It cascades wavelet trans form convolutions with nonlinear modulus and averaging op erators The 64257rst network layer outputs SIFTtype descriptors whereas t he next layers provide complementary invariant information which improv es classi64257ca ti ID: 24083
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1InvariantScatteringConvolutionNetworksJoanBrunaandSt´ephaneMallatCMAP,EcolePolytechnique,Palaiseau,France F AbstractAwaveletscatteringnetworkcomputesatranslationinvari-antimagerepresentation,whichisstabletodeformationsandpreserveshighfrequencyinformationforclassication.Itcascadeswavelettrans-formconvolutionswithnon-linearmodulusandaveragingoperators.TherstnetworklayeroutputsSIFT-typedescriptorswhereasthenextlayersprovidecomplementaryinvariantinformationwhichimprovesclassica-tion.Themathematicalanalysisofwaveletscatteringnetworksexplainimportantpropertiesofdeepconvolutionnetworksforclassication.AscatteringrepresentationofstationaryprocessesincorporateshigherordermomentsandcanthusdiscriminatetextureshavingsameFourierpowerspectrum.Stateoftheartclassicationresultsareob-tainedforhandwrittendigitsandtexturediscrimination,withaGaussiankernelSVMandagenerativePCAclassier.IndexTermsClassication,Convolutionnetworks,Deformations,In-variants,Wavelets1INTRODUCTIONAmajordifcultyofimageclassicationcomesfromtheconsiderablevariabilitywithinimageclassesandtheinabilityofEuclideandistancestomeasureimagesimi-larities.Partofthisvariabilityisduetorigidtranslations,rotationsorscaling.Thisvariabilityisoftenuninforma-tiveforclassicationandshouldthusbeeliminated.Intheframeworkofkernelclassiers[33],thedistancebetweentwosignalsxandx0isdenedasaEuclideandistancekx x0kappliedtoarepresentationxofeachx.Variabilityduetorigidtransformationsareremovedifisinvarianttothesetransformations.Non-rigiddeformationsalsoinduceimportantvari-abilitywithinobjectclasses[17],[3].Forinstance,inhandwrittendigitrecognition,onemusttakeintoac-countdigitdeformationsduetodifferentwritingstyles[3].However,afulldeformationinvariancewouldre-ducediscriminationsinceadigitcanbedeformedintoadifferentdigit,forexampleaoneintoaseven.Therep-resentationmustthereforenotbedeformationinvariant.Itshouldlinearizesmalldeformations,tohandlethemeffectivelywithlinearclassiers.LinearizationmeansthattherepresentationisLipschitzcontinuoustodefor-mations.Whenanimagexisslightlydeformedintox0thenkx x0kmustbeboundedbythesizeofthedeformation,asdenedinSection2. ThisworkisfundedbytheFrenchANRgrantBLAN012601.Translationinvariantrepresentationscanbecon-structedwithregistrationalgorithms[34],autocorrela-tionsorwiththeFouriertransformmodulus.However,Section2.1explainsthattheseinvariantsarenotstabletodeformationsandhencenotadaptedtoimageclas-sication.TryingtoavoidFouriertransforminstabilitiessuggestsreplacingsinusoidalwavesbylocalizedwave-formssuchaswavelets.However,wavelettransformsarenotinvariantbutcovarianttotranslations.Build-inginvariantrepresentationsfromwaveletcoefcientsrequiresintroducingnon-linearoperators,whichleadstoaconvolutionnetworkarchitecture.Deepconvolutionnetworkshavetheabilitytobuildlarge-scaleinvariants,whichseemtobestabletodefor-mations[20].Theyhavebeenappliedtoawiderangeofimageclassicationtasks.Despitethesuccessesofthisneuralnetworkarchitecture,thepropertiesandoptimalcongurationsofthesenetworksarenotwellunder-stood,becauseofcascadednon-linearities.Whyusemultiplelayers?Howmanylayers?Howtooptimizeltersandpoolingnon-linearities?Howmanyinternalandoutputneurons?Thesequestionsaremostlyan-sweredthroughnumericalexperimentationsthatrequiresignicantexpertise.Weaddressthesequestionsfrommathematicalandalgorithmicpointofviews,byconcentratingonapar-ticularclassofdeepconvolutionnetworks,denedbythescatteringtransformsintroducedin[24],[25].Ascat-teringtransformcomputesatranslationinvariantrepre-sentationbycascadingwavelettransformsandmoduluspoolingoperators,whichaveragetheamplitudeofit-eratedwaveletcoefcients.ItisLipschitzcontinuoustodeformations,whilepreservingthesignalenergy[25].ScatteringnetworksaredescribedinSection2andtheirpropertiesareexplainedinSection3.Theseproper-tiesguidetheoptimizationofthenetworkarchitecturetoretainimportantinformationwhileavoidinguselesscomputations.Anexpectedscatteringrepresentationofstationaryprocessesisintroducedfortexturediscrimination.Asop-posedtotheFourierpowerspectrum,itgivesinforma-tiononhigherordermomentsandcanthusdiscriminatenon-Gaussiantextureshavingthesamepowerspectrum.Scatteringcoefcientsprovideconsistentestimatorsofexpectedscatteringrepresentations. 2ClassicationapplicationsarestudiedinSection4.ClassiersareimplementedwithaGaussiankernelSVMandagenerativeclassier,whichselectsafnespacemodelscomputedwithaPCA.State-of-the-artresultsareobtainedforhandwrittendigitrecognitiononMNISTandUSPSdatabases,andfortexturediscrimination.Theseareproblemswheretranslationinvariance,station-arityanddeformationstabilityplayacrucialrole.Soft-wareisavailableatwww.cmap.polytechnique.fr/scattering.2TOWARDSACONVOLUTIONNETWORKSmalldeformationsarenearlylinearizedbyarepresenta-tioniftherepresentationisLipschitzcontinuoustotheactionofdeformations.Section2.1explainswhyhighfrequenciesaresourcesofinstabilities,whichpreventstandardinvariantstobeLipschitzcontinuous.Sec-tion2.2introducesawavelet-basedscatteringtransform,whichistranslationinvariantandLipschitzrelativelytodeformations.Section2.3describesitsconvolutionnetworkarchitecture.2.1FourierandRegistrationInvariantsArepresentationxisinvarianttoglobaltranslationsxc(u)=x(u c)byc=(c1;c2)2R2ifxc=x:(1)Acanonicalinvariant[17],[34]x=x(u a(x))registersxwithananchorpointa(x),whichistranslatedwhenxistranslated:a(xc)=a(x)+c.Itisthereforeinvariant:xc=x.Forexample,theanchorpointmaybealteredmaximuma(x)=argmaxujx?h(u)j,forsomelterh(u).TheFouriertransformmodulusisanotherexampleoftranslationinvariantrepresentation.Let^x(!)betheFouriertransformofx(u).Sincebxc(!)=e ic:!^x(!),itresultsthatjbxcj=j^xjdoesnotdependuponc.TheautocorrelationRx(v)=Rx(u)x(u v)duisalsotranslationinvariant:Rx=Rxc.Tobestabletoadditivenoisex0(u)=x(u)+(u),weneedaLipschitzcontinuityconditionwhichsupposesthatthereexistsC�0suchthatforallxandx0kx0 xkCkx0 xk;wherekxk2=Rjx(u)j2du.ThePlancherelformulaprovesthattheFouriermodulusx=j^xjsatisesthispropertywithC=2.Tobestabletodeformationvariabilities,mustalsobeLipschitzcontinuoustodeformations.Asmalldeforma-tionofxcanbewrittenx (u)=x(u (u)),where (u)isanon-constantdisplacementeldwhichdeformstheimage.Thedeformationgradienttensorr (u)isamatrixwhosenormjr (u)jmeasuresthedeformationamplitudeatuandsupujr (u)jistheglobaldefor-mationamplitude.Asmalldeformationisinvertibleifjr (u)j1[1].LipschitzcontinuityrelativelytodeformationsisobtainedifthereexistsC&#x-5.1;ä0suchthatforall andxkx xkCkxksupujr (u)j;(2)wherekxk2=Rjx(u)j2du.Thispropertyimpliesglobaltranslationinvariance,becauseif (u)=cthenr (u)=0,butitismuchstronger.IfisLipschitzcontinuoustodeformations thentheRadon-Nyko´ympropertyprovesthatthemapwhichtransforms intox isalmosteverywheredifferen-tiableinthesenseofGateau[22].Itmeansthatforsmalldeformations,x x iscloselyapproximatedbyaboundedlinearoperatorof ,whichistheGateauderivative.Deformationsarethuslinearizedby,whichenableslinearclassierstoeffectivelyhandledeforma-tionvariabilitiesintherepresentationspace.AFouriermodulusistranslationinvariantandstabletoadditivenoisebutunstabletosmalldeformationsathighfrequencies.Indeed,jj^x(!)j jcx (!)jjcanbearbi-trarilylargeatahighfrequency!,evenforsmalldefor-mationsandinparticularforasmalldilation (u)=u.Asaresult,x=j^xjdoesnotsatisfythedeformationcontinuitycondition(2)[25].Theautocorrelationx=RxsatisescRx(!)