基于小波修剪阈值法的消噪外文翻译全文.docx
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基于小波修剪阈值法的消噪外文翻译全文.docx
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基于小波修剪阈值法的消噪外文翻译全文
WAVELETDE-NOISINGBYMEANSOFTRIMMEDTHRESHOLDING
Abstract:
Waveletthresholdingde-noisingtechniquesprovideanewwaytoreducenoiseinsignal.However,thesoftthresholdingisbestinreducingnoisebutworstinpreservingedges,andhardthresholdingisbestinpreservingedgesbutworstinde-noising.Motivatedbyfindingamoregeneralcasethatincorporatesthesoftandhardthresholdingtoachieveacompromisebetweenthetwomethods,thetrimmedthresholdingmethodisproposedinthispaper.Finally,theexperimentresultsandthepowerspectralanalysisshowthatthetrimmedthresholdingissuperiortohardandsoftthresholdingmethods.
I.INTRODUCTION
De-noisingisapermanenttopicforengineersandappliedscientists.Inrecentyears,waveletde-noisinghasbeenmoreandmoreextensiveinsignalprocessing.Asanewsignalprocessingmethod,waveletanalysishascharacteristicsofmulti-resolutionandmulti-scale.Itcanmakeusobservethesignalprogressivelyfromcoarsetofineandhavetheabilitytoperformlocalsignalcharacteristicsinbothtimedomainandfrequencydomain.Wavelettransformde-noisingisanimportantaspecttomakewaveletanalysisappliedinengineeringpractice.Inprinciple,anyarithmeticthatcanusetheFouriertransformcanusewavelettransform,anditisnotlimitedbyshort-timewindow.Soitiswidelyusedinsignalprocessing.Therearemanywaysofwaveletde-noising,themoreinfluential,andmostcommonlyusedtwomethodsarewavelettransformmodulusmaximamethodofnoisereductionandnonlinearwaveletthresholdde-noisingmethod.Insignalprocessing,noisereductionwithasmallcrossinganalysishasbeenmorewidelyused,Ithasbeensuccessfullyappliedinmanyfields.Suchastheseismicsignalnoisereduction,remotesensingimagenoisereduction,speechsignalnoisereduction,noisereductionofnonlineartimeseries.II.PRINCIPLEOFWAVELETMULTIRESOLUTION
Waveletde-noisingisbasedonthemultiresolutionanalysis.S.Mallagaveamethodofwaveletdecompositionaccordingtotheprincipleofmultiresolution.Givenascalingfunction
itstranslatesanddilates
(
)generatesubspace
suchthat...
(1)
Thereexistsawavelet
ittranslatesanddilates
produceabasisofthe‘detail’subspace
togive
.Sowecanget
...,
Andasignalx(n)canbedecomposedby
(2)
(3)
Where
arethediscretedetailcoefficientsofthesignalatlevel
and
aretheapproximationcoefficientsatlevel
andarelow-passfilterandhigh-passfilterrespectivelycorrespondingtosomewaveletbasisandtheyareconnectedby
(4)
WhereNisthelengthofthefilters.
Thealgorithmofthereconstructofthesignalis
(5)
willbegotten,whichistheoriginalsignal
whenrepeatingthereconstructformula(5).
III.DE-NOISINGBYWAVELETTHRESHOLDING
Themethodofwaveletthresholdde-noisingisbasedontheprincipleofthe
multiresolutionanalysis.Thediscretedetailcoefficientsandthediscreteapproximationcoefficientscanbeobtainedbyamulti-levelwaveletdecompose[2].
Anoisyone-dimensionalsignalmodelcanbeexpressedasfollows:
Amongthem,
isthesignalwithnoise.
istheusefulsignal,
isthenoisesignal.Hereweconsider
asaGaussianwhitenoiseoflevel1.Itisusuallyahigh-frequencysignal.Butinengineeringpractice
isusuallyalow-frequencysignal,orsomestablesignal.Therefore,wecanusethefollowingmethodofde-noising:
Firstusewaveletdecompositiontothesignal(seeinFigure1),obtainedbythewaveletdecompositionlayerisapartofthesignalofseriesoflarge-scaleapproximationandthedetailssection.InFigure1CA3iscalledtheapproximationsignalorthesmoothingsignal,itiscorrespondedtothelowfrequencysignal.CD1,CD2,CD3arecalledthedetailsignal.Theyarecorrespondedtothesignalfrequencycomponents,thenoisepartisusuallyincludedintheCDI,CD2,CD3.Therefore,wecanprocessthewaveletcoefficientswiththresholdformandthenreconstructthesignaltoremovethenoise.Thepurposeofremovingnoisefrom
istosuppressthenoisepartofthesignaltorecoverthetruesignal
from
.
Figure1Threewaveletdecomposition
Waveletdecompositiontransformssignalfromtime-domaintotime-scaledomain,anditcandescribethelocalfeaturewellinbothtimedomainandfrequencydomain.Becausetheamplitudeofthediscretedetailcoefficientsofthenoisedecreaseswiththelevelincreasing,wecanselectathreshold,modifyandprocessallofthediscretedetailcoefficientsatallscalebythresholdmethodsoastoremovenoise.Thede-noisingprocedureproceedsinthreesteps:
(1)Decomposition
(2)Thresholddetailcoefficients
(3)Reconstruction.
