45 NO 5 JULY 1999 Design of Successively Re64257nable TrellisCoded Quantizers Hamid Jafarkhani Member IEEE and Vahid Tarokh Member IEEE Abstract We propose successively re64257nable trelliscoded quan tizers for progressive transmission Progress ID: 33147
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IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999DesignofSuccessivelyRenableTrellis-CodedQuantizersHamidJafarkhani,Member,IEEE,andVahidTarokh,Member,IEEEAbstractĂWeproposesuccessivelyrenabletrellis-codedquan-tizersforprogressivetransmission.(Progressivetransmissionisanessentialcomponentofimageandmultimediabrowsingsystems.)Anewtrellisstructurewhichisscalableisusedinthedesignofourtrellis-codedquantizers.Ahierarchicalsetpartitioningisdevelopedtopreservesuccessiverenability.Two IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999 Fig.3.Ahierarchicalsetpartitioningforatwo-levelSR-TCQ(Foratrainingsequencewith samples, wecanexpressthesample-averagedistortionas Foragivensetofcodebooks,usingtheViterbialgorithmwiththedistortionmeasurein(3)providesthebestoutputsatdifferentlevelsofrenement.Letusdene asthesetofalltrainingsampleswhichareencodedas .Thenifwereplaceeachcodevector withanewcodevector denedby theresultingsetofcodebooksprovidesalowerdistortionwhenthesamepathandcodewordswhichhavebeenusedfortheoldcodevectorsareutilized.Notethat doesnotnecessarilymeanthat istheclosestcodevectorto .Also, 'sarenotdisjoint.ThefollowingalgorithmcanbeusedtodesignSR-TCQ's:DesignAlgorithmforSR-TCQ's:‰0‹Initialization:(a)Pickasmallpositivenumber (b)Pickaninitialsetofcodebooks..(c)Set and ‰1‹EncodethetrainingsequenceusingtheViterbialgo-rithmandthedistortionmeasurein(3).Denotetheresultingdistortionas ‰2‹Updatethecodebooksbyusing(4)tondthebestsetofcodevectors.‰3‹If ,set andgoto‰1‹.Otherwise,stop.Sincethedistortionisreducedateachstepofthealgorithmandthedistortionislower-boundedbyzero,convergenceisC.OverlappingIntervalsInthissubsection,weexplainthereasonforoverlappingintervalsinourhierarchicalsetpartitioning.Asanexample,Fig.3demonstratesasetofcodebooksforTS-TCQandSR-TCQ( bit/sample).AsitisseenfromFig.3,theresultingcodevectorsdonotconstructatree-structuredscalarquantizerwithnonoverlappingintervals.Forexample, branchesintervene and branches.Thisisduetothefactthattheoutputcodevectoroftherststageisselected or .So,forexample,asampleclosetoacodevectorin maybeencodedtoacodevectorin viceversa.ThecrossinginFig.3allowstheencodertoxsuchshortcomingsatthesecondlevelofrenement.AnotherexampleisgiveninFig.4toshowthehierarchicalsetpartitioningwhentheincrementalratesaremorethanone.AscanbeseeninFig.4,thenumberofintervalscorrespondingtoeachpathinthetrellisismorethanonewhichresultsinparalleltransitionsfortrellispaths.V.COMPUTATIONALInthissection,wecomparethecomputationalcomplexityofTCQ,MS-TCQ,TS-TCQ,andSR-TCQforafour-statetrelliswitheachother.Thecomputationalcomplexityofthedecodersaremoreorlessthesameandmuchlowerthanthecomplexityofencoders.So,weonlyconsidertheencodersinouranalysis.Letusassumethatweonlyhavetwolevelsofrenementandeachcodebookispartitionedintofoursets.So,foradoubledcodebookTCQwithrate ,eachpartition codevectorscorrespondingto branches.Sinceparallelbranchescomefromthesamenode IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999TABLEIIISNRRESULTSOFEMORYLESS TABLEIVSNRRESULTSOFEMORYLESS the