EUROGRAPHICS’99 / P. BrunetandR. Scopigno(GuestEditors)

Volume18 (1999), Number3

A Shrink Wrapping Approachto Remeshing Polygonal Surfaces

Leif P. Kobbelt JensVorsatz Ulf Labsik Hans-PeterSeidel

ComputerGraphicsGroupMax-Planck-Institutfür Informatik

Im Stadtwald, 66123Saarbrücken,Germany

AbstractDueto their simplicityandflexibility, polygonalmeshesare aboutto becomethestandard representationfor sur-facegeometryin computergraphicsapplications.Somealgorithmsin thecontext of multiresolutionrepresentationandmodelingcanbeperformedmuch more efficientlyandrobustlyif theunderlyingsurfacetesselationshavethespecialsubdivisionconnectivity. In this paper, we proposea new algorithm for converting a givenunstructuredtriangle meshinto onehavingsubdivisionconnectivity. Thebasicideais to simulatetheshrinkwrappingprocessby adaptingthedeformablesurfacetechniqueknownfrom image processing. Theresultingalgorithm generatessubdivisionconnectivitymesheswhosebasemeshesonly havea very small numberof triangles.The iterativeoptimizationprocessthat distributesthemeshverticesover thegivensurfacegeometryguaranteeslow local dis-tortion of thetriangular faces.Weshowseveral examplesandapplicationsincludingtheprogressivetransmissionof subdivisionsurfaces.

1. Introduction

Unstructuredtrianglemeshesarethemostversatilesurfacerepresentationin computergraphicssince they enablethedescriptionof surfaceswith arbitrary shapeand topologywithout having to observe complicatedcompatibility con-ditions known from surfacepatchingbasedon splines7� 12.Thegenuinelynon-smoothcharacteristicsof piecewisepla-narsurfacescanbereducedby refining thetesselationuntila prescribedangletolerancebetweenadjacenttrianglesisreached.The prohibitive complexity of highly detailedtri-anglemeshescanbe controlledby introducinga hierarchi-cal level-of-detaildecompositionbasedon the intermediateresultsof a meshdecimationalgorithm2 � 26� 24� 11� 8 � 17.

However, especiallyin thecontext of multiresolutionrep-resentationand modeling,many algorithmsrequirea spe-cific structureof themeshes,namelysubdivisionconnectiv-ity. Thisspecialtypeof meshconnectivity is generatedby it-eratively applyingauniformsubdivisionoperatorto acoarsebasemesh

�0. Theuniformityof therefinementoperatorim-

plies that the resultingmeshis piecewise regular, i.e., eachsubmeshwhich topologicallyemergesfrom onesingle tri-

angleof thebasemeshhastheconnectivity of a regulargrid(cf. Fig. 1).

In this paper, we presenta new methodfor converting agiven meshwith arbitrary connectivity into one with sub-division connectivity. Thebenefitsfrom this conversionaremanifold.Theprimaryadvantageof subdivision connectiv-ity meshesis thedirectavailability of multiresolutionseman-tics sincethedifferentrefinementlevels provide thedyadiclevels of detail which enablethe generalizationof wavelettechniquesto parametericsurfaces21� 25� 27� 35. Moreover, theremeshingprocedureis equivalent to constructinga globalparameterizationof thegiven surfaceover a polyhedralpa-rameterdomain.Hence,the result provides the basis formany computergraphicsand engineeringalgorithmsliketexturing andmilling pathgeneration.In the result sectionwewill presentanotherinterestingapplicationin thecontextof progressive transmissionof surfacegeometry.

The paperis organizedasfollows. After properlydefin-ing theremeshingproblemin Section2 anddiscussingsomepreviouswork in Section3, weexplain thebasicprincipleofthealgorithmin Section4.Wethenidentify thecentralprob-lemswith thestraightforwardapproachin Section5andpro-

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Kobbeltet.al. / RemeshingPolygonalSurfaces

Figure 1: Mesheswith subdivisionconnectivityare generatedbyuniformlysubdividinga coarsebasemesh�

0. On therefinedmeshes

�m wecaneasilyidentifyregular submeshes(subdivisionpatches) which topologically correspondto onesingletriangle

of thebasemesh�

0 (right).

posesolutionswhich extendthebasicschemein two ways:In Section6atechniquefor findinganappropriatebasemeshwith a rathersmall numberof trianglesis describedandinSection7 we explain how to make the basicalgorithmnu-mericallymorestable.Finally in Section8 we illustratetheeffectivenessof ouralgorithmby severalexamplemeshes.

