Matching 3D Models with Shape Distributions

RobertOsada,ThomasFunkhouser,BernardChazelle,andDavid Dobkin

PrincetonUniversity

Abstract

Measuringthesimilarity between3D shapesis afundamentalprob-lem,with applicationsin computervision,molecularbiology, com-putergraphics,anda variety of otherfields. A challengingaspectof thisproblemis to find asuitableshapesignaturethatcanbecon-structedandcomparedquickly, while still discriminatingbetweensimilaranddissimilarshapes.

In this paper, we proposeandanalyzea methodfor computingshapesignaturesfor arbitrary(possiblydegenerate)3D polygonalmodels.Thekey ideais to representthesignatureof anobjectasashapedistributionsampledfrom ashapefunctionmeasuringglobalgeometricpropertiesof anobject. Theprimarymotivation for thisapproachis to reducethe shapematchingproblemto the compar-ison of probability distributions,which is a simplerproblemthanthecomparisonof 3D surfacesby traditionalshapematchingmeth-odsthatrequireposeregistration,featurecorrespondence,or modelfitting.

We find that the dissimilaritiesbetweensampleddistributionsof simpleshapefunctions(e.g.,the distancebetweentwo randompointson asurface)providea robustmethodfor discriminatingbe-tweenclassesof objects(e.g.,carsversusairplanes)in amoderatelysizeddatabase,despitethepresenceof arbitrarytranslations,rota-tions,scales,mirrors, tessellations,simplifications,andmodelde-generacies.They canbeevaluatedquickly, andthustheproposedmethodcouldbeappliedasapre-classifierin anobjectrecognitionsystemor in aninteractivecontent-basedretrieval application.

1 Introduction

Determiningthesimilaritybetween3Dshapesisafundamentaltaskin shape-basedrecognition,retrieval, clustering,andclassification.Its main applicationshave traditionally beenin computervision,mechanicalengineering,andmolecularbiology. However, duetothree recentdevelopments,we believe that 3D model databaseswill becomeubiquitous,and the applicationsof 3D shapeanaly-sis and matchingwill expandinto a wide variety of otherfields.First, improved modelingtools andscanningdevices are makingacquisitionof 3D modelseasierandlessexpensive,creatinga largesupplyof publically available3D datasets(e.g., the ProteinDataBank [30]). Second,the World Wide Web is enablingaccessto3D modelsconstructedby peopleall over the world, providing amechanismfor wide-spreaddistributionof highquality3D models(e.g.,avalon.viewpoint.com). Finally, 3D graphicshardware andCPUshavebecomefastenoughandcheapenoughthat3D datacanbeprocessedanddisplayedquickly on desktopcomputers,leadingto ahighdemandfor 3D modelsfrom awiderangeof sources.

Unfortunately, sincemost3D file formats(VRML, 3D Studio,etc.) have beendesignedfor visualization,they containonly ge-ometric and appearanceattributes,and usually lack semanticin-formationthatwould facilitateautomaticmatching.Althoughit ispossibleto includemeaningfulstructureandsemantictagsin some3D file formats(the “layer” field associatedwith entitiesin Auto-

Cadmodelsis a simpleexample),thevastmajority of 3D objectsavailablevia the World Wide Web will not have them,and therearefew standardsregardingtheir use. In general,3D modelswillbe acquiredwith scanningdevices,or outputfrom geometricma-nipulation tools (file format conversionprograms),and thus theywill haveonly geometricandappearanceinformation,usuallycom-pletelyvoid of structureor semanticinformation.Automaticshape-basedmatchingalgorithmswill beusefulfor recognition,retrieval,clustering,andclassificationof 3D modelsin suchdatabases.

Databasesof 3Dmodelshaveseveralnew andinterestingcharac-teristicsthatsignificantlyaffect shape-basedmatchingalgorithms.Unlike imagesandrangescans,3D modelsdo not dependon theconfigurationof cameras,light sources,or surroundingobjects(e.g.,mirrors).As aresult,they donotcontainreflections,shadows,occlusions,projections,or partialobjects,which greatlysimplifiesfinding matchesbetweenobjectsof thesametype. For example,itis plausibleto expectthatthe3D modelof ahorsecontainsexactlyfour legs of roughly equalsize. In contrast,any 2D imageof thesamehorsemay containfewer thanfour legs (if someof the legsareoccludedby tall grass),or it maycontain“extra legs” appearingastheresultof shadows on thebarnand/orreflectionsin a nearbypond,or someof the legs may appearsmallerthanothersduetoprojective distortions. Theseproblemsare vexing for traditionalcomputervision applications,but generallyabsentfrom 3D modelmatching.

In otherrespects,representingandprocessing3Dmodelsismorecomplicatedthanfor sampledmultimediadata.Themaindifficultyis that 3D surfacesrarely have simple parameterizations.Since3D surfacescan have arbitrary topologies,many useful methodsfor analyzingother media (e.g., Fourier analysis)have no obvi-ous analogsfor 3D surfacemodels. Moreover, the dimensional-ity is higher, which makes searchesfor poseregistration,featurecorrespondences,and modelparametersmoredifficult, while thelikelihoodof modeldegeneraciesis higher. In particular, most3Dmodelsin largedatabases,suchastheWorld WideWeb,arerepre-sentedonly by “polygonsoups”– unorganizedanddegeneratesetsof polygons. They seldomhave any topology or solid modelinginformation;they rarelyaremanifold; andmostarenot even self-consistent.We conjecturethatalmostevery3D computergraphicsmodelavailable today containsmissing, wrongly-oriented,inter-secting, disjoint,and/oroverlappingpolygons. As a few examples,the classicUtah teapotis missingits bottom, and the ubiquitousStanfordBunny [38] hasseveralholesalongits base.Theproblemwith thesedegeneraterepresentationsis thatmostinterestinggeo-metric featuresandshapesignaturesaredifficult to compute,andmany othersare ill-defined (e.g., what is the volume of a teapotwith no bottom?).Meanwhile,fixing thedegeneraciesin such3Dmodelsto form a consistentsolid region andmanifoldsurfaceis adifficult problem[11, 28, 42], often requiringhumaninterventionto resolveambiguities.

