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1、TheHaar–WaveletTransforminDigitalImageProcessing:ItsStatusandAchievementsPiotrPorwik,AgnieszkaLisowskaInstituteofInformatics,UniversityofSilesia,ul.B¸edzi´nska39,41-200Sosnowiec,Polande-mail:porwik@us.edu.plInstituteofMathematics,UniversityofSilesia,ul.Ba
2、nkowa14,40-007Katowice,Polande-mail:alisow@ux2.math.us.edu.plAbstract.Imageprocessingandanalysisbasedonthecontinuousordiscreteimagetransformsareclassictechniques.Theimagetransformsarewidelyusedinimagefiltering,datadescription,etc.Nowadaysthewavelettheorems
3、makeupverypopularmethodsofimageprocessing,denoisingandcompression.ConsideringthattheHaarfunctionsarethesimplestwavelets,theseformsareusedinmanymethodsofdiscreteimagetransformsandprocessing.Theimagetransformtheoryisawellknownareacharacterizedbyaprecisemath
4、ematicalbackground,butinmanycasessometransformshaveparticularpropertieswhicharenotstillinvestigated.ThispaperforthefirsttimepresentsgraphicdependencesbetweenpartsofHaarandwaveletsspectra.Italsopresentsamethodofimageanalysisbymeansofthewavelets–Haarspectrum
5、.SomepropertiesoftheHaarandwaveletsspectrumwereinvestigated.Theextractionofimagefeaturesimmediatelyfromspectralcoefficientsdistributionwereshown.Inthispaperitispresentedthattwo–dimensionalboth,theHaarandwaveletsfunctionsproductsmanbetreatedasextractorsofpar
6、ticularimagefeatures.Furthermore,itisalsoshownthatsomecoefficientsfrombothspectraareproportional,whichsimplifyslightlycomputationsandanalyses.Keywords:Wavelets,HaarTransform,multiresolution.1.IntroductionThecomputerandvideo–mediaapplicationshavedevelopedrap
7、idlythefieldofmulti-media,whichrequiresthehighperformance,speedydigitalvideoandaudiocapabilities.Nowadays,theimageprocessingandanalysisbasedoncontinuousordiscretetrans-formsaretheclassicprocessingtechniques[3,27,30,36,50].Digitalsignalprocessingiswidelyuse
8、dinmanyareasofelectronics,communicationandinformationtechniques[1,4,6,9,14,15,17,18,20,23,28,35].Inthesignalscompression,digitalfiltration,systemsidentification,thecommonlyusedtransformsarebasedonsinusoidalbasicfunc-t