2. Remeshing requirements

We startby fixing the notation.Let an arbitrary(manifold)trianglemesh� begivenwhich representsthegenus-zerosurfaceof a geometricmodel.Thetaskof remeshingthein-put data � meansto find a sequenceof meshes

�0 ����������

m

suchthateach�

i � 1 emergesfrom�

i by theapplicationof auniform subdivision operatorwhich performsa 1-to-4 splitoneverytriangularfaceof

�i (cf. Fig.1).Sincethe

�i should

be differently detailedapproximationsof � , the verticesp � i have to lie on the continuousgeometryof � butthey notnecessarilyhave to coincidewith � ’svertices.Weallow but do not requirethat the verticesof

�i area subset

of�

i � 1’s vertices(interpolatory/ non-interpolatorysubdi-vision).

In general,it would be enoughto generatethe mesh�m since the coarserlevels of detail

�i can be extracted

by subsampling.Nevertheless,building thewholesequence�0 ����������

m from coarseto fine often leadsto moreefficientmulti-level algorithms.

Thespecialconnectivity of�

m givesriseto a partitioninginto regular submeshes(subdivisionpatches) consistingof4m triangles.Eachsubdivision patch j � m correspondsto onetriangleTj of thebasemesh

�0 andcanbeparameter-

izednaturallyby assigningdyadicbarycentriccoordinatestotheverticesof j . Combiningthelocalparameterizationsofthesubdivision patchesyieldsa globalparameterizationfor�

m over thepolyhedralparameterdomain�

0.

The quality of a subdivision connectivity meshis mea-suredin two differentaspects.First, we want theabove pa-rameterizationwhich mapspoints from the basemesh

�0

to thecorrespondingpointson�

m to becloseto isometric,

i.e., the local distortionof thetrianglesshouldbesmallandevenlydistributedover eachpatch.To achieve this, it is nec-essaryto adapttheshapeof thetrianglesin thebasemesh

�0

carefully to the shapeof the correspondingsurfacepatchesin thegivenmesh� .

Figure 2: Minimizing the distortion of the mapfrom�

0 to�m. On theleft, thedistortionis minimizedwithin each base

triangle while on theright thedistortioncontrol is extendedacrossthebaseedges.

As mentionedin 19, distortion control can be extendedacrosstheedgesof theoriginalbasemesh(cf. Fig. 2).Whilethis leadsto an improved visual appearanceof the meshstructure,it is shown in 16 that the mathematicalquality intermsof parametricdistortionis notnecessarilyimproved.

The secondquality requirementratesthe basemesh�

0itself according to the usual quality criteria for trianglemeshes,i.e., uniform size and aspectratio of the basetri-angles.

While generalpurposemeshgenerationandmeshdeci-mationalgorithmstypically adaptthe sizeof the triangularfacesto the local curvatureof the underlyingsurface,onehasto follow adifferentconceptin thecontext of multireso-lution representationsandmodeling.In orderto havea clearrelationbetweenthesizeof a geometricfeatureandthe re-spectivesubdivisionlevel

�i onwhichit is introducedduring

refinement(space-frequencylocalization), it is morereason-able(to try) to make all trianglesfrom thesamerefinement

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Kobbeltet.al. / RemeshingPolygonalSurfaces

level to have thesamesize.Adaptionto the local curvaturecanthenbeachievedby selectively refiningthemeshwith ared-greentriangulationstrategy 31� 32, i.e. we locally switchto higherrefinementlevelsinsteadof adaptingthesizeof thetriangleson thesamelevel (cf. Fig. 3).

Figure 3: Two different strategies for adaptingthe sizeofthetrianglesto somelocal refinementcriterion. Left: thetri-angle size is continuouslyadaptedwithout classifyingtheverticesaccording to their respectivelevel of detail. Right:Adaptionis achievedby locally switching the (discrete)re-finementlevel.ThesurfacegapsresultingfromT-verticesatedgeswhere trianglesfromdifferent refinementlevelsmeet,are fixedby red-greentriangulation.

