gabor_filter_tutorial

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1、TutorialonGaborFiltersJavierR.Movellan1TheTemporal(1-D)GaborFilterGaborfilterscanserveasexcellentband-passfiltersforunidimensionalsignals(e.g.,speech).AcomplexGaborfilterisdefinedastheproductofaGaussiankerneltimesacomplexsinusoid,i.e.g(t)=kejθw(at)s(t)(1)where2−πtw(t)=e(2)s(t)=ej(2πfot)(3)

2、ejθs(t)ej(2πfot+θ)=sin(2πft+θ),jcos(2πft+θ)(4)ooHerek,θ,foarefilterparameters.WecanthinkofthecomplexGaborfilterastwooutofphasefilterscontinentlyallocatedintherealandcomplexpartofacomplexfunction,therealpartholdsthefiltergr(t)=w(t)sin(2πfot+θ)(5)andtheimaginarypartholdsthefiltergi(t)=w(t)cos

3、(2πfot+θ)(6)1.1FrequencyResponseTakingtheFouriertransformZ∞Z∞gˆ(f)=kejθe−j2πftw(at)s(t)dt=kejθe−j2π(f−fo)tw(at)dt(7)−∞−∞kjθf−fo=ewˆ()(8)aawhere2−πfwˆ(f)=w(f)=e(9)1.2GaborEnergyFiltersTherealandimaginarycomponentsofacomplexGaborfilterarephasesensitive,i.e.,asaconsequencetheirresponsetoasi

4、nusoidisanothersinusoid(seeFigure1.2).Bygettingthemagnitudeoftheoutput(squarerootofthesumofsquaredrealandimaginaryoutputs)wecangetaresponsethatphaseinsensitiveandthusunmodulatedpositiveresponsetoatargetsinusoidinput(seeFigure1.2).Insomecasesitisusefultocomputetheoveralloutputofthetwoout

5、ofphasefilters.Onecommonwayofdoingsoistoaddthesquaredoutput(theenergy)ofeachfilter,equivalentlywecangetthemagnitude.Thiscorrespondstothemagnitude(morepreciselythesquaredmagnitude)ofthecomplexGaborfilteroutput.Inthefrequencydomain,themagnitudeoftheresponsetoaparticularfrequencyissimplythema

6、gnitudeofthecomplexFouriertransform,i.e.kf−fokg(f)k=wˆ()(10)aaNotethisisaGaussianfunctioncenteredatf0andwithwidthproportionaltoa.10−102468500−5002468500−500246850002468Figure1:Top:Aninputsignal.Second:OutputofGaborfilter(cosinecarrier).Third:OutputofGaborFilterinquadrature(sinecarrier);F

7、ourth:OutputofGaborEnergyFilter1.2.1BandwidthandPeakResponseThusthepeakfilterresponseisatfo.Togetthehalf-magnitudebandwidth∆fnotef−fo−πf−fowˆ()=ea2=0.5(11)aThusthehalfpeakmagnitudeisachievedforpf−fo±a2log2π=0.4697a≈0.5a(12)Thusthehalf-magnitudebandwidthis(2)(0.4697)a)whichisappr