COMMUN.ATH.SCI.c\r2011InternationalPressVol.9,No.2,pp.413{457UNCONDITI - PDF document

COMMUN.ATH.SCI.c\r2011InternationalPressVol.9,No.2,pp.413{457UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGCAROLA-BIBIANESCHONLIEByANDANDREABERTOZZIzAbstract.Higherorderequations,whenappliedtoimageinpainting,havecertainadvantagesoversecondorderequations,suchascontinuationofbothedgeandintensityinformationoverlargerdistances.Discretizingafourthorderevolutionequationwithabruteforcemethodmayrestrictthetimestepstoasizeuptoorder
x4where
xdenotesthestepsizeofthespatialgrid.Inthisworkwepresentecientsemi-implicitschemesthatareguaranteedtobeunconditionallystable.WeexplainthemainideaoftheseschemesandpresentapplicationsinimageprocessingforinpaintingwiththeCahn-Hilliardequation,TV-H1inpainting,andinpaintingwithLCIS(lowcurvatureimagesimpliers).Keywords.Imageinpainting,higherorderequations,numericalschemes.AMSsubjectclassications.35G25;34K28.1.IntroductionAnimportanttaskinimageprocessingistheprocessofllinginmissingpartsofdamagedimagesbasedontheinformationgleanedfromthesurroundingareas.Itisessentiallyatypeofinterpolationandiscalledinpainting.Therebyonecouldrestoreimageswithdamagedpartsdueto,forinstance,intentionalscratching,aging,orweather.Oronecanrecoverobjectswhichareoccludedbyotherobjects,wherewithinthiscontexttheprocessiscalleddisocclusion.Infacttheapplicationsofimageinpaintingarecountless.Fromtherestorationofancientfrescoes[3],tothemedicalneedsofreducingartifactsinMRI-,CT-orPETimagingreconstructions[47],digitalimageinpaintingisubiquitousinourmoderncomputerizedsociety.SincetherstworksonimageinpaintingbyMumford,NitzbergandShiota[57],MasnouandMorel[52],Caselles,Morel,SbertandGillette[21],andBertalmioetal[10],mucheorthasgoneintodevelopingdigitalalgorithms.Thesemethodsincludethetexturesynthesisandexemplar-basedapproach(see,e.g.,[20,29,32,72])andanumberofvariational-andPDE-basedapproaches.Thispaperfocusesonthelatter.Inmathematicalterms,imageinpaintingcanbedescribedinthefollowingway:letfbethegivenimagedenedonanimagedomain\n.TheproblemistoreconstructtheoriginalimageuinthedamageddomainD\n,calledtheinpaintingdomain.Moreprecisely,let\nR2beanopenandboundeddomainwithLipschitzboundary,B1;B2twoBanachspacesandf2B1bethegivenimage.Ageneralvariationalapproachininpaintingcanbewrittenasmin2B2nE(u)=R(u)+k(fu)k2B1o;(1.1)whereR:B2Rand(x)=(0\nnD0D;(1.2) Received:September8,2009;accepted(inrevisedversion):September7,2010.CommunicatedbyMartinBurger.yInstituteforNumericalandAppliedMathematics,Georg-AugustUniversityofGottingen,Lotzestr.16-18,D-37083Gottingen,Germany(c.schoenlieb@math.uni-goettingen.de).zDepartmentofMathematics,UCLA(UniversityofCaliforniaLosAngeles),520PortolaPlaza,LosAngeles,CA90095-1555,USA(bertozzi@math.ucla.edu).413 414UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGisthecharacteristicfunctionof\nnDmultipliedbyaconstant01.R(u)denotestheregularizingtermandk(fu)kB1thesocalleddelitytermoftheinpaintingapproach.B2B1ingeneral,signifyingthesmoothingeectoftheregularizingtermontheminimizeru2B2.DependingonthechoiceoftheregularizingtermRandtheBanachspacesB1,B2,variousinpaintingapproacheshavebeendeveloped.Themostfamousmodelisthetotalvariation(TV)model,whereR(u)=R\njrujdxdenotesthetotalvariationofu,B1=L2(\n)andB2=BV(\n)thespaceoffunctionsofboundedvariation;cf.[23,25,61,60].AvariationalmodelwitharegularizingtermcontaininghigherorderderivativesistheEulerselasticamodel[26,27,52]whereR(u)=R\n(a+b2)jrujdxwithpositiveweightsaandb,andcurvature=r
(ru=jruj).Otherexamplestobementionedfor(1.1)aretheactivecontourmodelbasedonMumfordandShahssegmentation[68],theinpaintingschemebasedontheMumford-Shah-Eulerimagemodel[35],inpaintingwiththeNavier-Stokesequation[11],andwavelet-basedinpainting[28,30],onlytogivearoughoverview.ForamorecompleteintroductiontoimageinpaintingusingPDEswereferto[26,18,63].1.1.Second-versushigher-orderinpaintingapproaches.Secondordervariationalinpaintingmethods(wheretheorderofthemethodisdeterminedbythederivativesofhighestorderinthecorrespondingEuler-Lagrangeequation),likeTVinpainting,havedrawbacksasintheconnectionofedgesoverlargedistances(Con-nectivityPrinciple,cf.Figure1.1)andthesmoothpropagationoflevellines(setsofimagepointswithconstantgrayvalue)intothedamageddomain(CurvaturePreser-vation,cf.Figure1.2).Thisisduetothepenalizationofthelengthofthelevellineswithintheminimizingprocesswithasecondorderregularizer,connectinglevellinesfromtheboundaryoftheinpaintingdomainviatheshortestdistance(linearinterpo-lation).TheregularizingtermR(u)=R\njrujdxintheTVinpaintingapproach,forexample,canbeinterpretedviathecoareaformula,whichgivesminZ\njrujdx()minZ11length()d;where=fx2\n:u(x)=gisthelevellineforthegrayvalue.IfweconsiderontheotherhandtheregularizingtermintheEulerselasticainpaintingapproachthecoareaformulareadsminZ\n(a+b2)jrujdx()minZ11alength()+bcurvature2()d:(1.3)Thusnotonlythelengthofthelevellinesbutalsotheircurvatureispenalized(wherethepenalizationofeachdependsontheratiob=a).Thisresultsinasmoothcon-tinuationoflevellinesovertheinpaintingdomainalsooverlargedistances;compareFigures1.1and1.2.Theperformanceofhigherorderinpaintingmethodscanalsobeinterpretedviathesecondboundarycondition,whichisnecessaryforthewell-posednessofthecorrespondingEuler-Lagrangeequationoffourthorder.Notonlyarethegrayvaluesoftheimagespeciedontheboundaryoftheinpaintingdomain,butalsothegradientoftheimagefunction,namelythedirectionofthelevellines,isgiven.Inanattempttosolveboththeconnectivityprincipleandthestaircasingeectresultingfromsecondorderimagediusions,anumberofthirdandfourthorderdiusionshavebeensuggestedforimageinpainting.TherstworkconnectingimageinpaintingtoathirdorderPDE(partialdierentialequation)isthetransportprocess C.B.SCHONLIEBANDA.BERTOZZI415 Fig.1.1.TwoexamplesofcurvaturebasedinpaintingcomparedwithTVinpaintingfrom[26].InthecaseoflargeaspectratiostheTVinpaintingfailstocomplytotheConnectivityPrinciple. Fig.1.2.AnexampleofelasticainpaintingcomparedwithTVinpaintingfrom[27].Despitethepresenceofhighcurvature,TVinpaintingtruncatesthecircleinsidetheinpaintingdomain(linearinterpolationoflevellines,i.e.,CurvaturePreservation).DependingontheweightsaandbEulerselasticainpaintingreturnsasmoothlyrestoredobject,takingthecurvatureofthecircleintoaccount.ofBertalmioetal[10].Theimageinformation,modeledby
u,istransportedintotheinpaintingdomainalongthelevellinesoftheimage.TheresultingschemeisadiscretemodelbasedonthenonlinearPDEu=r?u
r
u;andissolvedinsidetheinpaintingdomainDusingtheimageinformationfromasmallstripearoundtheboundaryofD.Theoperatorr?denotestheperpendiculargradient(@y;@x).Duetothelackofcommunicationamongthelevellines,thetransportationmayresultinkinksorcontradictionsinsidetheinpaintingdomain.Thusin[10]theequationaboveisimplementedwithintermediatestepsofanisotropicdiusion.In[11]theauthorsdevelopatheoryfortheproperboundaryconditionsin[10]bymakingaconnectiontotheNavier-Stokesequations.Thetwoconditionsonthe\boundary"oftheinpaintingdomaincorrespondtothenoslipconditionforNavier-Stokes.AvariationalthirdorderapproachtoimageinpaintingisCDD(CurvatureDrivenDiusion)[24].Tosolvetheproblemofconnectinglevellinesalsooverlarge 416UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGdistances(connectivityprinciple),thelevellinesarestillinterpolatedlinearly.Thedrawbacksofthethird-orderinpaintingmodels[10]and[24]havedrivenChan,KangandShen[27]toareinvestigationoftheearlierproposalofMasnouandMorel[52]onimageinterpolationbasedonEulerselasticaenergy(1.3).ThefourthorderelasticainpaintingPDEcombinesCDD[24]andthetransportprocessofBertalmioetal[10],andisabletosolveboththeconnectivityprincipleandthestaircasingeect.OtherrecentlyproposedhigherorderinpaintingalgorithmsareinpaintingwiththeCahn-Hilliardequation[13,14],TV-H1inpainting[19,64]andcombinationsofsecondandhigherordermethods,e.g.[51].Inthispaperweareespeciallyinterestedinthree,rathernew,fourth-orderin-paintingschemes.Namely,weshalldiscussCahn-Hilliardinpainting,TV-H1inpaint-ing,andinpaintingwithLCIS(lowcurvatureimagesimpliers).Westartthediscus-sionwiththeinpaintingofbinaryimagesusingtheCahn-Hilliardequation[13,14].Theinpaintedversionuoff2L2(\n)isconstructedbyfollowingtheevolutionofu=

u+1 F0(u)+(fu);(1.4)whereF(u)isasocalleddouble-wellpotential,e.g.,F(u)=u2(u1)2.Theapplica-bilityoftheCahn-HilliardequationfortheinpaintingofbinaryimagesisduetothedoublewellpotentialF(u)intheequation.Thetwowellscorrespondtovaluesofuthataretakenbymostofthegrayscalevalues.Choosingapotentialwithwellsatthevalues0(black)and1(white),Equation(1.4)thereforeprovidesasimplemodelfortheinpaintingofbinaryimages.Theparameterdeterminesthesteepnessofthetransitionbetween0and1.Further,thefourthorderregularizingtermintheequa-tionprovidestheadvantagesofhigherorderinpaintingapproacheswhichhavebeendiscussedbefore,suchastheabilitytoconnectlevellinesalsooverlargedistances(cf.(1.3)).ThesecondmethodofinterestinthispaperisageneralizationoftheCahn-Hilliardinpaintingapproachtograyvalueimageswhichhasbeenrecentlyproposedin[19,64]andiscalledTV-H1inpainting.Thereintheinpaintedimageuoff2L2(\n)shallevolveviau=
p+(fu);p2@TV(u);(1.5)withTV(u)=(R\njrujdx;ifju(x)j1a.e.in\n;+;otherwise;where@TV(u)denotesthesubdierentialofthefunctionalTV(u).TobuildtheconnectiontoCahn-Hilliardinpaintingtheauthorsin[19]showthatsolutionsofanappropriatetime-discreteCahn-Hilliardinpaintingapproach-converge,as0,tosolutionsofanoptimizationproblemregularizedwiththeTVnorm.Asimilarformofthisapproachappearsinthecontextofdecompositionandrestorationofgrayvalueimages;seeforexample[49,58,70].Further,inBertalmioetal[12],anapplicationofthemodelfrom[70]toimageinpaintingisproposed.Incontrasttotheinpaintingapproach(1.5)theauthorsin[12]useamoregeneralformoftheTV-H1approachforadecompositionoftheimageintocartoonandtexturepriortotheinpaintingprocess.Thelatterisaccomplishedwiththemethodpresentedin[10].Moreover,we C.B.SCHONLIEBANDA.BERTOZZI417wouldliketomentionthatin[45]theauthorsconsideracomplexGinzburg-Landauenergyforinpaintingofgrayscale-andcolorimages.ThethirdinpaintingmodelwearegoingtodiscussisinpaintingwithLCIS(LowCurvatureImageSimplier).ThishigherorderinpaintingmodelismotivatedbytwofamoussecondordernonlinearPDEsinimageprocessing|theworksofRudin,OsherandFatemi[60]andPeronaMalik[59].Thesemethodsarebasedonanonlinearversionoftheheatequationu=r
