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附录A：英文原文LeastsquaresphaseunwrappinginwaveletdomainAbstract:Leastsquaresphaseunwrappingisoneoftherobusttechniquesusedtosolvetwo-dimensionalphaseunwrappingproblems.However,owingtoitssparsestructure,theconvergencerateisveryslow,andsomepracticalmethodshavebeenappliedtoimprovethiscondition.Inthispaper,anewmethodforsolvingtheleastsquarestwo-dimensionalphaseunwrappingproblemispresented.Thistechniqueisbasedonthemultiresolutionrepresentationofalinearsystemusingthediscretewavelettransform.Byapplyingthewavelettransform,theoriginalsystemisdecomposedintoitscoarseandfineresolutionlevels.Fastconvergenceinseparatecoarseresolutionlevelsmakestheoverallsystemconvergenceveryfast.1introductionTwo-dimensionalphaseunwrappingisanimportantprocessingstepinsomecoherentimagingapplications,suchassyntheticapertureradarinterferometry(InSAR)andmagneticresonanceimaging(MRI).Intheseprocesses,three-dimensionalinformationofthemeasuredobjectscanbeextractedfromthephaseofthesensedsignals,However,theobseryedphasedataarewrappedprincipalvalues,whicharerestrictedina2modulus,andtheymustbeunwrappedtotheirtrueabsolutephasevalues.Thisisthetaskofthephaseunwrapping,especiallyfortwo-dimensionalproblems.Thebasicassumptionofthegeneralphaseunwrappingmethodsisthatthediscretederivativesoftheunwrappedphaseatallgridpointsarelessthaninabsolutevalue.Withthisassumptionsatisfied,theabsolutephasecanbereconstructedperfectlybyintegratingthepartialderivativesofthewrappedphasedata.Inthegeneralcase,however,itisnotpossibletorecoverunambiguouslytheabsolutephasefromthemeasuredwrappedphasewhichisusuallycorruptedbynoiseoraliasingeffectssuchasshadow,layover,etc.Insuchcases,thebasicassumptionisviolatedandthesimpleintegrationprocedurecannotbeappliedowingtothephaseinconsistenciescausedbythecontaminations.AfterGoldstein-etalintroducedtheconceptofresiduesinthetwo-dimensionalphaseunwrappingproblemofInSAR,manyphaseunwrappingapproachestocopewiththisproblemhavebeeninvestigated.Path-following(orintegration-based)methodsandleastsquaresmethodsarethemostrepresentativetwobasicclassesinthisfield.TherehavealsobeensomeotherapproachessuchasGreenmethods,Bayesianregularizationmethods,imageprocessing-basedmethods,andmodel-basedmethods.Leastsquaresphaseunwrapping,establishedbyGhigliaandRomero,isoneofthemostrobusttechniquestosolvethetwo-dimensionalphaseunwrappingproblem.Thismethodobtainsanunwrappedsolutionbyminimizingthedifferencesbetweenthepartialderivativesofthewrappedphasedataandtheunwrappedsolution.Leastsquaresmethodisdividedintounweightedandweightedleastsquaresphaseunwrapping.Toisolatethephaseinconsistencies,aweightedleastsquaresmethodshouldbeused,whichdepressesthecontaminationeffectsbyusingtheweightingarrays.GreenmethodsandBayesianmethodsarealsobasedontheleastsquaresscheme.Butthesemethodsaredifferentfromthoseof,intheconceptofphaseinconsistencytreatment.Thus,thispaperconcernsonlytheleastsquaresphaseunwrappingproblemofGhigliascategory.Theleastsquaresmethodiswell-definedmathematicallyandequivalenttothesolutionofPoissonspartialdifferentialequation,whichcanbeexpressedasasparselinearequation.anteriormethodisusuallyusedtosolvethislargelinearequation.However,alargecomputationtimeisrequiredandthereforeimprovingtheconvergencerateisaveryimportanttaskwhenusingthismethod.Somenumericalalgorithmshavebeenappliedtothisproblemtoimproveconvergenceconditions.Anapproachforfastconvergenceofasparselinearequationistotransfertheoriginalequationsystemintoanewsystemwithlargersupports.Multiresolutionorhierarchicalrepresentationconceptshaveoftenbeenusedforthispurpose.Recently,wavelettransformhasbeeninvestigateddeeplyinscienceandengineeringfieldsasasophisticatedtoolforthemultiresolutionanalysisofsignalsandsystems.Itdecomposesasignalspaceintoitslow-resolutionsubspaceandthecomplementarydetailsubspaces.Inourmethod,thediscretewavelettransformisappliedtothelinearsystemofleastsquaresphaseunwrappingproblemtorepresenttheoriginalsysteminseparatemultiresolutionspaces.Inthisnewtransferredsystem,abetterconvergenceconditioncanbeachieved.Thismethodwasbrieflyintroducedinoutpreviouswork,wheretheproposedmethodwasappliedonlytotheunweightedproblem,Inthispaper,thisnewmethodisextendedtotheweightedleastsquaresproblem.Also,afulldescriptionoftheproposedmethodisgivenhere.2Weightedleastsquaresphaseunwrapping:areviewLeastsquaresphaseobtainsanunwrappedsolutionbyminimizingthe2L-normbetweenthediscretepartialderivativesofthewrappedphasedataandthoseoftheunwrappedsolutionfunction.Giventhewrappedphase,ijonanM×Nrectangulargrid(01iM,01jN),thepartialderivativesofthewrappedphasearedefinedas,1,xijijijW,1,yijijijW(1)WhereWisthewrappingoperatorthatwrapsthephaseintotheinterval,.Thedifferencesbetweenthepartialderivativesofthesolution,ijandthosein(1)canbeminimizedintheweightedleastsquaressense,bydifferentiatingthesum22,1,1,xxyyijijijijijijijijijijww(2)Withrespectto,ijandsettingtheresulttozero.In(2),thegradientweights,xijwand,yijw,areusedtopreventsomephasevaluescorruptedbynoiseoraliasingfromdegradingtheunwrapping,andaredefinedby22,1,min,xijijijwww,22,1,min,yijijijwww,01ijw(3)Theweightedleastsquaresphaseunwrappingproblemistofindthesolution,ijthatminimizesthesumof(2).Theinitialweightarray,ijwisuser-definedandsomemethodsfordefiningtheseweightsarepresentedin1,11.Whenalltheweights,1ijw,theaboveequationistheunweightedphaseunwrappingproblem.Sinceweightarrayisrelatedtotheexactitudeoftheresultantunwrappedsolution,itmustbedefinedproperly.Inthispaper,however,itisassumedthattheweightarrayisdefinedalreadyforthegivenphasedataandhowtodefineitisnotcoveredhere.Onlytheconvergenceratesissueoftheweightedleastsquaresphaseunwrappingproblemisconsideredhere.Theleastsquaressolutiontothisproblemyieldsthefollowingequation:,1,1,1,1,1,1,xxyyijijijijijijijijijijijijijwwww(4)Where,ijistheweightedphaseLaplaciandefinedby,1,1,1,1xxxxxxxxijijijijijijijijijwwww(5)Theunwrappedsolution,ijisobtainedbyiterativelysolvingthefollowingequation,1,1,1,1,1,1,1,1/xxyyxxyyijijijijijijijijijijijijijijwwwwwwww(6)Equation(4)istheweightedanddiscreteversionofthePoissonspartialdifferentialequation(PDE),2.Byconcatenatingallthenodalvariables,ijintoMN×1onecolumnvector,theaboveequationisexpressedasalinearsystemA(7)WherethesystemmatrixAisofsizeK×K(K=MN)andisacolumnvectorof,ij,Thatis,thesolutionoftheleastsquaresphaseunwrappingproblemcanbeobtainedbysolvingthislinearsystem,andforgivenAand,whicharedefinedfromtheweightarray,xijwandthemeasuredwrappedphase,ijtheunwrappedphasehastheuniquesolution1A,ButsinceAisaverylargematrix,thedirectinverseoperationispracticallyimpossible.ThestructureofthesystemmatrixAisverysparseandmostoftheoff-diagonalelementsarezero,whichisevidentfrom(4).DirectmethodsbasedonthefastFouriertransform(FFT)orthediscretecosinetransform(DCT)canbeappliedtosolvetheunweightedphaseunwrappingproblem.However,intheweightedcase,iterativemethodsshouldbeadopted.TheclassicaliterativemethodforsolvingthelinearsystemistheGauss-Seidelrelaxation,whichsolves(6)bysimpleiterationuntilitconverges.However,thismethodisnotpracticalowingtoitsextremelyslowconvergence,whichiscausedbythesparsecharacteristicsofthesystemmatrixA.Somenumericalalgorithmssuchaspreconditionedconjugategradient(PCG),ormultigridmethodwereappliedtoimplementtheweightedleastsquaresphaseunwrapping.ThePCGmethodconvergesrapidlyonunweightedphaseunwrappingproblemsorweightedproblemsthatdonothavelargephasediscontinuities.However,ondatawithlargediscontinuities,itrequiresmanyiterationstoconverge.ThemultigridmethodisanefficientalgorithmtosolvealinearsystemandperformsmuchbetterthantheGauss-SeidelmethodandthePCGmethodinsolvingtheleastsquaresphaseunwrappingproblem.However,intheweightedcase,themethodneedsanadditionalweightrestrictionoperation,Thisoperationisverycomplicatedandalthoughitisdesignedproperlyinsomebooks,theremaybesomeerrorsduringtherestriction.Thereareotherapproachestosolveasparselinearsystemproblemefficiently,Intheseapproaches,asystemisconvertedintoanotherequivalentsystemwithbetterconvergencecondition.TheconvergencespeedofthesystemischaracterisedbythesystemmatrixA.Thestructureofthesystemmatrixoftheleastsquaresphaseunwrappingproblemisverysparse.Intheiterativesolvingmethods,thelocalconnectionsbetweenthenodalvariablesslowdowntheprogressofthesolutioniniterationandresultinalowconvergencerate.Inotherwords,theGauss-Seidelmethodextractsthelocalhigh-frequencyinformationofthesurfacefromonlyfourneighboursofeachnodalvalue.Thus,thegloballow-frequencysurfaceinformationpropagatesveryslowly,whichisthemain