Wavelets. A Student Guide

271 Pages·2017·2.179 MB·english

by  Peter Nickolas

AustralianMathematicalSocietyLectureSeries:24
Wavelets: A Student Guide
PETER NICKOLAS
UniversityofWollongong,NewSouthWales
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107612518
©PeterNickolas2017
Firstpublished2017
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-Publicationdata
Names:Nickolas,Peter.
Title:Wavelets:astudentguide/PeterNickolas,UniversityofWollongong,
NewSouthWales.
Description:Cambridge:CambridgeUniversityPress,2017.|
Series:AustralianMathematicalSocietylectureseries;24|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016011212|ISBN9781107612518(pbk.)
Subjects:LCSH:Wavelets(Mathematics)–Textbooks.|Innerproduct
spaces–Textbooks.|Hilbertspace–Textbooks.
Classification:LCCQA403.3.N532016|DDC515/.2433–dc23LCrecordavailable
athttp://lccn.loc.gov/2016011212
ISBN978-1-107-61251-8Paperback
Contents
Preface pagevii
1 AnOverview 1
1.1 OrthonormalityinRn 1
1.2 SomeInfinite-DimensionalSpaces 6
1.3 FourierAnalysisandSynthesis 15
1.4 Wavelets 18
1.5 TheWaveletPhenomenon 25
Exercises 27
2 VectorSpaces 31
2.1 TheDefinition 31
2.2 Subspaces 34
2.3 Examples 35
Exercises 40
3 InnerProductSpaces 42
3.1 TheDefinition 42
3.2 Examples 43
3.3 SetsofMeasureZero 48
3.4 TheNormandtheNotionofDistance 51
3.5 OrthogonalityandOrthonormality 55
3.6 OrthogonalProjection 64
Exercises 73
4 HilbertSpaces 80
4.1 SequencesinInnerProductSpaces 80
4.2 ConvergenceinRn,(cid:2)2andtheL2Spaces 84
4.3 SeriesinInnerProductSpaces 90
4.4 CompletenessinInnerProductSpaces 94
4.5 OrthonormalBasesinHilbertSpaces 105
4.6 FourierSeriesinHilbertSpaces 108
4.7 OrthogonalProjectionsinHilbertSpaces 114
Exercises 117
5 TheHaarWavelet 126
5.1 Introduction 126
5.2 TheHaarWaveletMultiresolutionAnalysis 127
5.3 ApproximationSpacesandDetailSpaces 133
5.4 SomeTechnicalSimplifications 137
Exercises 138
6 WaveletsinGeneral 141
6.1 Introduction 141
6.2 TwoFundamentalDefinitions 145
6.3 ConstructingaWavelet 150
6.4 WaveletswithBoundedSupport 159
Exercises 168
7 TheDaubechiesWavelets 176
7.1 Introduction 176
7.2 ThePotentialScalingCoefficients 178
7.3 TheMomentsofaWavelet 179
7.4 TheDaubechiesScalingCoefficients 185
7.5 TheDaubechiesScalingFunctionsandWavelets 188
7.6 SomePropertiesof φ 200
2
7.7 SomePropertiesof φand ψ 205
N N
Exercises 209
8 WaveletsintheFourierDomain 218
8.1 Introduction 218
8.2 TheComplexCase 219
8.3 TheFourierTransform 227
8.4 ApplicationstoWavelets 232
8.5 ComputingtheDaubechiesScalingCoefficients 237
Exercises 242
Appendix: NotesonSources 251
References 259
Index 262
Preface
Overview
The overall aim of this book is to provide an introduction to the theory of
waveletsforstudentswithamathematicalbackgroundatseniorundergraduate
level.ThetextgrewfromasetoflecturenotesthatIdevelopedwhileteaching
acourseonwaveletsatthatleveloveranumberofyearsattheUniversityof
Wollongong.
Although the topic of wavelets is somewhat specialised and is certainly
not a standard one in the typical undergraduate syllabus, it is nevertheless
an attractive one for introduction to students at that level. This is for several
reasons, including its topicality and the intrinsic interest of its fundamental
ideas. Moreover, although a comprehensive study of the theory of wavelets
makes the use of advanced mathematics unavoidable, it remains true that
substantial parts of the theory can, with care, be made accessible at the
undergraduatelevel.
