From Fourier Analysis to Wavelets

216 Pages·2015·5.917 MB·English

by  Jonas Gomes, Luiz Velho

IMPA Monographs
Jonas Gomes
Luiz Velho
From Fourier
Analysis to
Wavelets
Instituto de Matemática Pura e Aplicada
IMPA Monographs
Volume 3
This series, jointly established by IMPA and Springer,publishes advanced mono-
graphsgivingauthoritativeaccountsofcurrentresearchinanyfieldofmathematics,
with emphasis on those fields that are closer to the areas currently supported at
IMPA.Theseriesgiveswell-writtenpresentationsofthe“state-of-the-art”infields
ofmathematicalresearchandpointerstofuturedirectionsofresearch.
SeriesEditors
EmanuelCarneiro,InstitutodeMatemáticaPuraeAplicada
SeverinoCollier,UniversidadeFederaldoRiodeJaneiro
ClaudioLandim,InstitutodeMatemáticaPuraeAplicada
PauloSad,InstitutodeMatemáticaPuraeAplicada
Moreinformationaboutthisseriesathttp://www.springer.com/series/13501

Jonas Gomes • Luiz Velho
From Fourier Analysis
to Wavelets
123
JonasGomes LuizVelho
BOZANOInvestimentos InstitutodeMatemáticaPuraeAplicada
OfficesShoppingLeblon RiodeJaneiro,Brazil
RiodeJaneiro,Brazil
IMPAMonographs
ISBN978-3-319-22074-1 ISBN978-3-319-22075-8 (eBook)
DOI10.1007/978-3-319-22075-8
LibraryofCongressControlNumber:2015945959
MathematicsSubjectClassification(2010):42-XX,42C40,65-XX,65T60
SpringerChamHeidelbergNewYorkDordrechtLondon
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To myfamily,thetrueforcethatpropelledmy
carrier.
—JonasGomes
To mybeloveddaughterAliceVelho
ChekrounandmygrandsonLevi Yitzchak
Chekroun
— LuizVelho.

Preface
From a digital viewpoint, most real-world applications can be reduced to the
problem of function representation and reconstruction. These two problems are
closelyrelatedtosynthesisandanalysisoffunctions.TheFouriertransformisthe
classical tool used to solve them. More recently, wavelets have entered the arena
providingmorerobustandflexiblesolutionstodiscretizeandreconstructfunctions.
Starting from Fourier analysis, the course guides the audience to acquire an
understandingof the basic ideas and techniques behind the wavelets. We start by
introducing the basic concepts of function spaces and operators, both from the
continuousanddiscreteviewpoints.WeintroducetheFourierandWindowFourier
Transform,theclassicaltoolsforfunctionanalysisinthefrequencydomain,andwe
usethemasaguidetoarriveattheWavelettransform.Thefundamentalaspectsof
multiresolutionrepresentationand its importanceto functiondiscretization and to
theconstructionofwaveletsarealsodiscussed.
Emphasiswillbegivenonideasandintuition,avoidingmostofthemathematical
machinery,whichareusuallyinvolvedinthestudyofwavelets.Becauseofthis,the
bookdemandsfromthereadersonlyabasicknowledgeoflinearalgebra,calculus,
andsomefamiliaritywithcomplexanalysis.Basicknowledgeofsignalandimage
processingwouldbedesirable.
This monographoriginated from the course notes of a very successful tutorial
given by the authors during the years of 1998 and 1999 at ACM SIGGRAPH -
theInternationalConferenceandExhibitiononComputerGraphicsandInteractive
Techniques.
ThenotesinEnglishfortheSIGGRAPHcoursehavebeenbasedonasetofnotes
in Portuguese that we wrote for a wavelet course on the Brazilian Mathematical
Colloquiumin1997atIMPA,RiodeJaneiro.WewishtothankSiomeGoldenstein
whocollaboratedwithustoproducethePortuguesenotes.
RiodeJaneiro,Brazil JonasGomes
May2015 LuizVelho
vii

Contents
1 Introduction ................................................................ 1
1.1 ComputationalMathematics......................................... 1
1.1.1 AbstractionLevels.......................................... 2
1.2 RelationBetweentheAbstractionLevels........................... 4
1.3 FunctionsandComputationalMathematics......................... 6
1.3.1 RepresentationandReconstructionofFunctions ......... 6
1.3.2 SpecificationofFunctions.................................. 7
1.4 WhatistheRelationwithGraphics?................................ 7
1.4.1 DescriptionofGraphicalObjects.......................... 8
1.5 WheredoWaveletsFit?.............................................. 9
1.5.1 FunctionRepresentationUsingWavelets.................. 9
1.5.2 MultiresolutionRepresentation............................ 10
1.6 AbouttheseBook .................................................... 10
1.7 CommentsandReferences........................................... 11
2 FunctionRepresentationandReconstruction .......................... 13
2.1 RepresentingFunctions.............................................. 13
2.1.1 TheRepresentationOperator............................... 14
2.2 BasisRepresentation................................................. 15
2.2.1 CompleteOrthonormalRepresentation.................... 15
2.3 RepresentationbyFrames ........................................... 16
2.4 RieszBasisRepresentation.......................................... 18
2.5 RepresentationbyProjection........................................ 19
2.6 GalerkinRepresentation ............................................. 20
2.7 Reconstruction,PointSamplingandInterpolation ................. 21
2.7.1 PiecewiseConstantReconstruction........................ 21
2.7.2 PiecewiseLinearReconstruction .......................... 22
2.8 MultiresolutionRepresentation...................................... 23
2.9 RepresentationbyDictionaries...................................... 25
ix