cam11-21Proximity Algorithms for Image Models II L1 TV Denoising

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1、ProximityAlgorithmsforImageModelsII:L1/TVDenoisingCharlesA.Micchelli∗LixinShen†YueshengXu†§‡XueyingZeng§AbstractThispaperintroducesaproximityoperatorframeworkforstudyingtheL1/TVimagede-noisingmodelwhichminimizesthesumofadatafidelitytermmeasuredinthe`1-no

2、rmandthetotal-variationregularizationterm.Bothtermsinthemodelarenon-differentiable.Thiscausesalgorithmicdifficultiesforitsnumericaltreatment.Toovercomethedifficulties,weformulatethetotal-variationasacompositionoftheconvexfunctionwiththe`1-normorthe`2-normand

3、thefirstorderdifferenceoperator,andthenexpressthesolutionofthemodelintermsoftheproximityoperatorofthecomposition.Bydevelopinga“chainrule”fortheproximityoperatorofthecomposition,weidentifythesolutionasafixedpointofamappingexpressedintermsoftheproximityopera

4、torofthe`1-normorthe`2-norm,eachofwhichisexplicitlygiven.Thisformulationnaturallyleadstofixed-pointalgorithmsforthenumericaltreatmentofthemodel.Weproposeanalternativemodelbyreplacingthenon-differentiableconvexfunctionintheformulationofthetotalvariationwit

5、hitsdifferentiableMoreauenvelopeanddevelopcorrespondingfixed-pointalgorithmsforsolvingthenewmodel.Whenpartialinformationoftheunderlyingimageisavailable,wemodifythemodelbyaddinganindicatorfunctiontotheminimizationfunctionalandderiveitscorrespondingfixed-poi

6、ntalgorithms.Weestablishcon-vergenceresultsoftheproposedfixed-pointalgorithmsbyshowingthatthemappingswhichdefinethefixed-pointiterationsarenonexpansive.Numericalexperimentsareconductedtotesttheapproximationaccuracyandcomputationalefficiencyoftheproposedalgor

7、ithms.Also,weprovideacomparisonofourapproachtotwostate-of-the-artalgorithmsavailableinthelitera-ture.Numericalresultsconfirmthatouralgorithmsperformfavorably,intermsofPSNR-valuesandCPU-time,incomparisontothetwoalgorithms.1IntroductionTotal-variationbased

8、variationalmodelsarewidelyusedinimagedenoising.Thewell-knownRudin-Osher-Fatemi(ROF)imagedenoisingmodel[38]seeksaminimizerofthesumofadatafidelitytermmeasuredinthesquareof`2-normandthetotal-variationregularizationterm.Thisminimizati