图像的傅里叶变换-第5讲ppt课件.ppt

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1、第5讲医学图像的变换TheFourierTransform1Inasense,theFouriertransformislikeasecondlanguagefordescribingfunctions.Bilingualpersonsfrequentlyfindonelanguagebetterthananotherforexpressingcertainideas.TheFouriertransformisaveryimportanttoolfordigitalimageprocessinga

2、ndanalysis,foranyonewhointendstousedigitalimageprocessingseriouslyintheirwork,however,thetimespentbecomingfamiliarwiththeFouriertransformiswellinvested.2f(t)F(S)FouriertransformSpatialdomainFrequencydomainWorkinginthefrequencydomaingivesmanymoreopport

3、unitiesfortheimageanalyst.ToconvertanimagebetweenthespatialandthefrequencydomainweusetheFouriertransform.ThisispracticalbecauseoftheFastFourierTransformalgorithm(FFT).Twoprincipleareasofapplicationarefilteringandfocuscorrection.3Fouriertransformhastwo

4、mainstrongpoints:1、canobtainthesignalmagnitudeateveryfrequencypoint.2、cantransformconvolutionoperationintoproductoperation.456磁共振成像(MRI,MagneticResonanceImaging)技术是研究以不同的射频(RF,RadioFrequency)脉冲序列对组织激励后，用线圈检测技术获得组织弛豫信息和质子密度信息，通过图像重建形成磁共振图像的方法和技术。71、the

5、Fouriertransformofone-dimensionalfunctions:theFouriertransformofone-dimensionalcontinuousfunctions8u=0,1,…,N-1x=0,1,…,N-1theFouriertransformofone-dimensionaldiscretefunctions9UseEulerFormulaF(u)canbedescribedasfollows:polarcoordinates1011Temporal----a

6、mplitudeFrequency----amplitude121314Everyperiodicfunctioncanberepresentedasthesumofsineandcosinefunctions(basisfunction)----atearly19th.(1807)---FourierseriesEachcoefficientdescribesthe1Dsinusoidalfunctionneededtoreconstructthefunctionf(x)Frequencydom

7、aincanbeappliedtoanon-periodicfunctionifitisnonzerooverafiniterange------Fouriertransform---Transformafunctionbetweenatimeandfreq.domainFouriertransform15Fourierseries其中an和bn表示第n个谐振函数的振幅大小。确定这些系数的过程称为Fourier分析。假设有下述函数：tF(t)A-AT16它是一个方波函数，即t=0至t=½T间F(t

8、)=+A而t=½T至t=T间F(t)=-A，函数的周期为T。它可以展开如下：1718TimeandFrequencyexample:sin(ωt)+(1/3)sin(3ωt)19TimeandFrequencyexample:F(t)=sin(ωt)+(1/3)sin(3ωt)=+20TimeandFrequencyexample:F(t)=sin(ωt)+(1/3)sin(3ωt)=+21TimeandFrequencyexample:g(t)={=+=1,a/2