=j^x(!)j2.ThePlancherelformulathusprovesthatithasthesameinstabilitiesasaFouriertransform:kRx Rx k=(2) 1kj^xj2 j^x j2k:Besidesdeformationinstabilities,aFouriermodulusandanautocorrelationloosestoomuchinformation.Forexample,aDirac(u)andalinearchirpeiu2aretotallydifferentsignalshavingFouriertransformswhosemoduliareequalandconstant.VerydifferentsignalsmaynotbediscriminatedfromtheirFouriermodulus.Aregistrationinvariantx(u)=x(u a(x))carriesmoreinformationthanaFouriermodulus,andcharac-terizesxuptoaglobalabsolutepositioninformation[34].However,ithasthesamehigh-frequencyinstabilityasaFouriertransform.Indeed,foranychoiceofanchorpointa(x),applyingthePlancherelformulaprovesthatkx(u a(x)) x0(u a(x0))k(2) 1kj^x(!)j j^x0(!)jk:Ifx =x ,theFouriertransforminstabilityathighfrequenciesimpliesthatx=x(u a(x))isalsounstablewithrespecttodeformations.2.2ScatteringWaveletsAwavelettransformcomputesconvolutionswithdi-latedandrotatedwavelets.Waveletsarelocalizedwave-formsandarethusstabletodeformations,asopposedtoFouriersinusoidalwaves.However,convolutionsaretranslationcovariant,notinvariant.Ascatteringtrans-formbuildsnon-linearinvariantsfromwaveletcoef-cients,withmodulusandaveragingpoolingfunctions.LetGbeagroupofrotationsrofangles2k=Kfor0kK.Two-dimensionaldirectionalwaveletsare 3obtainedbyrotatingasingleband-passlter byr2Ganddilatingitby2jforj2Z (u)=2 2j (2 jr 1u)with=2 jr:(3)IftheFouriertransform^ (!)iscenteredatafrequencythen^ 2 jr(!)=^ (2jr 1!)hasasupportcenteredat2 jr,andabandwidthproportionalto2 j.Theindex=2 jrgivesthefrequencylocationof anditsamplitudeisjj=2 j.Thewavelettransformofxisfx? (u)g.Itisaredundanttransformwithnoorthogonalityproperty.Section3.1explainsthatitisstableandinvertibleifthewaveletlters^ (!)coverthewholefrequencyplane.Ondiscreteimages,toavoidaliasing,weonlycapturefrequenciesinthecirclej!jinscribedintheimagefrequencysquare.Mostcameraimageshavenegligibleenergyoutsidethisfrequencycircle.Letu:u0andjujdenotetheinnerproductandnorminR2.AMorletwavelet isanexampleofcomplexwaveletgivenby (u)= (eiu: )e juj2=(22);where 1isadjustedsothatR (u)du=0.It'srealandimagepartsarenearlyquadraturephaselters.Figure1showstheMorletwaveletwith=0:85and=3=4,usedinallclassicationexperiments.Awavelettransformcommuteswithtranslations,andisthereforenottranslationinvariant.Tobuildatransla-tioninvariantrepresentation,itisnecessarytointroduceanon-linearity.IfQisalinearornon-linearoperatorwhichcommuteswithtranslations,thenRQx(u)duistranslationinvariant.ApplyingthistoQx=x? givesatrivialinvariantRx? (u)du=0forallxbecauseR (u)du=0.IfQx=M(x? )andMislinearandcommuteswithtranslationsthentheintegralstillvanishes.Thisshowsthatcomputinginvariantsrequirestoincorporateanon-linearpoolingoperatorM,butwhichone?ToguaranteethatRM(x? )(u)duisstabletodefor-mations,wewantMtocommutewiththeactionofanydiffeomorphism.TopreservestabilitytoadditivenoisewealsowantMtobenonexpansive:kMy Mzkky zk.IfMisanonexpansiveoperatorwhichcommuteswiththeactionofdiffeomorphismsthenonecanprove[7]thatMisnecessarilyapointwiseoperator.ItmeansthatMy(u)isafunctionofthevaluey(u)only.Tobuildinvariantswhichalsopreservethesignalenergyrequirestochooseamodulusoperatorovercomplexsignalsy=yr+iyi:My(u)=jy(u)j=(jyr(u)j2+jyi(u)j2)1=2:(4)TheresultingtranslationinvariantcoefcientsarethenL1(R2)normskx? k1=Zjx? (u)jdu:TheL1(R2)normsfkx? k1gformacrudesignalrepresentation,whichmeasuresthesparsityofwaveletcoefcients.Thelossofinformationdoesnotcomefromthemoduluswhichremovesthecomplexphaseofx? (u).Indeed,onecanprove[38]thatxcanbereconstructedfromthemodulusofitswaveletcoef-cientsfjx? (u)jg,uptoamultiplicativeconstant.Theinformationlosscomesfromtheintegrationofjx? (u)j,whichremovesallnon-zerofrequencies.Thesenon-zerofrequenciesarerecoveredbycalculatingthewaveletcoefcientsfjx? 1j? 2(u)g2ofjx? 1j.TheirL1(R2)normsdeneamuchlargerfamilyofinvariants,forall1and2:kjx? 1j? 2k1=Zjjx? 1(u)j? 2jdu:Moretranslationinvariantcoefcientscanbecom-putedbyfurtheriteratingonthewavelettransformandmodulusoperators.LetU[]x=jx? j.Anysequencep=(1;2;:::;m)denesapath,alongwhichiscomputedanorderedproductofnon-linearandnon-commutingoperators:U[p]x=U[m]:::U[2]U[1]x=jjjx? 1j? 2j:::j? mj;withU[;]x=x.Ascatteringtransformalongthepathpisdenedasanintegral,normalizedbytheresponseofaDirac: Sx(p)= 1pZU[p]x(u)duwithp=ZU[p](u)du:Eachscatteringcoefcient Sx(p)isinvarianttoatrans-lationofx.WeshallseethatthistransformhasmanysimilaritieswiththeFouriertransformmodulus,whichisalsotranslationinvariant.However,ascatteringisLipschitzcontinuoustodeformationsasopposedtotheFouriertransformmodulus.Forclassication,itisoftenbettertocomputelocalizeddescriptorswhichareinvarianttotranslationssmallerthanapredenedscale2J,whilekeepingthespatialvariabilityatscaleslargerthan2J.Thisisobtainedbylocalizingthescatteringintegralwithascaledspatialwindow 2J(u)=2 2J (2 Ju).Itdenesawindowedscatteringtransformintheneighborhoodofu:S[p]x(u)=U[p]x? 2J(u)=ZU[p]x(v) 2J(u v)dv;andhenceS[p]x(u)=jjjx? 1j? 2j:::j? mj? 2J(u);withS[;]x=x? 2J.Foreachpathp,S[p]x(u)isafunctionofthewindowpositionu,whichcanbesubsampledatintervalsproportionaltothewindowsize2J.Theaveragingby 2Jimpliesthatifxc(u)=x(u c)withjcj 2Jthenthewindowedscatteringisnearlytranslationinvariant:S[p]xS[p]xc.Section3.1provesthatitisalsostablerelativelytodeformations. 4 (a) (b) (c)Fig.1.ComplexMorletwavelet.(a):Realpartof (u).(b):Imaginarypartof (u).(c):Fouriermodulusj^ (!)j. m=0 m=1 m=2 m=3 x U[1]x ;]x=x? 1; S[1]x 1; Fig.2.AscatteringpropagatorfWappliedtoxcomputestherstlayerofwaveletcoefcientsmodulusU[1]x=jx? 1jandoutputsitslocalaverageS[;]x=x? 2J(blackarrow).ApplyingfWtotherstlayersignalsU[1]xoutputsrstorderscatteringcoefcientsS[1]=U[1]? 