数字图像处理-冈萨雷斯-课件(英文)Chapter04 频域图像增强

DigitalImageProcessingChapter4:ImageEnhancementintheFrequencyDomain22June2005,Background:FourierSeries,Anyperiodicsignalscanbeviewedasweightedsumofsinusoidalsignalswithdifferentfrequencies,Fourierseries:,FrequencyDomain:viewfrequencyasanindependentvariable,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,FourierTr.andFrequencyDomain,Time,spatialDomainSignals,FrequencyDomainSignals,FourierTr.,InvFourierTr.,1-D,Continuouscase,FourierTr.:,Inv.FourierTr.:,FourierTr.andFrequencyDomain(cont.),1-D,Discretecase,FourierTr.:,Inv.FourierTr.:,u=0,M-1,x=0,M-1,F(u)canbewrittenas,or,where,Exampleof1-DFourierTransforms,Noticethatthelongerthetimedomainsignal,TheshorteritsFouriertransform,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,RelationBetweenDxandDu,Forasignalf(x)withMpoints,letspatialresolutionDxbespacebetweensamplesinf(x)andletfrequencyresolutionDubespacebetweenfrequenciescomponentsinF(u),wehave,Example:forasignalf(x)withsamplingperiod0.5sec,100point,wewillgetfrequencyresolutionequalto,ThismeansthatinF(u)wecandistinguish2frequenciesthatareapartby0.02Hertzormore.,2-DimensionalDiscreteFourierTransform,2-DIDFT,2-DDFT,u=frequencyinxdirection,u=0,M-1v=frequencyinydirection,v=0,N-1,x=0,M-1y=0,N-1,ForanimageofsizeMxNpixels,F(u,v)canbewrittenas,or,where,2-DimensionalDiscreteFourierTransform(cont.),Forthepurposeofviewing,weusuallydisplayonlytheMagnitudepartofF(u,v),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,2-DDFTProperties,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,2-DDFTProperties(cont.),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,2-DDFTProperties(cont.),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,2-DDFTProperties(cont.),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ComputationalAdvantageofFFTComparedtoDFT,RelationBetweenSpatialandFrequencyResolutions,whereDx=spatialresolutioninxdirectionDy=spatialresolutioninydirectionDu=frequencyresolutioninxdirectionDv=frequencyresolutioninydirectionN,M=imagewidthandheight,(DxandDyarepixelwidthandheight.),HowtoPerform2-DDFTbyUsing1-DDFT,1-DDFTbyrow,F(u,y),1-DDFTbycolumn,F(u,v),HowtoPerform2-DDFTbyUsing1-DDFT(cont.),1-DDFTbyrow,F(x,v),1-DDFTbycolumn,F(u,v),Alternativemethod,Periodicityof1-DDFT,0,N,2N,-N,DFTrepeatsitselfeveryNpoints(Period=N)butweusuallydisplayitforn=0,N-1,Wedisplayonlyinthisrange,FromDFT:,ConventionalDisplayfor1-DDFT,0,N-1,TimeDomainSignal,DFT,f(x),0,N-1,Lowfrequencyarea,Highfrequencyarea,ThegraphF(u)isnoteasytounderstand!,0,-N/2,N/2-1,0,N-1,ConventionalDisplayforDFT:FFTShift,FFTShift:ShiftcenterofthegraphF(u)to0togetbetterDisplaywhichiseasiertounderstand.,Highfrequencyarea,Lowfrequencyarea,Periodicityof2-DDFT,ForanimageofsizeNxMpixels,its2-DDFTrepeatsitselfeveryNpointsinx-directionandeveryMpointsiny-direction.,Wedisplayonlyinthisrange,0,N,2N,-N,0,M,2M,-M,2-DDFT:,g(x,y),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ConventionalDisplayfor2-DDFT,Highfrequencyarea,Lowfrequencyarea,F(u,v)haslowfrequencyareasatcornersoftheimagewhilehighfrequencyareasareatthecenteroftheimagewhichisinconvenienttointerpret.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,2-DFFTShift:BetterDisplayof2-DDFT,2DFFTSHIFT,2-DFFTShiftisaMATLABfunction:ShiftthezerofrequencyofF(u,v)tothecenterofanimage.,Highfrequencyarea,Lowfrequencyarea,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Originaldisplayof2DDFT,0,N,2N,-N,0,M,2M,-M,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Displayof2DDFTAfterFFTShift,2-DFFTShift(cont.):Howitworks,Exampleof2-DDFT,Noticethatthelongerthetimedomainsignal,TheshorteritsFouriertransform,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Exampleof2-DDFT,NoticethatdirectionofanobjectinspatialimageandItsFouriertransformareorthogonaltoeachother.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Exampleof2-DDFT,Originalimage,2DDFT,2DFFTShift,Exampleof2-DDFT,Originalimage,2DDFT,2DFFTShift,BasicConceptofFilteringintheFrequencyDomain,FromFourierTransformProperty:,Wecamperformfilteringprocessbyusing,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,MultiplicationinthefrequencydomainiseasierthanconvolutioninthespatialDomain.,FilteringintheFrequencyDomainwithFFTshift,f(x,y),2DFFT,FFTshift,F(u,v),FFTshift,2DIFFT,X,H(u,v)(Userdefined),G(u,v),g(x,y),Inthiscase,F(u,v)andH(u,v)musthavethesamesizeandhavethezerofrequencyatthecenter.,MultiplicationinFreq.Domain=CircularConvolution,f(x),DFT,F(u),G(u)=F(u)H(u),h(x),DFT,H(u),g(x),IDFT,MultiplicationofDFTsof2signalsisequivalenttoperformcircularconvolutioninthespatialdomain.,f(x),h(x),g(x),