In this paper, we describeandanalyzea methodfor computing3Dshapesignaturesanddissimilaritymeasuresfor arbitraryobjectsdescribedby possiblydegenerate3D polygonalmodels. The keyideais to representthesignatureof anobjectasashapedistribution

sampledfrom ashapefunctionmeasuringglobalgeometricproper-tiesof theobject.Theprimarymotivationfor this approachis thatthe shapematchingproblemis reducedto the comparisonof twoprobabilitydistributions,whichis arelatively simpleproblemwhencomparedto themoredifficult problemsencounteredby traditionalshapematchingmethods,suchasposeregistration,parameteriza-tion, featurecorrespondence,andmodelfitting. Thechallengesofthis approachare to selectdiscriminatingshapefunctions,to de-velopefficientmethodsfor samplingthem,andto computethedis-similarity of probabilitydistributionsrobustly. This paperpresentsour initial stepsto addresstheseissues.For eachissue,wedescribeseveraloptionsandpresentexperimentalevaluationof their relativeperformance.Overall,wefind thattheproposedmethodis notonlyfastandsimpleto implement,but it alsoprovidesusefuldiscrim-ination of 3D shapesand thus is suitableas a pre-classifierfor arecognitionor similarity retrieval system.

The remainderof the paperis organizedas follows. The nextsectioncontainsa summaryof relatedwork. An overview of theproposedapproachappearsin Section3,whiledetaileddescriptionsof issuesandproposedsolutionsfor implementingourapproachap-pearin Section4. Section5 presentsresultsof experimentsaimedat evaluatingthe robustnessanddiscriminationof shapedistribu-tions. Finally, Section6 containsa summaryof our experiencesandproposestopicsfor futurework.

2 Related Work

Theproblemof determiningthesimilarity of two shapeshasbeenwell-studiedin several fields. For a broadintroductionto shapematchingmethods,pleaserefer to any of severalsurvey papers[2,7, 13, 41, 44, 56]. To briefly review, prior matchingmethodscanbeclassifiedaccordingto their representationsof shape:2D contours,3D surfaces,3D volumes,structuralmodels,or statistics.

The vast majority of work in shapematchinghas focusedoncharacterizingsimilarity betweenobjectsin 2D images(e.g.,[18,26, 34, 43]). Unfortunately, most 2D methodsdo not extenddi-rectly to 3D modelmatching. The main problemis boundarypa-rameterization.Althoughthe1D boundarycontoursof 2D shapeshaveanaturalarclengthparameterization,3D surfacesof arbitrarygenusdo not. As a result,commonrepresentationsof 2D contoursfor shapematching,suchasFourierdescriptors[5], turning func-tions[6], circularautoregressivemodels[36], bendingenergy func-tions[59], archheightfunctions[40], andsizefunctions[55], havenoanalogsfor 3D models.

Shapematchinghasalsobeenwell-studiedfor 3D objects.Forinstance,representationsfor registeringandmatching3D surfacesinclude ExtendedGaussianImages[31], SphericalAttribute Im-ages[20, 21], HarmonicShapeImages[60], andSpinImages[35].Unfortunately, thesepreviousmethodsusuallyassumethata topo-logically valid surfacemeshor an explicit volumeis availableforevery object. In addition,volumetricdissimilaritymeasuresbasedwavelets[27] or EarthMover’s Distance[48] usuallyrely uponapriori registrationof objects’coordinatesystems,which is difficultto achieve automaticallyandrobustly. Geometrichashing[39] isa potentialsolution,but it requiresa large amountof storageforcomplex models.

Another popular approachto shapeanalysisand matchingisbasedon comparinghigh-level representationsof shape. For in-stance,model-basedapproachesfirst decomposea 3D object intoa setof features(or parts),andthencomputea dissimilarity mea-surebetweenobjectsbasedon the differencesbetweentheir fea-turesand/ortheir spatialrelationships.Examplerepresentationsofthis type include generalizedcylinders [16], superquadrics[52],geons[58], deformableregions [12], shockgraphs[50], medial

axes[10], andskeletons[17, 23]. Thesemethodswork bestwhen3D modelscanbesegmentedinto a canonicalsetof featuresnatu-rally andcorrespondencescanbefoundbetweenfeaturesrobustly.Unfortunately, thesetasksaredifficult andnot alwayswell-definedfor arbitrary3Dpolygonalmodels(e.g.,whatis thecanonicalskele-ton for anunconnectedsetof polygons?).Moreover, featuredetec-tion andsegmentationalgorithmstendto besensitive to smallper-turbationsto the model,placingundueburdenon subsequentfea-ture correspondenceanddissimilarity computationsteps.Finally,the combinatorialcomplexity of finding correspondencesin largediscretemodelsusually leadsto long computationtimes and/orlargestoragerequirements.

Finally, shapeshave beencomparedon thebasisof their statis-tical properties.The simplestapproachof this type is to evaluatedistancesbetweenfeaturevectors[22] in amultidimensionalspacewherethe axesencodeglobalgeometricproperties,suchascircu-larity, eccentricity, or algebraicmoments[45, 53]. Othermethodshave compareddiscretehistogramsof geometricstatistics.For ex-ample,Thackeretal [1, 4, 8, 9, 24, 25, 47, 54], Huetetal. [32], andIkeuchietal. [33] haveall representedshapesin 2D imagesby his-togramsof anglesanddistancesbetweenpairsof 2D line segments.For 3D shapes,Ankerstetal. [3] hasusedshapehistogramsdecom-posingshellsandsectorsarounda model’s centroid.Besl [14] hasconsideredhistogramsof thecreaseanglefor all edgesin a 3D tri-angularmesh.Besl’s methodis the mostsimilar to our approach.However, it worksonly for manifoldmeshes,it is sensitiveto cracksin the modelsandsmall perturbationsto thevertices,andit is notinvariant underchangesto meshtessellation.Moreover, the his-togramof creaseanglesdoesnot alwaysmatchour intuitivenotionof rigid shape.For example,addingany extraarmto ahumanbodyresultsin the samechangeto a creaseanglehistogram,no matterwhetherthe new arm extendsfrom the body’s shoulderor the topof its head.