(g(jruj)ru);inwhichgissmallinregionsofsharpgradients.LCISrepresentafourthorderrelativeofthesenonlinearsecondorderapproaches.Theyareproposedin[69]andlaterusedbyBertozziandGreerin[15]forthedenoisingofpiecewiselinearsignals.InthispaperweconsiderLCISforimageinpainting.Withf2L2(\n)ourinpaintedimageuevolvesintimeasu=r
(g(
u)r
u)+(fu);withthresholdingfunctiong(s)=1 1+s2.Notethatwithg(
u)r
u=r(arctan(
u))theaboveequationcanberewrittenasu=
(arctan(
u))+(fu):(1.6)1.2.Numericalsolutionofhigher-orderinpaintingequations.Onemainchallengeininpaintingwithhigherorder\rowsistheireectivenumeri-calimplementation.Discretizingafourthorderevolutionequationwithabrute-forcemethodmayrestrictthetimestepstoasizeuptoorder
x4where
xdenotesthestepsizeofthespatialgrid.Suchabrute-forcemethodiscomputationallyprohibitiveandhenceitisessentiallyneverdone;see,e.g.,[65].Thenumericalsolutionofhigher-orderequations,likethinlms,phaseeldmod-els,surfacediusionequations,andmanymore,occupiedabigpartofresearchinnumericalanalysisinthelastdecades.In[31]theauthorsproposeasemiimplicit-nitedierenceschemeforthesolutionofsecondorderparabolicequations.Adiusiontermisaddedimplicitlyandsubtractedexplicitlyintimetothenumericalschemeinordertosuppressunstablemodes.Smerekausesthisideatosolvethefourth-ordersurfacediusionequation;cf.[65].ThesameideaisappliedbyGlasnertoaphaseeldapproachfortheHele-Shawinterfacemodel;cf.[40].Besidesthenitedier-enceapproximations,therealsoexistmanyniteelementalgorithmsforfourth-orderequations.Barrett,Blowey,andGarckepublishedaseriesofpapersonthesolutionofvariousCahn-Hilliardequations;cf.[5,6,7].ForthesharpinterfacelimitofCahn-Hilliard,i.e.,theHele-Shawmodel,FengandProhlanalyzeniteelementmethodsin[37,38].Finiteelementmethodsforthinlmequationsarestudied,forinstance,in[8,46].Forimageinpainting,ecientnumericalschemesforhigher-ordermethodsisanactiveareaofresearch.Asdiscussedin[26]oneofthemostinterestingopenproblemsindigitalinpaintingis,infact,thefastandexactdigitalrealization.InthecaseofCahn-Hilliardinpainting,in[13]theauthorsproposeasemi-implicitschemewhichconstitutesthecommonnumericalmethoddiscussedinthispaper.Theyverifyitscomputationalsuperioritycomparedwithcurrentlyusednumericalmethodsforthreecurvaturedrivenapproaches.ItturnsoutthatCahn-Hilliardinpaintingperformsatleastoneorderofmagnitudefasterthanthecurvaturemethods.In[33,34]Elliott 418UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGandSmithemanproposeaniteelementmethodforTV-H1minimizationinthecontextofimagedenoisingandcartoon/texturedecomposition.Theyalsoproverig-orousresultsabouttheapproximationandconvergencepropertiesoftheirscheme.AnextensionoftheirapproachtoTV-H1inpaintingwouldbeinteresting.Notethat,however,thedierenceoftheinpaintingapproachfromdenoisinganddecom-positionisthattheformerdoesnotfollowavariationalprincipleandthedelitytermislocallydependentonthespatialposition.AnotheralgorithmforTV-H1inpaintingisproposedbyoneoftheauthorsin[62].ThisworkgeneralizesthedualapproachofChambolle[22]andBectetal.[9]fromanL2delitytermtoanH1delityandextendsitsapplicationfromTV-H1denoising[1,2]toimageinpainting.Themainmotivationfortheworkin[62]isthatwiththeproposedalgorithmthedomaindecompositionapproachdevelopedin[39]canbeappliedtothehigher-ordertotalvariationcase.BeingabletoapplydomaindecompositionmethodstoTV-H1inpaintingcanresultinatremendousaccelerationofcomputationalspeedduetotheabilitytoparallelizethecomputation.Anotherveryrecentapproachinthisdirectionis[18],wheretheauthorsproposeamultigridapproachforinpaintingwithCDD.InthispaperwediscussanecientsemiimplicitapproachbasedonanumericalmethodpresentedinEyre[36](alsocf.[71])calledconvexitysplitting.Convexitysplittingwasoriginallyproposedtosolveenergyminimizingequations.Weconsiderthefollowingproblem:LetE2C2(RN;R)beasmoothfunctionalfromRNintoR,whereNisthedimensionofthedataspace.Let\nbethespatialdomainofthedataspace.Findu2RNsuchthat(u=rE(u);in\n;u(:;t=0)=u0;in\n;(1.7)withinitialconditionu02RN.ThebasicideaofconvexitysplittingistosplitthefunctionalEintoaconvexandaconcavepart.Inthesemiimplicitscheme,thecon-vexpartistreatedimplicitlyandtheconcaveoneexplicitlyintime.Underadditionalassumptionson(1.7),thisdiscretizationapproachisunconditionallystable,consis-tent,andrelativelyeasytoapplytoalargerangeofvariationalproblems.Moreoverweshallseethattheideaofconvexitysplittingcanbeappliedtomoregeneralevo-lutionequations,andinparticulartothosethatdonotfollowavariationalprinciple,especiallytotheinpaintingEquations(1.4)and(1.5).Convexitysplittingmethods,althoughpossiblynotunderthesamename,alreadyhavealongtraditioninseveralpartsofnumericalanalysis.Inniteelementapprox-imationsforPDEs,examplesforsuchnumericalschemescanbefoundintheworksofBarrett,Blowley,andGarcke;cf.[4]Equation(3.42)foranapplicationtoamodelforphaseseparation.In[35]anitedierenceschemeforsecond-orderparabolicequationsispresentedwhichalsousestheconvexitysplittingidea;cf.Equation(5.4)in[35].Furtherconvexitysplittingisalsodiscussedinamoregeneraloptimizationcontext;cf.[73]Chaptertwoforanoverviewonthistopic.Themainpartofthepaperillustratestheapplicationoftheconvexitysplittingideatothethreefourth-orderinpaintingapproaches(1.4),(1.5),and(1.6).Moti-vatedbytheanalysisin[17],weshowthatwiththisnumericalapproachweareableto(approximately)computestrongsolutionsofthecontinuousproblemwithanunconditionallystablenitedierencescheme.Thenumericalschemeissaidtobeunconditionallystableifallsolutionsofthedierenceequationareboundedindepen-dentlyfromthetimestepsize;cf.Denition2.2.Moreover,weproveconsistencyoftheseschemesandconvergencetotheexactsolution.Further,wepresentnu- C.B.SCHONLIEBANDA.BERTOZZI419mericalresultsdemonstratingtheeectofthehigherorderregularizingtermintheapproaches.InthecaseofTV-H1inpaintingandinpaintingwithLCISwedirectlycomparethevisualresultswiththesecondorderTVinpaintingmethod.Organizationofthepaper.InSection2theideaofconvexitysplittingispre-sented.AfteranintroductiontogradientsystemswestateandproveEyre'stheoremabouttheunconditionalstabilityoftheconvexitysplittingscheme.Sections3-5arededicatedtotheapplicationofconvexitysplittingtoCahn-Hilliardinpainting(1.4),TV-H1inpainting(1.5),andinpaintingwithLCIS(1.6).InthecaseofCahn-HilliardandTV-H1inpaintingthecorrespondingEquations(1.4)and(1.5)arenotstrictlygradient\rows,buttheirevolutionisthesumofthegradientsoftwodierentener-gies.Here,convexitysplittingisappliedtoeachoftheseenergiesandresultsinasemi-implicitschemeforthewholeevolution.Rigorousproofsfortheconsistencyofthenumericalscheme,theboundednessofnumericalsolutionsandtheirconvergencetotheexactsolutionaregiven.Foreachoftheseinpaintingalgorithmsnumericalresultsarepresented.Intheconclusionofthepaperopenproblemsarediscussed.Notation.Inthispaperwediscussthenumericalsolutionofevolutionarydif-ferentialequations.ThereforewehavetodistinguishbetweentheexactsolutionuofthecontinuousequationandtheapproximativesolutionUofthecorrespondingtimediscretenumericalscheme.WewritecapitalUkforthekthsolutionofthediscreteequationandsmalluk=u(k
t)forasolutionofthecontinuousinpaintingequationattimek
twithtimestepsize
t.Letekdenotethetemporaldiscretizationerrorgivenbyek=ukUk.Insubsectiontwo,uandUarevectorsinRN,whereNdenotesthedimensionofthedata.InallotherpartsofthispaperuandUareassumedtobeelementsinL2(\n).LetE2C2(H;R)denoteafunctionalfromasuitableHilbertspaceHtoR,andrE(u)itsrstvariationwithrespecttou.Inthediscreteset-tingH=RN.Throughoutthispaperk
kdenotesthenorminL2(\n)(ortheEuclideannorminthediscretesetting),andh
;
itheinnerproductinL2(\n)(orinRNinthedis-cretesetting).Finally,sinceweposeallthreeinpaintingapproaches(1.4)-(1.6)withNeumannboundaryconditions,wehavetodenethenon-standardspaceH1(\n)asH1(\n)=nF2H1(\n)jhF;1i(H1);H1=0o;withnormk
k1:=\r\rr
1
\r\rL2(\n).Therebytheoperator
1denotesthein-verseof
withNeumannboundaryconditions.Inmoredetail,letH1(\n):= 2H1(\n):R\n dx=0 .Thenu=
1F2H1(\n)istheuniqueweaksolutionofthefollowingproblem:
uF=0;in\n;ru
=0;on@\n:Foramoreelaboratederivationoftheabovespacewereferto[19],AppendixA.2.TheconvexitysplittingideaAsalreadydiscussedintheIntroduction,convexitysplittingmethodsareusedinawiderangeofoptimizationproblems;cf.Section1.2forrelevantreferences.Orig-inallydesignedtosolvegradientsystems,weshallseeinthispaperthatconvexitysplittingschemesarerelevantformoregeneralproblems,i.e.,forevolutionequationswhichdonotfollowavariationalprinciple.SeeSections3-5forourthreeinpainting 420UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGapproaches(1.4)-(1.6).Firstweintroducethenotionofgradient\rowsandtheapplicationofconvexitysplit-tingmethodsinthiscontext.TodosowefollowtheexplanationsandnotationsinEyre'swork[36].WeconsiderEquation(1.7).IfEfulllsthefollowingconditions;(i)E(u)0;8u2RN;(ii)E(u)!1askuk!1;(iii)hJ(rE)(u)u;ui8u2RN;(2.1)thenEquation(1.7)iscalledagradientsystemanditssolutionsarecalledgradient\rows.TherebyJ(rE)(u)istheJacobianofrEinu,2Randh:;:idenotestheinnerproductonRNwithcorrespondingnormkuk2=hu;ui.Allgradientsystemssatisfythedissipationproperty,i.e.,dE(u) dt=krE(u)k2;andthereforeE(u(t))E(u0)forallt0.IfE(u)isstrictlyconvex,i.e.,incondition(2.1)(iii)ispositive,thenonlyasingleequilibriumforthegradientsystemexists.Unconditionallystableanduniquelysolvablenumericalschemesexistfortheseequations(cf.[66]).IfE(u)isnotstrictlyconvex,i.e.,0,multipleminimizersmayexistandthegradient\rowcanpossiblyexpandinu(t).Thestabilityofanexplicitgradientdescentalgorithm,i.e.,Uk+1=Uk
