Framelets and Wavelets: Algorithms, Analysis, and Applications

750 Pages·2017·14.67 MB·English

by  Bin Han (auth.)

Applied and Numerical Harmonic Analysis
Bin Han
Framelets
and
Wavelets
Algorithms, Analysis, and
Applications

Applied and Numerical Harmonic Analysis
SeriesEditor
JohnJ.Benedetto
UniversityofMaryland
CollegePark,MD,USA
EditorialAdvisoryBoard
AkramAldroubi GittaKutyniok
VanderbiltUniversity TechnischeUniversitätBerlin
Nashville,TN,USA Berlin,Germany
DouglasCochran MauroMaggioni
ArizonaStateUniversity DukeUniversity
Phoenix,AZ,USA Durham,NC,USA
HansG.Feichtinger ZuoweiShen
UniversityofVienna NationalUniversityofSingapore
Vienna,Austria Singapore,Singapore
ChristopherHeil ThomasStrohmer
GeorgiaInstituteofTechnology UniversityofCalifornia
Atlanta,GA,USA Davis,CA,USA
StéphaneJaffard YangWang
UniversityofParisXII MichiganStateUniversity
Paris,France EastLansing,MI,USA
JelenaKovacˇevic´
CarnegieMellonUniversity
Pittsburgh,PA,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/4968
Bin Han
Framelets and Wavelets
Algorithms, Analysis, and Applications
BinHan
DepartmentofMathematicalandStatistical
Sciences
UniversityofAlberta
Edmonton,Alberta,Canada
ISSN2296-5009 ISSN2296-5017 (electronic)
AppliedandNumericalHarmonicAnalysis
ISBN978-3-319-68529-8 ISBN978-3-319-68530-4 (eBook)
https://doi.org/10.1007/978-3-319-68530-4
LibraryofCongressControlNumber:2017959559
MathematicsSubjectClassification(2010):42C40,42C15,41A05,41A30,65T60
©SpringerInternationalPublishingAG2017
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To mywifeShuang,daughterRachel,son
James,andparents
ANHA Series Preface
The Applied and Numerical Harmonic Analysis (ANHA) book series aims to
providetheengineering,mathematical,andscientificcommunitieswithsignificant
developments in harmonic analysis, ranging from abstract harmonic analysis to
basic applications. The title of the series reflects the importance of applications
and numerical implementation, but richness and relevance of applications and
implementation depend fundamentally on the structure and depth of theoretical
underpinnings. Thus, from our point of view, the interleaving of theory and
applicationsandtheircreativesymbioticevolutionisaxiomatic.
Harmonicanalysisisawellspringofideasandapplicabilitythathasflourished,
developed, and deepened over time within many disciplines and by means of
creative cross-fertilization with diverse areas. The intricate and fundamental rela-
tionship between harmonic analysis and fields such as signal processing, partial
differentialequations(PDEs),andimageprocessingisreflectedinourstate-of-the-
artANHAseries.
Our vision of modem harmonic analysis includes mathematical areas such as
wavelettheory,Banachalgebras,classicalFourieranalysis,time-frequencyanalysis,
andfractalgeometry,aswellasthediversetopicsthatimpingeonthem.
Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwith
some basic problems in digital signal processing, speech and image processing,
geophysics, pattern recognition, biomedical engineering, and turbulence. These
areas implement the latest technology from sampling methods on surfaces to fast
algorithmsand computervision methods. The underlyingmathematics of wavelet
theorydependsnotonlyonclassicalFourieranalysisbutalsoonideasfromabstract
harmonicanalysis,includingvonNeumannalgebrasandtheaffinegroup.Thisleads
toastudyoftheHeisenberggroupanditsrelationshiptoGaborsystems,andofthe
metaplectic group for a meaningful interaction of signal decomposition methods.
Theunifyinginfluenceofwavelettheoryintheaforementionedtopicsillustratesthe
justification for providinga means for centralizing and disseminating information
fromthebroader,butstillfocused,areaofharmonicanalysis.Thiswillbeakeyrole
of ANHA. We intend to publish with the scope and interactionthat such a hostof
issuesdemands.
vii
viii ANHASeriesPreface
Alongwithourcommitmentto publishmathematicallysignificantworksatthe
frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopublish
majoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysisplays
asubstantialrole:
Antennatheory Predictiontheory
Biomedicalsignalprocessing Radarapplications
Digitalsignalprocessing Samplingtheory
Fastalgorithms Spectralestimation
Gabortheoryandapplications Speechprocessing
Imageprocessing Time-frequencyand
Numericalpartialdifferentialequations time-scaleanalysis
Wavelettheory
TheabovepointofviewfortheANHAbookseriesisinspiredbythehistoryof
Fourieranalysisitself,whosetentaclesreachintosomanyfields.
