The_stability_of_a_direct_method_for_superresolution

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1、ProceedingsofICASSP-98,Seattle,Washington,May1998THESTABILITYOFADIRECTMETHODFORSUPERRESOLUTIONJoseM.N.VieiraandPauloJ.S.G.Ferreira´Dep.deElectr´onicaeTelecomunicac¸˜oesUniversidadedeAveiro3810Aveiro,PortugalE-mail:pjf@inesca.ptABSTRACTTheexistenceanduniquenessofthesolutiontothissu-perre

2、solution/extrapolationproblemisastraightforwardAdirectmethodforsuperresolutionrecentlyproposedbyconsequenceoftheanalyticityoftheFouriertransformofWalshandDelaneyisfurtheranalyzedfromthepointofacompactlysupportedobject.However,analyticcontinua-viewofnumericalstability.Themethodisbasedona

3、settiondoesnotseemtoleadtoanypracticalsuperresolutionoflinearequationsAx=b,whereAismn,andbalgorithms,andothersolutionsareneeded.isasubset(ofcardinaln)oftheFouriertransformoftheOneofthefirstmethodsproposedusesthedoubleor-object(whichhasatotalofsamples).Wegiveexactthogonalityoftheprolate

4、spheroidalfunctions[15].Mostandbestpossibleapproximateexpressionsforthedeter-othermethodstakeadvantageofthefollowingfacts:math-minantofA,whenm=n.Asacorollary,itisshownematically,thesetofsignalsthatsatisfythetimeorspacethatthesmallesteigenvalueofAinabsolutevaluesatis-domainconditionsment

5、ionedabove(compact,knownsup-fies(n1)=2jjk(n),wherek(n)(whichisinde-minport)isasubspaceofsomeparentHilbertspacependentofH(often)isexplicitlygiven.Themagnitudeofthe).Similarly,thesetofsignalsthatsatisfythefrequency-smallesteigenvalueof2Abecomesincreasinglysmallasdomainconditionsdefi

6、nesanothersubspaceofgrows,evenwhenthenumberofunknownsH.Suc-nremainscon-cessiveprojectionsontoeachofthesesubspacesleadtoastant.Whenm>nthesingularvaluesofAarestudied,andsequenceofapproximationstothesolutionoftheproblemrelatedtotheeigenvaluesofthematrixoftwootherdirect[8,13,19].Amoregenera

7、lalternativeisthePOCS(projec-methods.Asaresult,theconnectionbetweenthemethodtionsontoconvexsets)method[14,20].andtheotherdirectmethodsisclarified.Numericalalgorithmsrequirediscrete-timeordiscrete-spaceformulations,thatis,sampleddata.Thefinitedimen-1.INTRODUCTIONsionalequivalentof