cam12-02Augmented Lagrangian Method for A Mean Curvature.pdf

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1、AugmentedLagrangianMethodforAMeanCurvatureBasedImageDenoisingModelWeiZhu∗,Xue-ChengTai†,andTonyChan‡AbstractHighorderderivativeinformationhasbeenwidelyusedindevelopingvariationalmodelsinimageprocessingtoaccomplishmoreadvancedtasks.However,itisanontrivialissuetoconstr

2、uctefficientnumericalalgorithmstodealwiththeminimizationofthesevariationalmodelsduetotheassociatedhighorderEuler-Lagrangeequations.Inthispaper,weproposeanefficientnumericalmethodforameancurvaturebasedimagedenoisingmodelusingtheaugmentedLagrangianmethod.Wedescribehowtocon

3、structanaugmentedLagrangianfunctionalforthemodelanddetailtheproceduresoffindingtherelatedsaddle-pointsofthefunctional.Wepresentnumericalexperimentstoillustratetheeffectivenessandefficiencyoftheproposednumericalmethod,andshowafewimportantfeaturesoftheimagedenoisingmodelsu

4、chaskeepingcornersandimagecontrast.Moreover,acomparisonwiththegradientdescentmethodfurtherdemonstratestheefficiencyoftheproposedaugmentedLagrangianmethod.1IntroductionImagedenoisingprovidesacrucialsteptowardsthecapturingimagesignalsfromnoisyimageswithlotsofapplications

5、inmedicalimagingandvideomonitoringetc.Duringthelastthreedecades,numerousmethodshavebeenappliedtodealwiththisproblem,suchasvariationalmethods,partialdifferentialequations,statisticalmethodsandsoon[3,4,8,23,24,25,27,28,29,33].Amongthesevariationalmethodisoneofthemostpop

6、ularmethods.AclassicalvariationalmodelonimagedenoisingwasdevelopedbyRudin,Osher,andFatemi(ROF)[29],inwhichachievingacleanimagefromanoisyoneamountstotheminimizationofthefollowingfunctional:ZZE(u)=λ

7、∇u

8、+(f−u)2,(1)ΩΩwheref:Ω→RisagivennoisyimagedefinedonΩ(alwaysarectangle

9、inR2)andλ>0isapositiveconstant.TheROFmodelhasprovedtobeveryefficientinthenoiseremoval∗DepartmentofMathematics,UniversityofAlabama,Box870350,Tuscaloosa,AL35487.E-mail:wzhu7@bama.ua.edu.†DepartmentofMathematics,UniversityofBergen,NorwayandDivisionofMathematicalSciences,S

10、choolofPhysicalandMathematicalSciences,NanyangTechnologicalUniversity,637371,Singapore.Email:tai@math.uib.no.‡OfficeofthePresident,Ho