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oblem,althoughvarioussolutionsareproposed,thelargeputationisnotsolved.Therecentproposedaffinitypropagationclustering(AP)algorithm[8-11]andK-meansalgorithmbothbelongtotheKcentersclusteringmethod.However,itoverestheshortingsofK-meansanditdoesnotneedtoselecttheinitialclustercenters.APcontinuallysearchesfortherightclustercenterduringtheiterativeprocess,andfinallymakesthefitnessfunction(objectivefunction)ofclusteringmaximizes.Itavoidstheproblemsofselectinginitialvaluesandithasfastrunspeedonlarge-scaledata.Therefore,itisverysuitableforhighdimensionalandlarge-scaleclassificationissue.
Inthispaper,APisadoptedtoobtaintherepresentativemodelsfromeachmodelbase,andthenthequerymodelisdeterminedwhichthemostpossiblemodelbaseitbelongstobyparingwiththerepresentativemodels.Then,theabovesimilarityEqs.(1)-(4)areusedforthemostsimilarmodelretrievalfromthemodelbase.Itlimitstheretrievalscopeandimprovesretrievalspeedandaccuracy.
PrincipleandStepsofRay-BasedMethod(一级标题实词的首字母大写,四号粗体)
Thebasicideaoftheray-basedmethodisthat:thesamplingraysareemittedtosomedirectionsfromthecenterof3Dmodel.Iftheraysintersectthetriangularfacetswhichposethemodelsurface,themaximumdistancefromtheintersectionstothecenterisusedasafeatureof3Dmodel(asshowninFig.1.)
Fig.1Principleofray-basedmethod
(图和表格标题第一个单词首字母大写,小五粗体)
Thefeatureextractionprocessincludesthefollowingsteps:
1)Thepretreatmentofthe3Dmodel:themodel'scenterismovedtotheorigin,andthenthemodelisscaledtotheunitsize,andallthemodelsareputinthesamedirection,
2)Supposingthemodelissurroundedbytheunitballwhosecenteristheorigin,theraysareemittedarounduniformly,andthenthecoordinatesoftheintersectionpointarecalculated.
Aseriesofsamplingraysthroughtheorigininsphericalcoordinatescanbeexpressedas:
whereisthedirectionofray,isthelengthofray,andareshowninFig.2.
Fig.2Diagramofsphericalcoordinates
Thecoordinatesofapointonatriangularfacetcanbeexpressedas:
where,andaretheverticesofthetriangular.
If,theparametersu,vandtcanbecalculatedby:
If,thecorrespondingparametersandtaresaved.
3)Takingthemaximumdistancefromtheintersectionstotheoriginasafeatureofthemodel,andthenextractingtheallfeaturevectorsfrom3Dmodel.
ProjectRay-BasedMethod
3.1ProjectRelationshipAnalysisofBall-SectionandTriangularFacets(二级标题实词首字母大写,四号粗体,若还有三级标题,第一个单词首字母大写,其他小写,五号斜体)
Therayswithafixedandvariousposeaballcross-section,andthentheballcross-sectionisprojectedtothesectionthroughtheoriginandperpendiculartotheballcross-section.Theprojectionisalinewhoseequationis:
Thelocationsoftheballcross-sectionandtriangularfacetsareshowninFig.3.
Fig.3Relationsofcross-sectionandtriangularfacets
Becausethemodelislimitedintheunitball,theintersectionoftheballcross-sectionandthetriangularfacetscanbejudgedaccordingtowhethertheprojectlineofballcross-sectionintersectstheprojectionsofthetriangularfacets.TherelationshipisshowninFig.4.
Fig.4Projectionrelationshipofballcross-sectionandtriangularfacets
Figs.3and4describethelocationrelationshipbetweentheballcross-sectionandthespatialtrianglefacetsandthecorrespondingprojectionrelationship.Accordingtotheaboveanalysis,thelocationrelationshipbetweenthelineandthetriangleinthesameplanecanbeusedtodeterminewhetherthetriangleintersectstheballsection.Thentwojudgmentmethodsare:
1)Themethodbasedondistance.Ifatleastonevertexoftriangleisonthelineorontheothersideoftheline,thetriangleintersectstheline.ThedistanceequationfromapointP(x,y)toalinethroughtheoriginis,wherethesignofax+bycanbeusedtodeterminethelocationrelationshipbetweenthepointsandlines.Thenbyusingtherelationshipsbetweenallthepointsoftriangleandtheline,whethertheballcross-sectionandintersectthetriangularfacetcanbedetermined.Forexample,P1andP2areendpointsofasideL1oftriangle,D1andD2arethedistancesfromP1andP2toL2.If,L1intersectsL2,otherwise,theydonotintersect.ThelocationrelationshipisshowninFig.5.
Fig.5Locationrelationbetweentwolines
2)Methodbasedonintersectionangle.Astoatriangle,ifoneofthefollowingEq.(5)holds,itmeansthatthelineisthroughanysideofthetriangle,thenthelineandtriangleintersect.TheprocessisshowninFig.6.
(5)
whereistheanglebetweenthevectorsOAandOC,whichcanbegotbythecosineformula.
Fig.6Diagramofanglerelationship
3.2StepsofProjectRay-BasedAlgorithm
Animportantwaytoimprovetheefficiencyofray-basedmethodistoreducethenumberofunnecessaryintersection.Theideaisthat:1)gettingNplaneswithafixed,differentintheball,2)usingthemethodinSection3.1torecordthetriangularfacetsintersectingtheplanes.Becauseaplane(aballcross-section)includesaseriesofrays,iftriangularfacetsdonotintersecttheballcross-section,itdefinitelydoesnotintersecttheraysintheballcross-section.3)removingtherayswhichdonotintersectthetriangularfacets.Thestepsofprojectray-basedmethodareasfollows:
Thegraticulevariablesla
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