Generally,themostpopularthresholdingmethodsarehardandsoftthresholding.Wecanexpectthatthetechniqueofsoftthresholdingwouldintroducemoreerrororbiasthanhardthresholdingdoes.Butontheotherhand,softthresholdingismoreefficientinde-noising.Exampleswillbeillustratedlater.Toachieveacompromisebetweenthetwomethods,thetrimmedthresholdingmethodisproposedinthis
paper.
Grossmann[3]provedthatthevariancesandamplitudesofthedetailsofthewhitenoiseatthevariouslevelsdecreaseregularlyasthelevelincrease.Ontheotherhand,theamplitudeandvariancesofwavelettransformoftheavailablesignalarenotrelatedtothechangeofscale.Accordingtothepropertiesofwavelettransformofthenoiseandtheavailablesignal,wecanweaken,andevenremovenoise.Inthewaveletthresholdimgde-noising,weshouldfirstselectathresholdandprocessthecomponentsofwavelettransformofthenoisysignalinordertoimprovesignal-to-noiseratio(SNR).
D.L.Donohoproposedaverysimplemethodofwaveletthresholdde-noising.Themethodinthesenseofminimummeansquareerroriseffectiveandhasbettervisualeffect.Thebasicideaofthismethodis:
doafewcontinuouswaveletdecompositiontonoisysignal
iscorrespondedtothescalewaveletcoefficients
.Somespecificlocationshavealargevalue.Thesepointsarecorrespondedtothelocationoftheoriginalsignal
ofoddchangesandimportantinformation.Whilemostothersmallerlocationsofwaveletcoefficients.Forwhitenoise
itiscorrespondedtowaveletcoefficientsx.Theamplitudeofwaveletcoefficientsxindistributionofeachscaleisuniform.Anditdecreaseswiththescaleincreasing.Sothenumber
canbefoundasasuitablethreshold.When
islessthanthethreshold,weconsiderthat
ismainlycausedbynoise.Anditissettozeroandabandoned.When
islargerthanthethreshold,weconsiderthat
ismainlycausedbythesignal.Wecandirectlyretainthispartof
orshrinktozerobyafixedamounttogetwaveletcoefficients
andthenreconstructthenewwaveletcoefficients
togetthede-noisedsignal.Anditformsthehardandsoftthresholding.
A.Softandhardthresholding[4]
LetΔtdenotethegiventhreshold.Thesoftthresholdingisdefinedby
(6)
Forthehardthresholding
(7)
Hardthresholdingcanbedescribedastheusualprocessofsettingtozerotheelementswhoseabsolutevaluesarelowerthanthethreshold.Softthresholdingisanextensionofhardthresholding,firstsettingtozerotheelementswhoseabsolutevaluesarelowerthanthethreshold,andthenshrinkingthenonzerocoefficientstowards0.
B.Wavelettrimmedthresholding
Motivatedbyfindingamoregeneralcasethatincorporatesthesoftandhardthresholding,weproposedthefollowingthresholdingrule
(8)
Δtischosenasanestimateofnoiselevel.Whenα=1,itisequivalenttosoftthresholding;whenα→
itisequivalenttohardthresholding.Figure1graphicallyshowsitsrelationwithsoftandhardthresholding.Itcanbeclearlyseenthattrimmedthresholdingissomethingbetweenhardandsoftthresholding.Withcarefultuningofparameterαforaparticularsignal,onecanachievebestde-noisingeffectwithinthresholdingframework.
C.Thresholdselectionrule
Accordingtothenoisemodel,therearefourthresholdselectionrules[3].TheThresholdselectionrulebasedonStein’sunbiasedestimatedofriskisusedinthispaper.Letnbethelengthofvectorxinthealgorithmofthefollowingthresholdselectionrule.WegetanestimateofriskforaparticularthresholdvalueΔt.MinimizingtherisksinΔtgivesaselectionofthethresholdvalue.Sorttheabsolutevalueofthevectortobeestimatedfromminimumtomaximum,andthenextracttherootofthesortedvectorandanewvectorNVisobtained.Andthealgorithmofriskattheindexkis
(9)
Thecorrespondingthresholdis
.
Inthethresholdselecting,weshouldnotignorethedetailcoefficientsineverylevelthatprobablyinfluencetherobustnessofthethresholdestimating.Sowehavetorescaleaselectedthresholdinsomelevel.Inthispaper,thethresholdisdependentonthedetailcoefficientsateverylevel.
Figure2hard,softandtrimmedthresholdingfunction
IV.EXPERIMENTANDRESULT
Adiscretesignalsequenceintimedomain,whichisshowninFigure3,istheoriginalsignal.
Figure3originalsignal
Thissignalgeneratedfromthefunction
.Theoriginalsignalconsistsofslowchangeandfastchangecomponents,soit’sappropriatefortestingthede-noisingperformanceofhard,softandtrimmedthresholding.ThenoisysignalwithSNRof15dB,whichisshowninFigure3,isartificiallycontaminatedbystochasticnoisesgeneratedfromnormalGaussianwhitenoisewithzero-means
.
Figure4contaminatedsignalwiththeSNRof15dB
Inallexperiments,thesym5waveletischosen,performlevel5decomposition.Figure5showsthede-noisedsignalprocessedby
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