2J(blackarrows)andcomputesthepropagatedsignalU[1;2]xofthesecondlayer.ApplyingfWtoeachpropagatedsignalU[p]xoutputsS[p]x=U[p]x? 2J(blackarrows)andcomputesanextlayerofpropagatedsignals.2.3ScatteringConvolutionNetworkIfp=(1;:::;m)isapathoflengthmthenS[p]x(u)iscalledawindowedscatteringcoefcientoforderm.Itiscomputedatthelayermofaconvolutionnetworkwhichisspecied.Forlargescaleinvariants,severallayersarenecessarytoavoidlosingcrucialinformation.Forappropriatewavelets,rstordercoefcientsS[1]xareequivalenttoSIFTcoefcients[23].Indeed,SIFTcomputesthelocalsumofimagegradientamplitudesamongimagegradientshavingnearlythesamedirec-tion,inahistogramhaving8differentdirectionbins.TheDAISYapproximation[35]showsthatthesecoefcientsarewellapproximatedbyS[1]x=jx? 1j? 2J(u)where 1arepartialderivativesofaGaussiancom-putedatthenestimagescale,along8differentrota-tions.Theaveraginglter 2JisascaledGaussian.Partialderivativewaveletsarewelladaptedtodetectedgesorsharptransitionsbutdonothaveenoughfre-quencyanddirectionalresolutiontodiscriminatecom-plexdirectionalstructures.Fortextureanalysis,manyresearchers[21],[32],[30]havebeenusingaveragedwaveletcoefcientamplitudesjx? j? 2J(u),calculatedwithacomplexwavelet havingabetterfrequencyanddirectionalresolution.Ascatteringtransformcomputeshigher-ordercoef-cientsbyfurtheriteratingonwavelettransformsandmodulusoperators.Waveletcoefcientsarecomputeduptoamaximumscale2Jandthelowerfrequenciesarelteredby 2J(u)=2 2J (2 Ju).ForaMorletwavelet ,theaveraginglter ischosentobeaGaussian.Sinceimagesarereal-valuedsignals,itissufcienttoconsiderpositiverotationsr2G+withanglesin[0;):Wx(u)=nx? 2J(u);x? (u)o2P;(5)withanindexsetP=f=2 jr:r2G+;jJg.Letusemphasizethat2Jand2jarespatialscalevariables,whereas=2 jrisafrequencyindexgivingthelocationofthefrequencysupportof^ (!).Awaveletmoduluspropagatorkeepsthelow-frequencyaveragingandcomputesthemodulusofcom- 5plexwaveletcoefcients:fWx(u)=nx? 2J(u);jx? (u)jo2P:(6)IteratingonfWdenesaconvolutionnetworkillustratedinFigure2.ThenetworknodesofthelayermcorrespondtothesetPmofallpathsp=(1;:::;m)oflengthm.ThismthlayerstoresthepropagatedsignalsfU[p]xgp2PmandoutputsthescatteringcoefcientsfS[p]xgp2Pm.Foranyp=(1;:::;m)wedenotep+=(1;:::;m;).SinceS[p]x=U[p]x? 2JandU[p+]x=jU[p]x? jitresultsthatfWU[p]x=nS[p]x;U[p+]xo2P:ApplyingfWtoallpropagatedsignalsU[p]xofthemthlayerPmoutputsallscatteringsignalsS[p]xandcom-putesallpropagatedsignalsU[p+]onthenextlayerPm+1.AlloutputscatteringsignalsS[p]xalongpathsoflengthm marethusobtainedbyrstcalculatingfWx=fS[;]x;U[]xg2PandtheniterativelyapplyingfWtoeachlayerofpropagatedsignalsforincreasingm m.ThetranslationinvarianceofS[p]xisduetotheav-eragingofU[p]xby 2J.Ithasbeenargued[8]thatanaveragepoolinglosesinformation,whichhasmotivatedtheuseofotheroperatorssuchashierarchicalmaxima[9].Ascatteringavoidsthisinformationlossbyrecover-ingwaveletcoefcientsatthenextlayer,whichexplainstheimportanceofamultilayernetworkstructure.Ascatteringisimplementedbyadeepconvolutionnetwork[20],havingaveryspecicarchitecture.Asopposedtostandardconvolutionnetworks,outputscat-teringcoefcientsareproducedbyeachlayerasopposedtothelastlayer[20].Filtersarenotlearnedfromdatabutarepredenedwavelets.Indeed,theybuildinvariantsrelativelytotheactionofthetranslationgroupwhichdoesnotneedtobelearned.Buildinginvariantstootherknowngroupssuchasrotationsorscalingissimilarlyobtainedwithpredenedwavelets,whichperformcon-volutionsalongrotationorscalevariables[25],[26].Differentcomplexquadraturephasewaveletsmaybechosenbutseparatingsignalvariationsatdifferentscalesisfundamentalfordeformationstability[25].Usingamodulus(4)topulltogetherquadraturephaseltersisalsoimportanttoremovethehighfrequencyoscillationsofwaveletcoefcients.NextsectionexplainsthatitguaranteesafastenergydecayofpropagatedsignalsU[p]xacrosslayers,sothatwecanlimitthenetworkdepth.Foraxedpositionu,windowedscatteringcoef-cientsS[p]x(u)oforderm=1;2aredisplayedaspiecewiseconstantimagesoveradiskrepresentingtheFouriersupportoftheimagex.Thisfrequencydiskispartitionedintosectorsf\n[p]gp2Pmindexedbythepathp.TheimagevalueisS[p]x(u)onthefrequencysectors\n[p],showninFigure3.Form=1,ascatteringcoefcientS[1]x(u)dependsuponthelocalFouriertransformenergyofxoverthesupportof^ 1.Itsvalueisdisplayedoverasector\n[1]whichapproximatesthefrequencysupportof^ 1.For1=2 j1r1,thereareKrotatedsectorslocatedinanannulusofscale2 j1,correspondingtoeachr12G,asshownbyFigure3(a).Theirareaareproportionaltok 1k2K 12 j1.SecondorderscatteringcoefcientsS[1;2]x(u)arecomputedwithasecondwavelettransformwhichper-formsasecondfrequencysubdivision.Thesecoefcientsaredisplayedoverfrequencysectors\n[1;2]whichsubdividethesectors\n[1]oftherstwavelets^ 1,asillustratedinFigure3(b).For2=2 j2r2,thescale2j2dividestheradialaxisandtheresultingsectorsaresubdividedintoKangularsectorscorrespondingtothedifferentr2.Thescaleandangularsubdivisionsareadjustedsothattheareaofeach\n[1;2]isproportionaltokj 1j? 2k2.Figure4showstheFouriertransformoftwoimages,andtheamplitudeoftheirscatteringcoefcients.Inthiscasethe2Jisequaltotheimagesize.Thetopandbottomimagesareverydifferentbuttheyhavethesamerstorderscatteringcoefcients.Thesecondordercoefcientsclearlydiscriminatetheseimages.Section3.1showsthatthesecond-orderscatteringcoefcientsofthetopimagehavealargeramplitudebecausetheimagewaveletcoefcientsaremoresparse.Higher-ordercoef-cientsarenotdisplayedbecausetheyhaveanegligibleenergyasexplainedinSection3.3SCATTERINGPROPERTIESAconvolutionnetworkishighlynon-linear,whichmakesitdifculttounderstandhowthecoefcientvaluesrelatetothesignalproperties.Forascatter-ingnetwork,Section3.1analyzesthecoefcientprop-ertiesandoptimizesthenetworkarchitecture.Section3.2describestheresultingcomputationalalgorithm.Fortextureanalysis,thescatteringtransformofstationaryprocessesisstudiedinSection3.3.Section3.4showsthatacosinetransformfurtherreducesthesizeofascatteringrepresentation.3.1EnergyPropagationandDeformationStabilityAwindowedscatteringSiscomputedwithacascadeofwaveletmodulusoperatorsfW,anditspropertiesthusdependuponthewavelettransformproperties.Conditionsaregivenonwaveletstodeneascatter-ingtransformwhichisnonexpansiveandpreservesthesignalnorm.ThisanalysisshowsthatkS[p]xkdecreasesquicklyasthelengthofpincreases,andisnon-negligibleonlyoveraparticularsubsetoffrequency-decreasingpaths.Reducingcomputationstothesepathsdenesaconvolutionnetworkwithmuchfewerinternalandoutputcoefcients. 6 1]\n[1; (a)(b) Fig.3.Todisplayscatteringcoefcients,thediskcoveringtheimagefrequencysupportispartitionedintosectors\n[p],whichdependuponthepathp.(a):Form=1,each\n[1]isasectorrotatedbyr1whichapproximatesthefrequencysupportof^ 1.(b):Form=2,all\n[1;2]areobtainedbysubdividingeach\n[1]. (a) (b) (c) (d)Fig.4.(a)Twoimagesx(u).(b)Fouriermodulusj^x(!)j.(c)FirstorderscatteringcoefcientsSx[1]displayedoverthefrequencysectorsofFigure3(a).Theyarethesameforbothimages.(d)SecondorderscatteringcoefcientsSx[1;2]overthefrequencysectorsofFigure3(b).Theyaredifferentforeachimage.ThenormanddistanceonatransformTx=fxngnwhichoutputafamilyofsignalswillbedenedbykTx Tx0k2=Xnkxn x0nk2:Ifthereexists�0suchthatforall!2R21 j^ (!)j2+1 21Xj=0Xr2Gj^ (2jr!)j21;(7)thenapplyingthePlancherelformulaprovesthatifxisrealthenWx=fx? 2J;x? g2Psatises(1 )kxk2kWxk2kxk2;(8)withkWxk2=kx? 2Jk2+X2Pkx? k2:Inthefollowingwesupposethat1andhencethatthewavelettransformisanonexpansiveandinvertibleoperator,withastableinverse.If=0thenWisunitary.TheMorletwavelet showninFigure1togetherwith (u)=exp( juj2=(22))=(22)for=0:7satisfy(7)with=0:25.Thesefunctionsareusedinallclassi-cationapplications.Rotatedanddilatedcubicsplinewaveletsareconstructedin[25]tosatisfy(7)with=0.Themodulusisnonexpansiveinthesensethatjjaj jbjjja bjforall(a;b)2C2SincefW=fx? 2J;jx? jg2PisobtainedwithawavelettransformWfollowedbyamodulus,whicharebothnonexpansive,itisalsononexpansive:kfWx fWykkx yk:LetP1=[m2NPmbethesetofallpathsforanylengthm2N.ThenormofSx=fS[p]xgp2P1iskSxk2=Xp2P1kS[p]xk2:SinceSiterativelyappliesfWwhichisnonexpansive,itisalsononexpansive:kSx Sykkx yk:Itisthusstabletoadditivenoise. 