stages).ForanSR-TCQ,theencodingdecisionsaremadesimultaneouslyforbothstages.So,theunderlyingtrelliscontains16states.Weneed16multiplications,80additions, comparisonsperinputsamples.OneoftheadvantagesofTCQoverotherquantizationschemesisthefactthatitscomputationalcomplexityisroughlyindependentoftherate[12].Mostotherquantizationschemes(excludingscalarquantizers)sufferfromanexponentialgrowthincomputationalburdenwiththerate.Therate-scalableTCQ'sproposedinthisworkpreservethecomplexityadvantageofaTCQ,i.e.,theircomputationalcomplexityisroughlyindependentoftheencodingrates.VI.SIMULATIONESULTSANDInthissection,wepresentsimulationresultsandcom-pareourresultswiththoseofTCQ,MS-TCQ,Lloyd±Maxquantizer,andthedistortion-ratefunctionforGaussianandLaplacianmemorylesssources.WepresentresultsforTS-TCQ'sandSR-TCQ'swithtwolevelsofrenement(SR-TCQisdesignedfortwoequiprobablestages).Theproposedalgo-rithmsarenotrestrictedtotwostages;however,havingtwostagessimpliesthepresentationandallowsacleardiscussiononresults.TablesIandIIshowtheR-Dperformanceofdiffer-entquantizers(four-statetrellis)forzero-mean,unit-variancememorylessGaussianandLaplaciansources,respectively.TheacronymN/Astandsfornotavailablethroughoutthetables.TheTS-TCQresultsarebetterthanthoseofMS-TCQ.Thesignal-to-noiseratio(SNR)differenceismorethan2dBinonecase.ThecomputationalcomplexityisthesamealthoughthememoryrequirementsofTS-TCQismorethanthatofMS-TCQ.TheperformanceofthesecondstageofSR-TCQisafewtenthsofadecibelbetterthanthatofTS-TCQwhiletherststageofSR-TCQperformsalmostaswellastherststageofTS-TCQ.Oneinterestingobservationisthefactthatfora ,byincreasingthebitrateoftherststage ,theperformanceofthesecondstageofTS-TCQandSR-TCQisimproved.ThisisnotatrendforMS-TCQ.Also,notethatthereisalwaysacombinationofratesforwhichSR-TCQoutperformsTCQatrate (althoughthecomparisonisnotcompletelyfairsinceSR-TCQismorecomplexthanTCQ).Wealsoprovidethesimulationresultsforatwo-stageSR-TCQwhichuses insteadof asthedistortionmeasureinTablesIIIandIV.Theperformanceoftheresultingquantizer,denotedSR-TCQ2,isidenticaltothatofaTCQusingthetrellisofFig.2althoughtheencodinganddecodingprocessesaredifferentandSR-TCQ2providesanembeddedbitstream.ThemotivationbehindSR-TCQ2isthefactthatsomeoftheresultsinTablesIandIIaresogoodthatmakesthestudyofthebestpossiblelaststageperformanceofanSR-TCQanditscomparisonwithanonscalableTCQinteresting.Tohaveamoreconclusivecomparison,wealsoprovidetheperformanceofTCQ'swithfourand16states.TablesIIIandIVshowthatnotonlydoesSR-TCQ2providesomedegreesofrate-scalability,butalsoitsperformanceisbetterthanthatofa16-statedoubledcodebookTCQatallreportedrates.NotethatthecomputationalcomplexityofSR-TCQ2islessthanthatofSR-TCQbecausethereisnoneedtocalculatethedistortionsoftheintermediatestagesandmultiplythembydifferentweights.FortheLaplaciansource,theresultsofSR-TCQ2aremuchbetterthanthoseofadoubledcodebookTCQ.ThisisduetothefactthatdoublingthecodebookisnotenoughforaLaplaciansource[12].Quadrupledcodebooksprovideabetterperformancewhileincreasingthecomputationalcomplexityalmostbyafactoroftwo[12].InTableIV,wealsotabulatetheperformanceofthebestquadrupledcodebookresultsfrom[12].AsimilarquadrupledcodebookcanbeusedinthedesignofSR-TCQ's.Theproposedalgorithmsmaybeextendedtotrellis-codedvectorquantizerspresentedin[18].Anotherfutureworkin-cludesthedesignofentropyconstrainedsuccessivelyrenabletrellis-codedquantizers.Suchanextensionisachievablebycombiningtheschemesproposedin[10]andthetrellisintro-