FUZZY VQ ALGORITHMS FOR COLOR QUANTIZATION

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FUZZYVQALGORITHMSFORCOLORQUANTIZATIONDo˘gan¨OzdemirTechnicalSciencesDepartmentNavalAcademy81704Tuzla˙Istanbul,Turkeye-mail:otdemir@dho.edu.trTel:+902163952630x3561

LaleAkarunBo˘gazic¸iUniversityComputerEngineeringDepartment80815Bebek˙Istanbul,Turkeye-mail:akarun@boun.edu.trTel:+902122631500x1858

ABSTRACTTwonewextensionsofFuzzyC-means(FCM)algorithmwhichminimizeanobjectivefunctionincorporatingavalidityindexareproposed.Thesealgorithmsareappliedtocolorquantizationofimages.Inthefirstapproach,weminimizeanobjectivefunctionincludingatermforpartitionindex.Thisalgorithmattemptstoplacetheclustercenterssuchthatthemembershipvaluesofthepixelsaremaximized.Inthesecondapproach,weminimizeanobjectivefunctionincludinganinter-clusterseparationterm.Thegoalhereistomoveclustercentersapartfromeachothertowardstheconvexhullofthecolorspace,henceobtainingacolorpalettewhichismoresuitablefordithering,anoperationgenerallyap-pliedafterthequantizationoftheimages.

1.INTRODUCTIONManyimagedisplayandprintingdevicesallowonlyalimitednumberofcolorstobeused.Thesecolorsconstituteapalette,whichtypicallycontains256orfewerentries.Originalimagesrep-resenteachcolorcomponentwithonebyte,thereforetheycancon-tainupto16milliondifferentcolors,whichmustthenbemappedtotheavailablecolorsinthepalette.Thisprocessofselectingasuitablepaletteandmappingeachpixelintheoriginalimagetoanentryinthepaletteiscalledquantization.TheC-meansvectorquantizationalgorithmhasbeenlongap-pliedtoimagepalettedesign,wherepaletteisgenerallyreferencedascodebook[1].TheC-meansalgorithmpartitionsacollectionof3x1vectors,j=1,...,,intoclusterswhereisthecodebook

size.Thealgorithmfindsaclustercenterineachgroupsuchthatanobjectivefunctionisminimized.Euclideandistanceiscom-

monlychosenasthedissimilaritymeasure.Thisschemaisalsocalledhardquantizationsinceeachpixelisrepresentedbyonlyonecodebookentry.Fuzzyquantizationisageneralizationofhardquantizationschemaandthebestknownandmostwidelyusedfuzzyquantiza-tiontechniqueistheFuzzyC-means(FCM)algorithmdevelopedbyDunn[2]andrefinedbyBezdek[3].IntheFCMalgorithm,eachdatapointbelongstoaclusterwithadegreespecifiedbyamembershipgradebetween0and1.Thus,theFCMalgorithmpartitionsvectorsintofuzzygroups.Summationofmember-shipvaluesisequaltounity:

(1)

Theobjectivefunctionisdefinedas:(2)whereistheparameteroffuzzinessanddenotethesetof

quantizationcolors.Althoughnoformalmethodexiststodefinetheoptimalvalueof,intheliteratureitisgenerallychosen

around2.BothhardquantizationandFCMalgorithmuseaniter-ativeprocedureandproducelocallyoptimalcodebooksdependingoninitialcodevectorlocations.

Clustervaliditymeasureshavebeenusedtoevaluatethequal-ityoftheclustersinquantitativeandobjectivefashion.Thequalityofaclusteringisindicatedbyavalidityfunctionwhichassignsanumbertotheoutputofaclassifiertoassociatethedatapointstoclustercenters.Examplesofwellknownvaliditymeasuresarethepartitioncoefficientandclassificationentropy[3],proportionexponent[4],Dunn’sindex[2]andDavies-Bouldinindex[5].Validityindexesaregenerallyusedtodeterminethebestchoiceoftoidentifythestructureinthedata.In[6],itisalsousedforthere-clusteringofafixednumber()ofclustersthroughasplitand

mergeapproachtoobtainabettercodebook.

Inthispaper,weproposetwonewextensionsoftheFCMal-gorithmwhichminimizeanobjectivefunctionincorporatingava-lidityindex.Inthefirstapproach,weminimizeanobjectivefunc-tionincludingatermforpartitionindex.Thisalgorithmattemptstoplacetheclustercenterssuchthatthemembershipvaluesofthepixelsaremaximized.Althoughtheintendedapplicationdo-mainisimagequantization,itisapplicabletoanyclassificationschema.Inthesecondapproach,weminimizeanobjectivefunc-tionincludinganinter-clusterseparationterm.Thegoalhereistomoveclustercentersapartfromeachothertowardsthecon-vexhullofthecolorspace,henceobtainingacolorpalettewhichismoresuitablefordithering,anoperationgenerallyappliedaf-terthequantizationoftheimages[7].Althoughthisalgorithmisspecificallydesignedforcolorimagequantization,itmayalsobeappliedtogeneralclassificationproblems.InSectionII,weintroduceFuzzyQuantizationwithPartitionIndexMaximization(PIM).InSectionIII,FuzzyQuantizationwithInter-clusterSepa-ration(ICS)isintroduced.TheresultsofbothapproachesareinSectionIV.InSectionV,thecurrentstudyisevaluatedandfutureareasofresearcharepointedout.