In the last two centuries Fourier analysis has had a major impact on the
development of mathematics, on the understanding of many engineering and
scientificphenomena,andonthesolutionofsomeofthemostimportantproblems
in mathematics and the sciences. Historically, Fourier series were developed in
the analysis of some of the classical PDEs of mathematical physics; these series
were used to solve such equations. In order to understand Fourier series and the
kindsofsolutionstheycouldrepresent,someofthemostbasicnotionsofanalysis
were defined, e.g., the concept of “function.” Since the coefficients of Fourier
seriesareintegrals,itisnosurprisethatRiemannintegralswereconceivedtodeal
with uniqueness properties of trigonometric series. Cantor’s set theory was also
developedbecauseofsuchuniquenessquestions.
A basic problem in Fourier analysis is to show how complicated phenomena,
suchassoundwaves,canbedescribedintermsofelementaryharmonics.Thereare
twoaspectsofthisproblem:first,tofind,orevendefineproperly,theharmonicsor
spectrumofagivenphenomenon,e.g.,thespectroscopyprobleminoptics;second,
todeterminewhichphenomenacanbeconstructedfromgivenclassesofharmonics,
asdone,forexample,bythemechanicalsynthesizersintidalanalysis.
Fourieranalysisisalsothenaturalsettingformanyotherproblemsinengineer-
ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in
Fourieranalysisnotonlycharacterizesthebehavioroftheprimenumbersbutalso
provides the proper notion of spectrum for phenomena such as white light; this
latterprocessleadsto theFourieranalysisassociatedwithcorrelationfunctionsin
filtering and prediction problems,and these problems,in turn, deal naturally with
Hardyspacesinthetheoryofcomplexvariables.
Nowadays, some of the theory of PDEs has given way to the study of Fourier
integral operators. Problems in antenna theory are studied in terms of unimodu-
lar trigonometric polynomials. Applications of Fourier analysis abound in signal
processing, whether with the fast Fourier transform (FFT), or filter design, or the
ANHASeriesPreface ix
adaptivemodelinginherentintime-frequency-scalemethodssuchaswavelettheory.
The coherentstates of mathematicalphysicsare translated and modulatedFourier
transforms, and these are used, in conjunction with the uncertainty principle, for
dealing with signal reconstruction in communications theory. We are back to the
raisond’êtreoftheANHAseries!
UniversityofMaryland JohnJ.Benedetto
CollegePark,MD,USA SeriesEditor
Preface
As a rapidly growing, multidisciplinary field of mathematics, wavelet theory
providesthemajormathematicalmultiscalerepresentationforanalyzingfunctions/
data and has undergoneextensive developmentfor more than 30 years, including
numerous applications in science and industry. This book introduces readers to
the mathematicaltheoryof waveletsfrom the perspectiveof framelets(i.e., frame
wavelets) and discrete framelet transforms by systematically covering various
topics.Itcanservenotonlyasatextbookforgraduatestudents,seniorundergrad-
uate students, and beginningresearchers, offeringthem self-contained elementary
proofs and supplementary exercises, but also as an advanced reference guide for
experiencedresearchersinmathematics,physics,andengineeringbysystematically
addressing a wide range of topics on wavelet theory in depth. The book can
also be used to teach or study some specific topics in approximation theory,
Fourieranalysis,appliedharmonicanalysis,functionalanalysis,andwavelet-based
signal/image processing. It consists of seven chapters and an appendix with 362
exerciseproblemsand72figures.
The classical part of wavelet theory, in particular, orthogonal wavelets and
biorthogonal wavelets in the square integrable function space L2.R/ with filter
banks,multiresolutionanalysis,andhomogeneousaffinesystems:
AS.(cid:2)/WDf 2jIk WD2j=2 .2j(cid:2)(cid:3)k/ W j;k 2Z; 2(cid:2)g;
where (cid:2) (cid:4) L2.R/, has been extensively addressed in many books under the
framework of the space L2.R/. A closely related but much less studied system is
anonhomogeneousaffinesystem:
ASJ.˚I(cid:2)/WDf(cid:3)2JIk W k2Z;(cid:3) 2˚g[f 2jIk W k 2Z;j>J; 2(cid:2)g; J 2Z
with˚;(cid:2) (cid:4)L2.R/.Theword“framelet”isformedbycombining“framewavelet.”
Forthe purposeof thisbook,a framelet(ora wavelet)is a set offunctionswhose
generated affine system is a frame (or a basis), where a frame is a generalization
of a basis by allowing additional/extra elements. Hence, a wavelet is merely a
xi