Sincethe shapeof the trianglesin the basemeshdeter-minestheglobalparameterizationwith respectto which themultiresolutiondecompositionis defined,thenotionof uni-formity (inherent to all subdivision basedmultiresolutionrepresentations)is bestemulatedby equalizingthesizesofthe basetriangles.This is quite different from the generalsettingof thesurfacetesselationproblemwherethedistribu-tion of theverticesis usuallydrivenby thelocalapproxima-tion error in orderto minimizethetotal numberof triangleswhile meetinga prescribedtolerance.

In orderto satisfyboth quality criteria (low local distor-tion anduniformtrianglesize),aremeshingalgorithmhastostartby finding a fairly regularbasemesh

�0 suchthateach

subdivision patch j � m hasapproximatelythesamesur-facearea.Thisguaranteesequalvertex densityontherefinedmeshandhencea uniform samplingdensityof theoriginalgeometry. In termsof surfaceparameterizationit meansthatthelocal stretchingandcontractingis boundedsincethepa-rametertriangleshavesimilarareasaswell. Whenuniformlysubdividing the basemesh

�0, the local distortion can be

further minimizedby applyinga relaxationoperatorwhichtendsto locally equalizethelengthof theedgesandthean-glesbetweenthem(e.g.Laplacesmoothing28, Umbrellaal-gorithm18).

3. Previous work

In 6 aremeshingalgorithmis proposedwhichstartsby plac-ing the verticesof the basemesh

�0 randomlyon the in-

putmesh� . Their localdensityandtheconnectivity of the

basemesh�

0 is controlledby growing geodesicVoronoi-tiles aroundeachvertex. This guarenteesthatthebasemeshis optimalwith respectto theglobaldistributionsinceall tri-anglesof thebasemeshcover approximatelythesameareaelementof the given surface.The parameterizationwithineachtriangleof thebasemeshis foundby solvingadiscreteoptimizationproblemin orderto minimize the local distor-tion. Theresultingparameterizationis optimalfor eachbasetrianglebut notsmoothacrosstheboundarybetweensurfacepatcheswhich belongto adjacentbasetriangles.Theactualremeshingis doneby uniformly samplingthe obtainedpa-rameterizationat thedyadicbarycentriccoordinatesoneachbasetriangle.

In 19 thebasemeshis foundby applyinga meshdecima-tion algorithmto theoriginalmesh.Thisprovidesmorecon-trol onthegenerationof thebasemeshsincefeaturelinescanbetakeninto considerationandlocalcurvatureestimatescanbeusedto determineregionswheremoreor lessverticesareremoved.An initial parameterizationis constructedby incre-mentallyprojectingtheremovedverticesontotheremainingcoarsemesh.This parameterizationis, again,not globallysmoothbut only locally within eachpatch correspondingto one basetriangle.The actualremeshingis not donebydirectly samplingthe initial parameterizationat the dyadicbarycentricparametervaluesbut an additional smoothingstepbasedonavariantof Loop’ssubdivisionschemeis usedto shift thesamplingsiteswithin theparameterdomain.

In bothalgorithmsthesurfacesamplingis doneby solv-ing the point location problemfor the barycentriccoordi-natesin thebasetrianglesandthenfindingthecorrespondingpoint on the original mesh.In somesenseboth algorithmsaremoregeneralthan the shrink wrappingapproachsincethey canprocessmesheswith non-zerogenus.

4. Shrink wrapping

Thephysicalmodelbehindour algorithmfor remeshingar-bitrary triangle meshesis the processof shrink wrappingwherea plasticmembraneis wrappedaroundanobjectandshrunkeither by heatingthe materialor by evacuatingairfromthespaceinbetweenthemembraneandtheobject’ssur-face.At theendof this process,theplasticskin providesanexactimprint of thegivengeometry.

To simluatetheshrinkwrappingprocess,weapproximatethe plasticmembraneby a trianglemesh

�m with subdivi-

sion connectivity. During theprocesseachvertex is movedaccordingto aforceappliedto it. Theforceis acombinationof two components.Onecomponentis pulling eachvertexin the direction of the given surface(attracting force) andtheothercomponentis pushingtheverticesin orderto min-imize the local distortionenergy within the membrane(re-laxing force). Therelaxingforcesupportsthedistributionoftheverticesover thesurface.Without this internalforce lo-cal clusteringof verticesandself-intersectionsof the mesh(folding) couldnot beprevented.

