Chapter12

SamplingandInterpolationofContinuousImages1.Samplingandinterpolation2.Computingspectra3.Aliasing4.Truncation5.ControllingaliasingerrorMainContents1.Samplingandinterpolation ——theShahfunction

Beforewecandescribequantitatively

theeffectsofsampling,wemustestablishamathematicalprocedureformodelingtheprocess.Todothis,weuseaspecialfunctioncalledtheShahfunctionwhichisasfollows:

Theinfiniteimpulsetrain,Ⅲ(x),isaseriesofunit-amplitudeimpulsesthatoccuratunitspacingalongthex-axis.TransformationofShahfunction：xxthesimilaritytheoremTransformationofShahfunction：TheShahfunctionanditsspectrumSinceIII(x)isaninfinitetrainofequallyspacedimpulses,italsoexhibitsthisbehaviorunderstretchingandcompression.1.Samplingandinterpolation

——samplingwiththeShahfunction

Supposeafunctionf(x)isbandlimitedatafrequencys0;thatis,

ImpulseResponseofaLinearSystem

Noticethat：g(t)Ifwesamplef(x)atequalintervalsτ,wedestroyf(x)everywhereexpectatx=nτ.Thesampledfunctionisillustratedinthefollowingfigure.ThesampledfunctionanditsfrequencydomainSamplingbyShah

functionτIII(t/τ)FrequencyDomainF(s)-s0s0-1/τ-s0s01/τG(s)=τIII(τs)*F(s)Timedomain原始信號及其頻譜采樣后信號及其頻譜ReconstructionSamplingtheoremFreq.domain

Applyingtheconvolutiontheorem,productintimedomainequalsconvolutioninfrequencydomain,thatis

III(x/τ)f(x)=τIII(τs)*F(s)

Theconvolutionofafunctionwithanimpulseproducesmerelyacopyofthatfunction,thatis,F(x)isreplicatedevery1/τalongthes-axis.

Samplingmakefinitebandwidthfunctiontobethefunctionwithspectrumfromminusinfinitytoinfinity.

Anyfunctionsampledatequalintervalsτhasaspectrumthatisperiodicwithfrequency1/τ.

1.Samplingandinterpolation ——thesamplingtheoremNowthatthefunctionf(x)

hasbeensampled,theinformationbetweensamplepointshasbeenlost.Sorecovertheoriginalfunctionisquiteimportant.

Twonecessaryconditionsoftransform:（1）retaintheonethatiscenteredupontheorigin（2）eliminateallthereplicasofF(s)

Recoverprocedure：

Wehaverecoveredthespectrumoff(x)fromthespectrumofthesampledsignalg(x)Weconvolvethesampledfunctiong(x)withaninterpolatingfunctionoftheform

sinc(x)=sin(x)/xtogetf(x)Themethodaboveshowsthatwecanindeedrecoverf(x)fromg(x).Thisdevelopmentissubjecttotworestrictions.

（1）f(x)mustbebandlimitedatso（2）therelationshipbetweenτandthebandlimitsomustsatisfySamplingtheorem：afunctionsampledatuniformspacingτcanbecompletelyrecoveredfromthesamplevalues,providedthatSoisthebandlimit1.Samplingandinterpolation ——functioninterpolationConvolvingg(x)withtheinterpolatingfunctionsin(x)/xineffectreplicatesanarrowsin(x)/x

ateachsamplepoint

guaranteesthatthesummationoftheoverlappingsin(x)/xfunctionswilladduptoreproducetheoriginalfunctionexactly.

Thefigureaboveillustratesthecasewheres1=1/2,but

allowsconsiderablearbitrarinessinthefrequencyofthesin(x)/xfunctionifthereciprocalofthesamplingintervalisconsiderablylargerthanthebandlimit,so.1.Samplingandinterpolation ——undersamplingandaliasing

Wemustsampleafunctionfinelyifwewantittobetotallyrecoveredfromitssamplevalues.Supposethesamplinginterval>1/2s0.ThenwhenF(s)isreplicatedtoformG(s),theindividualreplicaswilloverlapandsumtogether.

Ifwetheninterpolate,usingthefunctionintheprecedingequation,wewillnotrecoverf(x)exactly,becauseTheeffectofoverlapofthespectral：

Energyabovethefrequencys1isfoldedbackbelows1andaddedtothespectrum.Thisfoldingbackofenergyiscalledaliasing,andthedifferencebetweenf(x)andtheinterpolatedfunctionisduetoaliasingerror.2.Computingspectra

——spectrainthetimedomainSupposeasignalisrepresentedbyNsamplepointsseparatedbyconstantspacing.Thetotalintervaloverwhichthesignalissampledis

Computingspectra2.Computingspectra

——spectrainthefrequencydomainSinceisasampledfunctionwithsamplespacing,itsspectrumisperiodicwithperiod.ItiscommonpracticetospreadtheNsamplepointsevenlyacrossthatcycleofwhichiscenteredupontheorigin.ThismeansthatwecomputepointononlyovertherangeIfwespreadNequallyspacedsamplepointsoveronecycleof,thenisthesamplespacinginthefrequencydomain.Thebestchoiceforcomputingthespectrumofistocomputepointswithequalspacing,givenbytheprecedingequation.

