The split Bregman alogrithm for L1.pdf

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1、THESPLITBREGMANMETHODFORL1REGULARIZEDPROBLEMSTOMGOLDSTEIN,STANLEYOSHERAbstract.Theclassofl1-regularizedoptimizationproblemshasreceivedmuchattentionre-centlybecauseoftheintroductionofcompressedsensing,"whichallowsimagesandsignalstobereconstructedfromsmallamountsofdata.Despitethisrecenta

2、ttention,manyl1-regularizedprob-lemsstillremaindiculttosolve,orrequiretechniquesthatareveryproblem-specic.Inthispaper,weshowthatBregmaniterationcanbeusedtosolveawidevarietyofconstrainedoptimizationprob-lems.Usingthistechnique,weproposeaSplitBregman"method,whichcansolveaverybroadclass

3、ofl1-regularizedproblems.WeapplythistechniquetotheROFfunctionalforimagedenoising,andtoacompressedsensingproblemthatarisesinMagneticResonanceImaging.Keywords.ConstrainedOptimization,l1regularization,compressedsensing,totalvariationdenoising.1.Introduction.Thecategoryofl1-regularizedprobl

4、emsincludesmanyim-portantproblemsinengineering,computer,andimagingscience.Thegeneralformforsuchproblemsis(1.1)minj(u)j+H(u)uwherej
jdenotesthel1-norm,andbothj(u)jandH(u)areconvexfunctions.Manyimportantproblemsinimagingscience(andothercomputationalareas)canbeposedasl1-regularizedoptimi

5、zationproblems.Somecommonexamplesofthisincludethefollowing:2(1.2)TV/ROFDenoising:"minkukBV+kufk2u22(1.3)BasisPursuit/CompressedSensing:"minJ(u)+kAufk2u2whereJ(u)issomeregularizingfunctional,usuallyintheformofaBVorBesovnorm.TheRudin-Osher-Fatemi(ROF)functional(1.2),despiteitssimple

6、form,hasprovedtobeverydiculttominimizebyconventionalmethods.Totalvariationbasedimagerestorationwasrstintroducedin[22].Inthatpaper,theauthorsproposetominimizethisenergyusingagradientprojectionmethod.Whilethisapproachissimple,thenon-linearityandpoorconditioningoftheproblemmakethisapproa

7、chveryslow.Severalauthorshaveproposedimprovedtime-steppingschemesthatresultinbetterperformance,suchasthosepresentedin[26,15].AmoreecientclassofsolversarethosebasedonNewton'smethod.Onesuchalgorithmwaspresentedin[9],inwhichthepreconditionedconjugategradientmethodisusedtoinvertth