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GeneralizedGradients:PriorsonMinimizationFlowsG.Charpiat,P.Maurel,J.-P.Pons,R.Keriven,O.FaugerasOdysseeLabENS/INRIA/ENPCParis/Sophia-Antipolis/Champs-sur-Marne,FranceAbstractThispapertacklesanimportantaspectofthevariationalproblemunderlyingactivecontours:optimizationbygradientflows.Classically,thedefinitionofagradientdependsdirectlyonthechoiceofaninnerproductstructure.Thisconsiderationislargelyabsentfromtheactivecontoursliterature.Mostauthors,explicitelyorimplicitely,assumethatthespaceofadmissibledeformationsisruledbythecanonicalL2innerprod-uct.Theclassicalgradientflowsreportedintheliteraturearerelativetothisparticularchoice.Here,weinvestigatetherelevanceofusing(i)otherinnerproducts,yieldingothergradientdescents,and(ii)otherminimizingflowsnotderivingfromanyinnerproduct.Inparticular,weshowhowtoinducedifferentde-greesofspatialconsistencyintotheminimizingflow,inordertodecreasetheprobabilityofgettingtrappedintoirrelevantlocalminima.Wereportnumericalexperimentsindicatingthatthesensitivityoftheactivecontoursmethodtoinitialconditions,whichseriouslylimitsitsapplicabilityandefficiency,isalleviatedbyourapplication-specificspatiallycoherentminimizingflows.Weshowthatthechoiceoftheinnerproductcanbeseenasaprioronthedeformationfieldsandwepresentanextensionofthedefinitionofthegradienttowardmoregeneralpriors.1.IntroductionManyproblemsincomputervisioncanadvantageouslybecastinavariationalform,i.e.asaminimizationofanenergyfunctional.Inthispaper,wefocusonvariationalmethodsdedicatedtotherecoveryofcontours.Inthiscase,theproblemamountstofindingacontourwhichcorrespondstoaglobalminimumoftheenergy.Unfortunately,inmostcases,theexactminimizationoftheenergyfunctionaliscomputationallyunfeasibleduetothehugenumberofunknowns.Thegraphcutsmethodisapowerfulenergyminimizationmethodwhichallowstofindaglobalminimumorastronglocalminimumofanenergy.Inthelastfewyears,thismethodhasbeensuccessfullyappliedtoseveralproblemsincomputervision,includingstereovision[17]andimagesegmentation[5].However,ithasaseverelimitation:itcannotbeappliedtoanarbitraryenergyfunction[18],and,whenapplicable,iscomputationallyexpensive.Hence,inmostcases,asuboptimalstrategymustbeadopted.Acommonminimizationprocedurecon-sistsinevolvinganinitialcontour,positionedbytheuser,inthedirectionofsteepestdescentoftheenergy.Thisapproach,knownintheliteratureasactivecontoursordeformablemodels,waspioneeredbyKass.etGuillaume.Charpiat@di.ens.fr,Pierre.Maurel@di.ens.fr,Jean-Philippe.Pons@sophia.inria.fr,Renaud.Keriven@certis.enpc.fr,Olivier.Faugeras@sophia.inria.fr1
al.in[16]forthepurposeofimagesegmentation.Since,ithasbeenappliedinmanydomainsofcomputervisionandimageanalysis(imagesegmentation[6],surfacereconstruction[35,11],stereoreconstruction[12,15,13],etc.).However,duetothehighlynon-convexnatureofmostenergyfunctionals,agradientdescentflowisverylikelytobetrappedinalocalminimum.Also,thislocalminimumdependsonthepositionoftheinitialcontour.Ifthelatterisfarfromtheexpectedfinalconfiguration,theevolutionmaybetrappedinacompletelyirrelevantstate.Thissensitivitytoinitialconditionsseriouslylimitstheapplicabilityandefficiencyoftheactivecontoursmethod.Wedetailinsection2thegeneralgradientdescentprocesssoastoemphasizethecrucialroleoftheinnerproduct.Afteranabstractstudyinsection3onhowtohandleinnerproductsandminimizingflows,wepropose,insection4,variousinnerproductsandshowhowtheyinducedifferentdegreesofspatialcoherenceintheminimizingflowwithnumericalexamplesofshapewarpinginsection5.Insection6,arewritingoftheusualdefinitionofthegradientshowshowthechoiceofaninnerproductcanbeseenasawaytointroduceaprioronthedeformationfields,andthisleadsustoanaturalextensionofthenotionofgradienttomoregeneralpriors.2.MinimizationandinnerproductInthefollowingweconsiderashapeΓ,seenasamanifoldofdimensionkembeddedinRn,forexampleaplanarcurveorasurfaceinthespaceR3.WedenotebyE(Γ)theenergyfunctionaltobeminimized.Inordertodenethegradientoftheenergyfunctional,therststepistocomputeitsGaˆteauxderivativesδE,v)inalldirections,i.e.foralladmissiblevelocityfieldsvdefinedontheshapeΓwithvaluesinRn.Thedeformationspace,setofallthesefieldsv,canbeseenasthetangentspaceofΓ,considereditselfasapointinthemanifoldofalladmissibleshapes.defE(Γ+v)E(Γ)δE,v)=lim0.(1)Then,wewouldliketopickthegradientasthedirectionofsteepestdescentoftheenergy.However,itisnotyetpossibleatthisstage:tobeabletoassessthesteepnessoftheenergy,thedeformationspaceneedsadditionalstructure,namelyaninnerproductintroducingthegeometricalnotionsofanglesandlengths.Thisconsiderationislargelyabsentfromtheactivecontoursliterature:mostauthors,explicitelyorimplicitely,assumethatthedeformationspaceisruledbythecanonicalL2innerproductonΓ,whichis,fortwodefor-mationfieldsuandv:Z1hu|viL2=|Γ|u(x)v(x)dΓ(x),ΓwheredΓ(x)standsfortheareaelementofthecontoursothattheintegraloverΓisintrinsicanddoesnotdependontheparametrization.Here,forsakeofgenerality,wemodelthespaceofadmissibledeformationsasaninnerproductspace(F,h|iF).IfthereexistsadeformationfielduFsuchthatvF,δE,v)=hu|viF,thenuisunique,wecallitthegradientofErelativetotheinnerproducth|iF,andwedenotebyu=rFE(Γ).TheexistenceofuisrelatedtothesmoothnessofE,ormoreexactlytothecontinuityofδE,v)withrespecttov(Rieszrepresentationtheorem,see[27]formoredetails).2
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