小波消噪英文文献.docx
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小波消噪英文文献.docx
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小波消噪英文文献
WaveletDe-noising
First,thewaveletthresholdde-noisingthesignalestimate
Signalprocessingsignalde-noisingisoneoftheclassic.De-noisingmethodsincludetraditionallinearfilteringmethodandnonlinearfilteringmethods,suchasmedianfilterandwienerfiltering.De-noisingmethodisnottraditionalistheentropyofthesignalincreasedaftertransformation,cannotdescribethecharacteristicsofnon-stationarysignalsandcannotgetthesignalcorrelation.Toovercometheseshortcomings,peoplebegantosignalde-noisingusingthewavelettransformtosolvetheproblem.
Wavelettransformhasthefollowingfavorablecharacteristics:
(1)LowEntropyof:
thesparsedistributionofwaveletcoefficients,sothatreducestheentropyofthetransformedsignal;
(2)Multi-resolutionfeatures:
Yutocharacterizethesignalcanbeverynon-stationaryfeaturessuchasedges,spikes,breakpoints,etc.;
(3)Torelevance:
therelevanceofthesignalcanberemoved,andthenoiseinwavelettransformhaswhiteningtrend,themorebeneficialthanthetime-domainde-noising;
(4)Selectedbasedflexibility:
theflexibilitytochoosethewaveletbasisfunctioncanthereforeberequiredaccordingtothesignalcharacteristicsandselecttheappropriatewaveletde-noising
Inthefieldofwaveletde-noisinghasbeenmorewidelyused.Thresholdingmethodisasimple,bettermethodsofwaveletde-noising.Thresholdingmethodistheideaoflayersofwaveletdecompositioncoefficientsofthemodelislargerthanandsmallerthanacertainthresholdvalueofthecoefficientoftreatment,andthenre-processedthewaveletcoefficientsofananti-transformation,throughthereconstructedde-noisedSignal.Thefollowingfunctionsfromthethresholdandthresholdestimationofboththresholdingmethodsareintroduced.
1.Thresholdfunction
Commonlyusedthresholdfunctionismainlyhardandsoftthresholdfunctionthresholdfunction.
(1)Hardthresholdfunction.Expressionis
η(w)=wI(∣w∣>T).
(2)Softthresholdfunction.Expressionis
η(w)=(w-sgn(w)T)I(∣w∣>T)
Ingeneral,thehardthresholdingmethodcanpreservethesignaledgeoftheotherlocalfeatures,softthresholdisrelativelysmooth,butwillcausetheedgeoftheblurringdistortion.Toovercometheseshortcomings,recentlyproposedasemi-softthresholdfunction.Itcantakeintoaccountthesoftthresholdandhardthresholdmethodhastheadvantage,anditsexpressionis
η(w)=sgn(w)
Thebasisofthesoftthreshold,youcanimprovethemwiththeirmoreadvanced.Itcanbeseeninthenoise(waveletcoefficients)andtheusefulsignal(waveletcoefficients)thereisasmoothtransitionbetweentheareas,moreinlinewiththenaturalsignal/imageofcontinuousfeatures.Itsexpressionis
η(w)=
2.ThresholdestimationDonohoproposedin1994VisuShrinkmethod(oruniformthresholdingmethod).Itisforthemulti-dimensionaljointdistributionofindependentnormalvariables,whenthedimensiontendstoinfinitytheconclusionsofthemaximumestimateoftheminimumconstraintsderivedoptimalthreshold.Thechoiceofthresholdsmeets:
T=
DonohoprovethatgivenestimatesofthesignalisBesovset,obtainedinanumberofriskssimilartotheidealfunctionoftheriskofnoisereduction.AunifiedmethodofDonohothresholdeffectinthepracticalapplicationunsatisfactory,resultinginthephenomenonofoverkill,putforwardin1997Janseunbiasedestimatebasedonthethresholdcalculation.Riskfunctionisdefinedas:
Orthogonalityofwavelettransform,theriskfunctioncanbewritteninthesameforminthewaveletdomain
Set
So
Finally,theexpressionofriskfunctioncanbeobtained:
Whereistheindicatorfunction,takingthenumberoftwosmall.Thus,thebestthresholdselectioncanbeobtainedbyminimizingtheriskfunction,i.e.
MATLABtoachievethethresholdofsignalde-noising,includingthethresholdandthethresholdingforthetwoparties.Thefollowingdescriptionofthem.
Second,thewaveletde-noisingfunctioninMATLAB
1)Thresholds
ImplementedinMATLABfunctionofsignalthresholdforaddencmp,thselect,wbmpenandwdcbm,followingtheuseoftheirsimpleinstructions.Ddencmpcalltheformatofthefollowingthree
(1)[THR,SORH,KEEPAPP,CRIT]=ddencmp(IN1,IN2,X)
(2)[THR,SORH,KEEPAPP,CRIT]=ddencmp(IN1,'wp',X)
(3)[THR,SORH,KEEPAPP]=ddencmp(IN1,'wv',X)
Functionddencmpusedtoobtainintheprocessofde-noisingorcompressionthedefaultthreshold.InputparameterXisoneortwodimensionalsignals;IN1valueforthe'den'or'crop',den,saidthede-noising,cropthatiscompressed;IN2valueforthe'wv'or'wp',wv,saidselectionofwavelet,wpsaidthechoiceofwaveletpackets.ReturnvalueisthereturnthresholdTHR;SORHissoftorhardthresholdthresholdselectionparameters;KEEPAPPthatkeptlowfrequencysignal;CRITistheentropyofname(onlyusedinthechoiceofwaveletpacket).
