FoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessingAgendaWaveletTransformsWaveletWaveletTransformsThefollowingexerciseillustrateonewayofviewingthecontinuouswavelettransform.WefirstdefinethegeneralwaveletbasisfunctionatscaleaasTheTwo-DimensionalcontinuouswaveletTheTwo-DimensionalcontinuouswavelettransformTheinverseTwo-DimensionalcontinuouswavelettransformCompactDyadicWaveletsIfwefurtherrestrictf(x)andthebasicwavelettofunctionsthatarezerooutsidetheinterval[0,1],thenthefamilyoforthonormalbasisfunctioncanbespecifiedbyasingleindex,n;thatis,Wherejandkareactuallyfunctionsofn,asfollows:forj=0,1….K=0,1,…PSF:theimageofpointsourceShiftinvarianceofopticalsystems:thePSFistheimpulseresponseofthepointsourcePointslinearity:theconvolutionofthelightsourcewiththePSFCoherentillumination:thefixedphaserelationshipofpointsourcepermitsstablepatternsofinterference.TheconvolutionoperationisoperatedontheamplitudeoftheelectromagneticwavesIncoherentillumination:

randomrelativephaserelationship

theconvolutionisperformedonanintensitybasisPointsourceSpotimageFocalplaneImageplaneLensAsimpleimagingsystemTheimageproducedbyapointsourceiscalledasthepoint-spreadfunction(PSF)inopticalterms.TheOTFisthetwo-dimensionalFouriertransformofthePSF.Itwilltakeonitssmallestpossiblesizeifthesystemisinfocus,thatis,ifIfthepointsourcemovesoffthez-axistoaposition,thespotimagemovestoanewpositiongivenbyThemagnificationofthesystemreturnThetruncatedsphericalexitwavePupilfunction:thespatialdistributionoftransmittanceintheplanecontainingtheapertureAperture:thesizeoflensDiffractioneffect:aconvergingsphericalexitwaveistheresponsetothedivergingsphericalwaveofapointsource.ImageplaneConvergingsphericalwaveAmplitudedistributionPupilplane

TheCoherentPointSpreadFunction（PSF）

WenowhavetheextremelyimportantresultthatthecoherentPSFis,asidefromacomplexcoefficient,merelythetwo-dimensionalFouriertransformofthepupilfunctionThecomplexexponentialcoefficientinEq.affectsonlythephaseintheimageplane,andthisisignoredbycommonlyusedimagesensors.Thus,forourpurposes,theterminfrontoftheintegralismerelyacomplexconstant.Wecansimplifythenotationconsiderablyifwegiveupabsoluteamplitudecalibrationandignorethetermsinfrontoftheintegral.ThenwecanwritetheconvolutionrelationbetweentheobjectandtheimageasWheretheimpulseresponseisgivenbythetermistheamplitudedistributionoftheobject,andistheobjectafterprojectionwithoutdegradationintotheimageplane.Thus,wecanconsiderimagingasatwo-stepprocess:geometricalprojection,followedbyconvolutionintheimageplanewiththePSF.TheCoherentOpticalTransferFunctionTransformingthefunctiontwicemerelyreflectsitabouttheorigin,sothecoherenttransferfunctionisgivenbyInthecommoncaseofsymmetricalapertures,the180-degreerotationhasnoeffect.Thus,thepupilfunction,properlyscaled,isthecoherentOTF.TheIncoherentPoint-SpreadFunction Inthecaseofincoherentlight,israndomateachpointandindependentofitsneighbors.Inthiscase,theobservedintensityatapointis

Itindicatesthat,underincoherentillumination,thesystemislinearinintensity,andthePSFisthesquaredmodulusofh(x,y),thecoherentPSF.TheIncoherentOpticalTransferFunction（OTF）

ThenormalizedFouriertransformoftheincoherentPSFiscalledtheincoherentOTF.SincetheincoherentPSFisthepowerspectrumofthepupilfunction,theautocorrelationtheoremimpliesthattheincoherentOTFisthenormalizedautocorrelationfunctionofthepupilfunctionFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessingAgendaPart3TechnologyandApplications

