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1、IEEESIGNALPROCESSINGLETTERS,VOL.18,NO.10,OCTOBER2011595OnVariableDensityCompressiveSamplingGillesPuy,PierreVandergheynst,SeniorMember,IEEE,andYvesWiaux,Member,IEEEAbstract—IncoherencebetweensparsitybasisandsensingbasisWealsodenote.Finally,weaimatrecov-isanessentialconceptforco
2、mpressivesampling.Inthiscontext,weeringbysolvingthe-minimization1problemadvocateacoherence-drivenoptimizationprocedureforvariabledensitysampling.Theassociatedminimizationproblemissolved(2)byuseofconvexoptimizationalgorithms.Wealsoproposeare-finementofourtechniquewhenpriorinform
3、ationisavailableonthesignalsupportinthesparsitybasis.TheeffectivenessoftheInthissetting,commonstrategiesfocusonuniformrandommethodisconfirmedbynumericalexperiments.Ourresultsalsoselectionoftheindices.Forsignalssparseintheprovideatheoreticalunderpinningtostate-of-the-artvariable
4、den-Diracbasis,auniformrandomselectionofFourierbasisvec-sityFouriersamplingproceduresusedinMRI.torsrepresentsthebestsamplingstrategy.Indeed,theDiracIndexTerms—Compressedsensing,magneticresonanceandFourierbasisareoptimallyincoherent.Naturalsignalsareimaging,variabledensitysampl
5、ing.howeverrathersparseinmultiscalebases,e.g.,waveletbases,notoptimallyincoherentwiththeFourierbasis.Manymeasure-I.INTRODUCTIONmentsarethusneededtoreconstructsuchsignalsaccurately.ThisisforexamplethecaseinmagneticresonanceimagingOMPRESSEDsensingdemonstratesthatsparsesignals(MR
6、I).Toreducethenumberofmeasurements,theauthorsinCcanbesampledthroughlinearandnon-adaptivemeasure-[3]relyonthefactthattheenergyofMRIsignalsisessentiallymentsatasub-Nyquistrate,andstillaccuratelyrecoveredbyconcentratedatlowfrequencies.Theythusproposetoselectmeansofnon-linearitera
7、tivealgorithms.ThetheoryrequiresFourierbasisvectorsaccordingtoavariabledensitysamplingincoherencebetweenthesensingandsparsitybasesandalotofprofileselectingmorelowfrequenciesthanhighfrequencies.workhasthusbeendedicatedtodesignsuchsensingsystemsThisapproachwasshowntodrasticallyen
8、hancethequalityof[1].thereconstructedsignals.Thismethodishowe
