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1、0MinimizationforWaveletFrameBasedImageRestorationYongZhang∗BinDong†ZhaosongLu‡May12,2011AbstractThetheoryof(tight)waveletframeshasbeenextensivelystudiedinthepasttwentyyearsandtheyarecurrentlywidelyusedforimagerestorationandotherimageprocessingandanalysisproblems.T
2、hesuccessofwaveletframebasedmodels,includingbalancedapproach[20,8]andanalysisbasedapproach[13,32,49],isduetotheircapabilityofsparselyapproximatingpiecewisesmoothfunctionslikeimages.Motivatedbythebalancedapproachandanalysisbasedapproach,weshallproposeawaveletframeba
3、sed0minimizationmodel,wherethe0oftheframecoefficientsarepenalized.Weadaptthepenaltydecomposition(PD)methodof[40]tosolvetheproposedoptimizationproblem.NumericalresultsshowedthattheproposedmodelsolvedbythePDmethodcangenerateimageswithbetterqualitythanthoseobtainedbye
4、itheranalysisbasedapproachorbalancedapproachintermsofrestoringsharpfeaturesaswellasmaintainingsmoothnessoftherecoveredimages.SomeconvergenceanalysisofthePDmethodwillalsobeprovided.Keywords:0minimization,waveletframe,imagerestoration.1IntroductionMathematicshasbeen
5、playinganimportantroleinthemoderndevelopmentsofimageprocessingandanalysis.Imagerestoration,includingimagedenoising,deblurring,inpainting,tomography,etc.,isoneofthemostimportantareasinimageprocessingandanalysis.Itsmajorpurposeistoenhancethequalityofagivenimagethatis
6、corruptedinvariouswaysduringtheprocessofimaging,acquisitionandcommunication,andenableustoseecrucialbutsubtleobjectsresidingintheimage.Therefore,imagerestorationisanimportantsteptotaketowardsaccurateinterpretationsofthephysicalworldandmakingoptimaldecisions.1.1Image
7、RestorationImagerestorationisoftenformulatedasalinearinverseproblem.Forthesimplicityofthenotations,wedenotetheimagesasvectorsinRnwithnequalstothetotalnumberofpixels.Atypicalimagerestorationproblemisformulatedasf=Au+η,(1.1)wheref∈Rdistheobservedimage(ormeasurements)
8、,ηdenoteswhiteGaussiannoisewithvarianceσ2,andA∈Rd×nissomelinearoperator.Theobjectiveistofindtheunknowntrueimageu∈Rnfromtheobservedimagef.Typically
