An approach to vectorial total variation based on geometric measure theory.pdf

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1、AnApproachtoVectorialTotalVariationbasedonGeometricMeasureTheoryBastianGoldlueckeandDanielCremersTUMunich,Germanybastian.goldluecke,daniel.cremers@in.tum.deAbstractWeanalyzeapreviouslyunexploredgeneralizationofthescalartotalvariationtovector-valuedfunctions,whichismotivatedbygeometricmeas

2、uretheory.Acompletemathematicalcharacterizationisgiven,whichNoisyResultOriginalprovesimportantinvariancepropertiesaswellasexis-tenceofsolutionsofthevectorialROFmodel.AsanFigure1:Forinverseproblemslikedenoising,inpaint-importantfeature,thereexistsadualformulationforingorsuperresolutionthet

3、otalvariationisamongthetheproposedvectorialtotalvariation,whichleadstoamostpowerfulregularizers.Weproposeanovelgen-fastandstableminimizationalgorithm.Themaindif-eralizationoftotalvariationtovector-valuedimagesferencetopreviousapproacheswithsimilarpropertieswhichnaturallyarisesinthecontext

4、ofgeometricmea-isthatwepenalizeacrossacommonedgedirectionforsuretheory.allchannels,whichisamajortheoreticaladvantage.onecanbeobtainedbyconsideringthemathematicalExperimentsshowthatthisleadstoasignificiantlybet-frameworkofgeometricmeasuretheory.terrestorationofcoloredgesinpractice.Vectorial

5、TV.Foragreyscaleimagemodeledasadifferentiable1.Introductionmfunctionu:Ω→RonadomainΩ⊂R,thescalartotalvariationTV(u)isdefinedastheintegralovertheRegularityisofcentralimportanceincomputervi-Euclideannorm

6、·

7、ofthegradient,sion.Manyproblems,likedenoising,deblurring,super-2resolutionandinpainting

8、,areill-posed,andrequiretheTV(u)=

9、∇u

10、dx.(1)choiceofagoodpriorinordertoarriveatsensibleso-2Ωlutions.Thisprioroftentakestheformofaregular-izationtermforanenergyfunctionalwhichistobeThedefinitioncanbeextendedtolocallyintegrablefunctionsu∈L1(Ω,R)usingadualformulation:Letminimized.Foroptimizati

11、onpurposes,itisimportantlocEmdenotetheclosedunitballinRm,thenthattheregularizerisconvex,sinceonlythenonecanhopetoalwaysfindaglobaloptimumoftheenergywithinreasonabletime.Furthermore,imagesintheTV(u)=supudiv(ξ)dx.(2)ξ∈Cc1(Ω,Em)Ωreal-worldcanbeobservedtogenerallybepi