Shearlets : multiscale analysis for multivariate data

346 Pages·2012·3.684 MB·English

by  Gitta Kutyniok, et al (eds.)

Applied and Numerical Harmonic Analysis
SeriesEditor
JohnJ.Benedetto
UniversityofMaryland
CollegePark,MD,USA
EditorialAdvisoryBoard
AkramAldroubi JelenaKovacˇevic´
VanderbiltUniversity CarnegieMellonUniversity
Nashville,TN,USA Pittsburgh,PA,USA
AndreaBertozzi GittaKutyniok
UniversityofCalifornia TechnischeUniversita¨tBerlin
LosAngeles,CA,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
Ste´phaneJaffard YangWang
UniversityofParisXII MichiganStateUniversity
Paris,France EastLansing,MI,USA
Forfurthervolumes:
http://www.springer.com/series/4968
Gitta Kutyniok • Demetrio Labate
Editors
Shearlets
Multiscale Analysis for Multivariate Data
Editors
GittaKutyniok DemetrioLabate
Institutfu¨rMathematik DepartmentofMathematics
TechnischeUniversita¨tBerlin UniversityofHouston
Berlin,Germany Houston,TX,USA
ISBN978-0-8176-8315-3 e-ISBN978-0-8176-8316-0
DOI10.1007/978-0-8176-8316-0
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2012932597
MathematicalSubjectClassification(2010):42C15,42C40,65T60,68U10,94A08
(cid:2)c SpringerScience+BusinessMedia,LLC2012
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia
(www.birkhauser-science.com)
ANHA Series Preface
TheAppliedandNumericalHarmonicAnalysis(ANHA)bookseriesaimstoprovide
theengineering,mathematical,andscientificcommunitieswithsignificantdevelop-
mentsinharmonicanalysis,rangingfromabstractharmonicanalysistobasicappli-
cations.Thetitleoftheseriesreflectstheimportanceofapplicationsandnumerical
implementation,butrichnessandrelevanceofapplicationsandimplementationde-
pendfundamentallyonthestructureanddepthoftheoreticalunderpinnings.Thus,
fromourpointofview,theinterleavingoftheoryandapplicationsandtheircreative
symbioticevolutionisaxiomatic.
Harmonic analysis is a wellspring of ideas and applicability that has flour-
ished, developed, and deepened over time within many disciplines and by means
ofcreativecross-fertilizationwithdiverseareas.Theintricateandfundamentalre-
lationship between harmonic analysis and fields such as signal processing,partial
differentialequations(PDEs),andimageprocessingisreflectedinourstate-of-the-
artANHAseries.
Our vision of modern harmonic analysis includes mathematical areas such as
wavelettheory,Banachalgebras,classicalFourieranalysis,time–frequencyanaly-
sis,andfractalgeometry,aswellasthediversetopicsthatimpingeonthem.
Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwith
somebasicproblemsindigitalsignalprocessing,speechandimageprocessing,geo-
physics, pattern recognition, biomedical engineering, and turbulence. These areas
implement the latest technologyfrom sampling methods on surfaces to fast algo-
rithmsandcomputervisionmethods.Theunderlyingmathematicsofwavelettheory
dependsnotonlyonclassicalFourieranalysis,butalsoonideasfromabstracthar-
monic analysis, includingvon Neumannalgebrasand the affine group.This leads
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
ofANHA.Weintendtopublishthescopeandinteractionthatsuchahostofissues
demands.
v
vi 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.
Inthe lasttwocenturies,Fourieranalysishashada majorimpactonthedevel-
opment of mathematics, on the understanding of many engineering and scientific
phenomena,andonthesolutionofsomeofthemostimportantproblemsinmathe-
maticsandthesciences.Historically,Fourierseriesweredevelopedintheanalysis
of some of the classical PDEs of mathematical physics;these series were used to
solve such equations. In order to understandFourier series and the kinds of solu-
tionstheycouldrepresent,someofthemostbasicnotionsofanalysisweredefined,
e.g.,theconceptof“function”.SincethecoefficientsofFourierseriesareintegrals,
it is no surprise that Riemann integrals were conceived to deal with uniqueness
propertiesoftrigonometricseries.Cantorssettheorywasalsodevelopedbecauseof
suchuniquenessquestions.
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,e.g.,bythemechanicalsynthesizersintidalanalysis.
Fourieranalysisisalsothenaturalsettingformanyotherproblemsinengineer-
ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in
Fourieranalysisnotonlycharacterizesthebehavioroftheprimenumbers,butalso
providesthepropernotionofspectrumforphenomenasuchaswhitelight;thislatter
processleadstotheFourieranalysisassociatedwithcorrelationfunctionsinfilter-
ingandpredictionproblems,andtheseproblems,inturn,dealnaturallywithHardy
spacesinthetheoryofcomplexvariables.
Nowadays, some of the theory of PDEs has given way to the study of Fourier
integraloperators.Problemsin antenna theoryare studied in terms of unimodular
trigonometricpolynomials.Applicationsof Fourieranalysisaboundin signalpro-
cessing,whetherwiththefastFouriertransform(FFT),orfilterdesign,ortheadap-
tivemodelinginherentintimefrequency-scalemethodssuchaswavelettheory.The
ANHASeriesPreface vii
coherentstatesofmathematicalphysicsaretranslatedandmodulatedFouriertrans-
forms, and these are used, in conjunctionwith the uncertainty principle, for deal-
ingwithsignalreconstructionincommunicationstheory.Wearebacktotheraison
d’eˆtreoftheANHAseries!
UniversityofMaryland JohnJ.Benedetto
CollegePark SeriesEditor