To our knowledge,thereis no existing methodthatcanrapidlyandrobustly matchlarge 3D polygonalmodelswith arbitraryde-generacies.Previous approacheshave difficulty with 3D polygonsoupsbecausethey invariably requirea solutionto at leastoneofthe following difficult problems:reconstruction,parameterization,registration,or correspondence.The motivation behindour workis to develop a fast, simple, and robust methodfor matching3Dpolygonalmodelswithoutsolvingtheseproblems.

3 Overview of Approach

Our approachis to representtheshapesignaturefor a 3D modelasaprobabilitydistributionsampledfrom ashapefunctionmeasuringgeometricpropertiesof the3Dmodel.Wecall thisgeneralizationofgeometrichistogramsa shapedistribution. For example,onesuchshapedistribution,whichwe call D2, representsthedistributionofEuclideandistancesbetweenpairsof randomlyselectedpointsonthe surfaceof a 3D model. Samplesfrom this distribution canbecomputedquickly andeasilyfrom any 3D surfacemodel,while ourhypothesisis thatthedistributiondescribestheoverall shapeof therepresentedobject.Oncewehavecomputedtheshapedistributionsfor two objects,thedissimilaritybetweentheobjectscanbeevalu-atedusingany metricthatmeasuresdistancebetweendistributions(e.g.,

���norm),possiblywith a normalizationstepfor matching

scales.Thekey ideais to transformanarbitrary3D modelinto aparam-

eterizedfunction that canbe comparedwith otherseasily. In ourcase,thedomainof theshapefunctionprovidestheparameteriza-tion (e.g.,the ��� shapedistribution is a1D functionparameterizedby distance),andrandomsamplingprovidesthetransformation.

The primary advantageof this approachis its simplicity. The

shapematchingproblemis reducedto sampling,normalization,andcomparisonof probabilitydistributions,whicharerelatively simpletaskswhencomparedto prior methodsthat requirereconstructinga solid objector manifold surfacefrom degenerate3D data,reg-isteringposetransformations,finding featurecorrespondences,orfitting high-level models.Our approachworksdirectly on theorig-inal polygonsof a 3D model,makingfew assumptionsabouttheirorganization,andthusit canbeusedfor similarity queriesin awidevarietyof databases,includingonescontainingdegenerate3Dmod-els,suchasthosecurrentlyavailableon theWorld WideWeb.

In spiteof its simplicity, we expectthatour approachis abletodiscriminatewholeobjectswith differentgrossshapesrathereffec-tively (resultsof experimentstestingthis hypothesisarepresentedin Section5). In addition, it hasseveral propertiesdesirableforsimilarity matching:

� Invariance: shapedistributionshave all the transformationinvariancepropertiesof the sampledshapefunction. For in-stance,the D2 shapefunction yields invarianceunderrigidmotions(i.e., translation,rotation,and mirror imaging). Inthis case,invarianceunderscalingcanbe addedby normal-ization of shapedistributionsbeforecomparingthemand/orby factoringout scaleduring the comparison. Other shapefunctionsthat measureanglesor ratios betweenlengthsareinvariantto all similarity transformations.

� Robustness:asabonus,randomsamplingensuresthatshapedistributionsareinsensitiveto smallperturbations.Intuitively,sinceevery point in a 3D model contributesequally to theshapedistribution,themagnitudeof changesto theshapedis-tribution aredirectly relatedto themagnitudeof thechangesto the3D model. For example,if 10%of a 3D modelis per-turbed(e.g.,by addingrandomnoise,by addingasmallbumpontoasurface,or by addingsmallobjectsarbitrarily through-out space),then a distribution of randomsamplesfrom themodelcanchangeby at most10%. This propertyprovidesinsensitivity to noise,blur, cracks,anddust in the input 3Dmodels.We conjecturethat the distributionsfor mostglobalshapefunctionsbasedon distancesand/oranglesalso varycontinuouslyandmonotonicallyfor local shapechanges.

� Metric: thedissimilaritymeasureproducedby our approachadoptsthe propertiesof the norm we useto compareshapedistributions. In particular, if the norm is a metric, so is ourdissimilaritymeasure.This propertyholdsfor mostcommonnorms,including

���norms,EarthMover’s Distance,etc.

� Efficiency: constructionof the shapedistributions for adatabaseof 3D modelsis generallyfastandefficient. For in-stance,thecomplexity of samplingtheD2 shapefunctionfora3D modelwith � trianglesand � samplesis ���� ������ . Theresultingshapedistributionscanbe approximatedconciselyby piecewise-linearfunctionswith constantcomplexity stor-ageandcomparisoncosts.

� Generality: shapedistributionsare independentof the rep-resentation,topology, or applicationdomainof the sampled3D models.As a result,our shapesimilarity methodcanbeappliedequallywell to databaseswith 3D modelsstoredaspolygonsoup,meshes,constructivesolidgeometry, voxels,orany othergeometricrepresentationaslongasasuitableshapefunction canbe computedfrom eachrepresentation.More-over, a single database(suchas the World Wide Web) cancontain3D modelsin a variety of different representationsandfile formats. Finally, shapedistributionscanbe usedin

many differentapplicationdomainsfor comparisonof natu-ral, deformableshapes(e.g.,animals)and/orman-madeob-jects(e.g.,machinedparts).

The interestingissuesto beaddressedin implementingthepro-posedshapematchingapproachare: 1) to selectdiscriminatingshapefunctions,2) to constructshapefunctionsfor each3D modelefficiently, and3) to computea dissimilarity measurefor pairsofdistributions. We addresstheseissuesin the following sections.The challengeis to find methodswhosecombinationproducesadissimilarity measurewith the desirablepropertieslisted above,while providing enoughdiscriminationbetweensimilaranddissim-ilar shapesto be useful for a particularapplication. We proposesuchmethodsandevaluatethemexperimentallyin Section5 for adatabaseof 3DpolygonalmodelsdownloadedfromtheWorldWideWeb.