trE(Uk),inthiscasemayrequireextremelysmalltimesteps,dependingofcourseonthefunctionalE.Forfourthorderinpaintingapproaches,forinstance,E(Uk)containssecondorderderivativesresultinginarestrictionof
tuptoorder(
x)4(where
xdenotesthestepsizeofthespatialdiscretization).Thereforethedevelopmentofstableandecientdiscretizationsfornon-convexfunctionalsEishighlydesirable.ThebasicideaofconvexitysplittingistowritethefunctionalEasE(u)=Ec(u)E(u);(2.2)whereEo2C2(RN;R)andEo(u)isstrictlyconvexforallu2RN;o2fc;eg:(2.3)Thesemi-implicitdiscretizationof(1.7)isthengivenbyUk+1Uk=
t(rEc(Uk+1)rE(Uk));(2.4)whereU0=u0.Remark2.1.WewanttoanticipatethatthesettingofEyre,andhencethesubsequentpresentationofconvexitysplitting,isapurelydiscreteone.Neverthelessitactuallyholdsinamoregeneralframework,i.e.,formoregeneralgradient\rows.InthecaseofanL2gradient\rowforexample,theJacobianJofthediscretefunctionalEjusthastobereplacedbythesecondvariationofthecontinuousfunctionalEinL2(\n).Inthefollowingweshowthatconvexitysplittingcanbeappliedtotheinpaintingapproaches(1.4),(1.5),and(1.6),andproducesunconditionallygradientstableorunconditionallystablenumericalschemes. C.B.SCHONLIEBANDA.BERTOZZI421Definition2.1.[36]Aone-stepnumericalintegrationschemeisunconditionallygradientstableifthereexistsafunctionE(:):RNRsuchthat,forall
t�0andforallinitialdata:(i)E(U)0forallU2RN,(ii)E(U)!1askUk!1,(iii)E(Uk+1)E(Uk)forallUk2RN,(iv)IfE(Uk)=E(U0)forallk0thenU0isazeroofrEfor(1.7)and(2.1).NotethatCahn-Hilliardinpainting(1.4)andTV-H1inpainting(1.5)arenotgivenbygradient\rows.Hence,inthecontextoftheseinpaintingmodelsthemeaningofunconditionalstabilityhastoberedened.Namely,inthecaseofanevolutionequationwhichdoesnotfollowagradient\row,acorrespondingdiscretetimesteppingschemeissaidtobeunconditionallystableifsolutionsofthedierenceequationareboundedwithinanitetimeinterval,independentlyofthestepsize
t.Definition2.2.LetubeanelementofasuitablefunctionspaceHdenedon\n[0;T],with\nR2openandbounded,andT�0.LetfurtherGbearealvaluedfunctionandu=G(u;Du)beapartialdierentialequationwithspacederivativesDu,=1;:::;4.AcorrespondingdiscretetimesteppingmethodUk+1=Uk+
tGk(Uk;Uk+1;DUk;DUk+1);(2.5)whereGkisasuitableapproximationofGinUkandUk+1isunconditionallystable,ifallsolutionsof(2.5)areboundedforall
t�0andallksuchthatk
tT.consistentiflim
!0k(
t)=0;wherek(
t)isthelocaltruncationerroroftheschemeanddenedask(
t)=uk+1uk
tGk(uk;uk+1;Duk;Duk+1);(2.6)anduk=u(k
t)istheexactsolutionattimet=k
t.Inwhatfollowsweabbreviatekfork(
t).Moreover,wedenetheglobaltruncationerrortobe(
t)=maxkkk(
t)kH:Anumericalschemeissaidtobeoforderpintimeif(
t)=O(
tp)for
t0:WestartwithatheoremofEyre[36].Theproofpresentedbelowfollowsthesameargumentsasin[36]withadditionaldetails.Theorem2.3([36]Theorem1).LetEsatisfy(2.1),andEcandEsatisfy(2.2)-(2.3).IfE(u)additionallysatiseshJ(rE)(u)u;ui(2.7) 422UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGwhen0in(2.1)(iii),thenforanyinitialconditionthenumericalscheme(2.4)isconsistentwith(1.7),gradientstableforall
t&#x-0.8;أ倀0,andpossessesauniquesolutionforeachtimestep.Thelocaltruncationerrorforeachstepisk=
t 2(J(rEc(^u))+J(rE(^u)))rE(u());forsome2(k
t;(k+1)
t)andforsome^uintheparallelopipedwithoppositeverticesatUkandUk+1.Remark2.2.Condition(2.7)inTheorem2.3isequivalenttotherequirementthatalltheeigenvaluesofJ(rE)dominatethelargesteigenvalueofJ(rE),i.e.,hJ(rE)(u)u;ui(2.7)(2.1)hJ(rE)(u)u;uiforallu2RN,or^jj;foralleigenvalues^�0ofE:(2.8)Proof.(Eyre[36]).Theunconditionalgradientstabilityof(2.4)inthesenseofDenition2.1isestablishedrst.Byourassumptionsin(2.1)properties(i)and(ii)inDenition2.1immediatelyfollow.Property(iv)followsfromthegeneralbehaviorofgradientsystems,i.e.,ifE(Uk)=E(U0)forallk0thenU0isan!-limitpointof(1.7)and(2.1)andhenceU0isazeroofrE(cf.[48]).Themainpartoftheproofconsistsofthevericationofproperty(iii).NamelywehavetoshowthatE(Uk+1)E(Uk);8Uk2RN:TodosoweconsiderthedierenceE(Uk+1)E(Uk).TheproofisbyrepeatedapplicationofTaylor'stheorem.WestartwithanexactexpansionofEaboutUk+1uptosecondorderandobtainE(Uk)=E(Uk+1)hrE(Uk+1);Uk+1Uki+1 2hJ(rE(Uk+1(Uk+1Uk)))Uk+1Uk;Uk+1Ukiforsome2(0;1).Thenbyassumption(iii)in(2.1)wegetE(Uk+1)E(Uk)hrE(Uk+1);Uk+1Uki+jjkUk+1Ukk2:By(2.2)and(2.4)thisisthesameasE(Uk+1)E(Uk)hrEc(Uk+1)rE(Uk+1);Uk+1Uki+jjkUk+1Ukk21
t(Uk+1Uk)+rEc(Uk+1)rE(Uk);Uk+1Uk=hrE(Uk+1)rE(Uk);Uk+1Uki+jj1
tkUk+1Ukk2:(2.9)Similarly,weTaylorexpandEaboutUk+1andUk,respectively,asE(Uk)=E(Uk+1)hrE(Uk+1);Uk+1Uki+1 2hJ(rE(Uk+11(Uk+1Uk)))Uk+1Uk;Uk+1Uki; C.B.SCHONLIEBANDA.BERTOZZI423andE(Uk+1)=E(Uk)+hrE(Uk);Uk+1Uki+1 2hJ(rE(Uk2(Uk+1Uk)))Uk+1Uk;Uk+1Uki;forsome1and2in(0;1).SinceEisconvex,thenJ(rE)ispositivedeniteanditseigenvaluesarepositive.ByboundingtheeigenvaluesofJ(rE)by^�0andaddingtheaboveexpressionswegethrE(Uk+1)rE(Uk);Uk+1Uki^kUk+1Ukk2:Substitutingthisin(2.9),weobtainE(Uk+1)E(Uk)^jj+1
tkUk+1Ukk2:Byapplyingcondition(2.7)(i.e.,(2.8))theresultfollowsforall
t0.Hencethemethodisunconditionallygradientstable.Toprovetheuniquesolvabilityof(2.4)weconsiderthenonlinearequationsUk+1+
trEc(Uk+1)=Rk;whichmustbesolvedateachstepforagivenRk.SinceEcisstrictlyconvex,1 2kUk+1k2+
tEc(Uk+1)hUk+1;RkihasauniqueminimuminUk+1forall
t,and(2.4)hasauniquesolutionforall
t0.Theconsistencyandthelocaltruncationerrorof(2.4)canbeestablishedbysimilarTaylorexpansionsastheoneswedidabovetoprovetheunconditionalstabilityofthescheme.MorepreciselyitconsistsofexpandingUk+1andUkaround(k+1=2)
t,andrEc(Uk+1)andrE(Uk)aroundUk+1=2.ThisnishestheproofofTheorem2.3. Inthefollowingweapplytheideaofconvexitysplittingtoourthreeinpaintingmodels(1.4),(1.5),and(1.6).Forthiswechangefromthediscretesettingtothecontinuoussetting,i.e.,consideringfunctionsuinasuitableHilbertspaceinsteadofvectorsuinRN.Althoughthersttwooftheseinpaintingapproaches,i.e.,Cahn-HilliardinpaintingandTV-H1inpainting,arenotgivenbygradient\rows,weshowthattheresultingnumericalschemesarestillunconditionallystable(inthesenseofDenition2.2)andthereforesuitabletosolvethemaccuratelyandreasonablyfast.ForinpaintingwithLCIS(1.6)theresultsofEyrecanbedirectlyapplied,eveninthecontinuoussetting;cf.Remark2.1.Nevertheless,alsoforthiscase,weadditionallypresentarigorousanalysis,similartotheonedoneforCahn-HilliardandTV-H1inpainting.3.Cahn-HilliardinpaintingInthissectionweshowtheapplicationofconvexitysplittingtoCahn-Hilliardinpainting(1.4).Recallthattheinpaintedversionu(x)off(x)isconstructedbyfollowingtheevolutionequationu=

u+1 F0(u)+(fu) 424UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGtosteadystate.ThismodiedCahn-Hilliardequationisintroducedin[13]fortheinpaintingofbinaryimages.Thelatter,mainlynumericalpaper,wasfollowedbyaverycarefulanalysisof(1.4)in[14].Tostartwith,theauthorsproveglobalex-istenceofauniqueweaksolutionoftheevolutionEquation(1.4).MorepreciselythesolutionuisproventobeanelementinC([0;T];L2(\n))\L2(0;T;V),whereV=2H2(\n)j@=@=0on@\n ,andistheoutwardpointingnormalon@\n.Underadditionalconditionsonthegivenimagef,theyalsoderivesomeveryin-terestingresultsconcerningthecontinuationofthegradientoftheimageintotheinpaintingdomain.Infact,in[14]theauthorsprovethatinthelimit0!1astationarysolutionof(1.4)solves

u1 F0(u)=0;inD;u=f;on@D;ru=rf;on@D;(3.1)forfregularenough(f2C2).Theexistenceofastationarysolutionof(1.4)isassuredin[19].Additionally,in[14]theauthorspresentnumericalexampleswhichshowthattheconnectivityprincipleisfullled,andcomputeabifurcationdiagramforstationarysolutionsof(1.4).Thissupportstheclaimthatfourth-ordermethodsaresuperiortosecond-ordermethodswithrespecttoasmoothcontinuationoftheimagecontentsintothemissingdomain.Theideatoapplyconvexitysplittinginordertosolve(1.4)numericallywasbornin[13].Thenumericalresultspresentedthereillustratetheusefulnessofthisscheme.Althoughtheauthorsdonotanalyzetheschemerigorously,basedontheirnumericalresultstheyconjectureunconditionalstability.Inthefollowingweshallpresentthisnumericalschemeandderivesomeadditionalpropertiesbasedonarigorousanalysisofthelatter.TheoriginalCahn-Hilliardequationisagradient\rowinH1fortheenergyE1(u)=Z\n 2jruj2+1 F(u)dx;whilethettingtermin(1.4)canbederivedfromagradient\rowinL2fortheenergyE2(u)=1 2Z\n(fu)2dx:However,notethatEquation(1.4)asawholeisnolongeragradientsystem.Hence,forthediscretizationintime,weapplytheconvexitysplittingdiscussedinSection2tobothfunctionalsE1andE2separately.Namely,wesplitE1asE1=E1cE1,withE1c(u)=Z\n 2jruj2+C1 2juj2dx;E1(u)=Z\n1 F(u)+C1 2juj2dx:ApossiblesplittingforE2isE2=E2cE2withE2c(u)=1 2Z\nC2juj2dx;E2(u)=1 2Z\n(fu)2+C2juj2dx:TomakesurethatE1c;E1andE2c;E2arestrictlyconvex,theconstantsC1andC2havetobechosensuchthatC1�1 ,C2�0;see[14]. C.B.SCHONLIEBANDA.BERTOZZI425Thentheresultingdiscretetime-steppingschemeforaninitialconditionU0=u0isgivenbyUk+1Uk
t=rH1(E1c(Uk+1)E1(Uk))rL2(E2c(Uk+1)E2(Uk));whererH1andrL2representgradientdescentwithrespecttotheH1innerprod-uctandtheL2innerproductrespectively.ThistranslatestoanumericalschemeoftheformUk+1Uk
t+

Uk+1C1
Uk+1+C2Uk+1=1 
F0(Uk)C1
Uk+(fUk)+C2Uk;in\n:(3.2)WeenforceNeumannboundaryconditionson@\n,i.e.,rUk+1
~n=r
Uk+1
~n=0;on@\n;(3.3)where~nistheoutwardpointingnormalon@\n,andcomputeUk+1in(3.2)inthespectraldomainusingthediscretecosinetransform(DCT).TheideatousespectralmethodsforequationsinvolvingLaplacianoperatorsisclassicalandisbasedonthefactthattheLaplacematrixisdiagonalizedinthespectraldomain.Hence,solvingtheseequationsinthespectraldomaincanbedonemuchfastersincematrixmul-tiplicationisreplacedbyscalarmultiplication(multiplyingwiththeelementsinthemaindiagonal).Sinceadditionallytherealsoexistfastnumericalmethodstocom-putethediscreteFourier/Cosinetransform(suchasthefastFouriertransform(FFT))thismethodhasanoverallcomputationaladvantage.Let^UbetheDCTofUwitheigenvalues.ThenEquation(3.2)in^Ureads^Uk+1(i;j)=(1C1
