阳明大学放射医学科学研究所优秀课件

Haar Wavelet Analysis吳育德陽明大學放射醫學科學研究所台北榮總整合性腦功能實驗室阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀A First Course in Wavelets with Fourier AnalysisAlbert Boggess Francis J.NarcowichPrentice-Hall,Inc.,2001阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀OutlineslWhy WaveletlHaar WaveletsThe Haar Scaling FunctionBasic Properties of the Haar Scaling FunctionThe Haar WaveletlHaar Decomposition and Reconstruction Algorithms Decomposition Reconstruction Filters and DiagramslSummary阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.1 Why WaveletlWavelets were first applied in geophysics to analyze data from seismic surveys.lSeismic surveygeophonesgeophonesseismic traceseismic traceSesimic traceSesimic traceDirect wave(along the surface)Direct wave(along the surface)Subsequent waves(rock layers below ground)Subsequent waves(rock layers below ground)阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀lFourier Transform (FT)is not a good tool gives no direct information about when an oscillation occurred.lShort-time FT:equal time interval,high-frequency bursts occur are hard to detect.lWavelets can keep track of time and frequency information.They can be used to“zoom in”on the short bursts,or to“zoom out”to detect long,slow oscillations阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀frequencyfrequency+time(equal time intervals)frequency+time阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.2 Haar Wavelets 4.2.1 The Haar Scaling Functionl lWavelet functionsWavelet functionsScaling function Scaling function (father wavelet)(father wavelet)Wavelet Wavelet (mother wavelet)(mother wavelet)These two functions generate a family of functions These two functions generate a family of functions that can be used to break up or reconstruct a signal that can be used to break up or reconstruct a signal l lThe Haar Scaling FunctionThe Haar Scaling FunctionTranslationTranslationDilationDilation阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀Using Haar blocks to approximate a signall lHigh-frequency noise shows up as tall,thin blocks.High-frequency noise shows up as tall,thin blocks.l lNeeds an algorithm that eliminates the noise and not Needs an algorithm that eliminates the noise and not distribute the rest of the signal.distribute the rest of the signal.l lDisadvantages of Harr wavelet:discontinuous and Disadvantages of Harr wavelet:discontinuous and does not approximate continuous signals very well.does not approximate continuous signals very well.Figure 2Figure 2阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀Daubechies 8Daubechies 8Dubieties 3Dubieties 3Daubechies 4Daubechies 4Chap 6阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.2.2 Basic Properties of the Haar Scaling Functionl lThe Haar Scaling function is defined asl l(x-k)(x-k):same graph but translated by to the right(if :same graph but translated by to the right(if k0)by k unitsk0)by k unitsl lLet VLet V0 0 be the space of all functions of the form be the space of all functions of the form阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lV V0 0 consists of all piecewise constant functions whose consists of all piecewise constant functions whose discontinuities are contained in the set of integers discontinuities are contained in the set of integers l lV V0 0 has compact support.has compact support.Typical element in V0Typical element in V0Figure 5Figure 5Figure 6Figure 6has discontinuitieshas discontinuities at x=0,1,3,and 4at x=0,1,3,and 4阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lLet VLet V1 1 be the space of piecewise constant functions of be the space of piecewise constant functions of finite support with discontinuities at the half integers finite support with discontinuities at the half integers has discontinuities at x=0,1/2,3/2,and 2has discontinuities at x=0,1/2,3/2,and 2阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lSuppose Suppose j j is any nonnegative integer.The space of step is any nonnegative integer.The space of step functions at level functions at level j j,denoted by,denoted by V Vj j,is defined to be the,is defined to be the space spanned by the setspace spanned by the setover the real numbers.over the real numbers.l lV Vj j is the space of piecewise constant functions of finite is the space of piecewise constant functions of finite support whose discontinuities are contained in the setsupport whose discontinuities are contained in the setl means no information is lost as the resolution gets finer.V Vj contains all relevant information up to a resolution scale order 2-j阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀lA function f(x)belongs to V0 iff f(2jx)belongs to Vj阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀lA function f(x)belongs to Vj iff f(2-jx)belongs to V0阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀How to decompose a signal into its Vj-componentsl lWhen j is large,the graph of When j is large,the graph of(2(2j j x)x)is similar to one of is similar to one of the spikes of a signal that we may wish to filter out.the spikes of a signal that we may wish to filter out.l lOne way is to construct an orthonormal basis for VOne way is to construct an orthonormal basis for Vj j using the Lusing the L2 2 inner productinner product阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lTheorem:Theorem:阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.2.4 The Haar WaveletlWe want to isolate the spikes that belong to Vj,but that are not members of Vj-1lThe way is to decompose V Vj j as an orthonormal sum of Vj-1 and its complement.lStart with V1,assume the orthonormal complement of Vo is generated by translates of some functions,we we needneed:阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀Harr wavelet 阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lTheorem 4.8(extend to VTheorem 4.8(extend to Vj j)阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀Decomposing Vj阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lTheorem:Theorem:阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.3 Haar Decomposition and Reconstruction Algorithms阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀ImplementationlStep 1:Approximate the original signal f by a step function of the form阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lExample 4.11Example 4.11阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀General decomposition schemeWWj-1j-1-component-component V Vj-1j-1-component-component 阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lTheorem 4.12(Haar Decomposition)阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lExample 4.13V V8 8-component-component V V7 7-component-componentV V6 6-component-componentV V4 4-component-componentWW7 7-component-component阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.3.2 Reconstruction阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀General reconstruction scheme阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀General reconstruction scheme阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lTheorem 4.14(Haar Reconstruction)阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀l lExample 4.1580%compression80%compression 90%compression90%compressionsample signalsample signal阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀4.3.3 Filters and Diagramslk=-1,0lk=-1,0lDecomposition algorithm阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀downsampling downsampling operatoroperator阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀lReconstructionlk=0,1lk=0,1阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀upsampling upsampling operatoroperator阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀Summary阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀阳明大学放射医学科学研究所优秀ShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E2B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUiRfOcKcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3B0y)v%s#pXlUiQfNcK8H5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*u$qZnVkShPdMdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(v%r#oWoWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%s#oXlTiQfNbK8H8G5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F7F4C0z)w&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3B0y)v%v%s#pXlUiQfNcK8H5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3B0y)v%s#pXlUiQfNcK8H5E2A+x*u$rZnWkThPeMeMaJ7G4C1z-w&t!pYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdPdMaI7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNcK8H5D2A+x*u$r$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%r%rZoWlThQeNbJ8G4D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcNcK8H5E2A+x(u$rZnWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B3B0y(v%r#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)z)v&s!pXmUjRfOcK9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK9H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z-w&w&t!pYmVjRgOcL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E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