10ImageRestorationChapter16

ImageRestorationFoundationCh1~4ImagingSystem,Digitalization,Display,SoftwareCh5~8Histogram,PointOperations,AlgebraicOperations,GeometricOperationsTheoryCh9~12LinearSystem,FourierFrequencyTransform,FilterDesign,DiscreteSamplingCh13~15OrthogonalRadicleTransform,WaveletTime-frequencyTransform,OpticalFunctionTransformApplicationCh16~20ImageRestoration,Compression,PatternRecognitionCh21~22ColorandMulti-SpectralImageProcessing,ThreeDimensionImageProcessingAgendaINTRODUCTION

ImageRestorationisoneofthemainelementsofimageprocessing.Forimagerestoration,itmeanstheremovalorreductionofdegradationswhichoccursintheobtaineddigitalimage.Thedegradationsincludetheblurringfromopticalsystemsorimagemotion,aswellasthenoisefromelectronicandphotometricsources.mathematicalmodelThemodelofimagedegradationandrestorationisasfollow.Imagedegradationiscausedbysystemperformanceandnoise.Restorationisachievedthroughrehabilitatingfilter(inversefilter).16.2Classicalrestoration16.3Linearalgebraicrestoration16.4restorationoflessrestricteddegradations16.5Superresolution16.6Systemidentification16.7Noisemodeling16.8Implementation16.2ClassicalRestorationClassificationoffilters

lowpassBasicfiltertypesbandpass,bandstop highpass

Wienerestimatordesignoftwolinearfiltersthematcheddetectornonlinearfilterhighpass/high-frequencyenhancement

Ideal：

sGeneral：

sWienerFilteringWienerFilteringadoptsestimationprocedureoftheminimummean-squareerrorandhasthetwo-dimensionaltransferfunctionandcanalsobewrittenas(11-72):16.2ClassicalRestorationandarethepowerspectraofthesignalandnoise,respectively3.OptimalitycriterionofWienerfilterUseminimizedmeansquareerrorasacriterionofoptimalityErrorsignal：meansquareerror：Points：Giventhepowerspectraofs(t)andn(t),wemustdeterminetheimpulseresponseh(t)thatminimizesthemeansquareerror;Themeansquareerrorisafunctionalofh(t),theimpulseresponse,sinceafunction,h(t),mapsintoarealnumber,MSE.ispositiveforbothpositiveandnegativeerrors.Squaringtheerrorcauseslargeerrorstobe“penalized”moreseverelythansmallerrors。6.TransferfunctionandMSEofWienerfilterundertheconditionofuncorrelatedsignalandnoiseUncorrelatedsignalandnoisemeansthat:MSEofWienerfilter’soutput：Basedonformula(2),wecanderiveandprovetheconclusionsbelow：ThetransferfunctionofWienerfilteris：7.WienerDeconvolutionFigure5WienerdeconvolutionThetransferfunctionoftheoptimaldeconvolutionfilterinthemeansquaresenseis:S(t)w(t)x(t)n(t)y(t)z(t)F(s)1/F(s)+G(s)PowerSpectrumEqualization thefilterwhichrestoresthepowerspectrumofthedegradedimagetoitsoriginalamplitudeis: Thispowerspectrumequalization(PSE)filterisphase-less.Itisapplicableforphase-lessblurringfunctions,orphasemaybedeterminedbyothermethods.GeometricMeanFiltersNoticethatif=1,Eqreducestoadeconvolutionfilter.If=1/2and=1,itreducestoPSEfilter.ItisthegeometricmeanbetweenordinarydeconvolutionandWienerone.Soitisalsocalledasgeometricmeanfilter.Itiscommonpractice,however,torefertothemoregeneralfiltermentionedaboveasthegeometricmeanfilter.If=0,theresultistheparametricWienerfilterIf=1,thisbecomestheWienerdeconvolutionfilter.While=0itreducestoapuredeconvolution.Ingeneral,maybeselectedforanyforWiener-typesmoothing.Theresearchshowthat,undertheseconditions,purede-convolutionisleastdesirable,andWienerde-convolutionproduceslow-passfilteringmoreseverethatthehumaneyedesires.TheparametricWienerfilterwithlessthanunityandthegeometricmeanfilterwiththesameconstraintseemtoproducemorepleasingresults.16.2ClassicalRestoration16.3LinearAlgebraicRestoration16.3LinearAlgebraicRestoration Itoffersaunifieddevelopmentofrestorationfilter,includingthosepreviouslymentioned,andityieldsinsightintothenumericalaspectsoftheimagerestorationproblem.ThemodelofLinearAlgebraicRestorationasfigure16-5.TheDiscreteRestorationModel