4 Method

In this section,we provide a detaileddescriptionof the methodswe useto build shapedistributionsfrom 3D polygonalmodelsandcomputeameasureof theirdissimilarities.

4.1 Selecting a Shape Function

Thefirst andmostinterestingissueis to selectafunctionwhosedis-tributionprovidesagoodsignaturefor theshapeof a3D polygonalmodel. Ideally, the distribution shouldbe invariantundersimilar-ity transformationsandtessellations,andit shouldbeinsensitive tonoise,cracks,tessellation,andinsertion/removal of smallpolygons.

In general,any functioncouldbesampledto form ashapedistri-bution,includingonesthatincorporatedomain-specificknowledge,visibility information(e.g., the distancebetweenrandombut mu-tually visible points),and/orsurfaceattributes(e.g.,color, texturecoordinates,normalsandcurvature).However, for thesake of clar-ity, we focuson purelygeometricshapefunctionsbasedon simplemeasurements(e.g.,angles,distances,areas,andvolumes).Specif-ically, wehaveexperimentedwith thefollowing shapefunctions:

� A3: Measurestheanglebetweenthreerandompointson thesurfaceof a3D model.

� D1: Measuresthedistancebetweenafixedpointandoneran-dompointonthesurface.Weusethecentroidof theboundaryof themodelasthefixedpoint.

� D2: Measuresthedistancebetweentwo randompointsonthesurface.

� D3: Measurestheareaof thetrianglebetweenthreerandompointson thesurface,andtakesthesquareroot.

� D4: Measuresthe volume of the tetrahedronbetweenfourrandompointson thesurface,andtakesthecuberoot.

Theseshapefunctionswerechosenmostly for their simplicityandinvariances.In particular, they areeasyto computeandproducedistributionsthatareinvariantto rigid motions.They areinvariantto tessellationof the3D polygonalmodel,sincepointsareselectedrandomlyfrom thesurface.They areinsensitive to smallperturba-tionsdueto noise,cracks,andinsertion/removal of polygons,sincesamplingis areaweighted. Finally, the ��� shapefunction is in-variantto scale,while the othershave to be normalizedto enablecomparisons.

Weexpectthesegeneral-purposeshapefunctionsto befairly dis-tinguishingassignaturesfor 3D shape,assignificantchangesto therigid structuresin the3D modelaffect thegeometricrelationshipsbetweenpoints on their surfaces. For instance,considerthe D2shapefunction,whosedistributionsareshown for a few cannonicalshapesin Figure1(a-f). Notehow each distribution is distinctive.Also, notehow continuouschangesto the3D modelaffect theD2distributions. For instance,Figure 1(g-h) shows the distancedis-tributionsof two unit spheresasthey move 0, 1, 2, 3, and4 unitsapart,respectfully. In eachdistribution, the first humpresemblesthe linear distribution of a sphere,while the secondhump is thecross-termof distancesbetweenthe two spheres.As the spheresmove fartherapart,theshapedistributionchangescontinuously.

a)Line segment.

c) Triangle.

e)Cylinder(withoutcaps).

g) Two adjacentunitspheres.

b) Circle.

d) Cube.

f) Sphere.

h) Two unit spheresseparatedby 1, 2, 3, and4 units.

Figure1: ExampleD2 shapedistributions.

4.2 Constructing Shape Distributions

Having chosena shapefunction, the next issueis to computeandstorea representationof its distribution. Analytic calculationofthe distribution is feasibleonly for certaincombinationsof shapefunctionsandmodels(e.g., the ��� function for a sphere).So, ingeneral,we employ stochasticmethods.Specifically, we evaluate� samplesfrom the shapedistribution andconstructa histogramby countinghow many samplesfall into eachof � fixedsizedbins.Fromthehistogram,wereconstructapiecewiselinearfunctionwith�

( ��� ) equallyspacedvertices,which formsour representationfor theshapedistribution. We computetheshapedistributiononcefor eachmodelandstoreit asasequenceof V integers.

Oneissuewe mustbeconcernedwith is samplingdensity. Themoresampleswe take, the moreaccuratelyandpreciselywe can

reconstructthe shapedistribution. On the otherhand,the time tosamplea shapedistribution is linearly proportionalto the numberof samples,so thereis an accuracy/time tradeoff in the choiceof� . Similarly, moreverticesyieldshigherresolutiondistributions,while increasingthestorageandcomparisoncostsof theshapesig-nature.In ourexperiments,wehavechosento erronthesideof ro-bustness,takingalargenumbersof samplesfor eachhistogrambin.Empirically, we have found that ����� ���"!$# samples,����� ���"!bins, and

� �&%�! verticesyields shapedistributions with lowenoughvarianceand high enoughresolutionto be useful for thedatabaseswe’ve tested.

A secondissueis samplegeneration.As our shapefunctionsaredescribedin termsof randompointson thesurfaceof a 3D model,weimplementedthefollowingmethodtogenerateunbiasedrandompointswith respectto thesurfaceareaof a polygonalmodel.First,weiteratethroughall polygons,splittingtheminto trianglesasnec-essary. Then,for eachtriangle,we computeits areaandstoreit inanarrayalongwith thecumulative areaof trianglesvisitedso far.Next, we selecta trianglewith probabilityproportionalto its areaby generatinga randomnumberbetween0 andthe total cumula-tiveareaandperformingabinarysearchon thearrayof cumulativeareas.For eachselectedtrianglewith vertices ���(')�*',+-� , we con-structa pointon its surfaceby generatingtwo randomnumbers,.$/and. # , between0 and1, andevaluatingthefollowing equation:

0 �1�2�4365 . / �)�6785 . / �2�439. # �)�:7:5 . / . # + (1)

Intuitively, 5 .$/ setsthepercentagefrom vertex A to theoppos-ing edge,while . # representsthe percentagealongthis edge(seeFigure2). Taking the square-rootof .$/ gives a uniform randompoint (with respectto surfacearea)in thetriangle.

A

B C

r1

r2

Figure2: Generatinga randompoint in a triangle.

