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1、832IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.49,NO.6,JUNE2002MultidimensionalCausal,Stable,PerfectReconstructionFilterBanksSankarBasu,Fellow,IEEEAbstract—Followingourearlierone-dmensional(1-D)work,erature(seee.g.,[9],[8]),thetechniqueofchoicehasbeentheweshowthat
2、itispossibletohavealargefamilyofbiorthogonaluseofseparablesubbandcodingschemeswith1-DunstableIIRperfectreconstructionmultidimensional(-D)subbandcodingfilters.Suchschemesmaybeimplementedbyfirstperformingfilterbanks,whicharecausalandIIRbothattheanalysisandatastable-antistablefactorizationandthenby
3、implementingthethesynthesisends.WhiletheneedforperfectreconstructionIIRfilterbanksaremoreapparentformultidimensionalsignalsthanstablepartinacausalandtheanti-stablepartinananti-causalfor1-Dsignals,lackofdesigntechniqueshavemadeitimpossiblefashion.Fornonseparablemultidimensional(D)schemes,tousesuc
4、hschemes.Wearguethatsuchfilterbanks,includinghowever,thestable-antistablefactorizationcannotbeperformed,theirmultichannelcounterparts,shouldnotonlyexistinalargeandasaresult,theabovestrategydoesnotapply.Whilethisvariety,butwealsoindicateanumberoftoolspotentiallyusefulisfeasibleforcertainpurelyspa
5、tialsignalprocessingtasksintheirdesign.Specifically,acompleteparameterizationofsuchfiltersleadingtodesignmethodsin2-Daregiven.Parameteriza-e.g.,processingofstillpictures,wherenoneofthevariablesistionsinfrequencydomaintermsaswellasintermsofstatespacetemporalandrealtimeprocessingisnotwarranted,the
6、reexistdescriptionsoffiltersareprovided.Theselattertechniqueshavesituationswheresuchdoubleendedprocessingisnotpermitted.potentialforbetternumericalimplementation.ThedevelopmentAnexampleofthisistherealtimethree-dimensional(3-D)pro-isvalidfortwo-bandaswellasformultibandsubbandcodingcessingofvideoi
7、mageswhereoneofthevariablesistemporal,schemes.Examplesofbi-orthogonalversionofcontinuousmul-tidimensionalwaveletsgeneratedbytheseiteratedfilterbanks,whereastheothertwovariablesarespatial.Thus,itseemsthatanthathavenotbeenprev
