dilation matrices for nonseparablenew

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1、DilationMatricesforNonseparableBidimensionalWaveletsAnaRuedinDepartamentodeComputaci´on,FacultaddeCienciasExactasyNaturales,UniversidaddeBuenosAiresCiudadUniversitaria,Pab.I.CP1428,CiudaddeBuenosAiresanita@dc.uba.arAbstract.Fornonseparablebidimensionalwav

2、elettransforms,thecho-iceofthedilationmatrixisall–important,sinceitgovernsthedownsam-plingandupsamplingsteps,determinesthecosetsthatgivethepositionsofthefilters,anddefinestheelementarysetthatgivesatesselationoftheplane.Weintroducenonseparablebidimensionalwa

3、velets,andgiveformulaefortheanalysisandsynthesisofimages.Weanalyzeseveraldilationmatrices,andshowhowthewavelettransformoperatesvisually.Wealsoshowsomedistorsionsproducedbysomeofthesematrices.Weshowthattherequirementoftheireigenvaluesbeinggreaterthan1inabs

4、olutevalueisnotenoughtoguaranteetheirsuitabilityforimagepro-cessingapplications,anddiscussotherconditions.Keywords:quincunx,nonseparable,dilation,wavelet.1IntroductionInthelast20years,wavelettransformshaveadquiredgreatimportance.Theyhaveprovedefficientinman

5、ydomains,andthenumberoftheirapplicationsiscontinuallyincreasing.Asatoolinimageprocessing,wavelettransformsarebeingappliedtoimagecompression(theyarethebasisofthestandardJPEG2000[1]),imagedenoising,textureanalysisandpatternrecognition.Waveletsfunctionsareba

6、seswhichhaveexcellenttime(orspace)localization,andgoodfrequencylocalization.Inonedimensiontheyaredilations(changesinscaleinpowersof2)andintegertranslationsofonesamefunctionΨ(x)calledwavelet.Representingasignalinthesebases,weobtainadecompositionofthesignal

7、insumsofdetailsofdifferentresolutions,plusacoarseapproximationofthesignal.Thewavelettransformcanbecalculatedbymeansofthefastwavelettrans-form,throughconvolutionsofthesignalwithtwofilters,onelowpassfilterandonehighpassfilter,followedbyadecimationby2,i.e.onlyth

8、eevencoefficientsareretained.Becauseofthepropertiesofthewavelet,thisoperationisreversible.Toprocessanimage,thetraditionalmanneristoapplytwiceaone-dimensionalwavelettransform,firstbyrowsandthenbycolumns.Thisiscalledthes