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The proposedapproachis very similar to the conceptofsnakesor deformablesurfacesknown in imageprocessingand computervision 3� 14� 23� 29� 30� 22. Thesetechniquesareusedto extract contoursfrom two- and three-dimensionalimagedataby optimizingtheshapeof a polygonor a trian-glemesh.Theoptimizationis controlledby aninternalstabi-lizationforce(bendingenergy minimization)andanexternalforcepulling theverticestowardsthecontour. Theexternalforceis usuallyderivedfrom thegradientinformationin theunderlyingtwo- or three-dimensionalscalarfield.

We could easily adaptthe deformablesurfaceapproachto our specificsettingby voxelizing the given object 13� 33,i.e. by samplingits characteristicfunctionona threedimen-sionalcartesiangrid. However, this methodis not appropri-ateheredueto theinsufficientspatialresolution.As thequal-ity requirementsfor theremeshingprocedureareratherhighin termsof geometrictolerance,we cannotapproximatethegivengeometryby aniso-surfacecontourof a piecewisetri-linearscalarfield.

We thereforesuggestto replacethe attractingforce by aprojectionoperatorP whichmapseachvertex of theshrink-ing mesh

�m onto the closestpoint of the given geometry

� . Thisoperatorhasasimilareffecton thevertex positionslike imposinganattractingforcebut theverticesareplacedexactlyon thetargetsurfaceandnot only nearby.

The projection is computedby shootingrays in normaldirectionfrom themesh

�m andcomputingtheintersections

with � . Wecanuseaspacepartitiontechniqueknown fromray-tracing9 to considerablyacceleratethisstep.

For theinternalrelaxingforceweusethedensityweightedumbrellaoperatorU 18 which iteratively minimizesthesur-faceareaof themesh(membraneenergy) by updatingeachvertex p accordingto

p � � 1 α � p � α∑ j d j

n � 1∑j � 0

d j q j (1)

with n beingthevalenceof thevertex p andq0 ��������� qn � 1 de-notingits adjacentneighborsin

�m. Thedensitycoefficients

d j aresetto theaveragelengthof theedgesemanatingfromq j respectively. Theeffectof thedensityweightsis thatclus-tering is preventedmoreeffectively andthevertex distribu-tion is performedmoreaggressively. Thedampingfactorisinitially setto α � 1

2 .

Thealgorithmthatsimulatestheshrinkwrappingis anit-erative processwherewe alternatetheP operatorandtheUoperator, i.e.,we alternateprojectionontothetargetsurface� andrelaxingthemesh.By slowly fadingout therelaxingforce(α � 0) from oneiterationto thenext, weguaranteeef-fective vertex distributionduringthefirst stepsandeventualconvergenceof theverticesto locationson � afterwards.

The iterative shrinkingalgorithmcanbe greatlyacceler-

atedby usingamulti-level approach10� 18 for solvingtheun-derlyingoptimizationproblem:Insteadof immediatelypro-jecting and relaxing the mesh

�m, we first apply the al-

gorithm to an intermediatemesh�

i with lower resolution.Oncetheschemeconverges(on the ith refinementlevel), wesubdivide the mesh

�i � � i � 1 andcontinuethe P � U iter-

ation on�

i � 1. The advantageof the multi-level strategy isthatwecanfind coarseapproximationswith meshesof mod-eratecomplexity (wherethecomputationalcostsof P andUare low). Whenwe eventually switch to higher refinementlevels,thecurrentstartingmeshis alreadycloseto thefinalsolution and only few additional iteration stepsare neces-sary.

5. Problems

The simple meshfitting schemeexplainedin the last sec-tion works very effectively on simplesurfaces.However, itdoesnot generatereasonableresultson more complicatedobjects.Onereasonfor the occurringmeshartifactscanbefoundby observingthebehavior of realshrinkwrappingma-terial. In fact,mostpackagesmadeby this techniquesufferfrom severewrinkles.Mathematicallythis effect canbeex-plainedby the fact that the object’s surfaceandthe plasticmembranehave ratherdifferentsurfacemetrics4, i.e., theirfirst fundamentalformsdiffer significantly. Hence,whenfit-ting themembrane,thereareregionswheretheplastichastobestretchedextremelyandregionswherethereis too muchmaterialandhencecreasesaregeneratedfrom superfluousmaterial.