Preface
The introduction of wavelets about 20 years ago has revolutionized applied
mathematics, computer science, and engineering by providing a highly effective
methodologyfor analyzing and processing univariate functions/signalscontaining
singularities. However, wavelets do not perform equally well in the multivariate
caseduetothefactthattheyarecapableofefficientlyencodingonlyisotropicfea-
tures.ThislimitationcanbeseenbyobservingthatBesovspacescanbeprecisely
characterizedbydecaypropertiesofsequencesofwaveletcoefficients,buttheyare
not capable of capturingthose geometricfeatures which could be associated with
edgesandotherdistributedsingularities.Indeed,suchgeometricfeaturesareessen-
tial in the multivariate setting, since multivariate problems are typically governed
byanisotropicphenomenasuchassingularitiesconcentratedonlowerdimensional
embeddedmanifolds.Todealwiththischallenge,severalapproacheswereproposed
intheattempttoextendthebenefitsofthewaveletframeworktohigherdimensions,
withtheaimofintroducingrepresentationsystemswhichcouldprovidebothopti-
mallysparse approximationsofanisotropicfeaturesanda unifiedtreatmentofthe
continuumanddigitalworld.Amongthevariousmethodologiesproposed,suchas
curveletsandcontourlets,theshearletsystem,whichwasintroducedin2005,stands
outasthefirstandsofartheonlyapproachcapableofsatisfyingthiscombination
ofrequirements.
Today,variousdirectionsofresearchhavebeenestablishedinthetheoryofshear-
lets. These include, in particular, the theory of continuous shearlets—associated
with a parameter set of continuous range—and its application to the analysis of
distributions.Anotherdirectionisthetheoryofdiscreteshearlets—associatedwith
adiscreteparameterset—andtheirsparseapproximationproperties.Thankstothe
fact that shearlets provide a unified treatment of the continuum and digital realm
throughtheutilizationoftheshearingoperator,digitalizationandhencenumerical
realizationscan be performedin a faithfulmanner,and this leads to very efficient
algorithms.Buildingontheseresults,severalshearlet-basedalgorithmsweredevel-
opedtoaddressarangeofproblemsinimageanddataprocessing.
Thisbookisthefirstmonographdevotedtoshearlets.Itisnotonlyaimedatand
accessibletoabroadreadershipincludinggraduatestudentsandresearchersinthe
ix