Thesamplingtheoremindicatesthatajudiciouschoiceofsamplespacingcancompletelyavoidaliasingwhenoneissamplingabandlimitedfunction.Ifweareforcedtoworkwithinherentlynon-bandlimitedfunctions,thenwearecondemnedtoworkintheshadowofunavoidablealiasing.

Sincetheconvolutionoftwofunctionscanbenonarrowerthaneither,weconcludethatthespectrumofthetruncatedfunctionisofinfiniteextentinthefrequencydomain.Thus,truncationdestroysbandlimitednessandcondemnsdigitalprocessingtoproducingaliasinginallcases.3.Aliasing ——theunavoidabilityofaliasing3.Aliasing

——boundingaliasingerrorThefollowingfigureillustrateshowdosealinearsystemcomputethespectrumofitsresponsetoarectangularpulse.

Iff(t)istheinputpulseandg(t)isthesystem’soutput,thetransferfunctionish(t)LinearsystemidentificationTheinputsignalanditsspectrumSinceF(S)extendsfromminustoplusinfinity,nochoiceofwillcompletelyavoidaliasing.SinceF(0)isunityandtheenvelopeis,wecanwriteanupperboundonaliasingas3.Aliasing

——spectralresolutionF(s)hassinusoidalvariationsoffrequencya.LetdenotebyMthenumberofsamplepointspercycleofF(s)onthecomputedspectrumanduseitasameasureofspectralresolution.Theperiodofthesinusoidalvariationis1/a,then

WemayhaveasmanysamplepointspercycleofF(s)asdesiredifwemakethesamplingperiodTlargecomparedtothehalf-widthofthepulse.4.Truncation

——computingthespectrumofanedgeThissamplecalculatesthespectrumofastepfunction’sedge.ThestepfunctionanditsspectrumInordertocalculatethespectrumofthefunctionsign(x),wemustfirsttruncatetoafiniteduration.IfwetruncatethefunctionwithatruncationwindowofwidthT,theresultingfunctionis:ThespectrumoftruncatedstepfunctionThetruncatedstepfunctionSincethetruncatedfunctionisanoddpairofrectangularpulses:Theprecedingequationproducesthespectrumofthetruncatededgefunction：thatis：ThespectrumofthetruncatedsignalisasinusoidenclosedunderanenvelopethatistwicethedesiredspectrumF(s)ThesamplepointsonwillbecomputedatdiscretefrequenciesThecosinetermtakesonthevalue±１foreveniand-1foroddI,soioddievenThecomputedspectrumofthestepfunction4.Truncation

——truncationeffectsNoticethecuriouseffectoftruncationintheprecedingexamples:

Theodd-numberedpointswerecorrect,whiletheeven-numberedpointswerezero.Thetruncationredistributedtheenergyamongtheoddandevenpoints.

Onecouldobtaintheexpectedresultbyconvolvingwithanarrow,triangularlocal-averagingfilter.Thiswouldavoidthediscontinuitiesatandpreventtruncationerror.5.Controllingaliasingerror

——theantialiasingfilterTherearetwoparametersthatwecanusetopreventaliasingfromcorruptingtheinformationthatisofinterestintheimage:thesamplingapertureandthesamplespacing.Thefigurebelowillustrateshowonecanreducealiasing:thewidthoftheapertureistwicethesamplespacing.Thisplacesthefirstzero-crossingofitstransferfunctionat.Thus,energyatfrequenciesabovewillbeattenuatedseverely.

ReductionofaliasingwitharectangularapertureThetriangularsamplingapertureusedinthefigurebelowisfoursamplepointswideandalsohasitszero-crossingat.Sinceitsspectrumdiesoutwithfrequencymorerapidlythanthatoftherectangularpulse,itismoreeffectiveagainstaliasing.Liketherectangularpulse,however,itreducestheenergyinbelow.Reductionofaliasingwithatriangularaperture5.Controllingaliasingerror

——oversampling

Ifwemakethesamplespacingsmall,wecanplacefNfarbeyondthefrequenciesofinterestinthespectrum.Then,whenaliasingcontaminatestheupperpartofthespectrum,itwillhavelittleornoeffectuponthedataofinterest.Thetruncationwindowshouldbelarge