On theshrunksubdivisionconnectivity mesh�

m wehavethesamedifficultieswith thedifferentsurfacemetrics.Thestrong stretchingin some regions results in significantlylarger trianglesand in regions with relatively high vertexdensity, we canobserve creasesandripples(cf. Fig. 4). Asexplainedin Sect.2 wehaveto adaptthebasemesh

�0 to the

shapeof � in orderto avoid theseartifactssinceextremestretchingandfolding (contraction)canonly be avoidedifall surfacepatches j correspondingto the trianglesTj ofthebasemesh

�0 have approximatelythesamearea.

Another severe problem with the projection/smoothingapproachis thephenomenonthatprojectionalgorithmscanfail or at leastproducecounter-intuitive resultsif theshapeof thestartingmeshandtheshapeof thetargetmesharetoodifferentfrom eachother. This is causedby the fact that inthis casethe normalvectorsof the startingmeshmight notintersectwith thetargetmesh.In orderto derive a robustal-gorithmfor theremeshing,we have to constructgoodstart-ing configurationswhich aresufficiently closeto the origi-nal geometry� suchthat theprojectionoperatorbecomeswell-definedandstable.

In the next two sections,we presentsolutionsfor theseproblemswhichextendthebasicshrinkwrappingalgorithm.Thekey ideais to find aparameterizationof thegivenmodel

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Figure 4: Behindtheears of the Stanford bunnyis a concaveregion where excessmaterial causeswrinklesin the shrinkingmembrane. Adaptingthe basemesh

�0 to thesurfacemetric of the givenmesh� removesthis artifact. (From left to right:

original mesh,non-adaptedmembrane, adaptedmembrane)

over theunit sphere.Insteadof usingtheoriginalmodel,thenecessaryoperationsfor the shrink wrappingcan then beperformedon thespherewhich is easiersincetheprojectionoperatorP becomestrivial.

6. Constructing the base mesh

The processof ”custom tayloring” the initial mesh�

0 fora givengeometry� requiresto find a coarsetriangulationwhich partitionsthe mesh � into triangularpatcheswithequalsurfacearea.This guaranteesan approximatelycon-stantstretchingfactorfor themap

�0 ��

m.

While thetechniquesproposedin 6� 19 operatedirectly onthe given trianglemesh � , we solve this problemby firstcomputingaparameterizationfor � overaboundingsphere(cf. Fig. 5). Thiswill make it easierto controlthenumberoftrianglesin thebasemesh

�0 whichwewantto keepassmall

aspossible.A collectionof algorithmshow to computealowdistortionprojectionmapfrom an arbitrarygenus-zeroob-ject to a boundingspherecanbefoundin 15. Thebasicideaof thesealgorithmsis to let theobjectgrow towardsits con-vex hull andthenprojectit directlyontothesphere.Withoutlossof generality, we canassumetheboundingsphereto betheunit sphere.

In our implementationwe usea variationof this princi-ple.Wefirst computeavoxelizationof thegivenmodel.Thegradientof the correspondingvolumetricdistancefunctionis thenusedto guidetheverticesof thegivenmeshtowardsa convex configuration.This processcan be consideredassomekind of ”inverse”deformablesurfacetechnique.

A parameterizationF of the mesh � over a boundingsphereis given by projectingevery vertex p �� onto apoint p � on the sphere.Using the connectivity of � , thepointsp � definea sphericalmesh � � wherethe edgesarereplacedby circular arcs.Evaluating the parameterizationF at somepoint q � on the sphererequiresto first find thatsphericaltriangle � � � A� � B� � C � � of � � which containsq � .Theprojectionq � of q � ontothecorrespondingplanetriangle

��� A� � B� � C ��� canbeexpressedin barycentriccoordinates

q � � αA� � βB� � γC � � α � β � γ � 1 �Applying thesamebarycentriccombinationto theassociatedverticesA, B, andC of theoriginalmesh� finally yieldsthefunctionvalueq � F � q ������ .

In orderto make the parameterizationwell-defined,it isnecessaryto guaranteethat the projectionoperator ������ doesnot flip any triangles.Moreover, for numericalro-bustnessof the evaluation routine, the projection methodshouldboundtheaspectratio of thetrianglesin � � .