Theformationofthemodelcanbeexpressedcompactlyas:g=Hf+nwhereg,fandnareN2–by-1columnvectorsandHisanN2×N2matrix.Iftheblurisshiftinvariant,Hisablock-circulantmatrix.UnconstrainedRestoration Ifn=0orifweknownothingaboutthenoise,wecansetuptherestorationasaleastsquareminimizationproblem.e(f)=g-Hf W(f)=(g-Hf)t(g-Hf)Settingtozerothederivativeoff,yieldsf=H-1gConstrainedLeastSquaresRestoration

Theminimizationconstraintthatthenormsofeachsideofg-Hf=nbethesame，thatis,Nowwecansetuptheminimizationproblemas

WhereQisamatrixtodefinesomelinearoperatoronfandλisaconstantcalledtheLagrangeFactor.TheabilitytospecifyQgivesusflexibilityinsettingthegoaloftherestoration.Asbefore,wesettozerothederivativeofW（f）withrespecttof:Whereareconstantsthatmustbeadjustedsothattheconstraintofaboveissatisfied.ThePseudoinverseFilter LetQ=I

minimizethenormoffTheParametricWienerFilterletQtobethenoise-to-signalratio

solvingforSmoothnessConstraintsInversefilteringoftenemphasizessmalldetails.Forthisreason,therestoredimagecansufferfromlargeartifactualoscillations.OnewaytocombatitistoselectQsoastoenforcesomedegreeofsmoothnessontherestoredimage.ThentheaboveEqseeksanestimatethatissmooth,unblurred,andnoisefree.16.2ClassicalRestoration16.3LinearAlgebraicRestoration16.4restorationoflessrestricteddegradationsSpatiallyVariantBlurringwhileopticaldefocusandlinearmotionblurarespatiallyinvariantlinearoperations,astigmatism,coma,curvatureoffield,androtarymotionblurarespatiallyvariant.Adirectandeffectiverestorationmethodforcorrectingthesedegradationsiscoordinatetransformationrestoration.Theapproachinvolvesusingageometrictransformationonthedegradedimagethatmakestheresultantblurringfunctionspatiallyinvariant.Thisisfollowedbyanordinaryspatiallyinvariantrestorationtechniqueandthenbyageometrictransformationthatinvertsthefirstsuchoperationandputstheimagebackintoitsoriginalformat

weconsidersituationthatarenotrestrictedtoshift-invariantblurringandstationarysignalsandnoise.TemporallyVariantBlurring Thediffraction-limitedresolutionofa200-inchtelescopeisapproximately0.05secondofarc.Underunfavorableconditions,however,atmosphericturbulencecanreducethisresolutiontoabout2secofarc.Viewingstarsthroughathroughaturbulentatmosphereissimilartowatchingapointsourceoflightthroughamovingtextured-glassshowerdoor.NonstationarySignalsandNoise Foranimagetobestationary,thelocallycomputedpowerspectrumwouldhavetobesameovertheentireimage.Unfortunately,thisisoftennottobethesameovertheentireimage.Mostimagesare,infact,highlynonstationary.MatrixFormulation

Imposingtheconstraintofshiftinvarianceallowedustoreducethesuperpositionintegraltoasimpleconvolution.Ifwedonotimposeshiftinvariance,thesuperpositionthatmodelsimagedegradationcanbewritteninmatrixnotationasW＝FS＋N

Aminimummeansquareestimatorcanbederivedfromthismatrixformulation.LocalStationarity Whileimagesareseldomstationaryinaglobalsense,theyfrequentlycanbeassumedlocallystationary.Thismeansthatthelocalpowerspectrumchangesslowlyasonemovesthewindowwithintheimage.Incertainimagesthisassumptionmightbequietgoodandinothersmarginalorquestionable,butitrepresentsasignificantimprovementovertheassumptionofglobalstationarity.PowerSpectrumParametersTherestorationmustbeprecededbyastepthatcomputesameanimageandavarianceimagefromtheinputimage.Thenwecangetthenoisepowerspectrumandthesignalpowerspectrum.Wecannowobtainasignal-dependentspatiallyvariantgeneralizedWienerfilterbysubstitutingtheimagetoEqs.Letusassumethatthenoiseislocallywhitewithzeromeanandpowerproportionaltolocalmeangraylevel.ThenthenoisepowerspectrumandvariancearerelatedbyWhereisaconstantandistheaveragegraylevelcomputedoversomelocalwindowcenteredon

Wecannowwriteasignal-dependentspatiallyvariantgeneralizedWienerfilterbysubstitutingtheimagetoEqs.ImagePartitioningAmorepracticalsolutionistoproduceatwo-dimensionalhistogramof,andlookforclustersofpixelsinmeanversusvariancespace.Thespacecanthenbepartitionedintoareascontainingtheseclusters.Theresultingregionscouldbemappedbackintotheimagetodefineregionsofrelativelyconstantmeanandvariance.Thenarestorationfiltercanbedesignedandimplementedoneachsuchregion.Inthisway,spatiallyvariantrestorationwouldbeonlyafewtimesmoreexpensivethansimplestationaryrestoration.NoisePowerRatiothegeneralizedWienerfilteronlyrespondstotheratioofnoise-to-signalpower.Asimplifiedrestorationprocedurethroughouttheimageisresultedifassumedthesignalandnoisepowerspectrachangeinamplitude,butnotinfunctionalform.Ifthenoiseislocallywhite,anditssignal-dependentamplitudeisgivenbythebelowEq.WherethenoisepowerratioNPR(X,Y)representsthespatialvariabilityofpowerspectraandiseasilycomputedfromthemeanandvarianceimageofthedegradedimage.OnecouldusethresholdsonNPR(X,Y)atseveralgraylevelstopartitionthedegradedimageintoregionsofroughlysimilarSNRs.Adifferentrestorationfiltercouldthenbeusedineachoftheseregions.LinearCombinationFiltersThereisanotherwaytousetheNPRimagetoguidespatiallyvariantrestoration.Supposewegenerateamaskfunctionm(x,y)bynormalizingNPR(x,y)totherange[0,1].Nextwedesigntworestorationfilters,g1(x,y)andg2(x,y)tocorrespondtolowandhighNPR(X,Y),respectively.z1(x,y)=w(x,y)*g1(x,y)z2(x,y)=w(x,y)*g2(x,y)Thefinalrestoredoutputisformedbylinearcombinationrestorationoftwofilters.16.2ClassicalRestoration16.3LinearAlgebraicRestoration16.4restorationoflessrestricteddegradations16.5Superresolution16.5Superresolution Incoherenttransferfunctionofanopticalsystemistheautocorrelationfunctionofthepupilfunction.Restorationprocedurestorecoverinformationbeyondthediffractionlimitarereferredtoassuperresolutiontechnique.

Usethesamplevaluesoff(t)tocomputepointsonitsspectrumF(s).FigureComputingspectraTheinputsignalanditsspectrumF(s)isenclosedinanenvelopeoftheform1/s,however,andthisassurethatthepeakamplitudeofthefunctiondiesoutwithincreasingfrequency.Wecantakethistobetheworstcaseforaliasinganddefine,asameasureofaliasingerror,theratioofF(s)toF(o).Figureillustratesthecasewheres1=1/2τ.Thatequationallowsustoplaces1anywherebetweens0and1/τ.ThentheinterpolatingfunctionbecomesAnalyticContinuationIfafunctionisspatiallybounded,itsspectrumisananalyticfunction.Awell-knownpropertyofanalyticfunctionisthatifsuchfunctionisknownoverafiniteinterval,itisknowneverywhere.Theprocessofreconstructingananalyticfunctioninitsentirety,giventhevaluesofthefunctionoveraspecifiedinterval,iscalledasanalyticcontinuation.Sinceanimageisspatiallybounded,itsspectrummustbeanalytic.Harris’Technique Harristhoughtthatitshouldbepossibletoreconstructthatobjectininfinitedetailfromitsdiffraction-limitedimage.Thetechniqueinvolvesapplyingthesamplingtheorem,withdomainsreversed,toobtainasystemoflinearequationsthatcanbesolvedforvaluesofthesignalspectrumoutsidethediffraction-limitedpassband.SuccessiveEnergyReductionItinvolvessuccessivelyenforcingspace-limitednessupontheimage,whilekeepingtheknownlow-frequencyportionofthespectrumintact.Noticethatbandlimitingthespectrumcauseg0(x)nolongertobespacelimited.Thefirststepoftherestorationisenforcingspace-limitednessupong0(x)bysettingittozerooutsidethedomainofthepulse.Thesecondstepinvolvesreplacing.Theconvergencegenerallybecomesratherslowafterthefirstfewsteps.PracticalConsiderationsAnydigitalimplementationoftheextrapolationofananalyticfunctionmustbedonecarefully.AndrewsandHuntreferto“themythofsuperresolution”andarguethatnoiseconstraintsprecludeanypracticalextensionofresolutionbeyondthediffractionlimit.Onlywithveryhighqualityimagedigitizationandcarefullycomputationcansignificantimprovementbeexprcted.16.2ClassicalRestoration16.3LinearAlgebraicRestoration16.4restorationoflessrestricteddegradations16.5Superresolution16.6SystemIdentification16.6SystemIdentificationBeforeimagerestorationcanbeaccomplished,thePSFoftheblurringfunctionmustbeknown.Insomecases,itisknowninadvance,butinothersitmustbedeterminedexperimentallyfromthedegradedimage.Inthissection,weconsidermethodsfordeterminingthePSFandMTFofanimagingsystem.SystemIdentificationbyCalibrationTargets SupposethatforthesysteminFigure,theimpulseresponseh(x,y)isunknownandmustbedetermined.Wecanfindthetransferfunctiondirectlyfrom