4.3 Comparing Shape Distributions

Having constructedthe shapedistributions for two 3D models,we are left with the task of comparingthem to producea dis-similarity measure. There are many standardways of compar-ing two functions. Examplesinclude the Minkowski

� �norms,

Kolmogorov-Smirnov distance,Kullback-Leiblerdivergencedis-tances[37], Matchdistances[49, 57], EarthMover’s distance[48],andBhattacharyyadistance[15]. Othermethods,perhapsbasedon2D curvematching,couldalsobeused.

In our implementation,we have experimentedwith six simpledissimilaritymeasuresbasedon

���normsof theprobabilityden-

sity functions(pdfs)andcumulativedistributionfunctions(cdfs)for���1�;'2�$'=< .1 In thedescriptionsbelow, assume> and representpdfsfor two models,while ?> and ? representthecorrespondingcdfs– i.e., ?>@��AB�C� DEGF > .

1Thecdf H / andthecdf H F normsarethe H / EarthMover’s [48] andtheKolmogorov-Smirnov distances,respectively.

� PDF�I�

: Minkowski�I�

normof thepdf:�J�)>G'K L�M�N� O�>P3Q BO

�� /SR

�.

� CDF� �

: Minkowski� �

normof thecdf:�J�)>G'K L�M�N� O ?>P3 ? TO

�� /SR

�.

Sinceeachshapedistribution is representedasa piecewise lin-earfunction,analyticcomputationof thesenormscanbedoneeffi-ciently in time proportionalto thenumberof verticesusedto storethedistributions.

For certainshapefunctions,wemustaddanormalizationsteptothecomparisonprocessto accountfor differencesin scale.Sofar,we have investigatedthreemethodsfor normalization:1) align themaximumsamplevalues,2) align themeansamplevalues,and3)searchfor thescalethatproducetheminimaldissimilaritymeasureduring eachcomparison.The first two of thesecanbe evaluatedanalytically, andthusarevery fast.However, they maynotproducetheminimaldissimilaritymeasuresdueto mismatchingscales.Thethird methodrequiresan optimizationprocedure.Specifically, if> and representshapedistributionsfor two models,theminimaldissimilaritymeasurewith normalizationis thendefinedas:

U(V�WX �J�)>@��AB�Y'YZ[ ��)Z[AB�2� (2)

We have implementeda simple methodto perform the searchover Z . First,we scaleour distributionssothattheaveragesamplein eachdistributionhasvalue1. Thenweevaluate�J�)>@��AB�Y',Z2 ��)Z[AB�2�for valuesof \�]"^_Z from -10 to 10, in 100equallyspacedintervals.Wereturntheminimumamongtheresultsasthedissimilaritymea-surefor thenormalizedshapedistributions.

5 Experimental Results

Themethodsdescribedin theprecedingsectionshave beenimple-mentedin C++ andincorporatedinto ashapematchingsystemthatrunson SiliconGraphicsandPC/Windowscomputers.

In orderto test the effectivenessof this system,we executedaseriesof shapematchingexperimentswith a databaseof 3D mod-els downloadedfrom a variety of siteson the World Wide Web.Themodelscomprisedsetsof independentpolygons,withoutstruc-ture,adjacency information,or registeredcoordinatesystems.Themodelscontainedanywhere from 20 to 186,000polygons,withthe averagemodel containingaround7,000polygons. Very fewof the modelsformed a single manifold surfaceor even a well-definedsolidregion. Instead,they almostall containedcracks,self-intersections,missingpolygons,one-sidedsurfaces,and/ordoublesurfaces– noneof which causedsignificantartifactsduring ren-deringwith a z-buffer, but all of which areproblematicfor most3D shapematchingalgorithms.Theexperimentswererun on a PCwith a400MHzPentiumII processorand256MBof memory.

5.1 Robustness Results

In our first experiment,we testedthe robustnessof our dissimilar-ity measureto transformationsandperturbationsof the3D models.Specifically, wechosetenrepresentative3D models(shown in Fig-ure5), andappliedsix transformationsto eachof them.Theresult-ing databasehadsevenversionsof eachmodel(theoriginalandsixtransformedvariants),making70 modelsin all. The transforma-tionswereasfollows:

� Scale:Grow by a factorof 10 in everydimension.� Rotate: Rotateby 45 degreesthreetimes,first aroundtheXaxis,thenaroundtheY axis,thenaroundtheZ axis.

� Mirr or: Mirror over the YZ plane,thenover the XZ plane,thenover theXY plane.� Noise:Perturbeachvertex of everypolygonrandomlyby 1%of thelongestlengthof themodel’sboundingbox. As verticeswerenot sharedby adjacentpolygons,this transformationin-troducedthin cracks(seeFigure3(a)).� Delete: Randomlyremove 5% of the polygons. (note theholesin thebumperandwindshieldof thecarin Figure3(b)).� Insert: Randomlyinsertcopiesof 5% of thepolygons.

a)1%Noise. b) 5% Deletion.

Figure3: A carmodelafter(a)perturbingall theverticesby 1%,or(b) removing 5%of thepolygons.

We testedtherobustnessof our samplingmethodby generatingtheD2 shapedistribution for eachmodel.Theresultingshapedis-tributionsfor all 70 modelsareplottedin Figure4 (scaledto aligntheir meanvalues).Note thatonly tendistinctcurvesareapparentin theplot. This is becauseeach“thick” curveappearsastheresultof sevennearlyoverlappingshapedistributionscomputedfor differ-entvariantsof thesamemodel. For instance,thecurve containingthetall spike in themiddleis drawn seventimes,oncefor eachvari-antof themugmodel. Theresultsof this experimentdemonstrateoursamplingmethod’s repeatabilityandits robustnessto similaritytransformations,noise,andsmallcracksandholes.

distance

probability

Figure4: D2 shapedistributionsfor sevenvariantsof tenmodels.