Once we have the parameterizationover the boundingsphere,we constructthe basemesh

�0. Our primary goal

is to usea minimum numberof triangles.We start with afairly regularconvex polyhedron� whoseverticeslie onthesphere(e.g.anicosahedron).Thesegmentationof thesphereinducedby the correspondingsphericalpolyhedron��� par-titions thetrianglesof � � into disjoint subsets.For eachofthesesubsets,we sumup thesurfaceareaof theassociatedoriginal trianglesin � � F �!�"�#� . By thisweassignanesti-matedsurfaceareato every sphericaltriangle(to every cell)in � � .

We canevaluatetheparameterizationF at theverticesof� to obtainthebasemesh

�0 � F �$�%� . Thesurfaceareasas-

sociatedwith thecellsof � thenprovide goodestimatesforthesurfaceareasof thesubdivision patchesassociatedwiththecorrespondingbasetriangleof

�0.

Sincethe basetrianglesof�

0 arelarge comparedto thetrianglesof � , we can neglect thosetriangleswhich aremappedontooneof theedges(greatcircles)of thesphericalpolyhedron��� .

We perform a simpleoptimizationprocedurewhich ap-plies relaxationand split operationsto � . The goal of therelaxationis to move the verticesin order to equalizethesurfaceareasassociatedwith eachcell. The split operationsupportsthis egalizationby introducingnew verticesin re-gionswith highsurfaceareapercell.

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Figure 5: An arbitrary genus-zero triangle mesh� can be parameterizedover the sphere by projecting its vertices.Sincebothmesheshavethesameconnectivityit is easyto establisha one-to-onecorrespondencebetweenthecontinuoussurfacesbybarycentricparameterizationof thetriangles.

The relaxationoperatoris a variantof the weightedum-brellaoperator(1). In this casethedensityd of a vertex p inthemesh� is setto thesumof thesurfaceareasassociatedwith its adjacentcells. Performingseveral iterationsof (1)quickly convergesto a stableconfigurationwith all surfaceareasbeingapproximatelyequal.Theparameterα actslikeadampingfactorto preventdegenerateconfigurations.Noticethattheverticesof � haveto beprojectedbackto thebound-ing sphereafter every relaxationstepandsurfaceareaspercell have to berecomputed.

If thesurfaceareavariesdrasticallyamongthecellsin theinitial mesh� , theresultof therelaxationcanhave stronglydistortedtriangles.To avoid these,weapplyasplit operationwhen the varianceof the associatedareasis too big. Thefollowing heuristicleadsto satisfactoryresults:Wepick thatedgefor which thesumof theareasassociatedwith its twoadjacentcells is a maximum.Insertingthemidpoint of thatedgetriggersa bisectionof thetwo adjacentcellsandhencegeneratesfour cellswith smallersurfacearea.

After several stepsof this procedurewe have a sphericalpolyhedron � � on the boundingspheresuch that the sub-meshesof � which are associatedwith the facesof thecorrespondingbasemesh

�0 � F �$� � � accordingto the pa-

rameterizationF : � � �&� have approximatelythe samesize.Thebasemesh

�0 is thenusedasthe initial meshfor

themulti-level shrinkwrapping,i.e.,thealternatedensityre-laxation,projection,anduniformsubdivision.

7. Stable projection

The seconddifficulty that madeour primitive shrink wrap-ping algorithmfail on complicatedmodelsis theprojectionof the verticesof

�m (or�

i during the multi-level shrinkwrapping)onto � . Thisprojectionfailsif theshapesof bothmeshesare too different from eachothersuchthat normalraysof

�i do not intersectwith � . To avoid suchproblems

we have to find goodstartingshapesfor the meshes�

i inorderto make theprojectionsave.

We do this by exploiting the sphereparameterizationF : � � ��� of the input mesh � that we alreadyusedfor thegenerationof thebasemesh

�0. Sincetheverticesof

all intermediatemeshes�

i lie onthesurface� , they inherit� ’ssphereparameterizationin auniqueway:For eachver-tex p � i we find thecorrespondingpoint p � on thebound-ing sphereby first representingp in barycentriccoordinateswith respectto a triangle �'� A � B � C � of � andthenapply-ing the samebarycentriccombinationto the correspondingtriangle �'� A� � B� � C � � of � � .