Functionthselectcallthefollowingformat:
THR=thselect(X,TPTR)
THR=thselect(X,TPTR)accordingtothedefinitionofthestringTPTRthresholdselectionrulestoselectthesignalXoftheadaptivethreshold.
Adaptivethresholdselectionrulesincludethefollowingfour.
TPTR='rigrsure',adaptivethresholdchoosetouseStein'sunbiasedriskestimateprinciple.
TPTR='heursure',usingtheheuristicthresholdselection.
TPTR='sqtwolog',thethresholdvalueisequaltosqrt(2*log(1ength(X))).TPTR='minimaxi',withtheminimaxprincipleofselectionthreshold.
Thresholdselectionrulebasedonthemodel,AistheGaussiannoiseN(O,1).
Functionwbmpencallthefollowingformat:
THR=wbmpen(C,L,SIGMA,ALPHA)
THR=wbmpen(C,L,SIGMA,ALPHA)returnstheglobalde-noisingthresholdTHR.THRbyagivenselectionrulescalculatedwaveletcoefficients,waveletcoefficientsselectionruleusingtheBirge-Massartpenaltyalgorithm.[C,L]isthede-noisingofthesignalorthewaveletdecompositionstructure;SIGMAisazeromeanGaussianwhitenoiseofstandarddeviation;ALPHAadjusttheparametersusedforpunishment,itmustbearealnumbergreaterthan1,aSharestakeALPHA=2.
Lett*isthecrit(t)=-sum(c(k)^2,k<=t)+2*SIGMA^2*t*(ALPHA+log(n/t))minimum,wherec(k)areorderedfromlargesttosmallestabsolutevalueofwaveletpacketcoefficients,nisthenumberofcoefficients,theTHR=c(t*).
wbmpen(C,L,SIGMA,ALPHA,ARG)calculatedthethresholdanddrawthethreecurves.
2*SIGMA^2*t*(ALPHA+10g(n/t))
Sum(c(k)^2,k<=t)
crit(t)
Functionwdcbmcallthefollowingtwoformats:
(1)[THR,NKEEP]=wdcbm(C,L,ALPHA)
(2)[THR,NKEEP]=wdcbm(C,L,ALPHA,M)
FunctionwdcbmusingBirge-Massartmethodforone-dimensionalwavelettransformtoobtainthethreshold.ReturnvalueTHRisthethresholdandscaleindependent,NKEEPisthenumberofcoefficients.[C,L]istocarryoutsignalde-noisingorcompressioninthej=length(L)-2layerbreakdownstructure;ALPHAandMmustbearealnumbergreaterthan1;THRisaboutjofthevector,THR(i)isthei-layerthreshold;NKEEPisavectoronthej,NKEEP(i)isthecoefficientofilayernumber.1.5forthegeneralcompressionALPHA,ALPHAde-noisingtake3.2)Signalthresholdde-noising
MATLAB,thethresholdforsignalde-noisingfunctionhaswden,wdencmp,wthresh,wthcoef,wpthcoefandwpdencmp.Followingtheusageoftheirbrief.Functionwdencallthefollowingtwoformats:
(1)[XD,CXD,LXD]=wden(X,TPTR,SORH,SCAL,N,'wname')
(2)[XD,CXD,LXD]=wden(C,L,TPTR,SORH,SCAL,N,'wname')
Functionwdenfortheautomaticone-dimensionalsignalde-noising.Xistheoriginalsignal,[C,L]forthesignaldecomposition,Nisthenumberoflayersofwaveletdecomposition.
TPTRthethresholdselectionrules,TPTRthefollowingfourvalues:
TPTR='rigrsure',bySteinunbiasedlikelihoodestimation.
TPTR='heursure',usingheuristicthresholdselection.
TPTR='sqtwolog',takeuniversalthreshold
TPTR='minimaxi',usingthemaximumthresholdfortheminimumvalueselection.SORHissoftorhardthresholdthresholdselection(correspondingto's'and'h').SCALreferstothethresholdusedbytheneedtore-adjust,includingthebottomthree:
SCAL='one',donotadjust.
SCAL='sln',accordingtothefirstlayeroftheestimatedcoefficientstoadjustthenoisefloorthreshold.
SCAL='mln',accordingtodifferentestimatestoadjustthenoiselevelthreshold.XDforthenoisedsignal,[CXD,LXD]forthesignalafterde-noisingwaveletdecompositionstructure.Format
(1)returnsthesignalXthroughNlayersdecomposedwaveletcoefficientsafterthresholdingandsignalde-noisingsignalXDXDthewaveletdecompositionstructure[CXD,LXD].Format
(2)returnparametersandformat
(1),butitsstructurebydirectdecompositionofthesignalstructureof[C,L]obtainedbythresholdprocessing.
Functionwdencmpcallthefollowingthreeformats:
(1)[XC,CXC,LXC,PERF0,PERFL2]=wdenemp('gbl',X,'wname',N,THR,SORH,KEEPAPP)
(2)[XC,CXC,LXC,PERF0,PERFL2]=wdencmp('1vd',X,'wname',N,THR,SORH)
(3)[XC,CXC,LXC,PERF0,PERFL2]=wdencmp('1vd',C,L,'wname',N,THR,SORH)
Functionwdencmpforoneortwodimensionalsignalde-noisingorcompression.wnamewaveletfunctionisused,gbl(globalabbreviation)thateachhaveadoptedathresholdforthesametreatment,lvdthateachusedifferentthresholdsfortreatment,Nsaidthatthenumberoflayersofwaveletdecomposition,THRisthethresholdvectorForFormat
(2)and(3)requireseachdepartmenthasathresholdvalue,sothethresholdvectorlengthTHRN,SORHthatchoiceofsoftorhardthresholdthreshold(value,respectively,forthe's'and'h),theparameterKEEPAPPvalue
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