信号与数据稀疏性建模.

信号与数据稀疏性建模丁兴号教授厦门大学通信工程系高维数据分析与处理研究室[e-mail]2012年9月24日混合噪声去除数字水印4InpaintingOutcomeSource5Inpainting6ImageCompressionResultsResultsfor820Bytespereachfile11.9910.8310.9310.498.928.718.817.898.615.564.825.58OriginalJPEGJPEG-2000Local-PCAK-SVD7Resultsfor550Bytespereachfile15.8114.6715.3013.8912.4112.5710.669.4410.276.605.496.36ImageCompressionResultsOriginalJPEGJPEG-2000Local-PCAK-SVD8Resultsfor400Bytespereachfile18.6216.1216.8112.3011.3812.547.616.317.20???ImageCompressionResultsOriginalJPEGJPEG-2000Local-PCAK-SVD超分辨率重建图3..动态场景背景与前景分离效果图视频背景前景分离鲁棒人脸识别目标跟踪•性能比较——car4过桥底车辆跟踪目标跟踪•性能比较——PkTest02红外图像车辆跟踪目标跟踪•性能比较——OneLeaveShop商场行人跟踪目标跟踪•性能比较——seq_mb人脸跟踪原始图像采样模板填0重建填0重建残差CSRMI重建27.59CSMRI重建残差BP重建32.90BP重建残差BP+WT重建33.34BP+WT重建残差BP+TV重建36.09BP+TV重建残差MRI重建（采样率10%）图像与视频去雾MORERESULTSMORERESULTS低照度图像与视频增强高动态范围图像与视频增强24TheArtofDataProcessingVoiceSignalRadarImagingStillImageStockMarketHeartSignalCTTrafficInformationWearesurroundedbyvarioussourcesofmassiveinformationofdifferentnature.Thesesourceshavesomeinternalstructure,whichcanbeexploited.25ADataModelanditsuseADatamodelanditsusePractically,almostanytaskinimage(data)processingrequiresamodel–thisistruefordenoising,deblurring,super-resolution,inpainting,compression,detectionofanomalies,synthesis,sampling,andmore.Thereisanewmodelintown–sparseandredundantrepresentation–wewillcallitSparseland.SignalModelsSignalmodelsareafundamentaltoolforsolvinglow-levelsignalprocessingtasksNoiseRemovalImageScalingCompressionDemosaicingInpaintingDeblurringTompgraphyreconstructionSuper-resolutionDetailenhancementMorphologicdecompositionSourceseparationDeinterlacingCompressivesensing横看成岭侧成峰远近高低各不同29Model?Effectiveremovalofnoise(andmanyotherapplications)reliesonanpropermodelingofthesignal30WhichModeltoChoose?Therearemanydifferentwaystomathematicallymodelsignalsandimageswithvaryingdegreesofsuccess.Thefollowingisapartiallistofsuchmodelsforimages:Goodmodelsshouldbesimplewhilematchingthesignals:Principal-Component-AnalysisAnisotropicdiffusionMarkovRandomFieldWiennerFilteringDCTandJPEGWavelet&JPEG-2000Piece-Wise-SmoothC2-smoothnessBesov-SpacesTotal-VariationBeltrami-FlowSimplicityReliability31AnExample:JPEGandDCT178KB–Rawdata4KB8KB12KB20KB24KBHow&whydoesitworks?DiscreteCosineTrans.Themodelassumption:afterDCT,thetopleftcoefficientstobedominantandtherestzeros.TheevolutionofthedataModelSignalModelsSmoothPiecewisesmoothSmoothwithpointsingularities?Signalmodel:amathematicaldescriptionofthebehaviorweexpectfroma“good”(uncontaminated)signalinoursystemMachineLearning34MathematicsSignalProcessingNewEmergingModelsSparselandandExample-BasedModelsWaveletTheorySignalTransformsMulti-ScaleAnalysisApproximationTheoryLinearAlgebraOptimizationTheoryDenoisingCompressionInpaintingBlindSourceSeparationDemosaicingSuper-Resolution35TheSparselandModelTask:modelimagepatchesofsize10×10pixels.Weassumethatadictionaryofsuchimagepatchesisgiven,containing256atomimages.TheSparselandmodelassumption:everyimagepatchcanbedescribedasalinearcombinationoffewatoms.α1α2α3Σ36TheSparselandModel/TransformWestartwitha10-by-10pixelspatchandrepresentitusing256numbers–Thisisaredundantrepresentation.However,outofthose256elementsintherepresentation,only3arenon-zeros–Thisisasparserepresentation.Bottomlineinthiscase:100numbersrepresentingthepatcharereplacedby6(3fortheindicesofthenon-zeros,and3fortheirentries).Propertiesofthismodel:SparsityandRedundancy.α1α2α3Σ37ProblemsWithSparselandSparseandRedundantRepresentationsTheoryNumericalProblemsApplications(imageprocessing)38DifficultiesWithSparselandProblem1:Givenanimagepatch,howcanwefinditsatomdecomposition?Asimpleexample:Thereare2000atomsinthedictionaryThesignalisknowntobebuiltof15atomspossibilitiesIfeachofthesetakes1nano-sectotest,thiswilltake~7.5e20yearstofinish!!!!!!Solution:Approximationalgorithmsα1α2α3Σ20002.4e3715α1α2α3Σ39DifficultiesWithSparseland0200400600800100012001400160018002000-2-1012Iteration00200400600800100012001400160018002000-2-1012Iteration10200400600800100012001400160018002000-2-1012Iteration20200400600800100012001400160018002000-2-1012Iteration30200400600800100012001400160018002000-2-1012Iteration40200400600800100012001400160018002000-2-1012Iteration50200400600800100012001400160018002000-2-1012Iteration6Variousalgorithmsexist.TheirtheoreticalanalysisguaranteestheirsuccessifthesolutionissparseenoughHereisanexample–theIterativeReweightedLS:40DifficultiesWithSparselandα1α2α3ΣProblem2:Givenafamilyofsignals,howdowefindthedictionarytorepresentitwell?Solution:Learn!Gatheralargesetofsignals(manythousands),andfindthedictionarythatsparsifiesthem.Suchalgorithmsweredevelopedinthepast5years(e.g.,K-SVD),andtheirperformanceissurprisinglygood.Thisisonlythebeginningofanewerainsignalprocessing…41DifficultiesWithSparselandα1α2α3ΣProblem3:Isthismodelflexibleenoughtodescribevarioussources?e.g.,Isitgoodforimages?Audio?Stocks?…Generalanswer:Yes,thismodelisextremelyeffectiveinrepresentingvarioussources.Theoreticalanswer:yettobegiven.Empiricalanswer:wewillseeinthiscourse