Image-restoration-by-sparse-3D-transform-domain-co

Imagerestorationbysparse3Dtransform-domaincollaborativefilteringKostadinDabov,AlessandroFoi,VladimirKatkovnik,andKarenEgiazarianDepartmentofSignalProcessing,TampereUniversityofTechnologyP.O.Box553,33101Tampere,Finlandfirstname.lastname@tut.fiABSTRACTWeproposeanimagerestorationtechniqueexploitingregularizedinversionandtherecentblock-matchingand3Dfiltering(BM3D)denoisingfilter.TheBM3Demploysanon-localmodelingofimagesbycollectingsimilarimagepatchesin3Darrays.Theso-calledcollaborativefilteringappliedonsucha3Darrayisrealizedbytransform-domainshrinkage.Inthiswork,weproposeanextensionoftheBM3Dfilterforcolorednoise,whichweuseinatwo-stepdeblurringalgorithmtoimprovetheregularizationafterinversionindiscreteFourierdomain.ThefirststepofthealgorithmisaregularizedinversionusingBM3Dwithcollaborativehard-thresholdingandthesecondsstepisaregularizedWienerinversionusingBM3DwithcollaborativeWienerfiltering.Theexperimentalresultsshowthattheproposedtechniqueiscompetitivewithandinmostcasesoutperformsthecurrentbestimagerestorationmethodsintermsofimprovementinsignal-to-noiseratio.Keywords:imagerestoration,deconvolution,deblurring,block-matching,collaborativefiltering1.INTRODUCTIONImageblurringisacommondegradationinimaging.Inmanycases,theblurringcanbeassumedspace-invariantandthusmodeledasaconvolutionofthetrueimagewithafixedpoint-spreadfunction(PSF).Suchamodelisgivenby}({)=(|~y)({)+({),(1)where|isthetrue(non-degraded)image,yisablurPSF,isi.i.d.Gaussiannoisewithzeromeanandvariance2,and{5[isa2Dcoordinateintheimagedomain[.Theinversionoftheblurringisingeneralanill-posedproblem;thus,evennoisewithverysmallmagnitude,suchastruncationnoiseduetolimited-precisionarithmetic,cancauseextremedegradationsafternaiveinversion.Regularizationisawellknownandextensivelystudiedapproachtoalleviatethisproblem.Itimposessomeregularityconditions(e.g.,smoothness)ontheobtainedimageestimateand/oronitsderivatives.Numerousapproachesthatemployregularizationhavebeenproposed;anintroductioncanbefoundforexampleinthebooks.1,2Inparticular,animagerestorationschemethatcomprisesofregularizedinversionfollowedbydenoisinghasbeenabasisofthecurrentbest-performingrestorationmethods.3,4Suchdenoisingaftertheinversioncanbeconsideredaspartoftheregularizationsinceitattenuatesthenoiseintheobtainedsolution(i.e.thesolutionissmoothed).Variousdenoisingmethodscanbeemployedtosuppressthenoiseaftertheinversion.Filteringinmultiresolu-tiontransformdomain(e.g.,overcompletewaveletandpyramidtransforms)wasshown4—6tobeeectiveforthispurpose.Inparticular,theSV-GSM,4whichemploysGaussianscalemixturesinovercompletedirectionalandmultiresolutionpyramids,isamongthecurrentbestimagedeblurringmethods.Anotherdenoisingtechniqueusedafterregularizedinversion3,7,8istheLPA-ICI9whichexploitsanon-parametriclocalpolynomialfitinanisotropicestimationneighborhoods.ThebestresultsofthemethodsbasedonLPA-ICIwereachievedbytheshape-adaptivediscretecosinetransform(SA-DCT)deblurring3wherethedenoisingisrealizedbyshrinkageoftheSA-DCTappliedonlocalneighborhoodswhosearbitraryshapesaredefinedbytheLPA-ICI.ThisworkwaspartlysupportedbytheAcademyofFinland,projectNo.213462(FinnishCentreofExcellenceprogram[2006-2011]);theworkofK.DabovwassupportedbytheTampereGraduateSchoolinInformationScienceandEngineering(TISE).Figure1.Flowchartoftheproposeddeconvolutionalgorithm.AfragmentofHouseillustratestheimagesaftereachoperation.Inthisworkwefollowtheaboverestorationscheme(regularizedinversionfollowedbydenoising)exploitinganextensionoftheblock-matchingand3Dfiltering10(BM3D).Thisfilterisbasedontheassumptionthatthereexistmutuallysimilarpatcheswithinanaturalimage–thesameassumptionusedinothernon-localimagefilterssuchas.11,12TheBM3Dprocessesanoisyimageinasliding-window(block)manner,whereblock-matchingisperformedtofindblockssimilartothecurrentlyprocessedone.Theblocksarethenstackedtogethertoforma3Darrayandthenoiseisattenuatedbyshrinkageina3D-transformdomain.Thisresultsina3Darrayoffilteredblocks.Adenoisedimageisproducedbyaggregatingthefilteredblockstotheiroriginallocationsusingweightedaveraging.Thisfilterwasshown10tobehighlyeectiveforattenuationofadditivei.i.d.Gaussian(white)noise.Thecontributionofthisworkincludes•extensionoftheBM3Dfilterforadditivecolorednoise,and•imagedeblurringmethodthatexploitstheextendedBM3DfilterforimprovingtheregularizationafterregularizedinversioninFouriertransformdomain.Thepaperisorganizedasfollows.ThedevelopedimagerestorationmethodandtheextensionoftheBM3DfilterarepresentedinSections2.SimulationresultsandabriefdiscussionaregiveninSection3andrelevantconclusionsaremadeinSection4.2.IMAGERESTORATIONWITHREGULARIZATIONBYBM3DFILTERINGTheobservationmodelgiveninEquation(1)canbeexpressedindiscreteFouriertransform(DFT)domainas]=\Y+˜,(2)where\,Y,and˜aretheDFTspectraof|,y,and,respectively.CapitallettersdenoteDFTofasignal;e.g.]=F{}},Y=F{y};theonlyexceptioninthatnotationisfor˜=F{}.DuetothenormalizationoftheforwardDFT,thevarianceof˜is|[|2,where|[|isthecardinalityoftheset[(i.e.,|[|isthenumberofpixelsintheinputimage).Giventheinputblurredandnoisyimage},theblurPSFy,andth