An Iteratively Weighted Norm Algorithm for Total Variation Regularization

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1、AnIterativelyReweightedNormAlgorithmforTotalVariationRegularizationPaulRodr´ıguezandBrendtWohlbergAbstract—TotalVariation(TV)regularizationhasbecomewhichefficient(time-performanceandmemoryrequirements)apopularmethodforawidevarietyofimagerestorationalgorithmdevelopmentisnot

2、welladvanced[7].problems,includingdenoisinganddeconvolution.Recently,aInthispaper,weintroducetheIterativelyReweightedNormnumberofauthorshavenotedtheadvantages,includingsuperior(IRN)algorithmforsolvingthegeneralizedTVfunctionalperformancewithcertainnon-Gaussiannoise,ofrepl

3、acingthestandard2datafidelitytermwithan1norm.Weproposeapq1λsimplebutveryflexibleandcomputationallyefficientmethod,J(u)=Au−b+(Dxu)2+(Dyu)2(3)pqtheIterativelyReweightedNormalgorithm,forminimizingapq2generalizedTVfunctionalwhichincludesboththe-TVand1forp≥1an

4、dq≥1(wehavefoundthealgorithmtoconvergeand-TVproblems.forsmallerpandqvalues,butcannotprovethatthealgorithmconvergesinthesecases).TheIRNalgorithmisasimplebutI.INTRODUCTIONcomputationallyefficientandveryflexiblemethodwhichiscompetitivewithexisting,well-established[4],[5]algor

5、ithmsTotalVariation(TV)regularizationwasfirstintroducedforfor2-TV,andwhilewehavenotcompareditwithalloftheimagedenoising[1],andhassinceevolvedintoamoregeneral1-TValgorithmsofwhichweareaware[7],[8],[9],[10],toolforsolvingawidevarietyofimagerestorationproblems,2[11],itissig

6、nificantlyfasterthanthosewithwhichwehaveincludingdeconvolutionandinpainting[2],[3].The-TVperformedcomparisons,andincontrasttomanyoftheseotherregularizedsolutionoftheinverseprobleminvolvingdatabmethods,easilyallowstheinclusionofforwardoperatorAforandforwardlinearoperatorA(

7、theidentityinthecaseofamoregeneralinverseproblemthandenoising.denoising,andaconvolutionforadeconvolutionproblem,forexample)istheminimumofthefunctionalII.ITERATIVELYREWEIGHTEDNORMAPPROACH21A.PreviousRelatedWorkJ(u)=Au−b+λ(Dxu)2+(Dyu)2(1)221TheIRNapproachis

8、closelyrelatedtotheIterativelyReweightedLeastSquares(IRLS)[12],[13]method,whichwhere(Du)2+(D