wavelet series” ofanyfiniteenergysignal,obtained by “discretizing” thecontinuous wavelet transform. Thecoefficientsofsucha waveletseriesofasignalcom- pletely capture the time-frequency characteristics of the signal, with each . coefficient corresponding to a discrete time and frequency window This feature gives the discrete wavelet transform a multiresolutional property, which makespossiblethestudyofasignalat varyingresolutions. However, thealgebraicpropertiesof thewavelet transform aresimilar tothoseof the Fouriertransform,andthediscretewavelettransformisdefinedandstudied IV with the help of the Fourier transform. Thus, while wavelet transforms do not supplant the Fourier transform, they are relatively superior and more natural toolsof analysis in certain settings. Thus, the utilityand versatilityof waveletanalysiscannot be ignored by thestudent in theengineeringand mathematicalsciences. Thereisalready a significant amount of material on the subject of wavelet analysis in the form of pioneeringresearchpapers,expositoryarticles, books,etc.,and the literature is growing rapidly. The books by C. K. Chui [3], I. Daubechies [9],and Y.Meyer[29]offerexcellentexpositionsof variousaspectsand per- spectivesof waveletanalysistothescientificcommunity. However,thereis stilla need for booksat thelevel of the undergraduateand graduatelevels forabasicexpositionofthisfascinatingsubject,sothat thestudentofengi- neeringgetsanearlystart in equipping himself/herself with thetechniques of wavelet analysis. Moreoften than not, thestudent of engineering is not exposed to a rigorous or full-blown course on Fourier or functional analy- sis, and this limits and impedes his/her access to the nuances of wavelet analysis. This book isan attempt tointroducethe basicaspectsof wavelet analysis to thestudents of electrical engineering,computer scienceand re- lated disciplinesasearlyas possiblein their curriculum. Thisbookevolved out of aseriesof informal lectures held by thefirst author toasmall group of graduate students interested in the theory and applications of wavelet analysis. Thefirst four chaptersintroducethe basictopicsof analysis that arevi- tal to understanding the mathematics of wavelet transforms. These chap- ters are by no means a course in analysis, but serve to explain concisely theconceptsof analysis needed to develop wavelet analysisin alegitimate fashion. Indeed, the authors feel that very often the student of engineer- ing is asked to make many simplifying assumptions to compensate for the lack of a strong mathematical foundation, which may lead to a shallow or less than useful idea of the subject. The first chapter introduces notation and set-theoreticconcepts used throughout the book. The second chapter briefly discusses thestructure and properties of infinite dimensional linear manifolds, Banach and Hilbert spaces. Chapter three quickly develops the theory of Lebesgue integral, and introduces the spaces of absolutely inte- grable and square integrable functions. Chapter four deals with Fourier series and Fourier transforms. Chapters five, six and seven are devoted to wavelet analysis and its ap- plicationstoimage processing. Thesechaptersdonot constituteacompre- hensivetreatment ofallaspectsof waveletanalysis, but traceanexpository - path through the major themes of wavelet transform theory and applica tions. Chapter five introduces the continuous wavelet transform in the V context of time-frequency analysis, and the discrete wavelet transform as a means of expressinga function in terms of the coefficients of a vector in a Hilbert space with respect to a Riesz basis. Orthonormal wavelet bases associated with multiresolution analyses are discussed, and Mallat’s fast wavelet algorithmfor decomposingand reconstructingasignalisobtained. Chaptersixisdedicated tomethodsofconstructionof wavelet baseswith various desirable properties. The Battle-Lemarie wavelets are constructed as an example of orthonormal wavelet bases associated with a