Wefurtherinvestigatedtherobustnessof ourmethodby testingitwith differentpolygontessellationsof two 3D shapes.WeusedtheSimplificationEnvelopessoftwareprovidedby Cohenet al. [19] toproduce8 versionsof theStanfordBunny [38] rangingfrom 70,000down to 600triangles,and6 versionsof asphererangingfrom 200down to 28 triangles.Then,we constructedD2 shapedistributionsfor eachof theseversions.Theresulting14curvesareshown in Fig-ure6. Notethat theshapedistributionsvary slightly from original

Car

Phone

Chair

Plane

Human

Skateboard

Missile

Sub

Mug

Table

Figure5: Imagesof theten3D modelsusedin our initial robustnessexperiments.

modelsto thesimplifiedversions,but not significantlywhencom-paredto differencesbetweentheoriginal models.This experimentcorroboratesourexpectationthatshapedistributionsareinsensitiveto changingtessellationand,morespecifically, stableundermodelsimplification.

distance

probability

SphereBunny

Figure6: D2 shapedistributionsfor simplifiedversionsof theStan-ford bunny andasphere.

5.2 Discrimination Results

In our secondexperiment,we investigatedtheability of our shapematching method to discriminateamong similar and dissimilarshapes.As a first steptowardsthisgoal,wecomputedour dissimi-larity measurefor all pairsof the70shapedistributionsdescribedintheprevioussection.During this shapematchingtest,thedistribu-tionswerecomparedwith thepdf

� / normandthey werescaledbyaligningtheirmeanvalues.Theresultingdissimilaritymeasuresareshown asa matrix in Figure7. In this visualizationof thematrix,the lightnessof eachelement ��`a')b�� is proportionalto the magni-tudeof thecomputeddissimilaritybetweenmodels andb . Thatis,eachrow andcolumnrepresentthedissimilaritymeasuresfor asin-gle modelwhencomparedto all othermodelsin thedatabase(thematrix is symmetric). Darker elementsrepresentbettermatches,while lighterelementsindicateworsematches.Theorderingof the

classesis alphabeticalsooneshouldnotexpectany particulardark-nesspatternexcepttheclearlyvisible7x7blocksof matrixelementswith indistinguishablecolors.Thispatterndemonstratestherobust-nessof our distribution comparisonmethod,asall variantsof thesamemodelproducealmostthe samedissimilaritymeasurewhencomparedto all variantsof every othermodel. Moreover, notethedarkerblocksof 7x7matrixelementsalongthemaindiagonal.Thispatternresultsfrom thefactthatall 7 variantsof everyshapematcheachotherbetterthan they matchany othershape. Accordingly,for this simpledatabase,our methodcouldbeusedto perfectlyas-signall 70modelsto oneof the10classesusinganearestneighborclassifier.

Althoughtheresultsof this experimentareencouraging,amoreinterestingand challenging test is to determinehow well ourmethodcandiscriminateclassesof shapesin a largerandmoredi-versedatabaseof 3D models.To investigatethis question,we ex-ecuteda seriesof testson a databaseof 133modelsretrievedfromtheWorld Wide Webandgroupedqualitatively (by functionmorethanby shape)into 25 classesby a third party. Figure9 summa-rizesthetypesandsizesof theseclasses.First,notethateachclasscontainsanarbitrarynumberof objects,usuallydeterminedby howmany modelswere found in a quick searchof the Web (e.g., theplaneclasshassignificantlymoremodelsthan the others). Sec-ond, note that the similarity betweenclassesvaried greatly. Forinstance,someclasseswerevery similar to oneanother(e.g.,pensandmissileslook alike), while somewerequite distinct from theothersin thedatabase(belts). Third, notethat thesimilarity of ob-jectswithineach classalsovaried.Someclasses(suchasball,mug,openbook,pen,andsub)contained3D modelswith shapesgreatlyresemblingeachother, while others(suchasanimal,boat,car, andplane)containedmodelswith a wide varietyof shapes.This diver-sity within classesis shown in Figure8, which containspicturesof threemodelsfrom themug,car, andboatclasses.Notethat themugsarevisually quite similar, while the carsandboatsaresig-nificantly different. Anotherclass,planes,is even more diverse,containingbiplanes,fighter jets,propellerplanes,andcommercialjets,all similar in function,but quitedifferentin shape.

To investigatetheability of our shapematchingmethodsto dis-

Car Sub Chair Plane Missile Mug Phone Skateboard Human Table

Car

Sub

Chair

Plane

Missile

Mug

Phone

Skateboard

Human

Table

Car Sub Chair Plane Missile Mug Phone Skateboard Human Table

Car

Sub

Chair

Plane

Missile

Mug

Phone

Skateboard

Human

Table

Figure7: Similarity matrix for seven variantsof ten 3D models.Lightnessindicatesthedissimilaritybetweenmodels.

criminatebetweenclassesof objects,werananexperimentby com-puting the D2 shapedistributions for all 133 models. They areshown in Figure9 – eachplot shows theD2 distribution for all 3Dmodelsof oneclass,normalizedby their means.Examiningthesedistributionsqualitatively, we find that the shapedistributionsformostobjectswithin asingleclassarehighly correlated,asmultiplecurvesappearalmoston topof oneanother. Moreover, many of theclasseshave a distinctive shapedistribution thatcouldbeusedforclassification.For instance,balls have a nearlylinear distributionwith a sharpfalloff on the right, mugshave onesharppeakin themiddle, beltshave a peakon the right, and lampshave two largepeakswith avalley in between.

To a limited degree,it is evenpossibleto infer thegrossshapeofsomeobjectsfrom theirD2 shapedistributions.For instance,refer-ring backto Figure1, ballshaveadistributionresemblingasphere,beltsresembleacircle,mugsresembleacylinder, andlampsresem-bletwospheresseparatedbysomedistance.Althoughseveralof theotherclasseshavevisually lessdistinctiveunimodalshapedistribu-tions,we canseethat the“humps” in thedistributionsof differentclassesareusuallydistinguishableby their locations,heights,andshapes,leadingusto believe thatshapedistributionscouldbeusedeffectively for objectclassification.

To test this hypothesisand to investigatewhich combinationsof shapefunctions,normalizationmethods,andcomparisonnormsprovided thebestclassificationmethods,we rana seriesof “leaveoneout” classificationtests.In every test,we comparedtheshapedistribution of eachmodel in the database(the “query” model)againstall others. The testwas repeated90 times for all combi-nationsof the 5 shapefunctions,3 normalizationmethods,and6comparisonnormsdescribedin Section4.

