201005-Compressed Sensing with Coherent and Redundant Dictionaries

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1、CompressedSensingwithCoherentandRedundantDictionariesEmmanuelJ.Candes1,YoninaC.Eldar2,andDeannaNeedell11DepartmentsofMathematicsandStatistics,StanfordUniversity,Stanford,CA943052DepartmentofElectricalEngineering,Technion-IsraelInstituteofTechnology,Haifa32000May17,2010Abs

2、tractThisarticlepresentsnovelresultsconcerningtherecoveryofsignalsfromundersampleddatainthecommonsituationwheresuchsignalsarenotsparseinanorthonormalbasisorincoherentdictionary,butinatrulyredundantdictionary.Thisworkthusbridgesagapintheliteratureandshowsnotonlythatcompresse

3、dsensingisviableinthiscontext,butalsothataccuraterecoveryispossibleviaan`1-analysisoptimizationproblem.Weintroduceaconditiononthemeasurement/sensingmatrix,whichisanaturalgeneralizationofthenowwell-knownrestrictedisometryproperty,andwhichguaranteesaccuraterecoveryofsignalsth

4、atarenearlysparsein(possibly)highlyovercompleteandcoherentdictionaries.Thisconditionimposesnoincoherencerestrictiononthedictionaryandourresultsmaybetherstofthiskind.Wediscusspracticalexamplesandtheimplicationsofourresultsonthoseapplications,andcomplementourstudybydemonstra

5、tingthepotentialof`1-analysisforsuchproblems.1IntroductionCompressedsensingisanewdataacquisitiontheorybasedonthediscoverythatonecanexploitsparsityorcompressibilitywhenacquiringsignalsofgeneralinterest,andthatonecandesignnonadaptivesamplingtechniquesthatcondensetheinformatio

6、ninacompressiblesignalintoasmallamountofdata[11,14,16].Inanutshell,reliable,nonadaptivedataacquisition,withfarfewermeasurementsthantraditionallyassumed,ispossible.Bynow,applicationsofcompressedsensingareabundantandrangefromimaginganderrorcorrectiontoradarandremotesensing,se

7、e[2,1]andreferencestherein.Inanutshell,compressedsensingproposesacquiringasignalx2Rnbycollectingmlinearmeasurementsoftheformyk=hak;xi+zk,1km,orinmatrixnotation,arXiv:1005.2613v1[math.NA]14May2010y=Ax+z;(1.1)Aisanmnsensingmatrixwithmtypicallysmallerthannbyoneorseveralorde

8、rsofmagnitude(indicatingsomesignicantundersampling)andzisanerrortermmodelingmeasu