Application of wavelets and neural networks to diagnostic system development, 1, feature extraction(4)

卷积神经网络和一些独立成分分析的外文文献

B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906901

Thestretchedandcompressedwaveletsthroughscal-ingoperationareusedtocapturethedifferentfre-quencycomponentsofthefunctionbeinganalysed.Thetranslationoperation,ontheotherhand,involvesshift-ingofthemotherwaveletalongthetimeaxistocapturethetimeinformationofthefunctiontobeanalysedatadifferentposition.Inthisway,afamilyofscaledandtranslatedwaveletscanbecreatedusingscalingandtranslationparametersaandb.Thisallowssignalsoccurringatdifferenttimesandhavingdifferentfre-quenciestobeanalysed.Incontrasttotheshort-timeFouriertransform,whichusesasingleanalysiswindowfunction,thewavelettransformcanuseshortwindowsathighfrequenciesorlongwindowsatlowfrequencies.Thuswavelettransformiscapableofzooming-inonshort-livedhighfrequencyphenomenaandzooming-outonsustainedlowfrequencyphenomena.Thisisthemainadvantageofthewaveletovertheshort-timeFouriertransform.

Wavelettransformcanbecategorisedintocontinu-ousanddiscrete.Continuous,inthecontextofwavelettransform,impliesthatthescalingandtranslationparametersaandbchangecontinuously.However,calculatingwaveletcoef cientsforeverypossiblescalecanrepresentaconsiderableeffortandresultinavastamountofdata.Thereforediscreteparameterwavelettransformisoftenused.Thediscreteparameterwavelettransformusesscaleandpositionvaluesbasedonpow-ersoftwo-so-calleddyadicscalesandpositionsandmakestheanalysismuchmoreef cient,whilstremain-ingaccurate.Todothis,thescaleandtimeparametersarediscretisedasfollows,a=am0,

b=nb0an0

m,nareintegers

(3)

Thefamilyofwavelets{ m,n(t)}isgivenby

m,n(t)=a 0m/2 (a 0

m

t nb0)(4)

resultinginadiscretewavelettransform(DWT)havingtheform

DWTf(m,n)= f, m,n

+

=a

0

m/2&

f(t) (a 0

m

t nb0)(5)

Mallat(1989)developedanapproachforimplement-ingthisusing lters.Formanysignals,thelowfre-quencycontentisthemostimportantpart.Thehigh

frequencycontent,ontheotherhandprovides avourornuance.Inwaveletanalysisthelowfrequencycon-tentiscalledtheapproximationandthehighfrequencycontentiscalledthedetail.The lteringprocessuseslowpassandhighpass lterstodecomposeanoriginalsignalintotheapproximationanddetailparts.Itisnotnecessarytopreservealltheoutputsfromthe lters.Normallytheyaredownsampledandkeeponlytheevencomponentsofthelowpassandhighpass lteroutputs.

Thedecompositioncanbeiterated,withsuccessive

approximationsbeingdecomposedinturn,sothatonesignalisbrokenintomanylower-resolutioncomponents.

Inthecaseofadiscretewavelettransform,recon-structionoftheoriginalsignalisnotguaranteed.Daubechies(1992)developedconditionsunderwhichthe{ m,n}ually,a0=2andb0=1areused,althoughanyvaluescanbeused.Inthiscase,boththetransformandreconstructionarecompletebecausethefamilyofwaveletsformanor-thonormalbasis.