Tables1-3 containthreecross-sectionsof the resultsmeasuredin thesetests. In eachtable, the first columnindicatesthe shapefunction,normalizationmethod,or comparisonnormused(unlessotherwisespecified,the D2 shapefunction, the MEANnormaliza-

Mug 1

Antiquecar

Galleon(with sails)

Mug 2

Convertible

Tugboat

Mug 3

Camaro

Warship

Figure8: Exampleclassesof shapesin our database:mugs(toprow), cars(middlerow), andboats(bottomrow).

tion method,andthePDF� / normwasused).Thesecondcolumn

(“First Tier”) lists thepercentageof top cd3:� matches(excludingthequery)from the query’s class,wherek is thesizeof theclass.This criteria is stringency, sinceeachmodel in the classhasonlyonechanceto bein thefirst tier. An idealmatchingwould give nofalsepositivesandreturnascoreof 100%.Thethird column(“Sec-ond Tier”) lists the sametype of result,but for the top ��)ce3f� �matches.The fourth columnlists the percentageof test in whichthe top match(“NearestNeighbor”) was from the query’s class.Finally, the right-mostcolumncontainsthecomputationtimesforsamplingandcomparisonof shapedistributions. Note that sam-pling timesarein seconds,while comparisontimesarein millisec-onds.

Fromtheresultsin thesetables,wemake thefollowing observa-tions.First, theD2 shapefunctionclassifiedobjectsbetterthantheothershapefunctionsin our tests.Thereareseveralplausibleinter-pretationsfor why othershapefunctionsprovedlessdiscriminatorythanD2. For one,shapefunctionssuchasD1 aredifficult to rep-resentaccurately, asempiricallythey containsharppeaks(e.g.,theD1 distribution for asphereis asingledeltafunction).For another,we noticeda trendthat shapedistributionsfor D3 andD4 appearsimilar for many typesof models.

Second,in our tests,theMAX scalingmethodis not asgoodasMEANor SEARCHasit suffersfrom thefollowing problem.If theidealizedshapedensityfalls off nearthe tail, thenthereis a rela-tively high variationin themaximumsamplefound(dueto anout-lier). Scalingtheentireshapedistributionbasedon thisonesampleaffects the signaturefor the entireobject. This result is intuitive,asthe meanis a morestablestatisticthanthe maximum. For theothernormalizationmethods(MEANandSEARCH) theclassifica-tion resultswereapproximatelythesame.We foundthatsearchinghelpsminimizethedifferencebetweenshapedistributions,but thisdid notimprovethediscriminabilityof themethodonthisdatabase.

Third, thePDF� / normperformedthebestfor comparingshape

Shape First Second Nearest SampleFunction Tier Tier Neighbor Time(s)

A3 38% 54% 55% 12.6D1 35% 48% 56% 8.6D2 49% 66% 66% 8.6D3 42% 58% 58% 13.5D4 32% 42% 47% 15.8

Table1: Comparisonof shapefunctions(usingMEANandPDF gih ).Scale First Second Nearest Compare

Method Tier Tier Neighbor Time(ms)MAX 41% 56% 63% 0.1

MEAN 49% 66% 66% 0.1SEARCH 49% 66% 68% 9.0

Table2: Comparisonof normalizationmethods(usingD2 andPDF gCh ).Norm First Second Nearest Compare

Method Tier Tier Neighbor Time(ms)PDF g h 49% 66% 66% 0.1PDF gj 47% 64% 62% 0.1PDF gBk 42% 59% 61% 0.1CDF gCh 46% 63% 59% 0.2CDF g j 44% 63% 59% 0.1CDF g k 43% 59% 57% 0.1

Table3: Comparisonof normmethods(usingD2 andMEAN).

distributionsin ourtest.In general,thepdfsdid betterthanthecdfs,possiblybecausepeaksandvalleys of pdf curvesareeasierto dis-criminateusing

� �normsthanthesteepareasandplateausof cdf

curves. Meanwhile,the�# and

� F normsperformedworsethanthe

� / norms,in general.For higher� , the� �

normsbecomelessforgiving of large differences,and thus perhapsour comparisonsbecamemoresensitive to outliersor normalizationerrors.

Finally, we examinethe utility of our dissimilarity measureforclassifyingobjectsfrom thedatabasewith 133models.In this test,we usedtheD2 shapefunction,MEANnormalizationmethod,and� / norm. Figure10 shows the similarity matrix for this test. Asin Figure7, the lightnessof eachelement ��`a')b�� is proportionaltothemagnitudeof thecomputeddissimilaritybetweenmodels andb (i.e.,darker elementsrepresentbettermatches).Thus,if thesim-ilarity metric were ideal (i.e., if it were able to readthe mind ofthehumanthat formedtheclasses),thedissimilaritymeasuresformodelsin the sameclasswould be less(appeardarker) than foronesin different classes.That is, we hopeto seea sequenceofdarker blocksalongthemaindiagonal,with sizescorrespondingtothenumbersof modelsin eachclass,with mostly lighter colorsinthe off-diagonalmatrix elements.Of course,given the ambiguityanddiversityof thedatabase,webelieve thatit wouldbeoptimisticto expectthis result,evenfor a humanassigningdissimilaritymea-sures.

Examiningthesimilaritymatrixin Figure10,weseethatthedis-similarity valuescomputedwith our methodarefairly discriminat-ing in this test.Therearemany darkblocksreadilyapparentalongthe main diagonalcorrespondingto groupsof objectswithin thesameclassthatproducegoodmatches(e.g.,mugs,phones,chairs,planes,spaceships,etc.). Meanwhile,mostelementsoff-diagonalare lighter shades,indicating relatively few falsepositives. Off-diagonalblocksof darkelementsoftenrepresentmatchesbetweenclasses(e.g.,spaceshipversusplane). Surprisingly, several of the

classeswith very diversemodels(e.g.,carsandplanes)canbedis-tinguishedvery clearly asdark blocksin this plot, indicatingthatour methodis usefulfor discriminatingthemfrom othermodelsinthe databasein spite of their diversity. On the otherhand,thereareotherclasseswhosemodelsdid notmatchwell in this test(e.g.,boats,helicopters,etc.). Someof thesefailuresaredueto the in-herentdifficultiesof shapematchingwithout realworld knowledge(e.g., the boatswere moresimilar in function thanshape),whileothersare probablydue to the limitations of our implementation(e.g.,

� �normsproducelargedissimilaritiesfor lamps). Overall,

for 66%of themodels,ourprototypesystemproducedatop-matchwithin thesameclass.Wearenotawareof anothershapematchingmethodthatcouldachieve this resultfor suchadifficult database.

