Compressed Sensing.pdf

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1、CompressedSensingDavidL.DonohoDepartmentofStatisticsStanfordUniversitySeptember14,2004AbstractmSupposexisanunknownvectorinR(dependingoncontext,adigitalimageorsignal);weplantoacquiredataandthenreconstruct.Nominallythis‘should’requiremsamples.Butsupposewekno

2、wapriorithatxiscompressiblebytransformcodingwithaknowntransform,andweareallowedtoacquiredataaboutxbymeasuringngenerallinearfunctionals–ratherthantheusualpixels.Ifthecollectionoflinearfunctionalsiswell-chosen,andweallowforadegreeofreconstructionerror,thesiz

3、eofncanbedramaticallysmallerthanthesizemusuallyconsiderednecessary.Thus,certainnaturalclassesofimageswithmpixelsneedonlyn=O(m1/4log5/2(m))nonadaptivenonpixelsamplesforfaithfulrecovery,asopposedtotheusualmpixelsamples.Ourapproachisabstractandgeneral.Wesuppo

4、sethattheobjecthasasparserep-resentationinsomeorthonormalbasis(eg.wavelet,Fourier)ortightframe(egcurvelet,Gabor),meaningthatthecoefficientsbelongtoan`pballfor0

5、1/p).Itispossibletodesignn=O(Nlog(m))nonadaptivemeasurementswhichcontaintheinformationnecessarytoreconstructanysuchobjectwithaccuracycomparabletothatwhichwouldbepossibleiftheNmostimportantcoefficientsofthatobjectweredirectlyobservable.Moreover,agoodapproxima

6、tiontothoseNimportantcoefficientsmaybeextractedfromthenmeasurementsbysolvingaconvenientlinearprogram,calledbythenameBasisPursuitinthesignalprocessingliterature.Thenonadaptivemeasurementshavethecharacterof‘random’linearcombinationsofbasis/frameelements.Theser

7、esultsaredevelopedinatheoreticalframeworkbasedonthetheoryofoptimalre-covery,thetheoryofn-widths,andinformation-basedcomplexity.Ourbasicresultsconcernpropertiesof`pballsinhigh-dimensionalEuclideanspaceinthecase0

8、giveacriterionfornear-optimalsubspacesforGel’fandn-widths,showthat‘most’subspacesarenear-optimal,andshowthatconvexoptimizationcanbeusedforprocessinginformationderivedfromthesenear-optimalsubspaces.Thetechniqu