Thebookassumesfamiliaritywithfinite-dimensionalvectorspacesandthe
elements of real analysis, but it does not assume exposure to analysis at an
advanced level, to functional analysis, to the theory of Lebesgue integration
and measure or to the theory of the Fourier integral transform. Knowledge
of all these topics and more is assumed routinely in all full accounts of
wavelet theory, which make heavy use of the Lebesgue and Fourier theories
inparticular.
The approach adopted here is therefore what is often referred to as ‘ele-
mentary’.Broadly, fullproofs ofresultsaregiven precisely totheextent that
theycanbeconstructedinaformthatisconsistentwiththerelativelymodest
assumptionsmadeaboutbackgroundknowledge.Anumberofcentralresults
in the theory of wavelets are by their nature deep and are not amenable in
anystraightforwardwaytoanelementaryapproach,andaconsequenceisthat
whilemostresultsintheearlierpartsofthebookaresuppliedwithcomplete
proofs, a few of those in the later parts are given only partial proofs or are
proved only in special cases or are stated without proof. While a degree of
intellectual danger is inherent in giving an exposition that is incomplete in
thisway,Iamcarefultoacknowledgegapswheretheyoccur,andIthinkthat
any minor disadvantages are outweighed by the advantages of being able to
introducesuchanattractivetopicattheundergraduatelevel.
StructureandContents
If a unifying thread runs through the book, it is that of exploring how the
fundamental ideas of an orthonormal basis in a finite-dimensional real inner
product space, and the associated expression of a vector in terms of its
projectionsontotheelementsofsuchabasis,generalisenaturallyandelegantly
tosuitableinfinite-dimensionalspaces.Thustheworkstartsinthefamiliarand
concreteterritoryofEuclideanspaceandmovestowardsthelessfamiliarand
moreabstractdomainofsequenceandfunctionspaces.
The structure and contents of the book are shown in some detail by the
Contents,butbriefcommentsonthefirstandlastchaptersspecificallymaybe
useful.
Chapter 1 is essentially a miniaturised version of the rest of the text. Its
inclusion is intended to allow the reader to gain as early as possible some
sense of what wavelets are, of how and why they are used and of the beauty
andunityoftheideasinvolved,withouthavingtowaitforthemoresystematic
developmentofwavelettheorythatstartsinChapter5.
Asnotedabove,theFouriertransformisanindispensabletechnicaltoolfor
the rigorous and systematic study of wavelets, but its use has been bypassed
inthistextinfavourofanelementaryapproachinordertomakethematerial
as accessible as possible. Chapter 8, however, is a partial exception to this
principle,sincewegivethereanoverviewofwavelettheoryusingthepowerful
extrainsightprovidedbytheuseofFourieranalysis.Forstudentswithadeeper
backgroundinanalysisthanthatassumedearlierinthetext,thischapterbrings
theworkofthebookmoreintolinewithstandardapproachestowavelettheory.
Inkeepingwiththis,weworkinthischapterwithcomplex-valuedratherthan
real-valuedfunctions.
Pathways
The book contains considerably more material than could be covered in a
typicalone-semestercourse,butlendsitselftouseinanumberofwaysforsuch
acourse.ItislikelythatChapters3and4oninnerproductspacesandHilbert
spaceswouldneedtobeincludedhoweverthetextisused,sincethismaterial
isrequiredfortheworkonwaveletsthatfollowsbutisnotusuallycoveredin
suchdepthintheundergraduatecurriculum.Beyondthis,coveragewilldepend
ontheknowledgethatcanbeassumed.Thefollowingthreepathwaysthrough
the material are proposed, depending on the assumed level of mathematical
preparation.
Atthemostelementarylevel,thebookcouldbeusedasanintroductionto
Hilbert spaces with applications to wavelets, by covering just Chapters 1–5,
perhaps with excursions, which could be quite brief, into Chapters 6 and 7.