h(x,y)f(x,y)g(x,y)PointSourceTargetsIfitwerepossibletoinputanimpulse(PSF),theoutputwouldbetheimpulseresponse.Whileanimpulseisphysicallyimpossible,wecouldgetbywithapulsethatisnarrowcomparedtothePSFitself.SineWaveTargets

iftheinputfunctionissinusoidalfunction,theoutputspectrumisG(u,v).Thisisanevenimpulsepairlocatedontheu-axisat

Byrepeatingthisprocedurewithmanydifferentfrequenciesatmanydifferentorientations,onecandeterminethetransferfunctiontoanydesireddegreeofaccuracy.LineTargetsIftheinputfunctionis,thusithastheeffectofintegratingoutthey-componentoftheimpulseresponse,thespectrumoftheoutputismerelythetransferfunctionevaluatedalongtheu-axis.Ifh(x,y)hascircularsymmetry,thenthetransferfunctionH(u,v)canbecompletelydeterminedfromtheline-spreadfunctionproducedbyaninputlineatanyorientation.Ifh(x,y)isseparableintoaproductofafunctionofxtimesafunctionofy,thentheverticalandhorizontalline-spreadfunctionofthesystemareadequatetodeterminethetransferfunction.Ifh(x,y)isasymmetrical,therotationpropertyofthetwo-dimensionalFouriertransformimpliesthatwecantakeline-spreadfunctionsateveryangleoforientation,transformthemtoobtainprofilesofH(u,v)ateveryangle,andthusreconstructthetransferfunction.EdgeTargetsIftheinputfunctionis.Itcontainsanabrupttransitionfromlowtohighamplitudealongthe-y-axis.Theedge-spreadfunctionissimilartheline-spreadfunction.Thusonecanproceedasbefore.FrequencySweepTargetsTheinputisharmonicsignalwhosefrequencyincreaseslinearlywithdistancefromtheorigin.AharmonicsignalwithfrequencyisgivenbyTheoutputreducestoWhichismerelytheinputinanenvelopethatisthetransferfunction.SystemIdentificationbyCross-CorrelationSupposewecross-correlatetheoutputofalinearsystemwithitsinput,asshowninFigure16-12.Thespectrumoftheoutputofthecross-correlationis

Ifisuncorrelatedwhitenoise,andisaconstant,theoutputofthecross-correlatorismerelytheimpulseresponseofthesystem.

Figure16-12Systemidentificationbycross-correlationIdentifyingtheSystemfromtheImageInsomecase,itisimpracticalorimpossibletocalibratetheimagingsystemunderthesameconditionsinwhichaparticulardegradedimagewasrecorded.Thisistrueformotionblurandstochasticdegradationsuchasatmosphericturbulence,andwhenaphotographistoberestoredandtheoriginalcamerasystemisunavailable.Insuchinstances,onemustattempttodeterminethedegradingPSFfromtheimagesitself.Iftheimagecontainsanyfeaturethatcanbemodeledanalytically,then,theoretically,thePSFcanbeobtainedbydeconvolutionofthemodel.PointSourcesIFonecanarrangeforthedegradedimagetoincludeapointsourceoflightoravanishinglysmalldarkspotonawhitebackground,thenthePSFisavailabledirectly.Ifthepointsourceorspotisofnonnegligibleextent,thenitcanbemodeledwithaGaussian,aflat-toppedcircularpulse,orsomeothersuitablefunctionthatcanbedeconvolvedtoyieldthePSF.Lines Undertheprojectiontheoremofthetwo-dimensionalFouriertransform,theFouriertransformoftheline-spreadfunctiongivesaone-dimensionalcomponentofthetwo-dimensionaltransferfunction.Theline-spreadfunctionapproachhastheadvantagethatalinesourceintheimagecanbeaveragedalongitsextenttogeneratearelativelynoise-freeestimate.Forhigh-qualitysystems,however,thelineobjectintheimagemustbeextremelythin.Thereforeitmustbeextremelybrightrelativetoitsbackgroundinordertocomethroughwithsufficientamplitude.Edges