t(1
x2+1
y2j)+C2
t)^Uk(i;j)+
\
F0(Uk)(i;j)+
t\(fUk) 1+C2
t+
t(1
x2+1
y2j)2C1
t(1
x2+1
y2j):3.1.RigorousEstimatesfortheScheme.FromTheorem2.3weknowthat(atleastinthespatiallydiscreteframework)theconvexitysplittingscheme(2.2)-(2.4)isunconditionallystable,i.e.,separatenumericalschemesforthegradient\rowsoftheenergiesE1(u)andE2(u)arenon{increasingforall
t�0.Butthisdoesnotguaranteethatthenumericalscheme(3.2)isunconditionallystable,sinceitcombinesthe\rowsoftwoenergies.Inthissectionweshallanalyzetheschemeinmoredetailandderivesomerigorousestimatesforitssolutions.Inparticularweshowthatthescheme(3.2)isunconditionallystableinthesenseofDenition2.2.Ourresultsaresummarizedinthefollowingtheorem.Theorem3.1.Letubetheexactsolutionof(1.4)anduk=u(k
t)theexactsolutionattimek
t,foratimestep
t�0andk2N.LetUkbethekthiterateof(3.2)withconstantsC1�1=,C2�0.Thenthefollowingstatementsaretrue:(i)Undertheassumptionthatkuk1andkr
uk2arebounded,thenumericalscheme(3.2)isconsistentwiththecontinuousEquation(1.4)andoforderoneintime. 426UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGUndertheadditionalassumptionthatF00(Uk1)K(3.4)foranonnegativeconstantK,wefurtherhave(ii)ThesolutionsequenceUkisboundedonanitetimeinterval[0;T],forall
t�0.Inparticularfork
tT,T�0xed,wehaveforevery
t�0krUkk22+
tK1k
Ukk22eK2TkrU0k22+
tK1k
U0k22+
tTC(\n;D;0;f);(3.5)forsuitableconstantsK1andK2,andconstantCdependingon\n;D;0;fonly.(iii)Thediscretizationerrorek,givenbyek=ukUk,convergestozeroas
t0.Inparticular,wehavefork
tT,T�0xed,thatkrekk22+
tC1 ~Ck
ekk22T ~CeK1T
C
(
t)2;(3.6)forsuitableconstantsC;~C;K1.Remark3.1.NotethatourassumptionsfortheconsistencyofthenumericalschemeonlyholdifthetimederivativeofthesolutionofthecontinuousEquation(1.4)isuniformlybounded.Thisistrueforsmoothandboundedsolutionsoftheequation.Further,sinceweareinterestedinboundedsolutionsUkofthediscreteEquation(3.2),itisnaturaltoassume(3.4),i.e.,thatthenonlinearityF00intheprevioustimestep(k1)
tisbounded.AlsonotethattheconstantKin(3.4)canbechosenarbitrarilylarge.TheproofofTheorem3.1isorganizedinPropositions3.2-3.4.Proposition3.2(Consistency(i)).UnderthesameassumptionsasinTheorem3.1,andinparticularundertheassumptionthatkuk1andkr
uk2arebounded,thenumericalscheme(3.2)isconsistentwiththecontinuousEquation(1.4)withkkk1=O(
t)as
t0,wherekisthelocaltruncationerrorasdenedinEquation(2.6)above.Proof.Letkbethelocaltruncationerrordenedasin(2.6).Thenk=1k+2k;with1k=uk+1uk
tu(k
t)2k=

(uk+1uk)C1
(uk+1uk)+C2(uk+1uk)=
t
2uk+1uk
tC1
t
uk+1uk
t+C2
tuk+1uk
t;i.e.,k=uk+1uk
t+
2uk+11 
F0(uk)(fuk)C1
(uk+1uk)+C2(uk+1uk):(3.7) C.B.SCHONLIEBANDA.BERTOZZI427UsingstandardTaylorseriesargumentsandassumingthatkuk1andkr
uk2areboundedwededucethattheglobaltruncationerrorisgivenby=maxkkkk1=O(
t)as
t0:(3.8) Proposition3.3(Unconditionalstability(ii)).UnderthesameassumptionsasinTheorem3.1andinparticularassumingthat(3.4)holds,thesolutionsequenceUkfullls(3.5).Thisgivesboundednessofthesolutionsequenceon[0;T].Proof.WeconsiderourdiscretemodelUk+1Uk
t+

Uk+1C1
Uk+1+C2Uk+1=1 
F0(Uk)C1
Uk+(fUk)+C2Uk;multiplytheequationwith
Uk+1andintegrateover\n.Weobtain1
tkrUk+1k22hrUk;rUk+1i2+kr
Uk+1k22+C1k
Uk+1k22+C2krUk+1k22=1 hF00(Uk)rUk;r
Uk+1i2+C1h
Uk;
Uk+1i2+hr(fUk);rUk+1i2+C2hrUk;rUk+1i2:UsingYoung'sinequalityweobtain1 2
tkrUk+1k22krUkk22+kr
Uk+1k22+C1k
Uk+1k22+C2krUk+1k221 2kF00(Uk)rUkk22+ 2kr
Uk+1k22+C1 2k
Ukk22+C1 2k
Uk+1k22+C2 2krUkk22+C2 2krUk+1k22+1 2kr(fUk)k22+1 2krUk+1k22:Usingtheestimatekr(fUk)k22220krUkk22+C(\n;D;0;f)andreorderingtheterms,weobtain1 2
t+C2 21 2krUk+1k22+C1 2k
Uk+1k22+ 2kr
Uk+1k221 2
t+C2 2+20krUkk22+1 2kF00(Uk)rUkk22+C1 2k
Ukk22+C(\n;D;0;f):Bychoosing=22,thethirdtermontheleftsideoftheinequalityiszero.BecauseofAssumption(3.4)weobtainthefollowingboundontherightsideoftheinequalitykF00(Uk)rUkk22K2krUkk22;andwehave1 2
t+C2 21 2krUk+1k22+C1 2k
Uk+1k221 2
t+C2 2+20+K2 43krUkk22+C1 2k
Ukk22+C(\n;D;0;f): 428UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGNowwemultiplytheaboveinequalityby2
tanddene~C=1+
t(C21);~~C=1+
tC2+220+K2 23:SinceC2ischosengreaterthan0�1,therstcoecient~Cispositiveandwecandividetheinequalitybyit.WeobtainkrUk+1k22+
tC1 ~Ck
Uk+1k22~~C ~CkrUkk22+
tC1 ~Ck
Ukk22+
tC(\n;D;0;f);whereweupdatedtheconstantC(\n;D;0;f)byC(\n;D;0;f)=~C.Since~~C ~C1,wecanmultiplythesecondtermontherightsideoftheinequalitybythisquotienttoobtainkrUk+1k22+
tC1 ~Ck
Uk+1k22~~C ~CkrUkk22+
tC1 ~Ck
Ukk22+
tC(\n;D;0;f):WededucebyinductionthatkrUkk22+
tC1 ~Ck
Ukk22 ~~C ~C!kkrU0k22+
tC1 ~Ck
U0k22+
tk1X=0 ~~C ~C!C(\n;D;0;f)=(1+K2
t)k (1+K1
t)kkrU0k22+
tC1 ~Ck
U0k22+
tk1X=0(1+K2
t) (1+K1
t)C(\n;D;0;f):Fork
tTwehavekrUkk22+
tC1 ~Ck
Ukk22e(K2K1)TkrU0k22+
tC1 ~Ck
U0k22+
tTe(K2K1)TC(\n;D;0;f)=e(K2K1)TkrU0k22+
tC1 ~Ck
U0k22+
tTC(\n;D;0;f);whichgivesboundednessofthesolutionsequenceon[0;T]foranyT�0,assumingthat(3.4)holds. Theconvergenceofthediscretesolutiontothecontinuousoneasthetimestep
t0isveriedinthefollowingproposition.Proposition3.4(Convergence(iii)).UnderthesameassumptionsasinTheorem3.1andinparticularunderAssumption(3.4)thediscretizationerrorekfullls(3.6). C.B.SCHONLIEBANDA.BERTOZZI429InordertoproveProposition3.4weneedthefollowingauxiliarylemma.Lemma3.2.TheerrorekbetweentheexactandapproximatesolutiondenedasinTheorem3.1fulllsZ\nekdx=O((
t)2):Proof.[ProofofLemma3.2]Becauseofthedelitytermin(1.4)and(3.2),solutionsoftheseequationsarenotmasspreserving,i.e.,R\nekdoesnotingeneralvanish.Infactwehave,forasolutionukof(1.4),d dtZ\nuk=Z\n
2uk+1 Z\n
F0(uk)+Z\n(fuk)=Z@\nr
uk
~n+1 Z@\nrF0(uk)
~n+Z\n(fuk);wherewehaveusedGaussdivergencetheoremtoobtaintheboundaryintegrals.AssumingzeroNeumannboundaryconditionsasin(3.3)thetwoboundaryintegralsvanish,andhenced dtZ\nuk=Z\n(fuk):Inparticulard dtZDuk=0:(3.9)Asimilarcomputationforthediscretesolutionof(3.2)showsthat1
t+C2Z\n(Uk+1Uk)=Z\n(fUk);andinparticular1
t+C2ZD(Uk+1Uk)=0:(3.10)Next,letusfollowthelinesoftheconsistencyproofin(3.7).Thenthediscretizationerroreksatisesek+1ek
t+
2ek+1C1
ek+1+C2ek+1=1
t(uk+1uk)1
t(Uk+1Uk)+
2uk+1
2Uk+1C1
uk+1+C1
Uk+1+C2uk+1C2Uk+1=1 
F0(Uk)C1
Uk+(fUk)+C2Uk+1 
F0(uk)+(fuk)C1
uk+C2uk+k=1 
(F0(Uk)F0(uk))C1
(Ukuk)+C2(Ukuk)(Ukuk)+k: 430UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGAsbefore,integratingover\n,applyingGaussdivergencetheoremandthezeroNeu-mannboundaryconditionsforukandUk,weget1
t+C2Z\n(ek+1ek)+Z\nek=Zk;(3.11)whereZ\nk=1
t+C2Z\n(uk+1uk)Z\n(uk)=1
t+C2Z\n(uk+
t(uk)+O((
t)2)uk)Z\n(uk)=O(
t):Now,toproveourclaimweapplyinductiononk.First,assumingthatu0=U0in\n,wehavethatZ\ne0=0;andhence1
t+C2Z\ne1=O(
t):(3.12)Assumingthatassertion(3.12)holdsforallindiceskandusing(3.9)and(3.10)wehave,for(3.11),1
t+C2Z\n(ek+1ek)+Z\nek=Zk1
t+C2Z\nek+1O(
t)+01
t+C21O(
t)=O(
t)1
t+C2Z\nek+1=O(
t);andhence(1+C2
t)Z\nek=O((
t)2)forallk0. WecontinuewiththeproofofProposition3.4.Proof.[ProofofProposition3.4]IntheproofofLemma3.2wehaveusedtheconsistencyresult(3.7)toshowthatthediscretizationerroreksatisesek+1ek
t+
2ek+1C1
ek+1+C2ek+1=1 
(F0(Uk)F0(uk))C1
(Ukuk)+C2(Ukuk)(Ukuk)+k: C.B.SCHONLIEBANDA.BERTOZZI431Multiplicationwith
ek+1leadsto1
thr(ek+1ek);rek+1i2+kr
ek+1k22+C1k
ek+1k22+C2krek+1k22=1 h
(F0(Uk)F0(uk));
ek+1i2C1h
(Ukuk);
ek+1i2+hr(Ukuk);rek+1i2C2hr(Ukuk);rek+1i2+\nr
1k;r
ek+12:Further,because1
tkrek+1k22hrek;rek+1i21 2
t(krek+1k22krekk22);weobtain1 2
t(krek+1k22krekk22)+kr
ek+1k22+C1k
ek+1k22+C2krek+1k221 h
(F0(Uk)F0(uk));
ek+1i2+C1h
ek;
ek+1i2hrek;rek+1i2+C2hrek;rek+1i2+\nr
1k;r
ek+12:ApplyingYoung'sinequalityleadsto1 2
t(krek+1k22krekk22)+kr
ek+1k22+C1k
ek+1k22+C2krek+1k221 h(F00(Uk)rUkF00(uk)ruk);r
ek+1i2+C1 21k
ekk22+C11 2k
ek+1k22+20 23krekk22+3 2krek+1k22+C2 22krekk22+C22 2krek+1k22+1 24kkk21+4 2kr
ek+1k22:Letusconsidertheremaininginnerproductinthelastinequality:1 h(F00(Uk)rUkF00(uk)ruk);r
ek+1i2=1 hF00(Uk)rek;r
ek+1i2+1 h(F00(uk)F00(Uk))ruk;r
ek+1i21 25kF00(Uk)jrekjk22+1 26k(F00(uk)F00(Uk))jrukjk22+5 2+6 2kr
ek+1k22:Nextweassumethat(3.4)holdsandthatrukisuniformlyboundedon[0;T]|inparticular,that9~~K�0suchthatkrukk2~~Kforallk
tT:(3.13)ThelatterassumptionwillbeproveninLemma3.3justaftertheendofthisproof.Moreover,sinceF00islocallyLipschitzcontinuousweobtain1 h(F00(Uk)rUkF00(uk)ruk);r
ek+1i2C 25krekk22+C 26kekk22+5 2+6 2kr
ek+1k22; 432UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGwherewehavesetCtobeauniversalconstantforallbounds.Further,usingLemma3.2andkekk22=kekO(
t)2+O(
t)2k222kekO(
t)2k22+2kO(
t)2k22;wecanapplythePoincareinequalitytotheL2normofek.Insumweget1 2
t+C212 23 2krek+1k22+C111 2k
ek+1k22+4 25+6 2kr
ek+1k221 2
t+20 23+C2 22+C 25+C 6krekk22+C1 21k
ekk22+1 24kkk21+C 6kO(
t)2k22:Nextwechoose1=1andmultiplytheinequalitywith2
t:(1+
t(C2(22)3))krek+1k22+
tC1k
ek+1k22+
t245+6 kr
ek+1k221+
t20 3+C2 2+C 5+2C 6krekk22+
tC1k
ekk22+
t 4kkk21+
t2C 6kO(
t)2k22:Let~C=1+
t(C2(22)3);~~C=1+
t20 3+C2 2+C 5+2C 6:Now,bychoosingallssuchthatthecoecientsofalltermsintheinequalityarenonnegativeandthequotient~~C=~C1,andbyestimatingthelasttermontheleftsidefrombelowbyzero,wegetkrek+1k22+
tC1 ~Ck
ek+1k22~~C ~Ckrekk22+
tC1 ~Ck
ekk22+
t ~C1 4kkk21+2C 6kO(
t)2k22;andbecause~~C=~C1wefurtherhavekrek+1k22+
tC1 ~Ck
ek+1k22~~C ~Ckrekk22+
tC1 ~Ck
ekk22+
t ~C1 4kkk21+2C 6kO(
t)2k22: C.B.SCHONLIEBANDA.BERTOZZI433Byinductiononkweobtainkrek+1k22+
tC1 ~Ck
ek+1k22 ~~C ~C!k+1kre0k22+
tC1 ~Ck
e0k22+
t ~CkX=0 ~~C ~C!
1 4maxkfkk21g+2C 6kO(
t)k22=
t ~CkX=0(1+K1
t)
1 4maxkfkk21g+2C 6kO(
t)2k22
t ~C
k
eK1k

1 4maxkfkk21g+2C 6kO(
t)2k22;wherewehaveusedthefactthate0=0and1~~C ~C=1+K1
t.Hence,byusingtheconsistencyresult(3.8)weconclude,fork
tT,thatkrekk22+
tC1 ~Ck
ekk22T ~CeK1T
C
(
t)2: From[13,14]weknowthatthesolutionuktothecontinuousequationgloballyexistsandisuniformlyboundedinL2(\n).Nextweshowthatassumption(3.13)holds.Lemma3.3.Letukbetheexactsolutionof(1.4)attimet=k
tandletT�0.ThenthereexistsaconstantC�0suchthatkrukk2Cforallk
tT.Proof.LetK(u)=
u+1 F0(u).WemultiplythecontinuousevolutionEqua-tion(1.4)withK(u)andobtainhu;K(u)i2=h
K(u);K(u)i2+h(fu);K(u)i2:LetusfurtherdeneE(u):= 2Z\njruj2dx+1 Z\nF(u)dx:Thenwehavehu;K(u)i2=u;
u+1 F0(u)2=hru;rui2+u;1 F0(u)2=d dtE(u);sinceusatisesNeumannboundaryconditions.Thereforewegetd dtE(u)=Z\njrK(u)j2dx+h(fu);
ui2+(fu);1 F0(u)2:(3.14) 434UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGSinceF(u)isboundedfrombelow,weonlyhavetoshowthatE(u)isuniformlyboundedon[0;T],andweautomaticallyhavethatjrujisuniformlyboundedon[0;T].Westartwiththelastterm,andrecallthefollowingboundsonF0(u)(cf.[67]):ThereexistpositiveconstantsC1;C2suchthatF0(s)sC1s2C2;8s2Rand,forevery�0,thereexistsaconstantC3suchthatjF0(s)jC1s2+C3();8s2R:Usingthelasttwoestimatesweobtainthefollowing:(fu);1 F0(u)20 Z\nnDF0(u)fdx0 Z\nnDF0(u)udx0 Z\nnDjF0(u)jdx
kfkL1(\n)0C1 Z\nnDu2dx0C2j\nnDj 0C(f;\n) C1 Z\nnDu2dxC3()j\nnDj !0C1 Z\nnDu2dx0C2j\nnDj 0C1 (1C(f;\n))Z\nnDu2dxC(0;;;\n;D;f);wherewechoose1=C(f;\n).Thereforeintegrating(3.14)overthetimeinterval[0;T]resultsinZT0d dtE(u(t))dtZT0Z\njrK(u)j2dxdt+ZT0h(fu);
ui2dt0C1 (1C(f;\n))ZT0Z\nnDu2dxdt+T
C(0;;;\n;D;f):Nextweconsiderthesecondtermontherightsideofthelastinequality.FromTheorem4.1in[13]weknowthatasolutionuof(1.4)isanelementinL2(0;T;H2(\n))forallT�0.Hence
u2L2(0;T;L2(\n))andthesecondtermisboundedbyaconstantdependingonT.Consequently,foreach0tT,wegetE(u(t))E(u(0))+C(T)+T
C(0;;;\n;D;f)ZT0"Z\njrK(u)j2dx+0C1 (1C(f;\n))Z\nnDu2dx#dt;andwiththis,foraxedT�0,thatjrujisuniformlyboundedin[0;T]. 3.2.Numericalresults.Inourcomputationstheoptimal
tturnedouttobe
t=1or10(dependingalsoonthesizeofand0).NumericalresultsoftheaboveschemearepresentedinFigures3.1,3.2and3.3.Inalloftheexampleswefollowtheprocedureof[13],i.e.,theinpaintedimageiscomputedinatwostepprocess.IntherststepCahn-Hilliardinpaintingissolvedwitharatherlargevalueof,e.g.,=0:1,untilthenumericalschemeisclosetosteadystate.Inthisstepthelevellinesarecontinuedintothemissingdomain.Inasecondstep,theresultofthe C.B.SCHONLIEBANDA.BERTOZZI435 Fig.3.1.BinaryimagewithunknowncenterandthesolutionofCahn-Hilliardinpaintingwith0=105andswitchingvalue:u(600)with=0:1,u(1000)with=0:01 Fig.3.2.Textremovalfromabinaryimage:thesolutionofCahn-Hilliardinpaintingwith0=109andswitchingvalue:u(200)with=0:8,u(500)with=0:01rststepisputasaninitialconditionintotheschemeforasmall,e.g.,=0:01,inordertosharpenthecontoursoftheimagecontents.Thereasonforthistwostepprocedureistwofold.Firstofallin[14]theauthorsgivenumericalevidencethatthesteadystateofthemodiedCahn-HilliardEquation(1.4)isnotunique,i.e.,itisdependentontheinitialconditioninsideoftheinpaintingdomain.Asaconsequence,computingtheinpaintedimagebytheapplicationofCahn-Hilliardinpaintingwithasmallonly,mightnotextendthelevellinesintothemissingdomainasdesired.Seealso[14]forabifurcationdiagrambasedonthenumericalcomputationsoftheauthors.ThesecondreasonforsolvingCahn-Hilliardinpaintingintwostepsisthatitiscomputationallylessexpensive.Solvingtheabovetime-marchingschemefor,e.g.,=0:1isfasterthansolvingitfor=0:01.ThisisbecauseofthedampingintroducedbyC1,i.e.,,intothescheme;cf.(3.2).Allnumericalexamplespresentedherehavebeencomputedinordersof10secondsona1.86GHzprocessorwith1GBRAM.ForafurtherdiscussiononcomputationaltimesfortheconvexitysplittingmethodappliedtoCahn-Hilliardinpaintingwereferto[13].OnepossiblegeneralizationofCahn-Hilliardinpaintingforgrayscaleimagesistosplitthegrayscaleimagebit-wiseintochannelsu(x)KXk=1uk(x)2(k1);whereK�0.TheCahn-Hilliardinpaintingapproachisthenappliedtoeachbinarychannelukseparately,compareFigure3.5.Attheendoftheinpaintingprocessthechannelsareassembledagainandtheresultistheinpaintedgrayvalueimageinlowergrayvalueresolution,compareFigure3.4.InFigure3.6theapplicationofbitwiseCahn-Hilliardinpaintingfortherestorationofsatelliteimagesofroadsis 436UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTING Fig.3.3.VandalizedbinaryimageandthesolutionofCahn-Hilliardinpaintingwith0=109andswitchingvalue:u(800)with=0:8,u(1600)with=0:01demonstrated.Onecanimaginethattheblackdotsintherstpicturerepresenttreesthatcoverpartsoftheroad.Theideaofbitwisebinaryinpaintingisproposedin[30]fortheinpaintingwithwaveletsbasedontheAllen-Cahnenergy. 20 40 60 80 100 120 20 40 60 80 100 120 140 160 Inpainted image 20 40 60 80 100 120 20 40 60 80 100 120 140 160 Fig.3.4.Cahn-HilliardbitwiseinpaintingwithK=8binarychannels(0=108,with=0:1untilt=800and=0:01untilt=1200)4.TV-H1inpaintingInthissectionwediscussconvexitysplittingforTV-H1inpainting(1.5).ToavoidnumericalandtheoreticaldicultiesweapproximateanelementpinthesubdierentialofthetotalvariationfunctionalTV(u)byasmoothedversionofr
(ru=jruj),thesquarerootregularizationforinstance.Withthelatterregu-larizationthesmoothedversionof(1.5)readsasu=
r
ru p jruj2+2!+(fu);(4.1)with01.Incontrasttoitssecond-orderanalogue,thewell-posednessof(1.5)stronglydependsonthesmoothingusedforr
(ru=jruj).Infacttherearesmoothingfunctionsforwhich(1.5)producessingularitiesinnitetime.Thisiscausedbythelackofmaximumprincipleswhichinthesecond-ordercaseguaranteethewell-posednessforallsmoothmonotoneregularizations.In[16]theauthorsconsider(1.5)with=0inallof\n,i.e.,thefourth-orderanaloguetoTV-L2denoising,whichwasoriginallyintroducedin[58].Theyproveglobalwell-posedness,inonespace C.B.SCHONLIEBANDA.BERTOZZI437 initial condition2 20 40 60 80 100 120 20 40 60 80 100 120 140 160 initial condition3 20 40 60 80 100 120 20 40 60 80 100 120 140 160 initial condition5 20 40 60 80 100 120 20 40 60 80 100 120 140 160 1200 iterations,2channel 20 40 60 80 100 120 20 40 60 80 100 120 140 160 1200 iterations,3channel 20 40 60 80 100 120 20 40 60 80 100 120 140 160 1200 iterations,5channel 20 40 60 80 100 120 20 40 60 80 100 120 140 160 Fig.3.5.Thegivenimage(rstrow)andtheCahn-Hilliardinpaintingresult(secondrow)forthechannels2;3and5. Fig.3.6.BitwiseCahn-HilliardinpaintingwithK=8binarychannelsappliedtoroadrestorationdimensionandforsmoothinitialdata,forthearctanregularization2 arctan(ux=)x;(4.2)where01.Forthesquarerootsmoothing ux p juxj2+2!x(4.3)theyconjecture,supportedbyempiricalevidence,thatsingularitiesoccurininnitetime,notnitetime.Thebehaviorofthefourth-orderPDEinonedimensionisalsorelevantfortwo-dimensionalimagessincealotofstructureinvolvesedgeswhichareone-dimensionalobjects.Intwodimensionssimilarresultsaremuchmorediculttoobtain,sinceenergyestimatesandtheSobolevlemmainvolvedinitsproofmightnot 438UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGholdinhigherdimensionsanymore.Wealsonotethatin[19]theauthorsprovetheexistenceofaweakstationarysolutionfor(1.5)intwospacedimensions.Inthefollowingwepresenttheconvexitysplittingmethodappliedto(1.5)forboththesquarerootandthearctanregularization.SimilarlytotheconvexitysplittingforCahn-Hilliardinpainting,weproposethefollowingsplittingfortheTV-H1inpaintingequation.Theregularizingtermin(1.5)canbemodeledbyagradient\rowinH1oftheenergyE1(u)=Z\njrujdx;wherejrujisreplacedbyitsregularizedversion,e.g., jruj2+2,�0.WesplitE1asE1cE1,withE1c(u)=Z\nC1 2jruj2dx;andE1(u)=Z\njruj+C1 2jruj2dx:ThettingtermissplitintoE2=E2cE2,analogoustoCahn-Hilliardinpainting.Theresultingtime-steppingschemeisgivenbyUk+1Uk
t+C1

Uk+1+C2Uk+1=C1

Uk
r
rUk jrUkj+C2Uk+(fUk):(4.4)WeassumethatUk+1satiseszeroNeumannboundaryconditionsandusetheDCTtosolve(4.4).TheconstantsC1andC2havetobechosensuchthatE1c;E1;E2c;E2areallstrictlyconvex.Inthefollowingwedemonstratehowtocomputetheappropriateconstants.LetusconsiderC1rst.ThefunctionalE1cisstrictlyconvexforallC1�0.ThechoiceofC1fortheconvexityofE1dependsontheregularizationofthetotalvariationweareusing.Weusethesquareregularization(4.3),i.e.,insteadofjrujwehaveZG(jruj)dx;withG(s)= s2+2:Settingy=jrujwehavetochooseC1suchthatC1 2y2G(y)isconvex.TheconvexityconditionforthesecondderivativegivesusthatC1�G00(y)()C1�2 (2+y2)3=2()C1�1 issucientas2 (2+y2)3=2hasitsmaximumvalueaty=0.Intheonedimensionalcase,wewouldliketocomparethiswiththearctanregularization(4.2),i.e.,replacingx jxjby2 arctan(x )asproposedin[16].HeretheconvexityconditionforthesecondderivativereadsC1d ds2 arctans �0:Thesignresultsfromtheabsentabsolutevalueintheregularizationdenition.WeobtainC12 1 (1+s2=2)�0: C.B.SCHONLIEBANDA.BERTOZZI439TheinequalitywithaplussigninsteadofistrueforallconstantsC1�0.IntheothercaseweobtainC1�2  2+s2;whichisfullledforalls2RifC1�2 .Notethatthisconditionisalmostthesameasinthecaseofthesquareregularization.NowweconsiderE2=E2cE2.ThefunctionalE2cisstrictlyconvexifC2�0.FortheconvexityofE2werewriteE2(u)=1 2Z\n(fu)2+C2juj2dx=ZDC2 2juj2dx+Z\nnD0 2(fu)2+C2 2juj2dx=ZDC2 2juj2dx+Z\nnDC2 20 2juj2+0fu0 2jfj2:ThisisconvexforC2�0,e.g.,withC2=0+1wecanwriteE2(u)=ZDC2 2juj2dx+Z\nnD1 2u+0f220+0 2jfj2dx:4.1.Rigorousestimatesforthescheme.AsinSection3.1forCahn-Hilliardinpainting,weproceedwithamoredetailedanalysisof(4.4).Throughoutthissectionweconsiderthesquare-rootregularizationofthetotalvariationbothinournumericalschemeandinthecontinuousevolutionEquation(1.5).Notethatsimilarresultsaretrueforothermonotoneregularizerssuchasthearctansmoothing.Ourresultsaresummarizedinthefollowingtheorem.Theorem4.1.Letubetheexactsolutionof(4.1)anduk=u(k
t)betheexactsolutionattimek
tforatimestep
t�0andk2N.LetUkbethekthiterateof(4.4)withconstantsC1�1=,C2�0.Thenthefollowingstatementsaretrue:(i)Undertheassumptionthatkuk1andkr
