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外文翻译--最小方波在小波领域的展开.doc

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外文翻译--最小方波在小波领域的展开.doc

附录A英文原文LeastsquaresphaseunwrappinginwaveletdomainAbstractLeastsquaresphaseunwrappingisoneoftherobusttechniquesusedtosolvetwodimensionalphaseunwrappingproblems.However,owingtoitssparsestructure,theconvergencerateisveryslow,andsomepracticalmethodshavebeenappliedtoimprovethiscondition.Inthispaper,anewmethodforsolvingtheleastsquarestwodimensionalphaseunwrappingproblemispresented.Thistechniqueisbasedonthemultiresolutionrepresentationofalinearsystemusingthediscretewavelettransform.Byapplyingthewavelettransform,theoriginalsystemisdecomposedintoitscoarseandfineresolutionlevels.Fastconvergenceinseparatecoarseresolutionlevelsmakestheoverallsystemconvergenceveryfast.1introductionTwodimensionalphaseunwrappingisanimportantprocessingstepinsomecoherentimagingapplications,suchassyntheticapertureradarinterferometryInSARandmagneticresonanceimagingMRI.Intheseprocesses,threedimensionalinformationofthemeasuredobjectscanbeextractedfromthephaseofthesensedsignals,However,theobseryedphasedataarewrappedprincipalvalues,whicharerestrictedina2modulus,andtheymustbeunwrappedtotheirtrueabsolutephasevalues.Thisisthetaskofthephaseunwrapping,especiallyfortwodimensionalproblems.Thebasicassumptionofthegeneralphaseunwrappingmethodsisthatthediscretederivativesoftheunwrappedphaseatallgridpointsarelessthaninabsolutevalue.Withthisassumptionsatisfied,theabsolutephasecanbereconstructedperfectlybyintegratingthepartialderivativesofthewrappedphasedata.Inthegeneralcase,however,itisnotpossibletorecoverunambiguouslytheabsolutephasefromthemeasuredwrappedphasewhichisusuallycorruptedbynoiseoraliasingeffectssuchasshadow,layover,etc.Insuchcases,thebasicassumptionisviolatedandthesimpleintegrationprocedurecannotbeappliedowingtothephaseinconsistenciescausedbythecontaminations.AfterGoldsteinetalintroducedtheconceptofresiduesinthetwodimensionalphaseunwrappingproblemofInSAR,manyphaseunwrappingapproachestocopewiththisproblemhavebeeninvestigated.Pathfollowingorintegrationbasedmethodsandleastsquaresmethodsarethemostrepresentativetwobasicclassesinthisfield.TherehavealsobeensomeotherapproachessuchasGreenmethods,Bayesianregularizationmethods,imageprocessingbasedmethods,andmodelbasedmethods.Leastsquaresphaseunwrapping,establishedbyGhigliaandRomero,isoneofthemostrobusttechniquestosolvethetwodimensionalphaseunwrappingproblem.Thismethodobtainsanunwrappedsolutionbyminimizingthedifferencesbetweenthepartialderivativesofthewrappedphasedataandtheunwrappedsolution.Leastsquaresmethodisdividedintounweightedandweightedleastsquaresphaseunwrapping.Toisolatethephaseinconsistencies,aweightedleastsquaresmethodshouldbeused,whichdepressesthecontaminationeffectsbyusingtheweightingarrays.GreenmethodsandBayesianmethodsarealsobasedontheleastsquaresscheme.Butthesemethodsaredifferentfromthoseof,intheconceptofphaseinconsistencytreatment.Thus,thispaperconcernsonlytheleastsquaresphaseunwrappingproblemofGhigliascategory.TheleastsquaresmethodiswelldefinedmathematicallyandequivalenttothesolutionofPoissonspartialdifferentialequation,whichcanbeexpressedasasparselinearequation.anteriormethodisusuallyusedtosolvethislargelinearequation.However,alargecomputationtimeisrequiredandthereforeimprovingtheconvergencerateisaveryimportanttaskwhenusingthismethod.Somenumericalalgorithmshavebeenappliedtothisproblemtoimproveconvergenceconditions.Anapproachforfastconvergenceofasparselinearequationistotransfertheoriginalequationsystemintoanewsystemwithlargersupports.Multiresolutionorhierarchicalrepresentationconceptshaveoftenbeenusedforthispurpose.Recently,wavelettransformhasbeeninvestigateddeeplyinscienceandengineeringfieldsasasophisticatedtoolforthemultiresolutionanalysisofsignalsandsystems.Itdecomposesasignalspaceintoitslowresolutionsubspaceandthecomplementarydetailsubspaces.Inourmethod,thediscretewavelettransformisappliedtothelinearsystemofleastsquaresphaseunwrappingproblemtorepresenttheoriginalsysteminseparatemultiresolutionspaces.Inthisnewtransferredsystem,abetterconvergenceconditioncanbeachieved.Thismethodwasbrieflyintroducedinoutpreviouswork,wheretheproposedmethodwasappliedonlytotheunweightedproblem,Inthispaper,thisnewmethodisextendedtotheweightedleastsquaresproblem.Also,afulldescriptionoftheproposedmethodisgivenhere.2WeightedleastsquaresphaseunwrappingareviewLeastsquaresphaseobtainsanunwrappedsolutionbyminimizingthe2Lnormbetweenthediscretepartialderivativesofthewrappedphasedataandthoseoftheunwrappedsolutionfunction.Giventhewrappedphase,ijonanMNrectangulargrid01iM,01jN,thepartialderivativesofthewrappedphasearedefinedas,1,,xijijijW,,,1,yijijijW1WhereWisthewrappingoperatorthatwrapsthephaseintotheinterval,.Thedifferencesbetweenthepartialderivativesofthesolution,ijandthosein1canbeminimizedintheweightedleastsquaressense,bydifferentiatingthesum22,1,,,,,1,,,,xxyyijijijijijijijijijijww2Withrespectto,ijandsettingtheresulttozero.In2,thegradientweights,,xijwand,yijw,areusedtopreventsomephasevaluescorruptedbynoiseoraliasingfromdegradingtheunwrapping,andaredefinedby22,1,,min,xijijijwww,22,,1,min,yijijijwww,,01ijw3Theweightedleastsquaresphaseunwrappingproblemistofindthesolution,ijthatminimizesthesumof2.Theinitialweightarray,ijwisuserdefinedandsomemethodsfordefiningtheseweightsarepresentedin1,11.Whenalltheweights,1ijw,theaboveequationistheunweightedphaseunwrappingproblem.Sinceweightarrayisrelatedtotheexactitudeoftheresultantunwrappedsolution,itmustbedefinedproperly.Inthispaper,however,itisassumedthattheweightarrayisdefinedalreadyforthegivenphasedataandhowtodefineitisnotcoveredhere.Onlytheconvergenceratesissueoftheweightedleastsquaresphaseunwrappingproblemisconsideredhere.Theleastsquaressolutiontothisproblemyieldsthefollowingequation,1,,1,,1,,,1,,1,,1,xxyyijijijijijijijijijijijijijwwww4Where,ijistheweightedphaseLaplaciandefinedby,,,1,1,,,,1,1xxxxxxxxijijijijijijijijijwwww5Theunwrappedsolution,ijisobtainedbyiterativelysolvingthefollowingequation,,1,1,1,,,1,1,1,,1,,,1/xxyyxxyyijijijijijijijijijijijijijijwwwwwwww6Equation4istheweightedanddiscreteversionofthePoissonspartialdifferentialequationPDE,2.Byconcatenatingallthenodalvariables,ijintoMN1onecolumnvector,theaboveequationisexpressedasalinearsystemA7WherethesystemmatrixAisofsizeKKKMNandisacolumnvectorof,ij,Thatis,thesolutionoftheleastsquaresphaseunwrappingproblemcanbeobtainedbysolvingthislinearsystem,andforgivenAand,whicharedefinedfromtheweightarray,xijwandthemeasuredwrappedphase,ijtheunwrappedphasehastheuniquesolution1A,ButsinceAisaverylargematrix,thedirectinverseoperationispracticallyimpossible.ThestructureofthesystemmatrixAisverysparseandmostoftheoffdiagonalelementsarezero,whichisevidentfrom4.DirectmethodsbasedonthefastFouriertransformFFTorthediscretecosinetransformDCTcanbeappliedtosolvetheunweightedphaseunwrappingproblem.However,intheweightedcase,iterativemethodsshouldbeadopted.TheclassicaliterativemethodforsolvingthelinearsystemistheGaussSeidelrelaxation,whichsolves6bysimpleiterationuntilitconverges.However,thismethodisnotpracticalowingtoitsextremelyslowconvergence,whichiscausedbythesparsecharacteristicsofthesystemmatrixA.SomenumericalalgorithmssuchaspreconditionedconjugategradientPCG,ormultigridmethodwereappliedtoimplementtheweightedleastsquaresphaseunwrapping.ThePCGmethodconvergesrapidlyonunweightedphaseunwrappingproblemsorweightedproblemsthatdonothavelargephasediscontinuities.However,ondatawithlargediscontinuities,itrequiresmanyiterationstoconverge.ThemultigridmethodisanefficientalgorithmtosolvealinearsystemandperformsmuchbetterthantheGaussSeidelmethodandthePCGmethodinsolvingtheleastsquaresphaseunwrappingproblem.However,intheweightedcase,themethodneedsanadditionalweightrestrictionoperation,Thisoperationisverycomplicatedandalthoughitisdesignedproperlyinsomebooks,theremaybesomeerrorsduringtherestriction.Thereareotherapproachestosolveasparselinearsystemproblemefficiently,Intheseapproaches,asystemisconvertedintoanotherequivalentsystemwithbetterconvergencecondition.TheconvergencespeedofthesystemischaracterisedbythesystemmatrixA.Thestructureofthesystemmatrixoftheleastsquaresphaseunwrappingproblemisverysparse.Intheiterativesolvingmethods,thelocalconnectionsbetweenthenodalvariablesslowdowntheprogressofthesolutioniniterationandresultinalowconvergencerate.Inotherwords,theGaussSeidelmethodextractsthelocalhighfrequencyinformationofthesurfacefromonlyfourneighboursofeachnodalvalue.Thus,thegloballowfrequencysurfaceinformationpropagatesveryslowly,whichisthemain

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