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1、CAPlets:waveletrepresentationswithoutwaveletsYoungmiHurAmosRonDepartmentofMathematicsComputerSciencesDepartmentUniversityofWisconsin-MadisonUniversityofWisconsin-Madison480LincolnDr.1210WestDaytonMadison,WI53706Madison,WI53706hur@math.wisc.eduamos@cs.wisc.edu
2、ABSTRACTMultiResolution(MR)isamongthemosteffectiveandthemostpopularapproachesfordatarepresentation.Inthatapproach,thegivendataareorganizedintoasequenceofresolutionlayers,andthenthe“difference”betweeneachtwoconsecutivelayersisrecordedintermsofdetailcoefficients.Wa
3、veletdecompositionisthebestknownrepresentationmethodologyintheMRcategory.Themajorreasonforthepopularityofwaveletdecompositionsistheirimplementationandinversionbyafastalgorithm,theso-calledfastwavelettransform(FWT).Anothercentralreasonforthesuccessofwaveletsis
4、thatthewaveletcoefficientscaptureveryaccuratelythesmoothnessclassofthefunctionhiddenbehindthedata.Thisisessentialfortheunderstandingoftheperformanceofkeywavelet-basedalgorithmsincompression,indenoising,andinotherapplications.Onthedownside,constructingwaveletswi
5、thgoodspace-frequencylocalizationpropertiesbecomesinvolvedasthespatialdimensiongrows.Analternativetothesometime-hard-to-constructwaveletrepresentationsisthealways-easy-to-construct(andslightlyolder)non-orthogonalpyramidalalgorithms.Similartowavelets,the(linea
6、r,regular,isotropic)pyramidalrepresentationsarebasedonsomemethodforlinearcoarsening(byadecompositionfilter)oftheirdata,andacomplementarymethodforlinearprediction(byapredictionfilter)oftheoriginaldatafromthecoarsenedone.Thefirststepcreatestheresolutionlayersandth
7、esecondallowsfortrivialextractionsofsuitabledetailcoefficients.Thedecompositionandreconstructionalgorithmsinthepyramidalapproachareasfastasthoseofwavelets.Incontrastwithorthonormalwavelets,therepresentationisredundant,viz.thetotalnumberofdetailcoefficientsexceeds
8、theoriginalsizeofthedata:denotingbystheratiobetweenthesizeofthedataattwoconsecutiveresolutionlayers,sthe“redundancyratio”inthepyramidalrepresentationis.s−1Inthispaper,weintroduceandstudya