uk2arebounded,thenumericalscheme(4.4)isconsistentwiththecontinuousEquation(1.5)andoforderoneintime.(ii)ThesolutionsequenceUkisboundedonanitetimeinterval[0;T],forall
t�0.Inparticular,fork
tT,T�0xed,wehaveforevery
t�0,krUkk22+
tK1kr
Ukk22eK2TkrU0k22+
tK1kr
U0k22+
tTC(\n;D;0;f)
(4.5)forsuitableconstantsK1,K2,andaconstantC,whichdependson\n;D;0;fonly.(iii)Letek=ukUk.ForsmoothsolutionsukandUk,theerrorekconvergestozeroas
t0.Inparticular,fork
tT,T�0xed,wehavekrekk22+
tM1kr
ekk22T M2eM3T(
t)2(4.6)forsuitablepositiveconstantsM1;M2andM3. 440UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGRemark4.1.FortheconvergenceresultinTheorem4.1(iii)weassumethatsmoothsolutionstoboththecontinuousintimeproblemandthediscreteintimeapproximationexist.Thevalidityofthisassumptionisnotknowningeneral.Notehoweverthattheglobalregularityresultsareknownin1Dforthearctansmooth-ing[16].Moreover,ournumericalresultsshownoindicationofsingularitiesin2D.Therefore,itisnotunreasonabletoanalyzetheconvergenceundertheseassumptions.TheproofofTheorem4.1issplitintothethreeseparatePropositions4.2-4.4.Proposition4.2(Consistency(i)).UnderthesameassumptionsasinThe-orem4.1andinparticularundertheassumptionthatkuk1andkr
uk2arebounded,thenumericalscheme(4.4)isconsistentwiththecontinuousEquation(4.1)withkkk1=O(
t)as
t0,wherekisthelocaltruncationerror.Proof.Thelocaltruncationerrorisdenedoveratimestepassatisfyingk=1k+2k;where1k=uk+1uk
tu(k
t);2k=C1
2(uk+1uk)+C2(uk+1uk);i.e.,k=uk+1uk
t+
r
ruk p jrukj2+2!!(fuk)+C1
2(uk+1uk)+C2(uk+1uk):(4.7)UsingstandardTaylorseriesargumentsandassumingthatkuk1,kr
uk2andkuk2areboundedwededucethatkkk1=O(
t)for
t0:(4.8) Proposition4.3(Unconditionalstability(ii)).Underthesameassump-tionsasinTheorem4.1thesolutionsequenceUkfullls(4.5).Thisgivesboundednessofthesolutionsequenceon[0;T].Proof.Ifwemultiply(4.4)with
Uk+1andintegrateover\nweobtain1
tkrUk+1k22hrUk;rUk+1i2+C2krUk+1k22+C1kr
Uk+1k22=*
r
0@rUk q jrUkj2+21A;
Uk+1+2+C1hr
Uk;r
Uk+1i2+hr((fUk));rUk+1i2+C2hrUk;rUk+1i2:ApplyingYoung'sinequalitytotheinnerproductsontherightandestimatingkr(fUk)k22220krUkk22+C(\n;D;0;f) C.B.SCHONLIEBANDA.BERTOZZI441resultsin1 2
tkrUk+1k22krUkk22+C2krUk+1k22+C1kr
Uk+1k22*
r
0@rUk q jrUkj2+21A;
Uk+1+2+C1 1kr
Ukk22+C11kr
Uk+1k22+220 2krUkk22+2krUk+1k22+C2 3krUkk22+C23krUk+1k22+C(\n;D;0;f):Now,thersttermontherightsideoftheinequalitycanbeestimatedasfollows*
r
0@rUk q jrUkj2+21A;
Uk+1+2=*rr
0@rUk q jrUkj2+21A;r
Uk+1+21 4\r\r\r\r\r\rrr
0@rUk q jrUkj2+21A\r\r\r\r\r\r22+4kr
Uk+1k22:ApplyingPoincare'sandCauchy'sinequalitytothersttermleadsto\r\r\r\r\r\rrr
0@rUk q jrUkj2+21A\r\r\r\r\r\r22C (krUkk22+k
Ukk22+kr
Ukk22):InterpolatingtheL2normof
ubytheL2normsofruandr
u,weobtain1 2
t+C2(13)2krUk+1k22+(C1(11)4)kr
Uk+1k221 2
t+220 2+C2 3+C(1=;\n) 4krUkk22+C1 1+C(1=;\n) 4kr
Ukk22+C(\n;D;0;f):For=1=2,i=1;:::;4weobtain1 2
t+C21 2krUk+1k22+C11 2kr
Uk+1k221 2
t+420+2(C2+C)krUkk22+2(C1+C)kr
Ukk22+C(\n;D;0;f):SinceC1andC2arechosensuchthatC1�1=�1andC2�0�1,thecoecientsintheinequalityabovearepositive.TherestoftheproofissimilartotheproofofProposition3.3.Wemultiplytheinequalityby2
tandsetCa=1+
t(C21);C=C11;Cc=1+2
t(420+2(C2+C));Cd=4(C1+C):WeobtainCakrUk+1k22+
tCkr
Uk+1k22CckrUkk22+
tCdkr
Ukk22+2
tC(\n;D;0;f): 442UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGDividingbyCa(whichis�0)wehavekrUk+1k22+
tCb Cakr
Uk+1k22Cc CakrUkk22+
tCd Cakr
Ukk222 Ca
tC(\n;D;0;f):WerewritetherighthandsideoftheinequalitysuchthatkrUk+1k22+
tC Cakr
Uk+1k22CcCd Ca1 CdkrUkk22+
t1 Cckr
Ukk22+2 Ca
tC(\n;D;0;f):SinceCd�1wecanmultiplythersttermwithinthebracketsontherighthandsideoftheinequalitywithCdandwillonlygetsomethingwhichislargerorequal.Forthesamereasonwecanmultiplythesecondtermwithinthebracketswith1CcC Ca=(1+2
t(420+2(C2+C)))(C11) 1+
t(C21)andgetkrUk+1k22+
tC Cakr
Uk+1k22CcCd CakrUkk22+
tC Cakr
Ukk22+2 Ca
tC(\n;D;0;f):ByinductionitfollowsthatkrUk+1k22+
tC Cakr
Uk+1k22CcCd CakkrU0k22+
tC Cakr
U0k22+
tk1X=0CcCd Ca2 CaC(\n;D;0;f):Thereforeweobtainfork
tTkrUkk22+
tCb Cakr
Ukk22eKTkrU0k22+
tCb Cakr
U0k22+
tT2 CaC(\n;D;0;f): Finallyweshowthatthediscretesolutionconvergestothecontinuousoneas
ttendstozero.Proposition4.4(Convergence(iii)).UnderthesameassumptionsasinTheorem4.1(iii)theerrorekfullls(4.6).Proof.ByourdiscreteApproximation(4.4)andtheconsistencycomputation C.B.SCHONLIEBANDA.BERTOZZI443(4.7),wehaveforek=ukUkek+1ek
t+C1
2ek+1+C2ek+1=1
t(uk+1uk)1
t(Uk+1Uk)+C1
2uk+1C1
2Uk+1+C2uk+1C2Uk+1=0@C1
2Uk
0@r
0@rUk q jrUkj2+21A1A+(fUk)+C2Uk1A0@
0@r
0@ruk q jrukj2+21A1A(fuk)C1
2ukC2uk1A+k="
0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A+C1
2(Ukuk)+C2(Ukuk)(Ukuk)#+k:Takingtheinnerproductwith
ek+1,wehave1
thr(ek+1ek);rek+1i2+C1kr
ek+1k22+C2krek+1k22=*
0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A;
ek+1+2+C1\n
2(Ukuk);
ek+12+hr(Ukuk);rek+1i2C2hr(Ukuk);rek+1i2\nr
1k;r
ek+12:UsingthesameargumentsasintheproofofProposition3.4weobtain1 2
t(krek+1k22krekk22)+C1kr
ek+1k22+C2krek+1k22*
0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A;
ek+1+2+C1 1kr
ekk22+C11kr
ek+1k22+20 3krekk22+3krek+1k22+C2 2krekk22+C22krek+1k22+1 4kkk21+4kr
ek+1k22:(4.9)Weconsiderthersttermontherightsideoftheaboveinequalityindetail,*
0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A;
ek+1+2=*r0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A;r
ek+1+2:(4.10) 444UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGWegetr
0@ru q jruj2+21A=
u q jruj2+2u2xuxx+2uxuyuxy+u2yuyy (jruj2+2)3=2:Next,weapplythegradienttothisexpressionandobtainr0@r
0@ru q jruj2+21A1A=r
u q jruj2+2
u 2(jruj2+2)3=2
rjruj2r(u2xuxx+2uxuyuxy+u2yuyy) (jruj2+2)3=2+3(u2xuxx+2uxuyuxy+u2yuyy) 2(jruj2+2)5=2
rjruj2;whererjruj2=0BB@2ru
uxxuyx2ru
uxyuyy1CCA=2uxuxx+uyuyxuxuxy+uyuyy;andr(u2xuxx+2uxuyuxy+u2yuyy)=2(uxuxx+uyuxy)rux+2(uxuxy+uyuyy)ruy+u2xruxx+2uxuyruxy+u2yruyy:Reorderingtheinvolvedtermswehaver0@r
0@ru q jruj2+21A1A=H1(ru)
r
u+H2(ux;uy;uxx;uxy;uyy)
rux+H3(ux;uy;uxx;uxy;uyy)
ruy+H4(ux;uy)
ruxx+H5(ux;uy)
ruxy+H6(ux;uy)
ruyy; C.B.SCHONLIEBANDA.BERTOZZI445whereH1(ru)=1 q jruj2+2;H2(ux;uy;uxx;uxy;uyy)=
uux+2(uxuxx+uyuxy) (jruj2+2)3=23(u2xuxx+2uxuyuxy+u2yuyy)ux (jruj2+2)5=2!;H3(ux;uy;uxx;uxy;uyy)=
uuy+2(uxuxy+uyuyy) (jruj2+2)3=23(u2xuxx+2uxuyuxy+u2yuyy)uy (jruj2+2)5=2!;H4(ux;uy)=u2x (jruj2+2)3=2;H5(ux;uy)=2uxuy (jruj2+2)3=2;H6(ux;uy)=u2y (jruj2+2)3=2:Nowwearegoingtoinsertthisinto(4.10).Foreaseofnotationwesuppressthetimeindexkfornow,i.e.,wedeneU:=Uk,u:=ukande:=ek.Weobtain*r0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A;r
ek+1+2=hH1(rU)
r
UH1(ru)
r
u;r
ek+1i2+hH2(Ux;Uy;Uxx;Uxy;Uyy)
rUxH2(ux;uy;uxx;uxy;uyy)
rux;r
ek+1i2+hH3(Ux;Uy;Uxx;Uxy;Uyy)
rUyH3(ux;uy;uxx;uxy;uyy)
ruy;r
ek+1i2+hH4(Ux;Uy)
rUxxH4(ux;uy)
ruxx;r
ek+1i2+hH5(Ux;Uy)
rUxyH5(ux;uy)
ruxy;r
ek+1i2+hH6(Ux;Uy)
rUyyH6(ux;uy)
ruyy;r
ek+1i2 446UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTING1 2kH1(rU)
(r
Ur
u)k22+1 2kr
u
(H1(ru)H1(rU))k22+1 2kH2(Ux;Uy;Uxx;Uxy;Uyy)
(rUxrux)k22+1 2krux
(H2(ux;uy;uxx;uxy;uyy)H2(Ux;Uy;Uxx;Uxy;Uyy))k22+1 2kH3(Ux;Uy;Uxx;Uxy;Uyy)
(rUyruy)k22+1 2kruy
(H3(ux;uy;uxx;uxy;uyy)H3(Ux;Uy;Uxx;Uxy;Uyy))k22+1 2kH4(Ux;Uy)
(rUxxruxx)k22+1 2kruxx
(H4(ux;uy)H4(Ux;Uy))k22+1 2kH5(Ux;Uy)
(rUxyruxy)k22+1 2kruxy
(H5(ux;uy)H5(Ux;Uy))k22+1 2kH6(Ux;Uy)
(rUyyruyy)k22+1 2kruyy
(H6(ux;uy)H6(Ux;Uy))k22+6kr
ek+1k22;forasuitableconstant�0.NextwewanttousethattheH'sareLipschitzcontin-uousin\n,withLipschitzconstantsL(1=),for&#x-0.8;أ倀0,whichgrowasdecreases.Forsimplicity,weonlypresenttheprooffortherstpartofH2,i.e.,forH12(ux;uy;uxx;uxy;uyy)=
uux+2(uxuxx+uyuxy) (jruj2+2)3=2=ux(3uxx+uyy)+2uyuxy (jruj2+2)3=2:Theothersfollowsimilarily.WehavekH12(ux;uy;uxx;uxy;uyy)H12(Ux;Uy;Uxx;Uxy;Uyy)k2\r\r\r\rux(3uxxuyy)+2uyuxy (jruj22)3=2Ux(3UxxUyy)+2UyUxy (jrUj22)3=2\r\r\r\r2\r\r\r\rux(3uxxuyy) (jruj22)3=2Ux(3UxxUyy) (jrUj22)3=2\r\r\r\r2+2\r\r\r\ruyuxy (jruj22)3=2UyUxy (jrUj22)3=2\r\r\r\r2\r\r\r\r(3uxxuyy)ux (jruj22)3=2Ux (jrUj22)3=2\r\r\r\r2\r\r\r\rUx (jrUj22)3=2(3exxeyy)\r\r\r\r2+2\r\r\r\ruxyuy (jruj22)3=2Uy (jrUj22)3=2\r\r\r\r2+2\r\r\r\rUy (jrUj22)3=2exy\r\r\r\r2:FromourassumptioninTheorem4.1(iii)wehaveacontinuousintimesmoothsolutionuonanitetimeinterval.Inparticularthisgivesusauniformboundforthesecondderivativesoftheexactsolutionu,i.e.,thereexistsaC�0suchthatkuxxk1+kuxyk1+kuyyk1Conanitetimeinterval[0;T].Further,withthefactthatthefunctionx (x2+y2+2)3=2isuniformlyboundedfor�0andforallx;y2RwehavekH12(ux;uy;uxx;uxy;uyy)H12(Ux;Uy;Uxx;Uxy;Uyy)k2C\r\r\r\r\rux (jruj2+2)3=2Ux (jrUj2+2)3=2\r\r\r\r\r2+Ck3exxeyyk2+2C\r\r\r\r\ruy (jruj2+2)3=2Uy (jrUj2+2)3=2\r\r\r\r\r2+2Ckexyk2; C.B.SCHONLIEBANDA.BERTOZZI447whereweusedauniversalconstantC�0fortheuniformbounds.Moreover,foraxedyand�0thefunctionx (x2+y2+2)3=2isLipschitzcontinuouswithconstantL(1=),whichisincreasingasdecreases.ByadditionallyapplyingthetriangularinequalityoncemoreweeventuallyhavekH12(ux;uy;uxx;uxy;uyy)H12(Ux;Uy;Uxx;Uxy;Uyy)k2CL(1=)(kexk2+keyk2+krek2)+C(kexxk2+kexyk2+keyyk2);andhencethatH12isLipschitzcontinuous.SimilarlyonecanshowthattheotherH'sareLipschitzcontinuous.LetusfurtherobservethatH1;H4;H5;H6areuniformlyboundedfor�0.Moreover,theuniformboundednessofH2andH3forthediscreteintimesolutionUonanitetimeintervalisgivenbythesmoothnessassumptioninTheorem4.1(iii)forU.Then,withtheLipschitzcontinuityandtheuniformboundednessoftheH'sonanitetimeinterval,andtheuniformboundednessonanitetimeintervalofruk,
uk,andr
ukfortheexactsolutionukgiveninTheorem4.1(iii),weeventuallyobtainanestimatefor(4.10):*r0@r
0@rUk q jrUkj2+21Ar
0@ruk q jrukj2+21A1A;r
ek+1+2C 2kr
ek22+CL3 krek22+C 2krexk22+C 2kreyk22+CL (kexxk22+kexyk22+keyyk22)+C 2krexxk22+C 2krexyk22+C 2kreyyk22+6kr
ek+1k22;(4.11)whereL=L(1=)denotesauniversalLipschitzconstantfortheH'sandCisauniversalconstantfortheinvolveduniformbounds.Further,havingassumedzeroNeumannboundaryconditionsfor(1.5)and(4.4),i.e.,ru
~n=rr
ru p jruj2+2!