7IfWisunitarythenfWalsopreservesthesignalnormkfWxk2=kxk2.TheconvolutionnetworkisbuiltlayerbylayerbyiteratingonfW.IffWpreservesthesignalnormthenthesignalenergyisequaltothesumofthescatteringenergyofeachlayerplustheenergyofthelastpropagatedlayer:kxk2= mXm=0Xp2PmkS[p]xk2+Xp2P m+1kU[p]k2:(9)Forappropriatewavelets,itisprovedin[25]thattheenergyofthemthlayerPp2PmkU[p]k2convergestozerowhenmincreases,aswellastheenergyofallscatteringcoefcientsbelowm.Thisresultisimportantfornumericalapplicationsbecauseitexplainswhythenetworkdepthcanbelimitedwithanegligiblelossofsignalenergy.Bylettingthenetworkdepth mgotoinnityin(9),itresultsthatthescatteringtransformpreservesthesignalenergykxk2=Xp2P1kS[p]xk2=kSxk2:(10)Thisscatteringenergyconservationalsoprovesthatthemoresparsethewaveletcoefcients,themoreenergypropagatestodeeperlayers.Indeed,when2Jincreases,onecanverifythatattherstlayer,S[1]x=jx? 1j? 2Jconvergestok k2kx? k21.Themoresparsex? ,thesmallerkx? k1andhencethemoreenergyispropagatedtodeeperlayerstosatisfytheglobalenergyconservation(10).Figure4showstwoimageshavingsamerstorderscatteringcoefcients,butthetopimageispiecewisereg-ularandhencehaswaveletcoefcientswhicharemuchmoresparsethantheuniformtextureatthebottom.Asaresultthetopimagehassecondorderscatteringcoefcientsoflargeramplitudethanatthebottom.Fortypicalimages,asintheCalTech101dataset[12],Table1showsthatthescatteringenergyhasanexponentialdecayasafunctionofthepathlengthm.Scatteringcoefcientsarecomputedwithcubicsplinewavelets,whichdeneaunitarywavelettransformandsatisfythescatteringenergyconservation(10).Asexpected,theenergyofscatteringcoefcientsconvergesto0asmincreases,anditisalreadybelow1%form3.ThepropagatedenergykU[p]xk2decaysbecauseU[p]xisaprogressivelylowerfrequencysignalasthepathlengthincreases.Indeed,eachmoduluscomputesaregularenvelopofoscillatingwaveletcoefcients.Themoduluscanthusbeinterpretedasanon-linearde-modulatorwhichpushesthewaveletcoefcientenergytowardslowerfrequencies.Asaresult,animportantportionoftheenergyofU[p]xisthencapturedbythelowpasslter 2JwhichoutputsS[p]x=U[p]x? 2J.Hencefewerenergyispropagatedtothenextlayer.Anotherconsequenceisthatthescatteringenergypropagatesonlyalongasubsetoffrequencydecreasingpaths.Sincetheenvelopejx? jismoreregularthanx? ,itresultsthatjx? (u)j? 0isnon-negligibleonlyTABLE1PercentageofenergyPp2Pm#kS[p]xk2=kxk2ofscatteringcoefcientsonfrequency-decreasingpathsoflengthm,dependinguponJ.TheseaveragevaluesarecomputedontheCaltech-101database,withzeromeanandunitvarianceimages. J m=0m=1m=2m=3m=4 m3 1 95.14.86--- 99.96 2 87.5611.970.35-- 99.89 3 76.2921.921.540.02- 99.78 4 61.5233.874.050.160 99.61 5 44.645.268.90.610.01 99.37 6 26.1557.0214.41.540.07 99.1 7 073.3721.983.560.25 98.91 if 0islocatedatlowerfrequenciesthan ,andhenceifj0jjj.Iteratingonwaveletmodulusoperatorsthuspropagatesthescatteringenergyalongfrequency-decreasingpathsp=(1;:::;m)wherejkjjk 1jfor1km.WedenotebyPm#thesetoffrequencyde-creasingpathsoflengthm.Scatteringcoefcientsalongotherpathshaveanegligibleenergy.ThisisveriedbyTable1whichshowsnotonlythatthescatteringenergyisconcentratedonlow-orderpaths,butalsothatmorethan99%oftheenergyisabsorbedbyfrequency-decreasingpathsoflengthm3.Numerically,itisthereforesufcienttocomputethescatteringtransformalongfrequency-decreasingpaths.Itdenesamuchsmallerconvolutionnetwork.Section3.2showsthattheresultingcoefcientsarecomputedwithO(NlogN)operations.Preservingenergydoesnotimplythatthesignalinfor-mationispreserved.Sinceascatteringtransformiscal-culatedbyiterativelyapplyingfW,invertingSrequirestoinvertfW.ThewavelettransformWisalinearinvert-ibleoperator,soinvertingfWz=fz? 2J;jz? jg2Pamountstorecoverthecomplexphasesofwaveletcoef-cientsremovedbythemodulus.ThephaseofFouriercoefcientscannotberecoveredfromtheirmodulusbutwaveletcoefcientsareredundant,asopposedtoFouriercoefcients.Forparticularwavelets,ithasbeenprovedthatthephaseofwaveletcoefcientscanberecoveredfromtheirmodulus,andthatfWhasacontinuousinverse[38].Still,onecannotexactlyinvertSbecausewediscardinformationwhencomputingthescatteringcoefcientsS[p]x=U[p]? 2JofthelastlayerP m.Indeed,thepropagatedcoefcientsjU[p]x? jofthenextlayerareeliminated,becausetheyarenotinvariantandhaveanegligibletotalenergy.Thenumberofsuchcoefcientsislargerthanthetotalnumberofscatteringcoefcientskeptatpreviouslayers.Initializingtheinversionbyconsideringthatthesesmallcoefcientsarezeropro-ducesanerror.ThiserrorisfurtherampliedastheinversionoffWprogressesacrosslayersfrom mto0.Numericalexperimentsconductedoverone-dimensionalaudiosignals,[2],[7]indicatethatreconstructedsig- 8nalshaveagoodaudioqualitywith m=2,aslongasthenumberofscatteringcoefcientsiscompara-bletothenumberofsignalsamples.Audioexamplesinwww.cmap.polytechnique.fr/scatteringshowthatrecon-structionsfromrstorderscatteringcoefcientsaretyp-icallyofmuchlowerqualitybecausetherearemuchfewerrstorderthansecondordercoefcients.Whentheinvariantscale2Jbecomestoolarge,thenumberofsecondordercoefcientsalsobecomestoosmallforaccuratereconstructions.Althoughindividualsignalscanbenotberecovered,reconstructionsofequivalentstationarytexturesarepossiblewitharbitrarilylargescalescatteringinvariants[7].Forclassicationapplications,besidescomputingarichsetofinvariants,themostimportantpropertyofascatteringtransformisitsLipschitzcontinuitytodeformations.Indeedwaveletsarestabletodeforma-tionsandthemoduluscommuteswithdeformations.Letx (u)=x(u (u))beanimagedeformedbythedisplacementeld .Letk k1=supuj (u)jandkr k1=supujr (u)j1.IfSxiscomputedonpathsoflengthm mthenitisprovedin[25]thatforsignalsxofcompactsupportkSx SxkC mkxk2 Jk k1+kr k1;(11)withasecondorderHessiantermwhichispartofthemetricdenitiononC2deformations,butwhichisnegligibleif (u)isregular.If2Jk k1=kr k1thenthetranslationtermcanbeneglectedandthetransformisLipschitzcontinuoustodeformations:kSx SxkC mkxkkr k1:(12)If mgoesto1thenC mcanbereplacedbyamorecom-plexexpression[25],whichisnumericallyconvergingfornaturalimages.3.2FastScatteringComputationsWedescribeafastscatteringimplementationoverfre-quencydecreasingpaths,wheremostofthescatteringenergyisconcentrated.Afrequencydecreasingpathp=(2 j1r1;:::;2 jmrm)satises0jkjk+1J.IfthewavelettransformiscomputedoverKrotationanglesthenthetotalnumberoffrequency-decreasingpathsoflengthmisKm Jm.LetNbethenumberofpixelsoftheimagex.Since 2Jisalow-passlterscaledby2J,S[p]x(u)=U[p]x? 2J(u)isuniformlysampledatintervals 2J,with =1or =1=2.EachS[p]xisanimagewith 22 2JNcoefcients.Thetotalnumberofcoefcientsinascatteringnetworkofmaximumdepth misthusP=N 22 2J mXm=0KmJm:(13)If m=2thenP' 2N2 2JK2J2=2.Itdecreasesexponentiallywhenthescale2Jincreases.Algorithm1describesthecomputationsofscatteringcoefcientsonsetsPm#offrequencydecreasingpathsoflengthm m.TheinitialsetP0#=f;gcorrespondstotheoriginalimageU[;]x=x.Letp+bethepathwhichbeginsbypandendswith2P.If=2 jrthenU[p+]x(u)=jU[p]x? (u)jhasenergyatfrequenciesmostlybelow2 j.Toreducecomputationswecanthussubsamplethisconvolutionatintervals 2j,with =1or =1=2toavoidaliasing. Algorithm1FastScatteringTransform form=1to mdoforallp2Pm 1#doOutputS[p]x( 2Jn)=U[p]x? 2J( 2Jn)endforforallp+m2Pm#withm=2 