“Wraparound”effect,H(u,v)GaussianLowpassFilterwithD0=5,Originalimage,Filteredimage(obtainedusingcircularconvolution),Incorrectareasatimagerims,MultiplicationinFreq.Domain=CircularConvolution,LinearConvolutionbyusingCircularConvolutionandZeroPadding,f(x),DFT,F(u),G(u)=F(u)H(u),h(x),DFT,H(u),g(x),IDFT,Zeropadding,Zeropadding,Concatenation,PaddingzerosBeforeDFT,Keeponlythispart,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,LinearConvolutionbyusingCircularConvolutionandZeroPadding,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,LinearConvolutionbyusingCircularConvolutionandZeroPadding,ZeropaddingareainthespatialDomainofthemaskimage(theideallowpassfilter),Filteredimage,Onlythisareaiskept.,FilteringintheFrequencyDomain:Example,Inthisexample,wesetF(0,0)tozerowhichmeansthatthezerofrequencycomponentisremoved.,Note:Zerofrequency=averageintensityofanimage,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,FilteringintheFrequencyDomain:Example,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,HighpassFilter,LowpassFilter,FilteringintheFrequencyDomain:Example(cont.),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ResultofSharpeningFilter,FilterMasksandTheirFourierTransforms,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,IdealLowpassFilter,whereD(u,v)=Distancefrom(u,v)tothecenterofthemask.,IdealLPFFilterTransferfunction,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ExamplesofIdealLowpassFilters,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ThesmallerD0,themorehighfrequencycomponentsareremoved.,ResultsofIdealLowpassFilters,Ringingeffectcanbeobviouslyseen!,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Howringingeffecthappens,IdealLowpassFilterwithD0=5,SurfacePlot,Abruptchangeintheamplitude,Howringingeffecthappens(cont.),SpatialResponseofIdealLowpassFilterwithD0=5,SurfacePlot,Ripplesthatcauseringingeffect,Howringingeffecthappens(cont.),(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ButterworthLowpassFilter,Transferfunction,WhereD0=Cutofffrequency,N=filterorder.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ResultsofButterworthLowpassFilters,Thereislessringingeffectcomparedtothoseofideallowpassfilters!,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,SpatialMasksoftheButterworthLowpassFilters,Someripplescanbeseen.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,GaussianLowpassFilter,Transferfunction,WhereD0=spreadfactor.,Note:theGaussianfilteristheonlyfilterthathasnorippleandhencenoringingeffect.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,GaussianLowpassFilter(cont.),GaussianlowpassfilterwithD0=5,SpatialresponesoftheGaussianlowpassfilterwithD0=5,Gaussianshape,ResultsofGaussianLowpassFilters,Noringingeffect!,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ApplicationofGaussianLowpassFilters,TheGLPFcanbeusedtoremovejaggededgesand“repair”brokencharacters.,BetterLooking,Originalimage,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ApplicationofGaussianLowpassFilters(cont.),Removewrinkles,Softer-Looking,Originalimage,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ApplicationofGaussianLowpassFilters(cont.),Removeartifactlines:thisisasimplebutcrudewaytodoit!,Filteredimage,Originalimage:ThegulfofMexicoandFloridafromNOAAsatellite.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,HighpassFilters,Hhp=1-Hlp,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,IdealHighpassFilters,whereD(u,v)=Distancefrom(u,v)tothecenterofthemask.,IdealLPFFilterTransferfunction,ButterworthHighpassFilters,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Transferfunction,WhereD0=Cutofffrequency,N=filterorder.,GaussianHighpassFilters,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,Transferfunction,WhereD0=spreadfactor.,GaussianHighpassFilters(cont.),GaussianhighpassfilterwithD0=5,SpatialresponesoftheGaussianhighpassfilterwithD0=5,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,SpatialResponsesofHighpassFilters,Ripples,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ResultsofIdealHighpassFilters,Ringingeffectcanbeobviouslyseen!,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ResultsofButterworthHighpassFilters,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,ResultsofGaussianHighpassFilters,(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.,LaplacianFilterintheFrequencyDomain,FromFourierTr.Property:,ThenforLaplacianoperator,Surfaceplot,Weget,Imageof(u2+v2),LaplacianFilterintheFrequencyDomain(cont.),Spatialresponseof(u2+v2),Crosssection,LaplacianmaskinChapter3,(ImagesfromRafaelC.Go