~n=0;on@\n;where~nistheoutwardpointingnormalon@\n,thesecondandthirdderivativesin(4.11)canbeboundedbykexxk22+kexyk22+keyyk22+krexxk22+krexyk22+kreyyk22B(k
ek22+kr
ek22);(4.12)forasuitableconstantB�0.BecauseoftheNeumannboundaryconditionswealsogetthatR\n
e=0.Hence,wecanapplyPoincare'sinequalitytok
ek2andobtain,for(4.9),1 2
tC2(12)3krek+1k22C1(11)46
kr
ek+1k221 2
tC2 220 33CL krekk22C1 1C 2BC5 2L1 kr
ekk221 4kkk21;wherewereintroducedtheindexnotationekfore.ThereforebyfollowingthelinesoftheproofofProposition3.4wenallyhave,fork
tT,krekk22+
tM1kr
ekk22T M2eM3T(
t)2; 448UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGforsuitablepositiveconstantsM1;M2andM3. Remark4.5.NotethattheLipschitzcontinuityoftheH's{necessaryfortheestimatesintheconvergenceproof{breaksdownif0,whereisthesmoothingparameterinthesquare-rootregularization(4.3)ofthetotalvariation.4.2.Numericalresults.NumericalresultsfortheTV-H1inpaintingap-proacharepresentedinFigures4.1and4.2.ForacomparisonofthehigherorderTV-H1inpaintingapproachwithitssecondordercousin,thestandardTV-L2in-paintingmethod,inFigure4.2weconsidertheperformanceofbothalgorithmsinasmallpartoftheimageinFigure4.1.InfacttheresultshowninFigures4.1and4.2stronglyindicatesthecontinuationofthegradientoftheimagefunctionintotheinpaintingdomain.Arigorousproofofthisobservation,astheoneforCahn-Hilliardinpainting(cf.Section3),isamatteroffutureresearch.Inbothexamplesthetotalvariationjrujisapproximatedby jruj2+andthetimestepsize
tischosentobeequaltoone.ThecomputationaltimefortheexampleinFigure4.1isoftheorderof100secondsona1.86GHzprocessorwith1GBRAM. Fig.4.1.TV-H1inpainting:u(1000)with0=103 Fig.4.2.(l.)u(1000)withTV-H1inpainting,(r.)u(5000)withTV-L2inpainting5.LCISinpaintingOurlastexamplefortheapplicabilityoftheconvexitysplittingmethodtohigher-orderinpaintingapproachesisinpaintingwithLCIS(1.6).Withf2L2(\n)ourin-paintedimageuevolvesintimeasu=
(arctan(
u))+(fu):Incontrasttotheothertwoinpaintingmethodsthatwediscussed,thisinpainting C.B.SCHONLIEBANDA.BERTOZZI449equationisagradient\rowinL2fortheenergyE(u)=Z\nG(
u)dx+1 2Z\n(fu)2;withG0(y)=arctan(y).ThereforeEyre'sresultinTheorem2.3canbeapplieddirectly.ThefunctionalE(u)issplitintoEcEwithEc(u)=Z\nC1 2(
u)2dx+1 2Z\nC2 2juj2dx;E(u)=Z\nG(
u)+C1 2(
u)2dx+1 2Z\n(fu)2+C2 2juj2dx:Theresultingtime-steppingschemeisUk+1Uk
t+C1
2Uk+1+C2Uk+1=
(arctan(
Uk))+C1
2Uk+(fUk)+C2Uk:(5.1)AgainweimposehomogeneousNeumannboundaryconditions,useDCTtosolve(5.1),andchoosetheconstantsC1andC2suchthatEcandEareallstrictlyconvexandcondition(2.7)issatised.ThefunctionalEcisconvexforallC1;C2�0.TherstterminEisconvexifC1�1.Thisfollowsfromitssecondvariation,namelyr2E1(u)(v;w)=d dsZ(C1
(u+sw)arctan(
(u+sw)))
vdxs=0=ZC11 1+(
u)2
v
wdx:ForE1tobeconvex,r2E1(u)(v;w)mustbe�0forallv;w2C1,andthereforeC11 1+(
u)2�0:Substitutings=
uweobtainC1�1 1+s28s2R:Thisinequalityisfullledforalls2RifC1�1.WeobtainthesameconditiononC1forG0(s)=arctan(s ).FortheconvexityofthesecondtermofE,thesecondconstanthastofulllC2�0;cf.thecomputationforthettingterminSection4.WiththesechoicesofC1andC2alsocondition(2.7)ofTheorem2.3isautomaticallysatised.5.1.Rigorousestimatesforthescheme.Finallywepresentrigorousresultsfor(5.1).IncontrasttotheinpaintingEquations(1.4)and(1.5),inpaintingwithLCISfollowsavariationalprinciple.Hence,bychoosingtheconstantsC1andC2appropriately,i.e.,C1�1,C2�0(cf.thecomputationsabove),Theorem2.3ensuresthattheiterativescheme(5.1)isunconditionallygradientstable.Inadditiontothisproperty,wepresentsimilarresultsasbeforeforCahn-HilliardandTV-H1inpainting.Theorem5.1.Letubetheexactsolutionof(1.6)anduk=u(k
t)theexactsolutionattimek
t,foratimestep
t�0andk2N.LetUkbethekthiterateof(5.1)withconstantsC1�1,C2�0.Thenthefollowingstatementshold: 450UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTING(i)Undertheassumptionthatkuk1andkr
uk2arebounded,thenumericalscheme(5.1)isconsistentwiththecontinuousEquation(1.6)andoforderoneintime.(ii)ThesolutionsequenceUkisboundedonanitetimeinterval[0;T]forall
t�0.Inparticular,fork
tT,T&#x-139;&#x.942;0xed,wehavekrUkk22+
tK1kr
Ukk22eK2TkrU0k22+
tK1kr
U0k22+
tTC(\n;D;0;f)
(5.2)forsuitableconstantsK1,K2,andaconstantCdependingon\n;D;0;fonly.(iii)Letek=ukUk.Ifkr
ukk22K;foraconstantK&#x-139;&#x.942;0;andforallk
tT(5.3)thentheerrorekconvergestozeroas
t0.Inparticular,fork
tT,T&#x-139;&#x.942;0xed,wehavekrekk22+
tM1kr
ekk22T M2eM3T(
t)2;(5.4)forsuitablenonnegativeconstantsM1;M2andM3.Remark5.1.AsinTheorem4.1(cf.alsoRemark4.1)theconvergenceoftheiteratesUktotheexactsolutionisprovenunderanassumptionontheexactsolution,i.e.,assumption(5.3),whosevalidityisunknowningeneral.However,previousresultsin[15]forthedenoisingcase,i.e.,for(x)=0inallof\n,andforsmoothinitialdataandsmoothf,suggesttheassumptionisalsoreasonablefortheinpaintingcase.TheproofofTheorem5.1isorganizedinthefollowingthreePropositions5.2-5.4.SincetheproofofconsistencyfollowsthelinesofProposition3.2andProposition4.2,wejuststatetheresult.Proposition5.2(Consistency(i)).UnderthesameassumptionsasinTheo-rem5.1andinparticularassumingthatkuk1andkr
uk2arebounded,wehavekkk1=O(
t)for
t0:Nextwewouldliketoshowtheboundednessofasolutionof(5.1)inthefollowingproposition.Proposition5.3.(Unconditionalstability(ii))UnderthesameassumptionsasinTheorem5.1thesolutionsequenceUkfullls(5.2).Thisgivesboundednessofthesolutionsequenceon[0;T].Proof.Ifwemultiply(5.1)with
Uk+1andintegrateover\n,weobtain1
tkrUk+1k22hrUk;rUk+1i2+C2krUk+1k22+C1kr
Uk+1k22=h
arctan(
Uk);
Uk+1i2+C1hr
Uk;r
Uk+1i2+hr((fUk));rUk+1i2+C2hrUk;rUk+1i2: C.B.SCHONLIEBANDA.BERTOZZI451UsingthesameargumentsasintheproofsofProposition3.3and4.3weobtain1 2
tkrUk+1k22krUkk22+C2krUk+1k22+C1kr
Uk+1k22h
arctan(
Uk);
Uk+1i2+C1 1kr
Ukk22+C11kr
Uk+1k22+20 22krUkk22+2krUk+1k22+C2 3krUkk22+C23krUk+1k22+C(\n;D;0;f):Now,thersttermontherightsideoftheinequalitycanbeestimatedasfollowsh
arctan(
Uk);
Uk+1i2=hrarctan(
Uk);r
Uk+1i2=1 1+(
Uk)2r
Uk;r
Uk+121 4\r\r\r\r1 1+(
Uk)2r
Uk\r\r\r\r22+4kr
Uk+1k221 4kr
Ukk22+4kr
Uk+1k22:(5.5)Fromthisweget1 2
t+C2(13)2krUk+1k22+(C1(11)4)kr
Uk+1k221 2
t+20 22+C2 3krUkk22+C1 1+1 4kr
Ukk22+C(\n;D;0;f):AnalogouslytoSection4.1,withCa=1+
t(C21);C=C11;Cc=1+2
t(20+2C2);Cd=4(C1+1);weobtainkrUkk22+
tCb Cakr
Ukk22eKTkrU0k22+
tCb Cakr
U0k22+
tT2 CaC(\n;D;0;f);whichgivesboundednessofthesolutionsequenceon[0;T]foranyT�0andany
t�0. Theconvergenceofthediscretesolutiontothecontinuousoneas
t0isveriedinthefollowingproposition.Proposition5.4(Convergence(iii)).UnderthesameassumptionsasinTheorem5.1andinparticularunderassumption(5.3),theerrorekfullls(5.4).Proof.Sinceallthecomputationsintheconvergenceprooffor(5.1)arethesameasinSection4.1for(4.4)exceptoftheestimatefortheregularizer
(arctan(
u)),weonlygivethedetailsforthelatterandleavetheresttothereader.Thus,fortheinnerproductinvolvingtheregularizerof(5.1)withintheconvergenceproof,weobtainh
(arctan(
Uk)arctan(
uk));
ek+1i2=hr(arctan(
Uk)arctan(
uk));r
ek+1i2=hw(
Uk)r
Ukw(
uk)r
uk;r
ek+1i2=hw(
Uk)r
ek;r
ek+1i2h(w(
Uk)w(
uk))r
uk;r
ek+1i21 2kw(
Uk)jr
ekjk22+1 21k(w(
uk)w(
Uk))jr
ukjk22++1 2kr
ek+1k22; 452UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGwherewehaveusedthatr(arctan(
u))=1 1+j
uj2r
u=w(
u)r
u:Usingtheuniformboundednessofw(s)foralls2R,theuniformboundonr
ukfromAssumption(5.3),andtheLipschitzcontinuityofw,wegeth
(arctan(
Uk)arctan(
uk));
ek+1i2C 2kr
ekk22+CL 21k
ekk22++1 2kr
ek+1k22:Moreover,becauseofthezeroNeumannboundaryconditionsfullledbysolutionsof(1.6)and(5.1),i.e.,ru
~n=r(arctan(
u))
~n=0;on@\n;where~nistheoutwardpointingnormalon@\n,
ekhaszeromeanandwecanapplyPoincare'sinequalitytoobtainh
(arctan(
Uk)arctan(
uk));
ek+1i2C 2CL 21kr
ekk221 2kr
ek+1k22:FollowingthesamestepsasintheproofofProposition4.4wenallyhave,fork
tT,krekk22+
tM1kr
ekk22T M2eM3T(
t)2forsuitablepositiveconstantsM1;M2andM3. 5.2.Numericalresults.ForthecomparisonwithTV-H1inpaintingweapply(5.1)tothesameimageasinSection4.2.ThisexampleispresentedinFigure5.1.InFigure5.2theLCISinpaintingresultiscomparedwithTV-H1-andTV-L2inpainting,forasmallpartinthegivenimage.Againtheresultofthiscomparisonindicatesthecontinuationofthegradientoftheimagefunctionintotheinpaintingdomainforthetwohigher-ordermethods.Arigorousproofofthisobservationisamatteroffutureresearch.Forthenumericalcomputationof(5.1)thearctan(s)wasregularizedbyarctan(s=),�0and
tchosentobeequalto0:01.TheinpaintedimageinFigure5.1hasbeencomputedinabout90secondsona1.86GHzprocessorwith1GBRAM.6.ConclusionInthispaperwepresentseveralhigherorderPDE-basedmethodsforimagein-painting,alongwithunconditionallystabletime-steppingschemesforthesolutionoftheseequations.SpecicexamplesdiscussedincludeCahn-Hilliardinpainting,TV-H1inpainting,andinpaintingwithLCIS.Theconstructionoftheseschemesisbasedontheideaofconvexitysplitting,alsointroducedinthispaper.Westudythenu-mericalanalysisoftheschemesincludingconsistency,unconditionalstability,andconvergence.Belowweconsidersomeopenproblemsforthisclassofmethods. C.B.SCHONLIEBANDA.BERTOZZI453 Fig.5.1.LCISinpaintingu(500)with=0:1and0=102. Fig.5.2.(l.)u(1000)withLCISinpainting,(m.)u(1000)withTV-H1inpainting,(r.)u(5000)withTV-L2inpaintingTheadvantageoffourthorderinpaintingmodels,overmodelsofseconddif-ferentialorder,isthesmoothcontinuationofimagecontents,includingdirec-tionofedges,acrossgapsintheimage.FourthorderPDEsrequireanextraboundaryconditioncomparedwithsecondorderequationsandthisisthemo-tivationforadditionalgeometriccontentprovidedbysuchmethods.However,ingeneral,theadditionalboundaryconditioncouldinvolveanyofthehigherderivatives,andforinpaintingisitdesirabletocontinuetherstderivativeaccrosstheinpaintingregion.Themethodsproposedhereareglobalmeth-odsbasedonanL2delitytermassociatedwiththeknowninformation.ForthespecialcaseoftheCahn-Hilliardequation[14],inthelimitas0!1astationarysolutionisprovedtosatisfypreciselythedesiredtwoboundaryconditions|matchingofgreyvalueandmatchingdirectionofedges.Weconjecturethatanalogousresultsaretruefortheothermethodspresentedherealthougharigorousproofisbeyondthescopeofthismanuscript.Fortheproofsofconvergenceofthediscretesolutiontotheexactsolution,i.e.,fortheproofsofTheorem4.4andTheorem5.4,wehadtoassumethattheexactsolutionisboundedonanitetimeintervalinacertainSobolevnorm.Aswealreadyarguedintheremarksafterthestatementofthetheorems,theseassumptionsseemtobeheuristicallyreasonableconsideringearlierre-sultsin[15,16].Neverthelessarigorousderivationofsuchboundsisstillmissing.Besidesthefactthatrigorousresultsforfourth-orderpartialdierentialequa-tionsarerareingeneral,anasymptoticanalysisofourthreeinpaintingmodelswouldbeofhigh(evenpractical)interest.MorepreciselytheconvergenceofasolutionoftheevolutionEquations(1.4),(1.5),and(1.6),toastation-arystateisstillopen.Sincetheinpaintedimageisthestationarysolution 454UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGofthoseevolutionequations,theasymptoticbehaviorisofcourseanissue.Also,inpracticethenumericalschemesaresolvedtosteadystate(uptoanapproximationalerror).Notethatinadditiontothefourthdierentialorder,adicultyintheconvergenceanalysisof(1.4)and(1.5)isthattheequationsdonotfollowavariationalprinciple.Thediscreteschemesproposedinthispaperareunconditionallystableandtheirnumericalperformanceisamatterof10to100secondsforsmalltomedium-sizedimages,i.e.,128128to256256pixels,andgapsthatcon-stituteaboutonetotenpercentoftheimagedomain.Fastnumericalsolversforhigherorderinpaintingmodelsisstillamostlyopeneldofresearch.AmongsuchfastsolverswefoundtherecentcontributionofBrito-LoezaandChen[18]veryinterestingandforward-looking,whouseamultigridmethodtosolveinpaintingwithCDD(CurvatureDrivenDiusion).Anotherap-proachistheSplitBregmanmethodofGoldsteinandOsher[41,42],whichsuggestsasplittingofahigher-ordervariationalproblemintwoconsecutivelyminimizedrst-orderproblems.Althoughnotdirectlyapplicabletothenon-variationalinpaintingtechniques(1.4)and(1.5),theirmethodpromisesanecientsolutionof,e.g.,(1.6)LCISinpainting.Acknowledgments.C.