WAVELETS FOR KIDS B.Vidakovic

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1、WAVEETSFR
DSATutorial
ntroductionByBraniVidakovicandeteruellerDukeUniversityStrictlyspeaking,waveletsaretopicofpuremathematics,howeverinonlyafewyearsofexistenceasatheoryoftheirown,theyhaveshowngreatpotentialandapplicabilityinmanyelds.Thereareseveralexcellentmonographsandarticlestalkingabout

2、wavelets,andthismodesttutorialdoesnotintendtocompetewithanyofthem.Ratheritisintendedtoserveasaveryrstreading,givingexamplesinterestingforthestatisticalcommunity.Wealsogivereferencesforfurtherreadingaswellassomemathematicado-it-yourselfprocedures.eywordsandphrases:Wavelets,ultiresolutionanalysi

3、s(mra),Haarwavelet,Thresholding.1991ASSubjectClassication:42A06,41A05,65D05.1WHATAREWAVEETS?21Whatarewavelets?Waveletsarefunctionsthatsatisfycertainrequirements.Theverynamewaveletcomesfromtherequirementthattheyshouldintegratetozero,waving"aboveandbelowthex-axis.Thediminutiveconnotationofwavel

4、etsuggestthefunctionhastobewelllocalized.therrequirementsaretechnicalandneededmostlytoinsurequickandeasycalculationofthedirectandinversewavelettransform.Therearemanykindsofwavelets.necanchoosebetweensmoothwavelets,com-pactlysupportedwavelets,waveletswithsimplemathematicalexpressions,waveletswit

5、hsimpleassociatedlters,etc.ThemostsimpleistheHaarwavelet,andwedis-cussitasanintroductoryexampleinthenextsection.Examplesofsomewavelets(fromthefamilyofDaubechieswavelets)aregiveninFigure1.ikesinesandcosinesinFourieranalysis,waveletsareusedasbasisfunctionsinrepresentingotherfunc-tions.ncethewave

6、let(sometimescalledthemotherwavelet)(x)isxed,onecanxb+formoftranslationsanddilationsofthemotherwaveletf();(a;b)2RRg.
tajisconvenienttotakespecialvaluesforaandbindeningthewaveletbasis:a=2jandb=k
2;wherekandjareintegers.Thischoiceofaandbiscalledcriticalsamplingandwillgiveasparsebasis.
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7、on,thischoicenaturallyconnectsmultiresolutionanalysisinsignalprocessingwiththeworldofwavelets.Waveletnovicesoftenask,whynotusethetraditionalFouriermethods?TherearesomeimportantdierencesbetweenFourieranalysisandwavelet