Application of wavelets and neural networks to diagnostic system development, 1, feature extraction(5)

3.2.Singularitydetectionusingwa6eletsforfeatureextraction

Singularitiesoftencarrythemostimportantinforma-tioninsignals.Singularitiesofasignalcanbeusedasthecompactrepresentation,i.e.thefeaturesoftheoriginalsignal.Mathematically,thelocalsingularityofafunctionismeasuredbyLipschitzexponents(Mallat&Hwang,1992).MallatandHwang(1992)provedthatthelocalmaximaofthewavelettransformmodulusdetectsthelocationsofirregularstructuresandpro-videsnumericalproceduresforcomputingtheLipschitzexponents.Withintheframeworkofscale-space lter-ing,in exionpointsoff(t)appearasextremafor(f(t)/(tandzerocrossingfor(2f(t)/(t2,soMallatandZhong(1992)suggestsusingawaveletwhichisthe rstderivativeofascalingfunctionF(t), (t)=

d (t)dt

withacubicspinebeingusedforthescalingfunction.BakshiandStephanopoulos(1996)usedthein exionpointsastheconnectionpointsofepisodesegmentsofasignal.

Thewaveletmodulusmaximaandzero-crossingrep-resentationsweredevelopedfromunderlyingcontinu-ous-timetheory.Forcomputerimplementation,thishastobecastindiscrete-timedomain.BermanandBaras(1993)provedthatwavelettransformextrema/zero-crossingprovidestablerepresentationsof nitelengthdiscrete-timesignals.Amorecompletediscrete-timeframeworkfortherepresentationofthewavelettransformwasdevelopedbyCvetkovicandVetterli(1995)andthereforeisusedinthisstudy.Theyde-signedanon-subsampledmulti-resolutionanalysis ingthis lterbank,thewaveletfunctioncanbeselectedfromawiderrangethantheB-splineinMallat’smethod.Non-subsampledmulti-resolutionanalysiswasusedtodeterminesingularitiesofasignal.Anoctavebandnon-subsampled lterbankwithanalysis ltersH0(z)andH1(z)isshowninFig.1.Inthismethod,awavelettransformreferstotheboundedlinearoperators

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