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1、AnApproachtoVectorialTotalVariationbasedonGeometricMeasureTheoryBastianGoldlueckeandDanielCremersTUMunich,Germanybastian.goldluecke,daniel.cremers@in.tum.deAbstractWeanalyzeapreviouslyunexploredgeneralizationofthescalartotalvariationtovector-valuedfunctions,whichi
2、smotivatedbygeometricmeasuretheory.Acompletemathematicalcharacterizationisgiven,whichNoisyResultOriginalprovesimportantinvariancepropertiesaswellasexis-tenceofsolutionsofthevectorialROFmodel.AsanFigure1:Forinverseproblemslikedenoising,inpaint-importantfeature,ther
3、eexistsadualformulationforingorsuperresolutionthetotalvariationisamongthetheproposedvectorialtotalvariation,whichleadstoamostpowerfulregularizers.Weproposeanovelgen-fastandstableminimizationalgorithm.Themaindif-eralizationoftotalvariationtovector-valuedimagesferen
4、cetopreviousapproacheswithsimilarpropertieswhichnaturallyarisesinthecontextofgeometricmea-isthatwepenalizeacrossacommonedgedirectionforsuretheory.allchannels,whichisamajortheoreticaladvantage.onecanbeobtainedbyconsideringthemathematicalExperimentsshowthatthisleads
5、toasignificiantlybet-frameworkofgeometricmeasuretheory.terrestorationofcoloredgesinpractice.VectorialTV.Foragreyscaleimagemodeledasadifferentiable1.Introductionmfunctionu:Ω→RonadomainΩ⊂R,thescalartotalvariationTV(u)isdefinedastheintegralovertheRegularityisofcentralim
6、portanceincomputervi-Euclideannorm
7、·
8、ofthegradient,sion.Manyproblems,likedenoising,deblurring,super-2resolutionandinpainting,areill-posed,andrequiretheTV(u)=
9、∇u
10、dx.(1)choiceofagoodpriorinordertoarriveatsensibleso-2Ωlutions.Thisprioroftentakestheformofaregular-iza
11、tiontermforanenergyfunctionalwhichistobeThedefinitioncanbeextendedtolocallyintegrablefunctionsu∈L1(Ω,R)usingadualformulation:Letminimized.Foroptimizationpurposes,itisimportantlocEmdenotetheclosedunitballinRm,thenthattheregularizerisconvex,sinceonlythenonecanhope
12、toalwaysfindaglobaloptimumoftheenergywithinreasonabletime.Furthermore,imagesintheTV(u)=supudiv(ξ)dx.(2)ξ∈Cc1(Ω,Em)Ωreal-worldcanbeobservedtogenerallybepi