-B.Schonliebacknowledgesthenancialsupportpro-videdbytheDFGGraduiertenkolleg1023IdenticationinMathematicalModels:SynergyofStochasticandNumericalMethods,supportbytheprojectWWTFFivesenses-Call2006,MathematicalMethodsforImageAnalysisandProcessingintheVi-sualArtsandbytheFFGprojectErarbeitungneuerAlgorithmenzumImageInpaint-ing,projectnumber813610.Further,thispublicationisbasedonworksupportedbyAwardNo.KUK-I1-007-43,madebyKingAbdullahUniversityofScienceandTech-nology(KAUST).Inaddition,C.-B.SchonliebthanksIPAM(InstituteforPureandAppliedMathematics),UCLA,forthehospitalityandthenancialsupportduringthepreparationofthiswork.BothauthorsacknowledgesupportfromtheNSFgrantBCS-0527388,ONRgrantN000140810363,andtheDepartmentofDefense.TheauthorsthanktherefereesandWenuahGao(UCLA)forusefulcommentsonthemanuscript.REFERENCES[1]J.-F.AujolandA.Chambolle,Dualnormsandimagedecompositionmodels,InternationalJournalofComputerVision,63(1),85{104,June,2005.[2]J.-F.AujolandG.Gilboa,ConstrainedandSNR-basedsolutionsforTV-Hilbertspaceimagedenoising,J.Math.ImagingandVision,26(1-2),217{237,November2006.[3]W.Baatz,M.Fornasier,P.MarkowichandC.B.Schonlieb,InpaintingofancientAustrianfrescoes,ConferenceProceedingsofBridges2008,Leeuwarden2008,150{156,2008.[4]J.W.BarrettandJ.F.Blowey,Finiteelementapproximationofamodelforphaseseparationofamulti-componentalloywithnon-smoothfreeenergy,Numer.Math.,77(1),1{34,1997.[5]J.W.BarrettandJ.F.Blowey,Finiteelementapproximationofamodelforphaseseparationofamulti-componentalloywithaconcentration-dependentmobilitymatrix,IMAJ.Numer.Anal.,18(2),287{328,1998.[6]J.W.BarrettandJ.F.Blowey,Finiteelementapproximationofamodelforphaseseparationofamulti-componentalloywithnonsmoothfreeenergyandaconcentrationdependentmobilitymatrix,Math.ModelsMethodsAppl.Sci.,9(5),627{663,1999.[7]J.W.BarrettandJ.F.Blowey,FiniteelementapproximationoftheCahn-Hilliardequationwithconcentrationdependentmobility,Math.Comput.,68(226),487{517,1999. C.B.SCHONLIEBANDA.BERTOZZI455[8]J.Becker,G.Grun,M.LenzandM.Rumpf,Numericalmethodsforfourthordernonlineardiusionproblems,ApplicationsofMathematics,47,517{543,2002.[9]J.Bect,L.Blanc-Feraud,G.AubertandA.Chambolle,A`1-uniedVariationalFrameworkforImageRestoration,T.PajdlaandJ.Matas(eds.),Proceedingsofthe8thEuropeanConferenceonComputerVisionIV,Prague,CzechRepublic,SpingerVerlag,2004.[10]M.Bertalmio,G.Sapiro,V.CasellesandC.Ballester,Imageinpainting,Siggraph2000,Com-puterGraphicsProceedings,417{424,2000.[11]M.Bertalmio,A.L.BertozziandG.Sapiro,Navier-Stokes,\ruiddynamics,andimageandvideoinpainting,Proceedingsofthe2001IEEEComputerSocietyConferenceonComputerVisionandPatternRecognition,1,355{362,2001.[12]M.Bertalmio,L.Vese,G.SapiroandS.Osher,Simultaneousstructureandtextureimageinpainting,IEEETrans.ImageProc.,12(8),882{889,2003.[13]A.Bertozzi,S.EsedogluandA.Gillette,InpaintingofbinaryimagesusingtheCahn-Hilliardequation,IEEETrans.ImageProc.,16(1),285{291,2007.[14]A.Bertozzi,S.EsedogluandA.Gillette,Analysisofatwo-scaleCahn-Hilliardmodelforimageinpainting,MultiscaleModelingandSimulation,6(3),913{936,2007.[15]A.BertozziandJ.B.Greer,Low-curvatureimagesimpliers:globalregularityofsmoothsolu-tionsandLaplacianlimitingschemes,Commun.PureAppl.Math.,LVII,764{790,2004.[16]A.Bertozzi,J.Greer,S.OsherandK.Vixie,NonlinearregularizationsofTVbasedPDEsforimageprocessing,AMSSeriesofContemporaryMathematics,G.-Q.Chen,G.Gasper,andJ.Jerome(eds.),371,29{40,2005.[17]A.Bertozzi,N.JuandH.W.Lu,Abiharmonic-modiedforwardtimesteppingmethodforfourthordernonlineardiusionequations,DiscreteandContinuousDynamicalSystems2009,submitted.UCLACAMreports08-18,2008.[18]C.Brito-LoezaandK.Chen,Multigridmethodforamodiedcurvaturedrivendiusionmodelforimageinpainting,J.Comput.Math.,26(6),856{875,2008.[19]M.Burger,L.HeandC.Schonlieb,Cahn-Hilliardinpaintingandageneralizationforgrayvalueimages,SIAMJ.ImagingSci.,2(4),1129{1167,2009.[20]F.Cao,Y.Gousseau,S.MasnouandP.Perez,Geometricallyguidedexemplar-basedinpainting,INRIApreprint,hal-00380394,2009..[21]V.Caselles,J.M.Morel,C.SbertandA.Gillette,Anaxiomaticapproachtoimageinterpola-tion,IEEETrans.onImageProc.,7(3),376{386,1998.[22]A.Chambolle,Analgorithmfortotalvariationminimizationandapplications,J.Math.Imag-ingVis.,20(1{2),89{97,2004.[23]T.F.ChanandJ.Shen,Mathematicalmodelsforlocalnon-textureinpaintings,SIAMJ.Appl.Math.,62(3),1019{1043,2001.[24]T.F.ChanandJ.Shen,Non-textureinpaintingbycurvaturedrivendiusions(CDD),J.VisualCommun.ImageRep.,12(4),436{449,2001.[25]T.F.ChanandJ.Shen,Variationalrestorationofnon-\ratimagefeatures:modelsandalgo-rithms,SIAMJ.Appl.Math.,61(4),1338{1361,2001.[26]T.F.ChanandJ.Shen,Variationalimageinpainting,Commun.PureAppliedMath.,58,579{619,2005.[27]T.F.Chan,S.H.KangandJ.Shen,Euler'selasticaandcurvature-basedinpainting,SIAMJ.Appl.Math.,63(2),564{592,2002.[28]T.F.Chan,J.H.ShenandH.M.Zhou,Totalvariationwaveletinpainting,J.Math.ImagingVis.,25(1),107{125,2006.[29]A.Criminisi,P.PerezandK.Toyama,Objectremovalbyexemplar-basedinpainting,IEEEInt.Conf.Comp.VisionandPatternRecog.2,721{728,2003.[30]J.A.DobrosotskayaandA.L.Bertozzi,AWavelet-laplacevariationaltechniqueforimagede-convolutionandinpainting,IEEETrans.Imag.Proc.,17(5),657{663,2008.[31]J.Jr.DouglasandT.Dupont,Alternating-directionGalerkinmethodsonrectanglesnumericalsolutionofpartialdierentialequations,II(SYNSPADE1970)(Proc.Symp.,UniversityofMaryland,CollegePark,Md.1970)(NewYork:Academic),133{214,1971.[32]A.A.EfrosandT.K.Leung,TextureSynthesisbyNon-parametricSampling,IEEEInterna-tionalConferenceonComputerVision,Corfu,Greece,September,1999.[33]C.M.ElliottandS.A.Smitheman,AnalysisoftheTVregularizationandH1delitymodelfordecomposinganimageintocartoonplustexture,Commun.PureAppl.Anal.,6(4),917{936,2007.[34]C.M.ElliottandS.A.Smitheman,NumericalanalysisoftheTVregularizationandH1delitymodelfordecomposinganimageintocartoonplustexture,IMAJournalofNumericalAnalysis,1{39,2008.[35]S.EsedogluandJ.H.Shen,DigitalinpaintingbasedontheMumford-Shah-Eulerimagemodel, 456UNCONDITIONALLYSTABLESCHEMESFORHIGHERORDERINPAINTINGEur.J.Appl.Math.,13(4),353{370,2002.[36]D.Eyre,Anunconditionallystableone-stepschemeforgradientsystems,Jun.,1998,unpub-lished.[37]X.FengandA.Prohl,ErroranalysisofamixedniteelementmethodfortheCahn-Hilliardequation,Numer.Math.,99(1),47{84,2004.[38]X.FengandA.Prohl,NumericalanalysisoftheCahn-HilliardequationandapproximationfortheHele-Shawproblem,InterfacesFreeBound.,7(1),1{28,2005.[39]M.FornasierandC.B.Schonlieb,Subspacecorrectionmethodsfortotalvariationandl1-min-imization,SIAMJ.Numer.Anal.,47(5),3397{3428,2009.[40]K.Glasner,AdiuseinterfaceapproachtoHele-Shaw\row,Nonlinearity,16,1{18,2003.[41]T.Goldstein,X.BressonandS.Osher,GeometricapplicationsofthesplitBregmanmethod:Segmentationandsurfacereconstruction,J.Sci.Comp.,45(1-3),pp.272{293,2010.[42]T.GoldsteinandS.Osher,ThesplitBregmanmethodforL1regularizedproblems,SIAMJournalonImagingSciences,2(2),323{343,2009.[43]J.B.GreerandA.Bertozzi,H1solutionsofaclassoffourthordernonlinearequationsforimageprocessing,DiscreteContin.Dyn.Syst.,10(1-2),349{366,2004.[44]J.B.GreerandA.Bertozzi,TravelingwavesolutionsoffourthorderPDEsforimageprocessing,SIAMJ.Math.Anal.,36(1),38{68,2004.[45]H.GrossauerandO.Scherzer,UsingthecomplexGinzburg-Landauequationfordigitalin-paintingin2Dand3D,ScaleSpaceMethodsinComputerVision,2695,225{236,2003.[46]G.GrunandM.Rumpf,Nonnegativitypreservingconvergentschemesforthethinlmequa-tion,Num.Math.,87,113{152,2000.[47]J.W.Gu,L.Zhang,G.Q.Yu,Y.X.XingandZ.Q.Chen,X-rayCTmetalartifactsreductionthroughcurvaturebasedsinograminpainting,JournalofX-RayScienceandTechnology,14(2),73{82,2006.[48]M.HirschandS.Smale,DierentialEquations,DynamicalSystemsandLinearAlgebra,Aca-demicPress,London,1974.[49]L.LieuandL.Vese,ImagerestorationanddecompostionviaboundedtotalvariationandnegativeHilbert-Sobolevspaces,AppliedMathematics&Optimization,58,167{193,2008.[50]M.Lysaker,A.LundervoldandX.C.Tai,Noiseremovalusingfourth-orderpartialdierentialequationswithapplicationstomedicalmagneticresonanceimagesinspaceandtime,IEEETrans.onImageProc.,12(12),1579{1590,2003.[51]O.M.LysakerandX.-C.Tai,Iterativeimagerestorationcombiningtotalvariationminimiza-tionandasecond-orderfunctional,Inter.J.Comput.Vis.,66(1),5{18,2006.[52]S.MasnouandJ.Morel,Levellinesbaseddisocclusion,5thIEEEInt'lConf.onImagePro-cessing,Chicago,IL,Oct.,4-7,259{263,1998.[53]L.ModicaandS.Mortola,Illimitenella-convergenzadiunafamigliadifunzionaliellittici,Boll.UnioneMat.Ital.,V.Ser.,A14,526{529,1977.[54]L.ModicaandS.Mortola,Unesempiodi-convergenza,Boll.UnioneMat.Ital.,V.Ser.,B14,285{299,1977.[55]M.J.NarashimaandA.M.Peterson,Onthecomputationofthediscretecosinetransform,IEEETrans.Commun.COM,26,934-936,June,1978.[56]L.K.Nielsen,X.C.Tai,S.I.AanonsenandM.Espedal,NoiseRemovalofSeismicDataUsingaFourth-OrderParabolicPDE,InternationalJ.Tomography&Statistics(IJTS),4(6),63,2006.[57]M.Nitzberg,D.MumfordandT.Shiota,Filtering,Segmentation,andDepth,Springer-Verlag,LectureNotesinComputerScience,662,1993.[58]S.Osher,A.SoleandL.Vese,ImagedecompositionandrestorationusingtotalvariationminimizationandtheH1norm,MultiscaleModelingandSimulation:ASIAMInterdis-ciplinaryJournal,1(3),349{370,2003.[59]P.PeronaandJ.Malik,Scale-spaceandEdgeDetectionUsingAnisotropicDiusion,IEEETransactionsonPatternAnalysisandMachineIntelligence,12(7),629{639,July,1990.[60]L.I.Rudin,S.OsherandE.Fatemi,Nonlineartotalvariationbasednoiseremovalalgorithms,PhysicaD,60,259{268,1992.[61]L.RudinandS.Osher,Totalvariationbasedimagerestorationwithfreelocalconstraints,Proc.1stIEEEICIP,1,31{35,1994.[62]C.B.Schonlieb,TotalvariationminimizationwithanH1constraint,CRMSeries9,Singu-laritiesinNonlinearEvolutionPhenomenaandApplicationsProceedings,ScuolaNormaleSuperiorePisa,2009,201{232.[63]C.B.Schonlieb,ModernPDETechniquesforImageInpainting,PhDThesis,UniversityofCambridge,265,June2009.[64]C.B.Schonlieb,A.Bertozzi,M.BurgerandL.He,ImageInpaintingUsingaFourth-Order C.B.SCHONLIEBANDA.BERTOZZI457TotalVariationFlow,Proc.Int.Conf.SampTA09,Marseilles,2009.[65]P.Smereka,Semi-implicitlevelsetmethodsforcurvatureandsurfacediusionmotion,SpecialissueinhonorofthesixtiethbirthdayofStanleyOsher,J.Sci.Comput.,19(1-3),439{456,2003.[66]A.M.StuartandA.R.Humphries,Modelproblemsinnumericalstabilitytheoryforinitialvalueproblems,SIAMRev.,36(2),226{257,1994.[67]R.Temam,Innite-DimensionalDynamicalSystemsinMechanicsandPhysics,Appl.Math.Sci.,Springer,NewYork,68,1988.[68]A.Tsai,J.A.YezziandA.S.Willsky,CurveevolutionimplementationoftheMumford-Shahfunctionalforimagesegmentation,denoising,interpolationandmagnication,IEEETrans.ImageProcess,10(8),1169{1186,2001.[69]J.TumblinandG.Turk,LCIS:ABoundaryhierarchyfordetail-preservingcontrastreduction,Siggraph,ComputerGraphicsProceedings,83{90,1999.[70]L.VeseandS.Osher,Modelingtextureswithtotalvariationminimizationandoscillatingpatternsinimageprocessing,J.Sci.Comput.,19(1-3),553{572,2003.[71]B.P.Vollmayr-LeeandA.D.Rutenberg,Fastandaccuratecoarseningsimulationwithanun-conditionallystabletimestep,Phys.Rev.E,68(0066703),1{13,2003.[72]L.Y.WeiandM.Levoy,Fasttexturesynthesisusingtree-structuredvectorquantization,InProc.SIGGRAPH'00,NewOrleans,USA,479{488,2000.[73]A.L.YuilleandA.Rangarajan,TheConcave-Convexprocedure(CCCP),NeuralComputation,15(4),915{936,April,2003.