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1、MULTISCALEMODEL.SIMUL.c2004SocietyforIndustrialandAppliedMathematicsVol.2,No.4,pp.554–579AMULTISCALEIMAGEREPRESENTATIONUSINGHIERARCHICAL(BV,L2)DECOMPOSITIONS∗EITANTADMOR†,SUZANNENEZZAR‡,ANDLUMINITAVESE‡Abstract.Weproposeanewmultiscaleimagedecompositionwhichoffersahierarchical,a
2、daptiverepresentationforthedifferentfeaturesingeneralimages.Thestartingpointisavari-ationaldecompositionofanimage,f=u0+v0,where[u0,v0]istheminimizerofaJ-functional,pJ(f,λ0;X,Y)=infu+v=fuX+λ0vY.Suchminimizersarestandardtoolsforimagema-nipulations(e.g.,denoising,deblurring,c
3、ompression);see,forexample,[M.MumfordandJ.Shah,ProceedingsoftheIEEEComputerVisionPatternRecognitionConference,SanFrancisco,CA,1985]and[L.Rudin,S.Osher,andE.Fatemi,Phys.D,60(1992),pp.259–268].Here,u0shouldcapture“essentialfeatures”offwhicharetobeseparatedfromthespuriouscomponent
4、sabsorbedbyv0,andλ0isafixedthresholdwhichdictatesseparationofscales.Toproceed,weiteratetherefinementstep[u,v]=arginfJ(v,λjkj+1j+1j02),leadingtothehierarchicaldecomposition,f=j=0uj+vk.Wefocusourattentionontheparticularcaseof(X,Y)=(BV,L2)decomposition.Theresultinghierarchicaldeco
5、mposition,f∼juj,isessentiallynonlinear.Thequestionsofconvergence,energydecomposition,localization,andadaptivityarediscussed.Thedecompositionisconstructedbynu-mericalsolutionofsuccessiveEuler–Lagrangeequations.Numericalresultsillustrateapplicationsofthenewdecompositiontosyntheti
6、candrealimages.Bothgreyscaleandcolorimagesareconsidered.Keywords.naturalimages,multiscaleexpansion,totalvariation,localization,adaptivityAMSsubjectclassifications.26B30,65C20,68U10DOI.10.1137/0306004481.Introductionandmotivations.ImagescouldberealizedasgeneralL2ob-jects,f∈L2(R2)
7、,representingthegreyscaleoftheobservedimage.Likewise,colorimagesaretypicallyrealizedintermsofvector-valuedfunctions,f=(f1,f2,f3)∈L2(R2)3,representingtheRGB-colorscales.Inpractice,themorenoticeablefeaturesofimagesareidentifiedwithinapropersubclassofallL2objects.Mostnoticeablearet
8、heedgesofanimage,whichareknowntobewellquantifiedwithint