jmrmdoComputeU[p+m]x( 2jmn)=jU[p]x? m( 2jmn)jendforendforforallp2P m#doOutputS[p]x( 2Jn)=U[p]x? 2J( 2Jn)endfor AtthelayermthereareKm JmpropagatedsignalsU[p]xwithp2Pm#.Theyaresampledatintervals 2jmwhichdependonp.Onecanverifybyinductiononmthatthelayermhasatotalnumberofsamplesequalto 2(K=3)mN.TherearealsoKm JmscatteringsignalsS[p]xbuttheyaresubsampledby2Jandthushavemuchlesscoefcients.ThenumberofoperationtocomputeeachlayeristhereforedrivenbytheO((K=3)mNlogN)operationsneededtocomputetheinternalpropagatedcoefcientswithFFT's.ForK�3,theoverallcomputa-tionalcomplexityisthusO((K=3) mNlogN).3.3ScatteringStationaryProcessesImagetexturescanbemodeledasrealizationsofsta-tionaryprocessesX(u).WedenotetheexpectedvalueofXbyE(X),whichdoesnotdependuponu.De-spitetheimportanceofspectralmethods,thepowerspectrumisoftennotsufcienttodiscriminateimagetexturesbecauseitonlydependsuponsecondordermoments.Figure5showsverydifferenttextureshavingsamepowerspectrum.Ascatteringrepresentationofstationaryprocessesdependsuponsecondorderandhigher-ordermoments,andcanthusdiscriminatesuchtextures.Moreover,itdoesnotsufferfromthelargevariancecurseofhighordermomentsestimators[37],becauseitiscomputedwithanonexpansiveoperator.IfX(u)isstationarythenU[p]X(u)remainsstationarybecauseitiscomputedwithacascadeofconvolutionsandmoduluswhichpreservestationarity.Itsexpectedvaluethusdoesnotdependuponuanddenestheexpectedscatteringtransform: SX(p)=E(U[p]X): 9 (a) (b) (c) (d)Fig.5.(a)RealizationsoftwostationaryprocessesX(u).Top:Brodatztexture.Bottom:Gaussianprocess.(b)Thepowerspectrumestimatedfromeachrealizationisnearlythesame.(c)FirstorderscatteringcoefcientsS[p]Xarenearlythesame,for2Jequaltotheimagewidth.(d)SecondorderscatteringcoefcientsS[p]Xareclearlydifferent.Awindowedscatteringgivesanestimatorof SX(p),calculatedfromasinglerealizationofX,byaveragingU[p]Xwith 2J:S[p]X(u)=U[p]X? 2J(u):SinceR 2J(u)du=1,thisestimatorisunbiased:E(S[p]X)=E(U[p]X)= SX(p).Forappropriatewavelets,itisprovedin[25]thatawindowedscatteringtransformconservesthesecondmomentofstationaryprocesses:Xp2P1E(jS[p]Xj2)=E(jXj2):(14)Thesecondordermomentsofallwaveletcoefcient,whichareusefulfortexturediscrimination,canalsoberecoveredfromscatteringcoefcients.Indeed,forp=(1;:::;m)ifwewrite+p=(;1;:::;m)thenS[p]jX? j=S[p]U[]X=S[+p]XandreplacingXbyjX? jin(14)givesXp2P1E(jS[+p]Xj2)=E(jX? j2):(15)However,ifphasalengthm,becauseofthemsuccessivemodulusnon-linearities,onecanshow[25]that SX(p)alsodependsuponnormalizedhighordermomentsofX,mainlyoforderupto2m.Scatteringcoefcientscanthusdiscriminatetextureshavingsamesecond-ordermomentsbutdifferenthigher-ordermoments.ThisisillustratedbythetwotexturesinFigure5,whichhavesamepowerspectrumandhencesamesecondordermoments.ScatteringcoefcientsS[p]Xareshownform=1andm=2withthefrequencytilingillustratedinFigure3.Thesquareddistancebetweentheorder1scatteringcoefcientsofthesetwotexturesisofordertheirvariance.Indeed,order1scatteringcoefcientsmostlydependuponsecond-ordermomentsandarethusnearlyequalforbothtextures.Onthecontrary,scatteringcoefcientsoforder2aredifferentbecausetheydependonmomentsupto4.Theirsquareddistanceismorethan5timesbiggerthantheirvariance.Highordermomentaredifculttouseinsignalprocessingbecausetheirestimatorshavealargevariance[37],whichcanintroduceimportanterrors.Thislargevariancecomesfromtheblowupoflargecoefcientout-liersproducedbyXqforq�2.Onthecontrary,ascatter-ingiscomputedwithanonexpansiveoperatorandthushasmuchlowervarianceestimators.Theestimationof SX(p)=E(U[p]X)byS[p]X=U[p]X? 2Jhasavari-ancewhichisreducedwhentheaveragingscale2Jin-creases.Forallimagetextures,itisnumericallyobservedthatthescatteringvariancePp2P1E(jS[p]X SX(p)j2decreasesexponentiallytozerowhen2Jincreases.Table2givesthedecayofthisscatteringvariance,computedonaverageovertheBrodatztexturedataset.Expectedscatteringcoefcientsofstationarytexturesarethusbetterestimatedfromwindowedscatteringtranformsatthelargestpossiblescale2J,equaltotheimagesize.Let P1bethesetofallpathsp=(1;:::;m)forallk=2jkrk22ZG+andalllengthm.Theconservationequation(14)togetherwiththescatteringvariancedecayalsoimpliesthatthesecondmomentisequaltotheenergyofexpectedscatteringcoefcientsin P1k SXk2=Xp2 P1j SX(p)j2=E(jXj2):(16) 10TABLE2NormalizedscatteringvariancePp2P1E(jS[p]X SX(p)j2)=E(jXj2),asafunctionofJ,computedonzero-meanandunitvarianceimagesoftheBrodatzdataset,withcubicsplinewavelets. J=1J=2J=3J=4J=5J=6J=7 0.850.650.450.260.140.070.0025 TABLE3PercentageofenergyPp2Pm#j SX(p)j2=E(jXj2)alongfrequencydecreasingpathsoflengthm,computedonthenormalizedBrodatzdataset,withcubicsplinewavelets. m=0m=1m=2m=3m=4 0741930.3 IndeedE(S[p]X)= SX(p)soE(jS[p]Xj2)= SX(p)2+E(jS[p]X E(S[p]X)j2):SummingoverpandlettingJgoto1gives(16).Table3givestheratiobetweentheaverageenergyalongfrequencydecreasingpathsoflengthmandsec-ondmoments,fortexturesintheBrodatzdataset.Mostofthisenergyisconcentratedoverpathsoflengthm3.3.4CosineScatteringTransformNaturalimageshavescatteringcoefcientsS[p]X(u)whicharecorrelatedacrosspathsp=(1;:::;m),atanygivenpositionu.Thestrongestcorrelationisbetweencoefcientsofasamelayer.Foreachm,scatteringcoef-cientsaredecorrelatedinaKarhunen-Lo evebasiswhichdiagonalizestheircovariancematrix.Figure6comparesthedecayofthesortedvariancesE(jS[p]X E(S[p]X)j2)andthevariancedecayintheKarhunen-Lo evebasiscomputedoverhalfoftheCaltechimagedataset,fortherstlayerandsecondcoefcients.ScatteringcoefcientsarecalculatedwithaMorletwavelet.Thevariancedecay(computedonthesecondhalfdataset)ismuchfasterintheKarhunen-Lo evebasis,whichshowsthatthereisastrongcorrelationbetweenscatteringcoefcientsofsamelayers.AchangeofvariablesprovesthatarotationandscalingX2lr(u)=X(2 lru)producesarotationandinversescalingonthepathvariable SX2lr(p)= SX(2lrp)where2lrp=(2lr1;:::;2lrm);and2lrk=2l jkrrk.Ifnaturalimagescanbecon-sideredasrandomlyrotatedandscaled[29],thenthepathpisrandomlyrotatedandscaled.Inthiscase,thescatteringtransformhasstationaryvariationsalongthescaleandrotationvariables.Thissuggestsapproximat-ingtheKarhunen-Lo evebasisbyacosinebasisalongthesevariables.Letusparametrizeeachrotationrbyitsangle2[0;2].Apathp=(2 j1r1;:::;2 jkrk)isthenparametrizedby((j1;1);:::;(jm;m)).Sincescatteringcoefcientsarecomputedalongfre-quencydecreasingpathsforwhich0jkjk+1J,toreduceboundaryeffects,aseparablecosinetransformiscomputedalongthevariablesl1=j1;l2=j2 j1;:::;lm=jm jm 1,andalongeachanglevariable1;2;:::;m.Cosinescatteringcoefcientsarebyap-plyingthisseparablediscretecosinetransformalongthescaleandanglevariablesofS[p]X(u),foreachuandeachpathlengthm.Figure6showsthatthecosinescatteringcoefcientshavevariancesform=1andm=2whichdecaynearlyasfastasthevariancesintheKarhunen-Lo evebasis.ItshowsthataDCTacrossscalesandorientationsisnearlyoptimaltodecorrelatescatteringcoefcients.Lower-frequencyDCTcoefcientsabsorbmostofthescatteringenergy.Onnaturalimages,morethan99.5%ofthescatteringenergyisabsorbedbythe1=2lowestfrequencycosinescatteringcoefcients.Wesawin(13)thatwithoutoversampling =1,when m=2,animageofsizeNisrepresentedbyP=N2 2J(KJ+K2J(J 