Mostimagesofordinaryscenecontainfeaturesthatcanbemodeledasidealedges.Suchanedgecanbeaveragedalongitsextenttoproducearelativelynoise-freeestimateofthesystemedge-spreadfunctioninaparticulardirection.Thiscanthenbedifferentiatedtoproducetheline-spreadfunction,andthatcanbetransformedtoproduceacomponentofthetransferfunction.16.216.2ClassicalRestoration16.3LinearAlgebraicRestoration16.4restorationoflessrestricteddegradations16.5Superresolution16.6SystemIdentification16.7NoiseModeling16.7NoiseModelingElectronicNoise Electronicnoiseduetotherandomthermalmotionofelectronsinresistivecircuitelementsisthesimplestofthethreesourcetomodel.Thistypeofnoisehasbeensuccessfullymodeledbycircuitdesignersforalongtime.ItisusuallymodeledasspectrallywhiteGaussiannoisewithzeromeanvalue.Thus,ithasaGaussianhistogramandaflatpowerspectrum.ItiscompletelyspecifiedbyitsRMSvalue.Sometimeelectroniccircuitsexhibitso-calledone-over-fnoise.Thisisrandomnoisewithanintensitythatdiesoutinverselywithfrequency.However,image-processingproblemsseldomrequiremodelingofthe1/fcomponentofthenoise.PhotoelectronicNoise

Photoelectronicnoiseisduetothestatisticalnatureoflightandofthephotoelectronicconversionprocessthattakesplaceinimagesensors.Atlowlightlevels,wheretheeffectisrelativelysevere,phtoelectronicnoiseisoftenmodeledasrandomwithaPoissondensityfunction.Thestandarddeviationofthisdistributionisequaltothesquarerootofthemean.FilmGrainNoiseFormostpracticalpurposes,filmgrainnoisecanbeeffictivelymodeledasaGaussianprocess.16.216.2ClassicalRestoration16.3LinearAlgebraicRestoration16.4restorationoflessrestricteddegradations16.5Superresolution16.6SystemIdentification16.7NoiseModeling16.8Implementation16.9Summary16.8ImplementationTransformDomainFiltering

Insomecases,therestorationisbetterdoneusingadiscretetransformotherthantheFouriertransform.

Large-KernelConvolution

Iftherestorationisalinear,shift-invariantoperation,itcanbeimplementedbyconvolutioninthespatialdomain.DiscreteTwo-DimensionalConvolution

Accountingtotwo-dimensionsituationthegeneralformtransformingamatrixwithN-by-NtomatrixwithN-by-N

SinusoidalDecompositionDiscreteFouriertransform（DFT）One-dimensiontransform

One-dimensioninversetransform

where

Two-dimensiontransformTwo-dimensioninversetransformTheHaarTransformTheHarrfunction

ifweletx=i/Nfori=0,1…N-1,thisgivesrisetoasetofbasisfunctions,eachofwhichisanoddrectangularpulsepair,exceptfork=0,which,asinthecaseofmanyoftheothertransformsdiscussedhere,isconstantTheHaarTransform（N=8）Theeight-by-eightunitarykernelmatrixfortheHaartransformis

TheHaarTransform（N=8）ThebasisimagesforN=8appearinFigure.Noticethatthelowerrightquadrantsearchesforsmallfeaturesatalldifferentlocationsintheimage.Small-KernelConvolution Unlesstheimageisseverelyoversampled,thesignalspectrum,andconsequentlytherestorationMTF,willnormallyextendmostofthewaytothefoldingfrequencybeforeitdiesout.FromthesimilaritytheoremoftheFouriertransform,weknowthatifthetransferfunctionisabroad,theimpulseresponsewillbenarrow.Thus,theconvoluti