6 Discussion and Conclusion

Themaincontributionof thispaperis theideaof usingrandomsam-pling to produceacontinuousprobabilitydistribution to beusedasa signaturefor 3D shape.The key featureof this approachis thatit provides a framework within which arbitrary and possiblyde-generate3D modelscanbetransformedinto functionswith naturalparameterizations,allowing simple function comparisonmethodsto producerobustdissimilaritymetrics.

Our initial experiencesverify many of the expectedfeaturesofthis approach.First, it is simple to implement– e.g., our wholesystemrequiresaround2000linesof C++ code.Second,it is fast–e.g.,thesystemtakesaroundtensecondsto constructashapedistri-butionfor typical3D modelscontainingthousandsof polygons,andit computesthe dissimilaritymeasurefor any pair of shapedistri-butionsin lessthanamillisecond.Third, invarianceandrobustnesspropertiescanbeensuredby choosingshapefunctionsandnormswith thedesiredproperties– e.g.,the ��� shapefunction is invari-antto rigid bodyandmirror transformations,andit is insensitive tonoise,blur, cracks,tessellation,anddust in the input 3D models.Normalizationof shapedistributionsprovidesinvarianceto scale,andusingthe

�I�normfor comparisonof distributionsensuresthat

ourdissimilaritymeasureis ametric.Our experimentalresultsdemonstratethat shapedistributions

canbefairly effectiveatdiscriminatingbetweengroupsof 3D mod-els.Overall,weachieved66%accuracy in ourclassificationexper-imentswith a diversedatabaseof degenerate3D modelsassignedto functionalgroups. Unfortunately, it is difficult to evaluatethequality of this resultascomparedto othermethods,asit dependslargelyonthedetailsof our testdatabase.However, webelieve thatit demonstratesthatour methodis usefulfor thediscriminationof3D shapes,at leastfor preclassificationprior to moreexactsimilar-ity comparisonswith moreexpensive methods.

An importantissuefor further researchin 3D shapematchingis developmentof benchmarkdatabasescontainingdegenerate3Dpolygonalmodelsso thatdifferentshapeanalysismethodscanbecompared.Of course,therearealsomany improvementsthatcouldbemadetoourinitial prototypesystemin futurework. For instance,onecouldinvestigatemoreefficientshapedistributionsamplingandreconstructionmethods,possiblybasedon adaptive strategies. Or,onecouldalsolookatcombiningthedistributionsof multipleshapefunctionsinto a singleclassifier, or combiningshapedistributionswith other attributes(e.g., surfacecolors) for improved discrim-inability. We arecurrently investigatinguserinterfacesfor speci-fying 3D shape-basedqueriesin aninteractive retrieval system.

Anotherimportanttopic for futurework is to studythetheoreti-cal propertiesof shapedistributions.For instance,it wouldbeniceto develop a theoryconcerningwhich shapefunctionsandnormswill be good classifiersof shape. We are investigatingprovablepropertiesfor the ��� shapefunction. Uniquenesspropertiesfor

homometricdiscrete point sets(oneswith the samedistancedis-tribution) have beenproven by Skienaet al. [51]. They developedupperandlowerboundsonthenumberof non-congruenthomomet-ric discretepoint setsin arbitrarydimensions.Propertieshave alsobeenproven for theRadontransformfor a convex region + in theplane[46, 29]. This transformmapsany orientedline to thelengthof its intersectionwith + . It completelyspecifiestheregion + , andit canbeinvertedfairly efficiently. TheRadontransformhasfoundmany usesin X-ray tomography. Proving uniquenesspropertiesfor othercontinuousshapefunctions,suchas ��� , mayhave simi-lar implicationsfor reconstructionandmanipulationof 3D modelsrepresentedby shapedistributions.

Finally, it would be interestingto investigatewhetherthe pro-posedshapematchingmethodis useful in other applicationdo-mains,suchascharacterrecognition,sign-languagerecognition,ormolecularbiology.

Acknowledgements

We would like to thankPatrick Min, David Jacobs,AdamFinkel-stein,andAdam’s dadfor their valuablediscussions.Patrick alsohelpedby assemblinga bibliographyon 3D shapematching,andMao Chenretrievedthe3D modelsfrom theWorld Wide Webforour experiments.Thanksalsogo to Alfred P. SloanFoundationforpartialfundingof thiswork.

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5 Animals

6 Cars

3 Lamps

4 Pens

4 Sofas

4 Balls

8 Chairs

3 Lightnings

4 Phones

6 Spaceships

2 Belts

3 Claws

6 Missiles

27Planes

3 Subs

3 Blimps

4 Helicopters

4 Mugs

4 Rifles

4 Tables

3 Boats

11 Humans

4 Openbooks

3 Skateboards

5 Tanks

Figure9: D2 shapedistributionsfor 133modelsgroupedinto 25 classes.

al bl btbp bt cr cr cw hr hn lp lg me mg ok pn pe pe re sd sa sp sb te tk

animal

ball

beltblimp

boat

car

chair

claw

helicopter

human

lamp

lightning

missle

mug

openbook

pen

phone

plane

rifle

skateboard

sofa

spaceship

sub

table

tank

al bl btbp bt cr cr cw hr hn lp lg me mg ok pn pe pe re sd sa sp sb te tk

animal

ball

beltblimp

boat

car

chair

claw

helicopter

human

lamp

lightning

missle

mug

openbook

pen

phone

plane

rifle

skateboard

sofa

spaceship

sub

table

tank

Figure10: Similarity matrix for ourdatabaseof 1333D models.