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1、AFastMethodforFindingtheGlobalSolutionoftheRegularizedStructuredTotalLeastSquaresProblemforImageDeblurringAmirBeck∗AharonBen-Tal†andChristianKanzow‡November15,2007AbstractGivenalinearsystemAx≈bovertherealorcomplexfieldwherebothAandbaresubjecttonoise,thet
2、otalleastsquares(TLS)problemseekstofindacorrectionmatrixandacorrectionrighthandsidevectorofminimalnormwhichmakesthelinearsystemfeasible.Toavoidill-posedness,aregularizationtermisaddedtotheobjectivefunction;thisleadstotheso-calledregularizedTLS(RTLS)probl
3、em.Afurthercom-plicationariseswhenthematrixAandcorrespondinglythecorrectionmatrixmusthaveaspecificstructure.ThisismodelledbytheregularizedstructuredTLS(RSTLS)problem.Ingeneralthisproblemisnonconvexandhencedifficulttosolve.However,theRSTLSproblemarisingfrom
4、imagedeblurringapplicationsunderreflexiveorpe-riodicboundaryconditionspossessaspecialstructurewhereallrelevantmatricesaresimultaneouslydiagonalizable(SD).Inthispaperweintroduceanalgorithmforfind-ingtheglobaloptimumoftheRSTLSproblemwiththisSDstructure.Thed
5、evisedmethodisbasedondecomposingtheproblemintosinglevariableproblemsandthentransformingthemintoone-dimensionalunimodalreal-valuedminimizationproblemswhichcanbesolvedglobally.BasedonuniquenessandattainmentpropertiesoftheRSTLSsolutionweshowthataconstraine
6、dversionoftheproblempossessastrongdualityresultandcanthusbesolvedviaasequenceofRSTLSproblems.1IntroductionGivenalinearsystemAx≈bovertherealorcomplexfieldwhereboththematrixAandtherighthandsidevectorbaresubjectedtonoise,thetotalleastsquares(TLS)problemseek
7、sto∗DepartmentofIndustrialEngineering,Technion—IsraelInstituteofTechnology,Haifa32000,Israel.E-mail:becka@ie.technion.ac.il.†MINERVAOptimizationCenter,DepartmentofIndustrialEngineering,Technion—IsraelInstituteofTechnology,Haifa32000,Israel.E-mail:abenta
8、l@ie.technion.ac.il.PartlysupportedbytheTechnionVPRfundforpromotionofResearch,grant#2005519‡UniversityofW¨urzburg,InstituteofMathematics,AmHubland,97074W¨urzburg,Germany.E-mail:kanzow@mathematik.uni-wuerzburg.de1minimizethesumofs