Let p beavertex of�

i andq0 �������(� qn � 1 itsadjacentneigh-bors.By samplingthesphereparameterizationof � weob-tain thecorrespondingpre-imagesp � andq �0 ��������� q �n � 1.

The reasonfor the instability of the smooth/projectit-erationis that after applying the umbrellarule (1) the re-projectionof the updatedvertex p may fail. We avoid thisproblemby deriving a modifiedumbrellarule which is ap-plied to the pre-imagesinstead.The projectionof the up-datedpre-imagep � backonto the boundingsphereis trivaland always works correctly. Finally, evaluatingthe sphereparameterizationF at the resultinglocation yields the up-datedpositionp which automaticallylies on the surfaceof� . Hence,by doingtherelaxationstepin theparameterdo-main �)� , we solve the problemof relaxing the mesh

�i

while keepingtheverticeson thegeometryof � in a moreelegantfashion.

It is obvious that thequality of the relaxationstepin theparameterdomainstronglydependson thedistortionof themap � � �)� . With thetechniqueproposedin 15 we min-imize this distortionbut we still have to solve the problemthat the metric of the given surface � and its boundingspherediffer considerably. To copewith this situation,wemodify the umbrellarule suchthat the weight coefficients

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reflectthesurfacemetricon � while it is appliedto pointson ��� .

Therearetwo differentwaysto accomplishthis.Thefirsttechniquehasalreadybeenusedfor thegenerationof theop-timized basemesh

�0 in the last section.We usea density

weightedumbrellarelaxationwherethedensitycoefficientsfor eachvertex arefoundby summinguptheassociatedsur-faceareasfor all adjacenttriangles.

As the verticesaremoving on the boundingsphere,theassociatedsurfaceareashave to be recomputedafter everyiteration.In Sect.6 the areaestimationhasbeensimplifiedby ignoringthosetrianglesof � � whichareintersectedby acell boundaryof F � 1 � � 0 � . This simplificationis acceptableas long as the cells are large comparedto the size of theindividual triangles.However for higher refinementlevels,it may occur that somecells of F � 1 � � i � areempty. If thishappens,weswitchto thesecondrelaxationtechniquewhichis still a variantof the densityweightedumbrellabut nowthe densityvaluesd aresimply obtainedby computingtheaverageedgelengths* F � p � �+ F � q � �,* ontheoriginalsurface� (just like in (1).

With this modified relaxing rule, we can distribute themeshverticesp � on the boundingspherewhile in fact us-ing the surfacemetric of the original input mesh � . Theoverallprocedurefor computingthemesh

�m is describedin

Sect.4 asa multi-level processwherewe startwith theini-tial basemesh

�0. We subdivide to obtain

�1 andperform

alternateprojectionandsmoothinguntil we reacha stableconfiguration.Thenwe subdivide againto go to thenext re-finementlevel. In general,we usethe result from level i tofind agoodstartingmeshfor level i � 1. Thisacceleratestheconvergencesincemuch fewer iterationsare necessaryoneachlevel.

As the projectiononto the meshworks savely only forhigher refinementlevels (whereoriginal geometry� andremeshedgeometry

�i are sufficiently close),we use the

modifiedrelaxingrule for the lower levels.Hence,we per-form theshrinkwrappingonthemeshes

� �0 �������-�� �i in thepa-

rameterdomain�)� . Oncetheapproximationis tightenoughto make theprojectiononto � stable,we switch to

�i and

proceedwith�

i ����������

m on � .

In ourexperiments,wefoundthedecisionwhento switchfrom the relaxing in the parameterdomainto the relaxingon theoriginalgeometryratherdifficult andstronglydepen-denton the actualmodel.The two relevant criteria arethatweshouldnotswitchtooearlyto avoid wrongprojectionbutwe alsoshouldnot switch too late sinceotherwisethefinalmeshquality is notoptimalor hasto beimprovedby a largenumberof relaxingsteps.This diminishesthe performancegain of the multi-level strategy. The reasonfor this behav-ior is that themetric in 3-spaceandthesimulatedmetric inthe parameterdomainare similar but not exactly identicalandhencethe relaxationon the boundingsphereis a goodapproximationof theoriginal operatorbut notoptimal.

