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Image+Processing+Fundamentals.pdf

Image+Processing+Fundamentals

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简介:本文档为《Image+Processing+Fundamentalspdf》,可适用于IT/计算机领域

FundamentalsofImageProcessingIanTYoungJanJGerbrandsLucasJvanVlietCIPDATAKONINKLIJKEBIBLIOTHEEK,DENHAAGYoung,IanTheodoreGerbrands,JanJacobVanVliet,LucasJozefFUNDAMENTALSOFIMAGEPROCESSINGISBN–––NUGISubjectheadings:DigitalImageProcessingDigitalImageAnalysisAllrightsreservedNopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeanselectronic,mechanical,photocopying,recording,orotherwisewithoutthepriorwrittenpermissionoftheauthorsVersionCopyright©,,byITYoung,JJGerbrandsandLJvanVlietCoverdesign:ITYoungPrintedinTheNetherlandsattheDelftUniversityofTechnologyIntroductionDigitalImageDefinitionsToolsPerceptionImageSamplingNoiseCamerasDisplaysIanTYoungAlgorithmsJanJGerbrandsTechniquesLucasJvanVlietAcknowledgmentsDelftUniversityofTechnologyReferencesIntroductionModerndigitaltechnologyhasmadeitpossibletomanipulatemultidimensionalsignalswithsystemsthatrangefromsimpledigitalcircuitstoadvancedparallelcomputersThegoalofthismanipulationcanbedividedintothreecategories:•ImageProcessingimageinfiimageout•ImageAnalysisimageinfimeasurementsout•ImageUnderstandingimageinfihighleveldescriptionoutWewillfocusonthefundamentalconceptsofimageprocessingSpacedoesnotpermitustomakemorethanafewintroductoryremarksaboutimageanalysisImageunderstandingrequiresanapproachthatdiffersfundamentallyfromthethemeofthisbookFurther,wewillrestrictourselvestotwo–dimensional(D)imageprocessingalthoughmostoftheconceptsandtechniquesthataretobedescribedcanbeextendedeasilytothreeormoredimensionsReadersinterestedineithergreaterdetailthanpresentedhereorinotheraspectsofimageprocessingarereferredtoWebeginwithcertainbasicdefinitionsAnimagedefinedinthe“realworld”isconsideredtobeafunctionoftworealvariables,forexample,a(x,y)withaastheamplitude(egbrightness)oftheimageattherealcoordinateposition(x,y)Animagemaybeconsideredtocontainsubimagessometimesreferredtoas…ImageProcessingFundamentalsregions–of–interest,ROIs,orsimplyregionsThisconceptreflectsthefactthatimagesfrequentlycontaincollectionsofobjectseachofwhichcanbethebasisforaregionInasophisticatedimageprocessingsystemitshouldbepossibletoapplyspecificimageprocessingoperationstoselectedregionsThusonepartofanimage(region)mightbeprocessedtosuppressmotionblurwhileanotherpartmightbeprocessedtoimprovecolorrenditionTheamplitudesofagivenimagewillalmostalwaysbeeitherrealnumbersorintegernumbersThelatterisusuallyaresultofaquantizationprocessthatconvertsacontinuousrange(say,betweenand)toadiscretenumberoflevelsIncertainimageformingprocesses,however,thesignalmayinvolvephotoncountingwhichimpliesthattheamplitudewouldbeinherentlyquantizedInotherimageformingprocedures,suchasmagneticresonanceimaging,thedirectphysicalmeasurementyieldsacomplexnumberintheformofarealmagnitudeandarealphaseFortheremainderofthisbookwewillconsideramplitudesasrealsorintegersunlessotherwiseindicatedDigitalImageDefinitionsAdigitalimageam,ndescribedinaDdiscretespaceisderivedfromananalogimagea(x,y)inaDcontinuousspacethroughasamplingprocessthatisfrequentlyreferredtoasdigitizationThemathematicsofthatsamplingprocesswillbedescribedinSectionFornowwewilllookatsomebasicdefinitionsassociatedwiththedigitalimageTheeffectofdigitizationisshowninFigureTheDcontinuousimagea(x,y)isdividedintoNrowsandMcolumnsTheintersectionofarowandacolumnistermedapixelThevalueassignedtotheintegercoordinatesm,nwith{m=,,,…,M–}and{n=,,,…,N–}isam,nInfact,inmostcasesa(x,y)whichwemightconsidertobethephysicalsignalthatimpingesonthefaceofaDsensorisactuallyafunctionofmanyvariablesincludingdepth(z),color(l),andtime(t)Unlessotherwisestated,wewillconsiderthecaseofD,monochromatic,staticimagesinthischapter…ImageProcessingFundamentalsRowsColumnsValue=a(x,y,z,l,t)Figure:DigitizationofacontinuousimageThepixelatcoordinatesm=,n=hastheintegerbrightnessvalueTheimageshowninFigurehasbeendividedintoN=rowsandM=columnsThevalueassignedtoeverypixelistheaveragebrightnessinthepixelroundedtothenearestintegervalueTheprocessofrepresentingtheamplitudeoftheDsignalatagivencoordinateasanintegervaluewithLdifferentgraylevelsisusuallyreferredtoasamplitudequantizationorsimplyquantizationCOMMONVALUESTherearestandardvaluesforthevariousparametersencounteredindigitalimageprocessingThesevaluescanbecausedbyvideostandards,byalgorithmicrequirements,orbythedesiretokeepdigitalcircuitrysimpleTablegivessomecommonlyencounteredvaluesParameterSymbolTypicalvaluesRowsN,,,,,ColumnsM,,,,GrayLevelsL,,,,,Table:CommonvaluesofdigitalimageparametersQuitefrequentlyweseecasesofM=N=Kwhere{K=,,}Thiscanbemotivatedbydigitalcircuitryorbytheuseofcertainalgorithmssuchasthe(fast)Fouriertransform(seeSection)…ImageProcessingFundamentalsThenumberofdistinctgraylevelsisusuallyapowerof,thatis,L=BwhereBisthenumberofbitsinthebinaryrepresentationofthebrightnesslevelsWhenB>wespeakofagraylevelimagewhenB=wespeakofabinaryimageInabinaryimagetherearejusttwograylevelswhichcanbereferredto,forexample,as“black”and“white”or“”and“”CHARACTERISTICSOFIMAGEOPERATIONSThereisavarietyofwaystoclassifyandcharacterizeimageoperationsThereasonfordoingsoistounderstandwhattypeofresultswemightexpecttoachievewithagiventypeofoperationorwhatmightbethecomputationalburdenassociatedwithagivenoperationTypesofoperationsThetypesofoperationsthatcanbeappliedtodigitalimagestotransformaninputimageam,nintoanoutputimagebm,n(oranotherrepresentation)canbeclassifiedintothreecategoriesasshowninTableOperationCharacterizationGenericComplexityPixel•Point–theoutputvalueataspecificcoordinateisdependentonlyontheinputvalueatthatsamecoordinateconstant•Local–theoutputvalueataspecificcoordinateisdependentontheinputvaluesintheneighborhoodofthatsamecoordinateP•Global–theoutputvalueataspecificcoordinateisdependentonallthevaluesintheinputimageNTable:TypesofimageoperationsImagesize=N´Nneighborhoodsize=P´PNotethatthecomplexityisspecifiedinoperationsperpixelThisisshowngraphicallyinFigureabPointabLocalabGlobal=m=mo,n=noFigure:Illustrationofvarioustypesofimageoperations…ImageProcessingFundamentalsTypesofneighborhoodsNeighborhoodoperationsplayakeyroleinmoderndigitalimageprocessingItisthereforeimportanttounderstandhowimagescanbesampledandhowthatrelatestothevariousneighborhoodsthatcanbeusedtoprocessanimage•Rectangularsampling–Inmostcases,imagesaresampledbylayingarectangulargridoveranimageasillustratedinFigureThisresultsinthetypeofsamplingshowninFigureab•Hexagonalsampling–AnalternativesamplingschemeisshowninFigurecandistermedhexagonalsamplingBothsamplingschemeshavebeenstudiedextensivelyandbothrepresentapossibleperiodictilingofthecontinuousimagespaceWewillrestrictourattention,however,toonlyrectangularsamplingasitremains,duetohardwareandsoftwareconsiderations,themethodofchoiceLocaloperationsproduceanoutputpixelvaluebm=mo,n=nobaseduponthepixelvaluesintheneighborhoodofam=mo,n=noSomeofthemostcommonneighborhoodsaretheconnectedneighborhoodandtheconnectedneighborhoodinthecaseofrectangularsamplingandtheconnectedneighborhoodinthecaseofhexagonalsamplingillustratedinFigureFigureaFigurebFigurecRectangularsamplingRectangularsamplingHexagonalsamplingconnectedconnectedconnectedVIDEOPARAMETERSWedonotproposetodescribetheprocessingofdynamicallychangingimagesinthisintroductionItisappropriategiventhatmanystaticimagesarederivedfromvideocamerasandframegrabberstomentionthestandardsthatareassociatedwiththethreestandardvideoschemesthatarecurrentlyinworldwideuse–NTSC,PAL,andSECAMThisinformationissummarizedinTable…ImageProcessingFundamentalsStandardNTSCPALSECAMPropertyimagessecondmsimagelinesimage(horizvert)=aspectratio:::interlace:::µslineTable:StandardvideoparametersInaninterlacedimagetheoddnumberedlines(,,,…)arescannedinhalfoftheallottedtime(egmsinPAL)andtheevennumberedlines(,,,…)arescannedintheremaininghalfTheimagedisplaymustbecoordinatedwiththisscanningformat(SeeSection)ThereasonforinterlacingthescanlinesofavideoimageistoreducetheperceptionofflickerinadisplayedimageIfoneisplanningtouseimagesthathavebeenscannedfromaninterlacedvideosource,itisimportanttoknowifthetwohalfimageshavebeenappropriately“shuffled”bythedigitizationhardwareorifthatshouldbeimplementedinsoftwareFurther,theanalysisofmovingobjectsrequiresspecialcarewithinterlacedvideotoavoid“zigzag”edgesThenumberofrows(N)fromavideosourcegenerallycorrespondsone–to–onewithlinesinthevideoimageThenumberofcolumns,however,dependsonthenatureoftheelectronicsthatisusedtodigitizetheimageDifferentframegrabbersforthesamevideocameramightproduceM=,,orcolumns(pixels)perlineToolsCertaintoolsarecentraltotheprocessingofdigitalimagesTheseincludemathematicaltoolssuchasconvolution,Fourieranalysis,andstatisticaldescriptions,andmanipulativetoolssuchaschaincodesandruncodesWewillpresentthesetoolswithoutanyspecificmotivationThemotivationwillfollowinlatersectionsCONVOLUTIONThereareseveralpossiblenotationstoindicatetheconvolutionoftwo(multidimensional)signalstoproduceanoutputsignalThemostcommonare:c=aÄb=a*b()…ImageProcessingFundamentalsWeshallusethefirstform,c=aÄb,withthefollowingformaldefinitionsInDcontinuousspace:c(x,y)=a(x,y)Äb(x,y)=a(c,z)b(xc,yz)dcdz¥¥ò¥¥ò()InDdiscretespace:cm,n=am,nÄbm,n=aj,kbmj,nkk=¥¥åj=¥¥å()PROPERTIESOFCONVOLUTIONThereareanumberofimportantmathematicalpropertiesassociatedwithconvolution•Convolutioniscommutativec=aÄb=bÄa()•Convolutionisassociativec=aÄ(bÄd)=(aÄb)Äd=aÄbÄd()•Convolutionisdistributivec=aÄ(bd)=(aÄb)(aÄd)()wherea,b,c,anddareallimages,eithercontinuousordiscreteFOURIERTRANSFORMSTheFouriertransformproducesanotherrepresentationofasignal,specificallyarepresentationasaweightedsumofcomplexexponentialsBecauseofEuler’sformula:ejq=cos(q)jsin(q)()wherej=,wecansaythattheFouriertransformproducesarepresentationofa(D)signalasaweightedsumofsinesandcosinesThedefiningformulasfortheforwardFourierandtheinverseFouriertransformsareasfollowsGivenanimageaanditsFouriertransformA,thentheforwardtransformgoesfromthe…ImageProcessingFundamentalsspatialdomain(eithercontinuousordiscrete)tothefrequencydomainwhichisalwayscontinuousForward–A=Fa{}()TheinverseFouriertransformgoesfromthefrequencydomainbacktothespatialdomainInverse–a=FA{}()TheFouriertransformisauniqueandinvertibleoperationsothat:a=FFa{}{}andA=FFA{}{}()ThespecificformulasfortransformingbackandforthbetweenthespatialdomainandthefrequencydomainaregivenbelowInDcontinuousspace:Forward–A(u,v)=a(x,y)ej(uxvy)dxdy¥¥ò¥¥ò()Inverse–a(x,y)=pA(u,v)ej(uxvy)dudv¥¥ò¥¥ò()InDdiscretespace:Forward–A(W,Y)=am,nej(WmYn)n=¥¥åm=¥¥å()Inverse–am,n=pA(W,Y)ej(WmYn)dWdYppòppò()PROPERTIESOFFOURIERTRANSFORMSThereareavarietyofpropertiesassociatedwiththeFouriertransformandtheinverseFouriertransformThefollowingaresomeofthemostrelevantfordigitalimageprocessing…ImageProcessingFundamentals•TheFouriertransformis,ingeneral,acomplexfunctionoftherealfrequencyvariablesAssuchthetransformcanbewrittenintermsofitsmagnitudeandphaseA(u,v)=A(u,v)ejj(u,v)A(W,Y)=A(W,Y)ejj(W,Y)()•ADsignalcanalsobecomplexandthuswrittenintermsofitsmagnitudeandphasea(x,y)=a(x,y)ejJ(x,y)am,n=am,nejJm,n()•IfaDsignalisreal,thentheFouriertransformhascertainsymmetriesA(u,v)=A*(u,v)A(W,Y)=A*(W,Y)()Thesymbol(*)indicatescomplexconjugationForrealsignalseq()leadsdirectlyto:A(u,v)=A(u,v)j(u,v)=j(u,v)A(W,Y)=A(W,Y)j(W,Y)=j(W,Y)()•IfaDsignalisrealandeven,thentheFouriertransformisrealandevenA(u,v)=A(u,v)A(W,Y)=A(W,Y)()•TheFourierandtheinverseFouriertransformsarelinearoperationsFwawb{}=Fwa{}Fwb{}=wAwBFwAwB{}=FwA{}FwB{}=wawb()whereaandbareDsignals(images)andwandwarearbitrary,complexconstants•TheFouriertransformindiscretespace,A(W,Y),isperiodicinbothWandYBothperiodsarepA(Wpj,Ypk)=A(W,Y)j,kintegers()•Theenergy,E,inasignalcanbemeasuredeitherinthespatialdomainorthefrequencydomainForasignalwithfiniteenergy:…ImageProcessingFundamentalsParseval’stheorem(Dcontinuousspace):E=a(x,y)dxdy¥¥ò¥¥ò=pA(u,v)dudv¥¥ò¥¥ò()Parseval’stheorem(Ddiscretespace):E=am,nn=¥¥åm=¥¥å=pA(W,Y)dWdYppòppò()This“signalenergy”isnottobeconfusedwiththephysicalenergyinthephenomenonthatproducedthesignalIf,forexample,thevalueam,nrepresentsaphotoncount,thenthephysicalenergyisproportionaltotheamplitude,a,andnotthesquareoftheamplitudeThisisgenerallythecaseinvideoimaging•Giventhree,multidimensionalsignalsa,b,andcandtheirFouriertransformsA,B,andC:c=aÄb«FC=A•Bandc=a•b«FC=pAÄB()Inwords,convolutioninthespatialdomainisequivalenttomultiplicationintheFourier(frequency)domainandviceversaThisisacentralresultwhichprovidesnotonlyamethodologyfortheimplementationofaconvolutionbutalsoinsightintohowtwosignalsinteractwitheachotherunderconvolutiontoproduceathirdsignalWeshallmakeextensiveuseofthisresultlater•Ifatwodimensionalsignala(x,y)isscaledinitsspatialcoordinatesthen:Ifa(x,y)®aMx•x,My•y()ThenA(u,v)®AuMx,vMyæèçöø÷Mx•My()•Ifatwodimensionalsignala(x,y)hasFourierspectrumA(u,v)then:A(u=,v=)=a(x,y)dxdy¥¥ò¥¥òa(x=,y=)=pA(u,v)dxdy¥¥ò¥¥ò()…ImageProcessingFundamentals•Ifatwodimensionalsignala(x,y)hasFourierspectrumA(u,v)then:¶a(x,y)¶x«FjuA(u,v)¶a(x,y)¶y«FjvA(u,v)¶a(x,y)¶x«FuA(u,v)¶a(x,y)¶y«FvA(u,v)()ImportanceofphaseandmagnitudeEquation()indicatesthattheFouriertransformofanimagecanbecomplexThisisillustratedbelowinFiguresacFigureashowstheoriginalimageam,n,Figurebthemagnitudeinascaledformaslog(|A(W,Y)|),andFigurecthephasej(W,Y)FigureaFigurebFigurecOriginallog(|A(W,Y)|)j(W,Y)BoththemagnitudeandthephasefunctionsarenecessaryforthecompletereconstructionofanimagefromitsFouriertransformFigureashowswhathappenswhenFigureaisrestoredsolelyonthebasisofthemagnitudeinformationandFigurebshowswhathappenswhenFigureaisrestoredsolelyonthebasisofthephaseinformation…ImageProcessingFundamentalsFigureaFigurebj(W,Y)=|A(W,Y)|=constantNeitherthemagnitudeinformationnorthephaseinformationissufficienttorestoretheimageThemagnitude–onlyimage(Figurea)isunrecognizableandhasseveredynamicrangeproblemsThephaseonlyimage(Figureb)isbarelyrecognizable,thatis,severelydegradedinqualityCircularlysymmetricsignalsAnarbitraryDsignala(x,y)canalwaysbewritteninapolarcoordinatesystemasa(r,q)WhentheDsignalexhibitsacircularsymmetrythismeansthat:a(x,y)=a(r,q)=a(r)()wherer=xyandtanq=yxAsanumberofphysicalsystemssuchaslensesexhibitcircularsymmetry,itisusefultobeabletocomputeanappropriateFourierrepresentationTheFouriertransformA(u,v)canbewritteninpolarcoordinatesA(wr,x)andthen,foracircularlysymmetricsignal,rewrittenasaHankeltransform:A(u,v)=Fa(x,y){}=pa(r)Jowrr()rdr¥ò=A(wr)()wherewr=uvandtanx=vuandJo(•)isaBesselfunctionofthefirstkindoforderzeroTheinverseHankeltransformisgivenby:a(r)=pA(wr)Jowrr()wrdwr¥ò()…ImageProcessingFundamentalsTheFouriertransformofacircularlysymmetricDsignalisafunctionofonlytheradialfrequency,wrThedependenceontheangularfrequency,x,hasvanishedFurther,ifa(x,y)=a(r)isreal,thenitisautomaticallyevenduetothecircularsymmetryAccordingtoequation(),A(wr)willthenberealandevenExamplesofDsignalsandtransformsTableshowssomebasicandusefulsignalsandtheirDFouriertransformsInusingthetableentriesintheremainderofthischapterwewillrefertoaspatialdomaintermasthepointspreadfunction(PSF)ortheDimpulseresponseanditsFouriertransformsastheopticaltransferfunction(OTF)orsimplytransferfunctionTwostan

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