At a somewhat higher level of sophistication, the coverage could largely
or completely bypass Chapter 1, survey the examples of Chapter 2 briefly
and then proceed through to the end of Chapter 7. A third pathway through
the material is possible for students with a more substantial background in
analysis, including significant experience with the Fourier transform and,
preferably, Lebesgue measure and integration. Here, coverage could begin
comfortablywithChapter3andcontinuetotheendofChapter8.
Exercises
The book contains about 230 exercises, and these should be regarded as an
integralpartofthetext.Althoughtheyarereferredtouniformlyas‘exercises’,
they range substantially in difficulty and complexity: from short and simple
problems to long and difficult ones, some of which extend the theory or
provide proofs that are omitted from the body of the text. In the longer and
more difficult cases, I have generally provided hints or outlined suggested
approachesorbrokenpossibleargumentsintostepsthatareindividuallymore
approachable.
Sources
Becausethebookisanintroductoryone,Idecidedagainstincludingreferences
to the literature in the main text. At the same time, however, it is certainly
desirable to provide such references for readers who want to consult source
material or look more deeply into issues arising from the text, and the
Appendixprovidesthese.
Acknowledgements
Theinitialsuggestionofconvertingmylecturenotesonwaveletsintoabook
came from Jacqui Ramagge. Adam Sierakowski read and commented on
the notes in detail at an early stage and Rodney Nillsen likewise read and
commented on substantial parts of the book when it was in something much
closer to its final form. I am indebted to these colleagues, as well as to two
anonymousreferees.
1
An Overview
This chapter gives an overview of the entire book. Since our focus turns
directlytowaveletsonlyinChapter5,abouthalfwaythrough,beginningwith
anoverviewisusefulbecauseitenablesusearlyontoconveyanideaofwhat
waveletsareandofthemathematicalsettingwithinwhichwewillstudythem.
The idea of the orthonormality of a collection of vectors, together with
some closely related ideas, is central to the chapter. These ideas should be
familiar at least in the finite-dimensional Euclidean spaces Rn (and certainly
inR2andR3inparticular),butafterrevisingtheEuclideancasewewillgoon
to examine the same ideas in certain spaces of infinite dimension, especially
spacesoffunctions.Inmanyways,thecentralchaptersofthebookconstitutea
systematicandgeneralinvestigationoftheseideas,andthetheoryofwavelets
is,fromonepointofview,anapplicationofthistheory.
Because this chapter is only an overview, the discussion will be rather
informal, and important details will be skated over quickly or suppressed
altogether,butbytheendofthechapterthereadershouldhaveabroadideaof
theshapeandcontentofthebook.
1.1 OrthonormalityinRn
Recallthattheinnerproduct(cid:2)a,b(cid:3)oftwovectors
⎛ ⎞ ⎛ ⎞
a=⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ aa...12 ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ and b=⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ bb...12 ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
a b
n n
2 1 AnOverview
inRnisdefinedby
(cid:8)n
(cid:2)a,b(cid:3)=a b +a b +···+a b = ab.
1 1 2 2 n n i i
i=1
Also,thenorm(cid:4)a(cid:4)ofaisdefinedby
(cid:9) (cid:10)(cid:8)n (cid:11)1
2
(cid:4)a(cid:4)= a21+a22+···+a2n = a2i =(cid:2)a,a(cid:3)12.
i=1
➤ Afewremarksonnotationandterminologyareappropriatebeforewegofurther.
First,inelementarylinearalgebra,specialnotationisoftenusedforvectors,especially
vectorsinEuclideanspace:avectormight,forexample,besignalledbytheuseofbold
face,asina,orbytheuseofaspecialsymbolaboveorbelowthemainsymbol,asin→a
ora.Thisnotationalcomplication,however,islogicallyunnecessary,andisnormally
∼
avoidedinmoreadvancedwork.
Second, you may be used to the terminology dot product and the corresponding
notationa·b,butwewillalwaysusethephraseinnerproductandthe‘angle-bracket’
notation(cid:2)a,b(cid:3).Similarly,(cid:4)a(cid:4)issometimesreferredtoasthelengthormagnitudeofa,
butwewillalwaysrefertoitasthenorm.