1)=2)scatteringcoefcients.NumericalcomputationsareperformedwithK=6rota-tionanglesandtheDCTreducesatleastby2thenumberofcoefcients.AtasmallinvariantscaleJ=2,theresultingcosinescatteringrepresentationhasP=3N=2coefcients.Asamatterofcomparison,SIFTrepresentssmallblocksof42pixelswith8coefcients,andadenseSIFTrepresentationthushasN=2coefcients.WhenJincreases,thesizeofacosinescatteringrepresentationdecreaseslike2 2J,withP=NforJ=3andPN=40forJ=7.4CLASSIFICATIONAscatteringtransformeliminatestheimagevariabilityduetotranslationsandlinearizessmalldeformations.ClassicationisstudiedwithlineargenerativemodelscomputedwithaPCA,andwithdiscriminantSVMclassiers.State-of-the-artresultsareobtainedforhand-writtendigitrecognitionandfortexturediscrimination.ScatteringrepresentationsarecomputedwithaMorletwavelet.4.1PCAAfneSpaceSelectionAlthoughdiscriminantclassierssuchasSVMhavebetterasymptoticpropertiesthangenerativeclassiers[28],thesituationcanbeinvertedforsmalltrainingsets.WeintroduceasimplerobustgenerativeclassierbasedonafnespacemodelscomputedwithaPCA.ApplyingaDCTonscatteringcoefcientshasnoeffectonanylinearclassierbecauseitisalinearorthogonaltrans-form.Keepingthe50%lowerfrequencycosinescatteringcoefcientsreducescomputationsandhasanegligibleeffectonclassicationresults.Theclassicationalgo-rithmisdescribeddirectlyonscatteringcoefcientstosimplifyexplanations.Eachsignalclassisrepresented 11 0 2 4 6 8 10 12 14 16 18 10-3 10-2 10-1 100 101 102 order 1 A B C 0 20 40 60 80 100 120 10-6 10-4 10-2 100 102 order 2 A B C Fig.6.(A):Sortedvariancesofscatteringcoefcientsoforder1(left)andorder2(right),computedontheCalTech101database.(B):Sortedvariancesofcosinetransformscatteringcoefcients.(C):SortedvariancesinaKarhunen-Lo evebasiscalculatedforeachlayerofscatteringcoefcients.byarandomvectorXk,whoserealizationsareimagesofNpixelsintheclass.EachscatteringvectorSXkhasPcoefcients.LetE(SXk)betheexpectedvectoroverthesignalclassk.ThedifferenceSXk E(SXk)isapproximatedbyitsprojectioninalinearspaceoflowdimensiond P.ThecovariancematrixofSXkhasP2coefcients.LetVkbethelinearspacegeneratedbythedPCAeigenvectorsofthiscovariancematrixhavingthelargesteigenvalues.Amongalllinearspacesofdimensiond,itisthespacewhichapproximatesSXk E(SXk)withthesmallestexpectedquadraticerror.Thisisequivalenttoapproxi-matingSXkbyitsprojectiononanafneapproximationspace:Ak=EfSXkg+Vk:Theclassierassociatestoeachsignalxtheclassindex^kofthebestapproximationspace:^k(x)=argminkCkSx PAk(Sx)k:(17)Theminimizationofthisdistancehassimilaritieswiththeminimizationofatangentialdistance[14]inthesensethatweremovetheprincipalscatteringdirectionsofvariabilitiestoevaluatethedistance.HoweveritismuchsimplersinceitdoesnotevaluateatangentialspacewhichdependsuponSx.LetV?kbetheorthogonalcomplementofVkcorrespondingtodirectionsoflowervariability.ThisdistanceisalsoequaltothenormofthedifferencebetweenSxandtheaverageclasstemplateE(SXk),projectedinV?k:kSx PAk(Sx)k=\r\r\rPV?kSx E(SXk)\r\r\r:(18)MinimizingtheafnespaceapproximationerroristhusequivalenttondingtheclasscentroidE(SXk)whichistheclosesttoSx,withouttakingintoaccounttherstdprincipalvariabilitydirections.ThedprincipaldirectionsofthespaceVkresultfromdeformationsandfromstructuralvariability.TheprojectionPAk(Sx)istheoptimumlinearpredictionofSxfromthesedprincipalmodes.Theselectedclasshasthesmallestpredictionerror.ThisafnespaceselectioniseffectiveifSXk E(SXk)iswellapproximatedbyaprojectioninalow-dimensionalspace.ThisisthecaseifrealizationsofXkaretranslationsandlimiteddeformationsofasingletemplate.Indeed,theLipschitzcontinuityimpliesthatsmalldeformationsarelinearizedbythescatteringtrans-form.Hand-writtendigitrecognitionisanexample.ThisisalsovalidforstationarytextureswhereSXkhasasmallvariance,whichcanbeinterpretedasstructuralvariability.ThedimensiondmustbeadjustedsothatSXkhasabetterapproximationintheafnespaceAkthaninafnespacesAlofotherclassesl=k.Thisisamodelselectionproblem,whichrequirestooptimizethedimensiondinordertoavoidover-tting[5].Theinvariancescale2Jmustalsobeoptimized.Whenthescale2Jincreases,translationinvarianceincreasesbutitcomeswithapartiallossofinformation,whichbringstherepresentationsofdifferentsignalscloser.Onecanprove[25]thatthescatteringdistancekSx Sx0kde-creaseswhen2Jincreases,anditconvergestoanon-zerovaluewhen2Jgoesto1.Toclassifydeformedtemplatessuchashand-writtendigits,theoptimal2Jisoftheorderofthemaximumpixeldisplacementsduetotranslationsanddeformations.Inastochasticframeworkwherexandx0arerealizationsofstationaryprocesses,SxandSx0convergetotheexpectedscatteringtransforms Sxand Sx0.Inordertoclassifystationaryprocessessuchastextures,theoptimalscaleisthemaximumscaleequaltotheimagewidth,becauseitminimizesthevarianceofthewindowedscatteringestimator.Across-validationprocedureisusedtondthedi-mensiondandthescale2Jwhichyieldthesmallestclassicationerror.Thiserroriscomputedonasubsetofthetrainingimages,whichisnotusedtoestimatethecovariancematrixforthePCAcalculations.AsinthecaseofSVM,theperformanceoftheafnePCAclassierareimprovedbyequalizingthedescriptorspace.Table1showsthatscatteringvectorshaveunequalenergydistributionalongitspathvariables,inparticularastheordervaries.Arobustequalizationisobtainedby 12dividingeachS[p]X(u)by\r(p)=maxxiXujS[p]xi(u)j21=2;(19)wherethemaximumiscomputedoveralltrainingsig-nalsxi.Tosimplifynotations,westillwriteSXthevec-torofnormalizedscatteringcoefcientsS[p]X(u)=\r(p).Afnespacescatteringmodelscanbeinterpretedasgenerativemodelscomputedindependentlyforeachclass.AsopposedtodiscriminativeclassierssuchasSVM,wedonotestimatecross-correlationinteractionsbetweenclasses,besidesoptimizingthemodeldimen-siond.Suchestimatorsareparticularlyeffectiveforsmallnumberoftrainingsamplesperclass.Indeed,iftherearefewtrainingsamplesperclassthenvariancetermsdominatebiaserrorswhenestimatingoff-diagonalcovariancecoefcientsbetweenclasses[4].Anafnespaceapproximationclassiercanalsobeinterpretedasarobustquadraticdiscriminantclassierobtainedbycoarselyquantizingtheeigenvaluesoftheinversecovariancematrix.Foreachclass,theeigenval-uesoftheinversecovariancearesetto0inVkandto1inV?k,wheredisadjustedbycross-validation.Thiscoarsequantizationisjustiedbythepoorestimationofcovarianceeigenvaluesfromfewtrainingsamples.Theseafnespacemodelsarerobustwhenappliedtodistributionsofscatteringvectorshavingnon-Gaussiandistributions,whereaGaussianFisherdiscriminantcanleadtosignicanterrors.4.2HandwrittenDigitRecognitionTheMNISTdatabaseofhand-writtendigitsisanexam-pleofstructuredpatternclassication,wheremostoftheintra-classvariabilityisduetolocaltranslationsanddeformations.Itcomprisesatmost60000trainingsam-plesand10000testsamples.Ifthetrainingdatasetisnotaugmentedwithdeformations,thestateoftheartwasachievedbydeep-learningconvolutionnetworks[31],deformationmodels[17],[3],anddictionarylearning[27].Theseresultsareimprovedbyascatteringclassier.AllcomputationsareperformedonthereducedcosinescatteringrepresentationdescribedinSection3.4,whichkeepsthelower-frequencyhalfofthecoefcients.Table4computesclassicationerrorsonaxedsetoftestimages,dependinguponthesizeofthetrainingset,fordifferentrepresentationsandclassiers.Theafnespaceselectionofsection4.1iscomparedwithanSVMclassierusingRBFkernels,whicharecomputedus-ingLibsvm[10],andwhosevarianceisadjustedusingstandardcross-validationoverasubsetofthetrainingset.TheSVMclassieristrainedwitharenormalizationwhichmapsallcoefcientsto[ 