A novel 3D mesh compression using mesh segmentation with multiple principal plane analysis(3)

Meshsegmentationisthefirststeptowardsregion-based3Dmeshcompression[19].Segmentationismorecommonintheimageprocessingarea,andhasbeenrecentlyintroducedintothe3Dmesharea[20–25].Amongexistingalgorithms,thewatershed-basedap-proachhasreceivedmoreattention.Thisapproachusesdiscretecurvatureateachmeshvertexastheheightfieldthatdriveswa-tershedsegmentation.Thecurvatureestimationfor3Dmeshesiscomputationallyexpensive[20–22],asitismathematicallydefinedforasmoothsurfaceonly.Featuresensitivemeshsegmentationthatmaintainssalientfeaturesisimportantformanycomputergraphicsandgeometricmodelingapplications[23].Chengetal.[24]proposedapatch-growingapproachusingashortest-pathlabelingtechniquetosegmentthemodelintomultipleregions.Zhang[25]proposedasimplesegmentationalgorithmusingGaussiancurva-tureanalysis,efficientforcertain3Dmodels.However,similartootherexistingmethods,thisalgorithmisdefectivewhenprocessingahigh-resolution3Dmodel,asthegeometriccharacteristicsofad-jacentpolygonsinsuchamodelaretooclosetobedifferentiated.Therefore,anefficientandrobustalgorithmisneededfor3Dmeshsegmentation.

Thecurrentworkproposesanefficientsegmentation-based3Dmeshcompressionsystem,whichprovidesprogressivetransmissionfor3Dmodelsoverabandwidth-limitednetwork.Thisalgorithmfirstperformsameshsegmentationscheme,basedonfusionofthewell-knownk-meansclusteringandtheproposedmultipleprinci-palplaneanalysistoseparatetheinput3Dmeshintoasetofdis-jointedpolygonalregions.Theboundaryindexingschemeforthewholeobjectiscreatedbyassemblinglocalregions.Finally,apro-posedtriangletraversalschemeencodesconnectivityandgeome-tryinformationsimultaneouslyforeveryregionundertheguidanceofboundaryindexing.Atthedecoder,basedonboundaryindexing,thisworkobtainseachindividualregionandreconstructsthewholeobjectusingconnectivityandgeometrycodinginformation.Simula-tionresultsdemonstratethattheproposedalgorithmobtainsgoodcompressionperformance.

Theremainderofthispaperisorganizedasfollows.Section2presentsthemethodtoseparatetheinputtrianglemeshintomul-tipleuniformregionsusingak-meansclusteringwiththeprincipalplaneanalysistechnique.Thecurrentapproachto3DmeshmodelcompressionisdescribedindetailinSection3.Section4presentsexperimentalresultsobtainedfromapplyingtheproposedmethodtotest3Dmodels.Section5outlinesabriefconclusion.2.Clustering-based3Dmeshsegmentation

Thissectionpresentsaclustering-based3Dmeshsegmentationmethodbasedonk-meansclustering[25–27]withtheproposedprincipalplaneanalysis[28].Toclearlyintroducetheproposedseg-mentationscheme,thek-meansmethodandprincipalplaneanalysisarefirstdescribed.2.1.k-meansclustering

In1967,MacQueenproposedthek-means[26]algorithmasoneofthesimplestunregulatedlearningalgorithmstosolvetheclus-teringproblem.Theprocedurefollowsasimpleandeasymethodtoclassifyagivendatasetintokofgroups.Themainideaistodefinekcentroids,oneforeachgroup.Thesecentroidsshouldbeproperly

placedbecausedifferentlocationscausedifferentresults.Choosingthemasfarawayaspossiblefromeachotheristhebetterchoice.Thenextsteptakeseachpointbelongingtoagivendatasetandassociatesittothenearestcentroid.Whennopointispending,thefirststepisaccomplishedwithanearlygrouping.Atthisstage,knewcentroidsofgroupsresultingfromthepreviousstepneedre-calculating.Afterobtainingknewcentroids,anewbindingmusttakeplacebetweenapointinthesamedatasetandthenearestnewcentroid,generatingaloop.Thisloopmaycausekcentroidstochangetheirlocationstepbystepuntilnomorechangesareaccom-plished.Inotherwords,centroidsdonotchangeanymore.Finally,thisalgorithmaimsatminimizinganobjectivefunction,inthiscaseasquarederrorfunction.TheobjectivefunctionJ k=

n x(j)

i uj 2

(1)

j=1i=1

where x(j)

(j)

i uj 2isachosendistancemeasurebetweenadatapointxiandthegroupcentroiduj,isanindicatorofthedistanceofndatapointsfromtheirrespectivegroupcenters.

Thealgorithmiscomposedofthefollowingsteps:(1)Datapointsareassignedatrandomtothekgroups.Thecentroid

iscomputedforeachgroup.

(2)Eachpointisassignedtothegroupwiththeclosestcentroid.(3)Whenallpointshavebeenassigned,recalculatethepositionsof

thekcentroids.

(4)RepeatSteps(2)and(3)untilcentroidsstopchanging.Thispro-ducesaseparationofthepointsintokgroupsfromwhichthemetrictobeminimizedcanbecalculated.k-meansisasimplealgorithmthathasbeenadaptedtomanyproblemdomains.Asthefollowingshows,itisagoodcandidateforextendingworkwith3Dmeshsegmentation.2.2.Principalplaneanalysisfor3Dmodels

In[29],thisworkproposesa3Dmodelretrievalmethodusingprincipalplaneanalysisthatdefinestheprincipleplaneasaskele-tonrepresentationcorrespondingtothesymmetricsurfacefora3Dobject.

Theprincipalplanecanbeconvenientlyrepresentedintermsofmoments.Inthecaseof3DfeaturespaceS3,theprincipalplaneHcanberepresentedasAx+By+Cz=D

(2)

whereA,B,andCarethedirectionalnumbersofHthatsatisfythefollowingrelationship:A2+B2+C2=1.

(3)

Thedistancefroma3Dvector c=(x,y,z)toHisgivenbyf(x,y,z)=Ax+By+Cz D.

(4)

Letthecentroid¯c

ofthe3Dspacebedefinedas N

(¯x,y¯,z¯)= 1 1 N1

NNxi,yi,zi

(5)

i=1

Ni=1

N

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