2010_Compressive Sensing and Structured Random Matrices_TOFC

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1、CompressiveSensingandStructuredRandomMatricesHolgerRauhutHausdorCenterforMathematics&InstituteforNumericalSimulationUniversityofBonnRICAMLinzSummerSchoolTheoreticalFoundationsandNumericalMethodsforSparseRecoveryAugust31{September4,2009InstituteforNumericalSimulationRheinis

2、cheFriedrich-Wilhelms-UniversitätBonnHolgerRauhutHausdorCenterforMathematics&InstituteforNumericalSimulationUniversityofBonnCompressiveSensingandStructuredRandomMatricesPurposeofthiscourseIntroductiontoCompressiveSensingStructuredRandomMatricesMethodsforprovingprobabili

3、sticconditionnumberestimatesforstructuredrandommatricesHolgerRauhut,UniversityofBonnCompressiveSensing2OverviewLecture1:CompressiveSensingMotivationandIntroductionRecoveryconditionsfor`1-minimizationRestrictedisometryconstants,CoherenceGaussianrandommatricesLecture2:Bou

4、ndedOrthonormalSystemsExamples(randompartialFouriermatrices)RecoveryResultsToolsfromProbabilityTheoryILecture3:ToolsfromProbabilityTailsandMomentsSymmetrizationScalarandnoncommutativeKhintchineinequalitiesRudelson'sLemmaProofofNonuniformrecoveryresultHolgerRauhut,Un

5、iversityofBonnCompressiveSensing3OverviewIILecture4:ProofofRIPestimateforBoundedOrthonormalSystemsDudley'sinequalityDeviationinequalityforsupremaofempiricalprocessesProofofRIPestimateLecture5:PartialrandomcirculantmatricesNoncommutativeKhintchineinequalityforRademacherc

6、haosConditionnumberestimateforpartialrandomcirculantmatricesProofofnonuniformrecoveryresultHolgerRauhut,UniversityofBonnCompressiveSensing4AnExperimentinComputerTomographyduetoCandes,RombergandTaoShepp-Loganphantomandradiallinesinthefrequencyplane(covering2%ofallpixels).

7、HolgerRauhut,UniversityofBonnCompressiveSensing5ReconstructionfromincompleteFouriersamplesReconstructionbyback-projection(`2-minimization)HolgerRauhut,UniversityofBonnCompressiveSensing6ReconstructionfromincompleteFouriersamplesReconstructionbyback-Exactreconstructionbytota

8、lprojection(`2-minimization)variationminimization!HolgerRauhut,UniversityofBonnCom