1;1].ThePCAclassieristrainedwiththerenormalisationfactors(19).ThersttwocolumnsofTable4showthatclassicationerrorsaremuchsmallerwithanSVMthanwiththePCAalgorithmifapplieddirectlyontheimage.The3rdand4thcolumnsgivetheclassicationerrorobtainedwithaPCAoranSVMclassicationappliedtothemodulusofawindowedFouriertransform.Thespatialsize2Jofthewindowisoptimizedwithacross-validationwhichyieldsaminimumerrorfor2J=8.Itcorrespondstothelargestpixeldisplacementsduetotranslationsordeformationsineachclass.RemovingthecomplexphaseofthewindowedFouriertransformyieldsalocallyinvariantrepresentationbutwhosehighfrequenciesareunstabletodeformations,asexplainedinSection2.1.Suppressingthislocaltranslationvariabilityimprovestheclassicationratebyafactor3foraPCAandbyalmost2foranSVM.ThecomparisonbetweenPCAandSVMconrmsthefactthatgenerativeclassierscanoutperformdiscriminativeclassierswhentrainingsamplesarescarce[28].Asthetrainingsetsizeincreases,thebias-variancetrade-offturnsinfavorofthericherSVMclassiers,independentlyofthedescriptor.Columns6and8givethePCAclassicationresultappliedtoawindowedscatteringrepresentationfor m=1and m=2.Thecrossvalidationalsochooses2J=8.Figure7displaysthearraysofnormalizedwindowedscatteringcoefcientsofadigit`3'.TherstandsecondordercoefcientsofS[p]X(u)aredisplayedasenergydistributionsoverfrequencydisksdescribedinSection2.3.Thespatialparameteruissampledatintervals2JsoeachimageofNpixelsisrepresentedbyN2 2J=42translateddisks,bothfororder1andorder2coefcients.Increasingthescatteringorderfrom m=1to m=2reduceserrorsbyabout30%,whichshowsthatsecondordercoefcientscarryimportantinformationevenatarelativelysmallscale2J=8.However,thirdordercoefcientshaveanegligibleenergyandincludingthembringsmarginalclassicationimprovements,whilein-creasingcomputationsbyanimportantfactor.Asthelearningsetincreasesinsize,theclassicationimprove-mentofascatteringtransformincreasesrelativelytowindowedFouriertransformbecausetheclassicationisabletoincorporatemorehighfrequencystructures,whichhavedeformationinstabilitiesintheFourierdo-mainasopposedtothescatteringdomain.Table4thatbelow5000trainingsamples,thescatter-ingPCAclassierimprovesresultsofadeep-learningconvolutionnetworks,whichlearnsallltercoefcientswithaback-propagationalgorithm[20].Asmoretrain-ingsamplesareavailable,theexibilityoftheSVMclas-sierbringsanimprovementoverthemorerigidafneclassier,yieldinga0:43%errorrateontheoriginaldataset,thusimprovinguponpreviousstateoftheartmethods.Toevaluatetheprecisionofafnespacemodels,wecomputeanaveragenormalizedapproximationerrorofscatteringvectorsprojectedontheafnespaceoftheirownclass,overallclassesk2d=C 1CXk=1E(kSXk PAk(SXk)k2) E(kSXkk2):(20)Anaverageseparationfactormeasurestheratiobetween 13 (a) (b) (c)Fig.7.(a):ImageX(u)ofadigit'3'.(b):ArraysofwindowedscatteringcoefcientsS[p]X(u)oforderm=1,withusampledatintervalsof2J=8pixels.(c):WindowedscatteringcoefcientsS[p]X(u)oforderm=2.TABLE4PercentageoferrorsofMNISTclassiers,dependingonthetrainingsize. Training x Wind.Four. Scat. m=1 Scat. m=2 Conv. size PCASVM PCASVM PCASVM PCASVM Net. 300 14:515:4 7:357:4 5:78 4:75:6 7:18 1000 7:28:2 3:743:74 2:354 2:32:6 3:21 2000 5:86:5 2:992:9 1:72:6 1:31:8 2:53 5000 4:94 2:342:2 1:61:6 1:031:4 1:52 10000 4:553:11 2:241:65 1:51:23 0:881 0:85 20000 4:252:2 1:921:15 1:40:96 0:790:58 0:76 40000 4:11:7 1:850:9 1:360:75 0:740:53 0:65 60000 4:31:4 1:800:8 1:340:62 0:70:43 0:53 TABLE5ForeachMNISTtrainingsize,thetablegivesthecross-validateddimensiondofafneapproximationspaces,togetherwiththeaverageapproximationerror2dandseparationratio2dofthesespaces. Training d22 300 5310 12 5000 100410 23 40000 140210 24 theapproximationerrorintheafnespaceAkofthesignalclassandtheminimumapproximationerrorinanotherafnemodelAlwithl=k,forallclassesk2d=C 1CXk=1E(minl=kkSXk PAl(SXk)k2) E(kSXk PAk(SXk)k2):(21)Forascatteringrepresentationwith m=2,Table5givesthedimensiondofafneapproximationspacesoptimizedwithacrossvalidation.Itvariesconsiderably,rangingfrom5to140whenthenumberoftrainingexamplesgoesfrom300to40000.Indeed,manytrainingsamplesareneededtoestimatereliablytheeigenvectorsofthecovariancematrixandthustocomputereliableafnespacemodelsforeachclass.Theaverageap-proximationerror2dofafnespacemodelsisprogres-sivelyreducedwhiletheseparationratio2dincreases.ItexplainsthereductionoftheclassicationerrorrateobservedinTable4,asthetrainingsizeincreases.TABLE6PercentageoferrorsforthewholeUSPSdatabase. Tang. Scat. m=2 Scat. m=1 Scat. m=2 Kern. SVM PCA PCA 2:4 2:7 3:24 2:6=2:3 TheUS-PostalServiceisanotherhandwrittendigitdataset,with7291trainingsamplesand2007testimages1616pixels.Thestateoftheartisobtainedwithtangentdistancekernels[14].Table6givesresultsobtainedwithascatteringtransformwiththePCAclassierfor m=1;2.Thecross-validationsetsthescatteringscaleto2J=8.AsintheMNISTcase,theerrorisreducedwhengoingfrom m=1to m=2butremainsstablefor m=3.Differentrenormalizationstrategiescanbringmarginalimprovementsonthisdataset.Iftherenormalizationisperformedbyequalizingusingthestandarddeviationofeachcomponent,theclassicationerroris2:3%whereasitis2:6%ifthesupremumisnormalized.Thescatteringtransformisstablebutnotinvarianttorotations.StabilitytorotationsisdemonstratedovertheMNISTdatabaseinthesettingdenedin[18].Adatabasewith12000trainingsamplesand50000testimagesisconstructedwithrandomrotationsofMNISTdigits.ThePCAafnespaceselectiontakesintoaccounttherotationvariabilitybyincreasingthedimensiondoftheafneapproximationspace.Thisisequivalenttoprojectingthedistancetotheclasscentroidonasmallerorthogonalspace,byremovingmoreprincipal 14TABLE7PercentageoferrorsonanMNISTrotateddataset[18]. Scat. m=1 Scat. m=2 Conv. PCA PCA Net. 8 4:4 8:8 TABLE8Percentageoferrorsonscaled/rotatedMNISTdigits Transformations Scat. m=1 Scat. m=2 onMNISTimages PCA PCA None 1:6 0:8 Rotations 6:7 3:3 Scalings 2 1 Rot.+Scal. 12 5:5 components.TheerrorrateinTable7ismuchsmallerwithascatteringPCAthanwithaconvolutionnetwork[18].Muchbetterresultsareobtainedforascatteringwith m=2thanwith m=1becausesecondordercoefcientsmaintainenoughdiscriminabilitydespitetheremovalofalargernumberdofprincipaldirections.Inthiscase, m=3marginallyreducestheerror.Scalingandrotationinvarianceisstudiedbyintro-ducingarandomscalingfactoruniformlydistributedbetween1=p 2andp 2,andarandomrotationbyauni-formangle.Inthiscase,thedigit`9'isremovedfromthedatabaseastoavoidanyindeterminationwiththedigit`6'whenrotated.Thetrainingsethas9000samples(1000samplesperclass).Table8givestheerrorrateontheoriginalMNISTdatabasewhentransformingthetrainingandtestingsampleseitherwithrandomrotations,scal-ings,orboth.Scalingshaveasmallerimpactontheerrorratethanrotationsbecausescaledscatteringvectorsspananinvariantlinearspaceoflowerdimension.Second-orderscatteringoutperformsrst-orderscattering,andthedifferencebecomesmoresignicantwhenrotationandscalingarecombined.Secondordercoefcientsarehighlydiscriminativeinpresenceofscalingandrotationvariability.4.3TextureDiscriminationVisualtexturediscriminationremainsanoutstandingimageprocessingproblembecausetexturesarerealiza-tionsofnon-Gaussianstationaryprocesses,whichcannotbediscriminatedusingthepowerspectrum.TheafnePCAspaceclassierremovesmostofthevariabilityofSX