(Leonid-I.-Rudin-1992-PPT)1992Nonlinear-total-vari

1/25JohannRadonInstituteforComputationalandAppliedMathematics:•TotalVariation–Rudin–Osher–Fatemi(ROF)Model–Denoising–Edgepreserving–Energyminimizing–BoundedvariationTakenfrom:2/25JohannRadonInstituteforComputationalandAppliedMathematics:•Lettheobservedintensityfunctionu0(x,y)denotethepixelvaluesofanoisyimageforx,yœW.Letu(x,y)denotethedesiredcleanimage,sowithnadditivewhite(0,s)noise.•Theconstrainedminimizationproblemis:Min(H|I1,I2)=Min(H-l1I1-l2I2)3/25JohannRadonInstituteforComputationalandAppliedMathematics:•WearriveattheEuler-Lagrangeequations0=dH-l1dI1-l2dI2•IntegratingtheexpressionoverWgivesl1=0.Sotheaverageintensityiskept.4/25JohannRadonInstituteforComputationalandAppliedMathematics:•Thesolutionprocedureusesaparabolicequationwithtimeasanevolutionparameter,orequivalently,thegradientdescentmethod.Thismeansthatwesolve5/25JohannRadonInstituteforComputationalandAppliedMathematics:•Wemustcomputel(t).WemerelymultiplytheequationinWby(u-u0)andintegratebypartsoverW.Wethenhave6/25JohannRadonInstituteforComputationalandAppliedMathematics:•Thenumericalmethodintwospatialdimensionsisasfollows:7/25JohannRadonInstituteforComputationalandAppliedMathematics:•Thenumericalapproximationis8/25JohannRadonInstituteforComputationalandAppliedMathematics:•andlnisdefineddiscreetlyvia•Astepsizerestrictionisimposedforstability:9/25JohannRadonInstituteforComputationalandAppliedMathematics:::•Original•Noisy•Wiener•TV12/25JohannRadonInstituteforComputationalandAppliedMathematics:•Doitslightlysloppy:•demoLt=∇·
∇L−1∇L+λ(L−L0)δL=C·D−i,j
D+i,jL−1D+i,jL+λ(L−L0)13/25JohannRadonInstituteforComputationalandAppliedMathematics:•Write•ThenI(u)isaconstantofmotion:I(u0)=I(ut):•Thenthepdeconvergestoaminimumonthemanifoldgivenbytheconstraints:ut=−(δH−λδI)∂tI(u)=δI,ut=0∂tH(u)=−u2t≤0λ=δH,δIδI,δIA,B=ΩA·BdΩ14/25JohannRadonInstituteforComputationalandAppliedMathematics:•Ablurrednoisyimage:u0=(Au)(x,y)+n(x,y)whereAisacompactoperatoronL2.•Multiplicativenoise&blurringu0=[(Au)(x,y)]n(x,y)u0=(Au)(x,y)+[u(x,y)n(x,y)]•Smarterfunctional:H=Ωφ(|∇L|)dΩ15/25JohannRadonInstituteforComputationalandAppliedMathematics:•T.Chan,S.Esedoglu,F.Park,A.YipHandbookofMathematicalModelsinComputerVisionSincetheirintroductioninaclassicpaperbyRudin,OsherandFatemi,totalvariationminimizingmodelshavebecomeoneofthemostpopularandsuccessfulmethodologyforimagerestoration.Morerecently,therehasbeenaresurgenceofinterestandexcitingnewdevelopments,someextendingtheapplicabilitiestoinpainting,blinddeconvolutionandvector-valuedimages,whileothersofferimprovementsinbetterpreservationofcontrast,geometryandtextures,inamelioratingthestaircasingeffect,andinexploitingthemulti-scalenatureofthemodels.Inaddition,newcomputationalmethodshavebeenproposedwithimprovedcomputationalspeedandrobustness.16/25JohannRadonInstituteforComputationalandAppliedMathematics:•TotalvariationbasedimagerestorationmodelswerefirstintroducedbyRudin,Osher,andFatemi(ROF)intheirpioneeringworkonedgepreservingimagedenoising.•ItisoneoftheearliestandbestknownexamplesofPDEbasededgepreservingdenoising.•Itwasdesignedwiththeexplicitgoalofpreservingsharpdiscontinuities(edges)inimageswhileremovingnoiseandotherunwantedfinescaledetail.•Beingconvex,theROFmodelisoneofthesimplestvariationalmodelshavingthismostdesirableproperty.•Therevolutionaryaspectofthismodelisitsregularizationtermthatallowsfordiscontinuitiesbutatthesametimedisfavorsoscillations.17/25JohannRadonInstituteforComputationalandAppliedMathematics:•Theconstraintoftheoptimizationforcestheminimizationtotakeplaceoverimagesthatareconsistentwiththisknownnoiselevel.•Theobjectivefunctionalitselfiscalledthetotalvariation(TV)ofthefunctionu(x);forsmoothimagesitisequivalenttotheL1normofthederivative,andhenceissomemeasureoftheamountofoscillationfoundinthefunctionu(x).18/25JohannRadonInstituteforComputationalandAppliedMathematics:•Thestepfromtoisnottrivial!19/25JohannRadonInstituteforComputationalandAppliedMathematics:•Thespaceoffunctionswithboundedvariation(BV)isanidealchoiceforminimizerstotheROFmodelsinceBVprovidesregularityofsolutionsbutalsoallowssharpdiscontinuities(edges).ManyotherspacesliketheSobolevspaceW1,1donotallowedges.•BeforedefiningthespaceBV,weformallystatethedefinitionofTVas:whereandisaboundedopenset.•WecannowdefinethespaceBVas•Thus,BVfunctionsamounttoL1functionswithboundedTVsemi-norm.20/25JohannRadonInstituteforComputationalandAppliedMathematics:•W