Continuous Primal-Dual Methods for Image Processing

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1、ContinuousPrimal-DualmethodsforImageProcessingM.Goldman∗July30,2011AbstractInthisarticlewestudyacontinuousPrimal-DualmethodproposedbyAppletonandTalbotandgeneralizeittootherproblemsinimageprocessing.WeinterpretitasanArrow-Hurwiczmethodwhichleadstoabet

2、terdescriptionofthesystemofPDEsobtained.Weshowexistenceanduniquenessofsolutionsandgetaconvergenceresultforthedenoisingproblem.Ouranalysisalsoyieldsnewaposterioriestimates.Acknowledgements.IwouldliketowarmlythankmyPhDadvisorAntoninChambolleforsuggesti

3、ngmethisproblemandforourfruitfuldis-cussions.ThisresearchwaspartiallysupportedbyANRprojectMICA(2006-2009).1IntroductionInimaging,dualityhasbeenrecognizedasafundamentalingredientforde-signingnumericalschemessolvingvariationalproblemsinvolvingatotalvar

4、i-ationterm.Primal-DualmethodswereintroducedintheeldbyChan,GolubandMuletin[12].Afterwards,ChanandZhu[20]proposedtorewritethediscreteminimizationproblemasamin-maxandsolveitusinganArrow-Hurwicz[5]algorithmwhichisagradientascentinonedirectionandagra-di

5、entdescentintheother.Justasforthesimplegradientdescent,onecan∗CMAP,CNRSUMR7641,EcolePolytechnique,91128PalaiseauCedex,France.e-mail:goldman@cmap.polytechnique.fr1thinkofextendingthismethodtothecontinuousframework.Thisisinfactwhatdoesthealgorithmprevi

6、ouslyproposedbyAppletonandTalbotin[4]derivedbyanalogywithdiscretegraphcutstechniques.ThersttonoticethelinkbetweentheirmethodandPrimal-DualschemeswereChambolleetal.in[9].Besidesitsintrinsictheoreticalinterest,consideringthecontinuousframe-workhasalso

7、praticalmotivations.Indeed,asillustratedbyAppletonandTalbotin[4],thisapproachleadstohigherqualityresultscomparedwithfullydiscreteschemessuchasthoseproposedbyChanandZhu.Wewillnumericallyillustratethisinthenalpartofthispaper.Thispaperproposestostudyth

8、econtinuousPrimal-Dualalgorithmfollow-ingthephilosophyoftheworkdoneforthegradientowbyCasellesanditscollaborators(seethebookofAndreuetal.[2]andthereferencestherein).WegivearigorousdenitionofthesystemofPDEswhichisobtainedandshowexistenceanduniqueness