EfSXgwithineachclass.ThisvariabilityisduetotheresidualstochasticvariabilitywhichdecaysasJincreases,andtovariabilityduetoillumination,rotation,scaling,orperspectivedeformationswhentexturesaremappedonsurfaces.TextureclassicationistestedontheCUReTtexturedatabase[21],[36],whichincludes61classesofimagetexturesofN=2002pixels.Eachtextureclassgivesimagesofthesamematerialwithdifferentposeandilluminationconditions.Specularities,shadowingandsurfacenormalvariationsmakeclassicationchalleng-ing.Posevariationrequiresglobalrotationandillumi-nationinvariance.Figure8illustratesthelargeintra-classvariability,afteranormalizationofthemeanandvarianceofeachtexturedimage.Table9compareserrorratesobtainedwithdifferentimagerepresentations.Thedatabaseisrandomlysplitintoatrainingandatestingset,with46trainingimagesforeachclassasin[36].Resultsareaveragedover10differentsplits.APCAafnespaceclassierapplieddirectlyontheimagepixelsyieldsalargeclassicationerrorof17%.Thelowestpublishedclassicationerrorsobtainedonthisdatasetare2%forMarkovRandomFields[36],1:53%foradictionaryoftextons[15],1:4%forBasicImageFeatures[11]and1%forhistogramsofimagevariations[6].APCAclassierappliedtoaFourierpowerspectrumestimatoralsoreaches1%error.ThepowerspectrumisestimatedwithwindowedFouriertransformscalculatedoverhalf-overlappingwin-dows,whosesquaredmodulusareaveragedoverthewholeimagetoreducetheestimatorvariance.Across-validationoptimizesthewindowsizeto2J=32pixels.ForthescatteringPCAclassier,thecrossvalidationchoosesanoptimalscale2Jequaltotheimagewidthtoreducethescatteringestimationvariance.Indeed,contrarilytoapowerspectrumestimation,thevarianceofthescatteringvectordecreaseswhen2Jincreases.Fig-ure9displaysthescatteringcoefcientsS[p]Xoforderm=1andm=2ofaCureTtexturedimageX.APCAclassicationwithonlyrstordercoefcients(mmax=1)yieldsanerror0:5%,althoughrst-orderscatteringco-efcientsarestronglycorrelatedwithsecondordermo-ments,whosevaluesdependontheFourierspectrum.TheclassicationerrorisimprovedrelativelytoapowerspectrumestimatorbecauseSX[1]X=jX? 1j? 2Jisanestimatorofarstordermoment S[1]X=E(jX? 1j)andthushasalowervariancethansecondordermomentestimators.APCAclassicationwithrstandsecondorderscatteringcoefcients(mmax=2)reducestheerrorto0:2%.Indeed,scatteringcoefcientsoforderm=2dependuponmomentsoforder4,whicharenecessarytodifferentiatetextureshavingsamesecondordermomentsasinFigure5.Moreover,theestimationof S[1;2]X=E(jjX? 1j? 2j)hasalowvariancebecauseXistransformedbyanonexpansiveoperatorasopposedtoXqforhighordermomentsq2.For m=2,thecrossvalidationchoosesafnespacemodelsofsmalldimensiond=16.However,theystillproduceasmallaverageapproximationerror(20)2d=2:510 1andtheseparationratio(21)is2d=3.ThePCAclassierprovidesapartialrotationinvari-ancebyremovingprincipalcomponents.Itmostlyaver-agesthescatteringcoefcientsalongrotatedpaths.Therotationofp=(2 j1r1;:::;2 jmrm)byrisdenedbyrp=(2 j1rr1;:::;2 jmrrm).Thisrotationinvarianceob-tainedbyaveragingcomesatthecostofareducedrep-resentationdiscriminability.Asinthetranslationcase,a 15 Fig.8.ExamplesoftexturesfromtheCUReTdatabasewithnormalizedmeanandvariance.Eachrowcorrespondstoadifferentclass,showingintra-classvariabilityintheformofstochasticvariabilityandchangesinposeandillumination. (a) (b) (c)Fig.9.(a):ExampleofCureTtextureX(u).(b):FirstorderscatteringcoefcientsS[p]X,for2Jequaltotheimagewidth.(c):SecondorderscatteringcoefcientsS[p]X(u).TABLE9PercentageofclassicationerrorsofdifferentalgorithmsonCUReT. Training X MRF Textons BIF Histo. Four.Spectr. Scat. m=1 Scat. m=2 size PCA [36] [15] [11] [6] PCA PCA PCA 46 17 2 1:5 1:4 1 1 0:5 0:2 multilayerscatteringalongrotationsrecoverstheinfor-mationlostbythisaveragingwithwaveletconvolutionsalongrotationangles[26].Itpreservesdiscriminabilitybyproducingalargernumberofinvariantcoefcientstotranslationsandrotations,whichimprovesrotationin-varianttexturediscrimination[26].Thiscombinedtrans-lationandrotationscatteringyieldsatranslationandrotationinvariantrepresentation,whichremainsstabletodeformations[25].5CONCLUSIONAscatteringtransformisimplementedbyadeepconvo-lutionnetwork.Itcomputesatranslationinvariantrepre-sentationwhichisLipschitzcontinuoustodeformations,withwaveletltersandamoduluspoolingnon-linearity.Averagedscatteringcoefcientsareprovidedbyeachlayer.TherstlayergivesSIFT-typedescriptors,whicharenotsufcientlyinformativeforlarge-scaleinvariance.Thesecondlayerprovidesimportantcoefcientsforclassication.Thedeformationstabilitygivesstate-of-the-artclas-sicationresultsforhandwrittendigitrecognitionandtexturediscrimination,withSVMandPCAclassiers.IfthedatasethasothersourcesofvariabilityduetotheactionofanotherLiegroupsuchasrotations,thenthisvariabilitycanalsobeeliminatedwithaninvariantscatteringcomputedonthisgroup[25],[26].IncompleximagedatabasessuchasCalTech256orPascal,importantsourcesofimagevariabilitydonotresultfromtheactionaknowngroup.Unsupervisedlearningisthennecessarytotakeintoaccountthisunknownvariability.Fordeepconvolutionnetworks,itinvolveslearningltersfromdata[20].Awaveletscatteringtransformcanthenprovidethersttwolayersofsuchnetworks.Iteliminatestranslationorrotationvariability,whichcanhelplearningthenextlayers. 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JoanBrunaJoanBrunagraduatedfromUniver-sitatPolitecnicadeCatalunyainbothMathemat-icsandElectricalEngineering,in2002and2004respectively.HeobtainedanMScinappliedmathematicsfromENSCachanin2005.From2005to2010,hewasaresearchengineerinanimageprocessingstartup,developingrealtimevideoprocessingalgorithms.Heiscurrentlypur-suinghisPhDdegreeinAppliedMathematicsatEcolePolytechnique,Palaiseau.Hisresearchin-terestsincludeinvariantsignalrepresentations,stochasticprocessesandfunctionalanalysis. St´ephaneMallatSt´ephaneMallatreceivedanengineeringdegreefromEcolePolytechnique,Paris,aPh.D.inelectricalengineeringfromtheUniversityofPennsylvania,Philadelphia,in1988,andanhabilitationinappliedmathematicsfromUniversit´eParis-Dauphine.In1988,hejoinedtheComputerScienceDe-partmentoftheCourantInstitueofMathematicalScienceswherehewasAssociateProfessorin1994andProfesssorin1996.From1995to2012,hewasafullProfessorintheAppliedMathematicsDepartmentatEcolePolytechnique,Paris.From2001to2008hewasaco-founderandCEOofastart-upcompany.Since2012,hejoinedthecomputersciencedepartmentofEcoleNormaleSup´erieure,inParis.Dr.MallatisanIEEEandEURASIPfellow.Hereceivedthe1990IEEESignalProcessingSociety'spaperaward,the1993AlfredSloanfellowshipinMathematics,the1997OutstandingAchievementAwardfromtheSPIEOpticalEngineeringSociety,the1997BlaisePascalPrizeinappliedmathematicsfromtheFrenchAcademyofSciences,the2004EuropeanISTGrandprize,the2004INIST-CNRSprizeformostcitedFrenchresearcherinengineeringandcomputerscience,andthe2007EADSprizeoftheFrenchAcademyofSciences.Hisresearchinterestsincludecomputervision,signalprocessingandharmonicanalysis.