Theinnerproductcanbeusedtodefinethecomponentandtheprojectionof
onevectoronanother.Geometrically,thecomponentofaonbisformedby
projectingaperpendicularlyontobandmeasuringthelengthoftheprojection;
informally,thecomponentmeasureshowfara‘sticksout’inthedirectionofb,
or ‘how long a looks’ from the perspective of b. Since this quantity should
depend only on the direction of b and not on its length, it is convenient to
defineitfirstwhenbisnormalised,oraunitvector,thatis,satisfies(cid:4)b(cid:4)=1.
Inthiscase,thecomponentofaonbisdefinedsimplytobe
(cid:2)a,b(cid:3).
If b is not normalised, then the component of a on b is obtained by applying
(cid:12) (cid:13)
thedefinitiontoitsnormalisation 1/(cid:4)b(cid:4) b,whichisaunitvector,givingthe
expression
(cid:14) (cid:15)
1 1
a, b = (cid:2)a,b(cid:3)
(cid:4)b(cid:4) (cid:4)b(cid:4)
forthecomponent(notethatwemustassumethatbisnotthezerovectorhere,
toensurethat(cid:4)b(cid:4)(cid:2)0).
Sincethecomponentisdefinedasaninnerproduct,itsvaluecanbeanyreal
number – positive, negative or zero. This implies that our initial description
ofthecomponentabovewasnotquiteaccurate:thecomponentrepresentsnot
justthelengthofthefirstvectorfromtheperspectiveofthesecond,butalso,
accordingtoitssign,howthefirstvectorisorientedwithrespecttothesecond.
(Forthespecialcasewhenthecomponentis0,seebelow.)
1.1 OrthonormalityinRn 3
Itisnowsimpletodefinetheprojectionofaonb;thisisthevectorwhose
direction is given by b and whose length and orientation are given by the
componentofaonb.Thusif(cid:4)b(cid:4)=1,thentheprojectionofaonbisgivenby
(cid:2)a,b(cid:3)b,
andforgeneralnon-zerobby
(cid:14) (cid:15)
1 1 1
a, b b= (cid:2)a,b(cid:3)b.
(cid:4)b(cid:4) (cid:4)b(cid:4) (cid:4)b(cid:4)2
Vectors a and b are said to be orthogonal if (cid:2)a,b(cid:3) = 0, and a collection
ofvectorsisorthonormalifeachvectorinthesethasnorm1andeverytwo
distinct elements from the set are orthogonal. If a set of vectors is indexed
by the values of a subscript, as in v ,v ,..., say, then the orthonormality of
1 2
the set can be expressed conveniently using the Kronecker delta: the set is
orthonormalifandonlyif
(cid:2)v,v (cid:3)=δ foralli, j,
i j i,j
wheretheKroneckerdeltaδ isdefinedtobe1wheni = jand0otherwise
i,j
(overwhateveristherelevantrangeofiand j).
All the above definitions are purely algebraic: they use nothing more than
thealgebraicoperationspermittedinavectorspaceandinthefieldofscalarsR.
However, the definitions also have clear geometric interpretations, at least in
R2 and R3, where we can visualise the geometry (see Exercise 1.2); indeed,
we used this fact explicitly at a number of points in the discussion (speaking
of‘howlong’onevectorlooksfromanother’sperspectiveandofthe‘length’
and‘orientation’ofavector,forexample).Wecanattemptasomewhatmore
comprehensivelistofsuchgeometricinterpretationsasfollows.
⎛ ⎞
• Avectora = ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ aa...12 ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠isjustapointinRn (whosecoordinatesareofcourse
a
n
thenumbersa ,a ,...,a ).
1 2 n
• The norm(cid:4)a(cid:4)isthe Euclidean distance of thepoint afromtheorigin,and
moregenerally(cid:4)a−b(cid:4)isthedistanceofthepointafromthepointb.(Given
the formula defining the norm, this is effectively just a reformulation of
Pythagoras’theorem;seeExercise1.2.)
• For a unit vector b, the inner product (cid:2)a,b(cid:3) gives the magnitude and
orientationofawhenseenfromb.