multireso- lution analysis. Next, a signal processing perspective is adapted to moti- vate the construction of compactly supported orthonormal and biorthog- onal wavelet bases. This describes the work of Daubechies [8] and Cohen [6]. Chapter seven illustrates two applications of wavelet transform theory in image processing attributable to Mallat et al. [27, 28,15]. Thechoice of image processing for illustrating the applications of wavelets was dictated by the authors’ research interest in the area. VI Acknowledgments Thisbook wasmade possiblethankstogeneroussupportfromtheOffice of Naval Research Division of Electronics. Our approach to wavelet analysis and its applications have been deeply influenced by the contributions of C. K. Chui, I. Daubechies, S. Mallat, and others in the field. Detailed comments from Dr. R. N. Madan and Professor R.L.Kashyaphelped ustounderstand many of the implications of our work. Invaluablecommentsfrom Professor K.R.Sreenivasan (Yale University) and ProfessorP. Vaidyanathan (CaliforniaInstituteof Technology) guided us in improving the content and scopeof the book. WearegratefultothefollowingindividualsintheLSURoboticsResearch Laboratory who have commented and contributed to earlier drafts of the book: RamanaRao, DarylThomas, Amit Nanavati,and John M. Zachary. Also, a special thanks goes to John M. Zachary, a research associate in the LSU Robotics Research Laboratory, for his contribution to parts of Chapter 7and his technical assistancein preparing the book in DT]EX. Over the years, we have received support in the form of grants from the Office of Naval Research Division of Electronics, the Naval Research Laboratory RemoteSensing Office, and the LEQSF-Board of Regents. We are deeply gratefulfor their support. L. Prasad S.S.Iyengar Contents 1 Introduction and Mathematical Preliminaries 1 1.1 Notation and Abbreviations 1 1.2 BasicSet Operations 3 1.3 Cardinality of Sets-Finite, Countable, and UncountableSets 6 1.4 Rings and Algebrasof Sets 9 2 Linear Spaces, Metric Spaces, and Hilbert Spaces 11 2.1 Linear Spaces 11 2.1.1 Subspaces 14 2.1.2 Factor spaces (quotient spaces) 15 2.1.3 Linear functionals 15 2.1.4 Null space (kernel) of afunctional-hyperplanes . . 16 2.1.5 Geometric interpretation of linear functions 16 2.1.6 Normed linear spaces 17 2.2 Metric Spaces 18 2.2.1 Continuous mappings 18 vii CONTENTS Vlll 2.2.2 Convergence 19 2.2.3 Dense subsets 20 2.2.4 Closed sets 21 2.2.5 Open sets 21 2.2.6 Complex metricspaces 22 2.2.7 Completion of metricspaces 22 2.2.8 Norm-induced metricand Banach spaces 23 2.3 Euclidean Spaces 23 2.3.1 Scalar products, orthogonalityand bases 23 2.3.2 Existenceof an orthogonal basis 25 2.3.3 Bessel’s inequality and closed orthogonalsystems . . 26 2.3.4 CompleteEuclideanspacesandtheRiesz-Fischerthe- orem 28 2.4 Hilbert Spaces 29 2.4.1 Subspaces,orthogonal complements, and direct sums 30 2.5 Characterization of Euclidean spaces 33 3 Integration 37 3.1 The Riemann Integral 37 3.1.1 Upper and lower Riemann integrals 38 3.1.2 Riemann integration vs Lebesgueintegration . . . . 41 3.2 The Lebesgue Measureon R 42 3.3 Measurable Functions 49 CONTENTS ix 3.3.1 Simple functions 55 3.4 Convergenceof Measurable Functions 56 3.5 Lebesgue Integration 58 3.5.1 Some propertiesof the Lebesgueintegral 68 4 Fourier Analysis 69 4.1 The Spaces L1(x) and L2(x) 70 4.1.1 The space Lx(x) 71 4.1.2 The space L2(x) 72 4.2 Fourier Series 79 4.2.1 Fourier series of squareintegrablefunctions 80 4.2.2 Fourier seriesof absolutely integrablefunctions . . . 86 4.2.3 The convolution product on L1(51) 89 4.3 Fourier Transforms 90 4.3.1 Fourier transformsof functions in L2(R) 96 4.3.2 Fourier transformsof functions in LX(R) 96 4.3.3 Poisson summation formula 99 5 Wavelet Analysis 101 - 5.1 Time-Frequency Analysisand the Windowed Fourier Trans form 101 5.1.1 Heisenberg’s Uncertainty Principle 103 5.2 The Integral Wavelet Transform 105 5.3 The Discrete Wavelet Transform 112
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