Model Based Compressive Sensing1

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1、Model-basedCompressiveSensingRichardBaraniukRiceUniversityChinmayHegdeVolkanCevherMarcoDuarteCompressiveSensing•Sensingviarandomizeddimensionalityreductionrandomsparsemeasurementssignalnonzeroentries•Recovery:solveanill-posedinverseproblemexploitthegeometricalstructureofsparse/compressi

2、blesignalsRestrictedIsometryProperty(RIP)•Preservethestructureofsparse/compressiblesignalsK-dimensionalsubspacesRestrictedIsometryProperty(RIP)•Preservethestructureofsparse/compressiblesignals•RIPoforder2Kimplies:forallK-sparsexandx12K-dimensionalsubspacesBeyondSparseModels•Sparse/compr

3、essiblesignalmodelcapturessimplisticprimarystructurepixels:wavelets:Gaboratoms:backgroundsubtractednaturalimageschirps/tonesimagesBeyondSparseModels•Sparse/compressiblesignalmodelcapturessimplisticprimarystructure•Moderncompression/processingalgorithmscapturerichersecondarycoefficientst

4、ructurepixels:wavelets:Gaboratoms:backgroundsubtractednaturalimageschirps/tonesimagesSparseSignals•Defn:K-sparsesignalscompriseaparticularsetofK-dimcanonicalsubspacesModel-SparseSignals•Defn:AK-sparsesignalmodelcomprisesaparticular(reduced)setofK-dimcanonicalsubspaces[BlumensathandDavie

5、s]•Fewersubspaces<>relaxedRIP<>stablerecoveryusingfewermeasurementsMModel-basedCSRunningExample:Tree-SparseSignalsWaveletSparse•Typicalofwavelettransformsofnaturalsignalsandimages(piecewisesmooth)Tree-Sparse•Model:K-sparsecoefficients+significantcoefficientslieonarootedsubtree•Typicalof

6、wavelettransformsofnaturalsignalsandimages(piecewisesmooth)Tree-Sparse•Model:K-sparsecoefficients+significantcoefficientslieonarootedsubtree•Sparseapprox:findbestsetofcoefficients–sorting–hardthresholding•Tree-sparseapprox:fidfindbdbestrootedsubbtreeofcoefficients–CSSA[[]B]–dynamicprogr

7、amming[Donoho]WaveletSparse•Model:K-sparsecoefficients+significantcoefficientslieonarootedsubtree•RIP:stableembeddingK-planesTree-Sparse•Model:K-sparsecoefficients+significantcoefficientslieonarootedsubtree•Tree-RIP:stableembedding[[]BlumensathandDavies]K-planesTree-Sparse•Mode