8. Results

Weimplementedtheremeshingschemeandappliedit tosev-eralpolygonalmodels.Theresultsareshown in theFigs.6–8. Noticetheextremly smallnumberof trianglesin thebasemesh.This is achieved by the optimization techniquede-scribedin Sect.6.

8.1. Progressive transmission

With theincreasingimportanceof distributeddatabases,theneedfor effective algorithmswhich enablethetransmissionof geometricinformation via low-bandwidthdataconnec-tions(like theinternet)is becomingparamount.For unstruc-turedtrianglemeshes,theeminenttechniqueof progressivemeshes11 hasestablishedadefactostandardwhichprovidesdatareductionpotentialof several ordersof mangitudefortheinitial shapewhich canbedisplayedimmediatelyby thereceiver. As moredetailinformationis arriving, thegeomet-ric modelcanberefineduntil thecompletemodelhasbeentransmitted.A slight disadvantageof this techniqueis theblocky natureof thecoarsemeshmodels(from anestheticalpoint of view) andthelackingrobustnessagainstthelossofdatapackages.

In 1� 21� 25, wavelet techniqueshave beenexploredfor theincrementaltransmissionand load adaptive displayof tex-tured geometricmodelswith subdivision connectivity. Inthesepapers,thecompressionratiosobtainedfor particularchoicesof thescalingfunctionsφi andthewaveletbasisψi . jarediscussed.

Theconversionof arbitrarygeometricmodelsinto mesheswith subdivision connectivity enablestheeffective progres-sive transmissionof surfacegeometrysincea smoothap-proximation of the original surface can be reconstructedfrom a rathersmallamountof data.We implementeda sim-ple methodfor the progressive transmissionof subdivisionconnectivity meshesbasedon theLoop subdivision scheme20 and the Butterfly algorithm 5� 34. We transmit the basemeshfirst suchthat the receiver can reconstructa smoothmodelby applyingseveral subdivision steps.As moreandmoredetail information is transmitted,the meshis refinedadaptively. Themeshesvertices,for which thedetailcoeffi-cientshave not beenreceivedyet, areplacedat thepositionpredictedby thesubdivision rule.

It turnsout thatwe obtaina ratherconvincing reconstruc-tion with only a minimal numberof detail coefficients in-cluded into the model. Fig. 9–11 show several examples.Noticethatprogressive transmissionof subdivision surfacesis robustagainstthe lossandpermutationsof datapackagessincethe detail coefficients are indexed independentfromeachother.

9. Conclusion

Wepresentedaveryeffectiveapproachfor convertingpolyg-onalsurfacesinto mesheswith subdivisionconnectivity. Our

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algorithm is basedon a simulationof the shrink wrappingprocess.By usinga low-distortion parameterizationof theoriginal modelover a boundingsphere,we areableto findvery coarsebasemeshes

�0. The sphereparameterization

alsoenablesa stablemulti-level relaxationtechniquewhichleadsto goodstartingmeshesfor theactualshrinkwrappingalgorithm.This is necessaryto make theprojectionstepro-bust.

Thealgorithmis quitedifferentfrom previousapproachessincewedonotconstructanexplicit parameterizationwhichis sampledat the dyadic barycentricparametervaluesoneachbasetriangle.Insteadwe distribute the meshverticesevenly over the given geometryby an iterative relaxationprocess.

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Figure 6: RemeshingtheStanford bunny. Theoriginal datasethas70K triangles.Our remeshingalgorithmgeneratesa basemeshwith 38 triangles(left). Thecenterandright image showsthe3rd andthe5th refinementlevel.

Figure 7: Theoriginal bustmodelhas61K triangles.Thebasemeshwith 72 trianglesis subdividedthreetimesto generatethecentermeshand5 timesto generatetheright image.

Figure 8: A brain datasetextractedfromCT volumedatais remeshed.Thebasemeshhasonly 20 triangles.In thecenter, the4th refinementlevel is shownandthe6th level on theright side.

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Figure 9: Progressivetransmissionof thebunny. Theleft image showstheLoopsubdivisionsurfacegeneratedfromthebasemeshwhich is transmittedfirst.Theothermodelsare obtainedafter including1%,3%,and10%of thedetail coefficients.

Figure 10: ProgressiveTransmissionof thebustmodel.Fromleft to right: 0%,1%,3%,and10%of thedetail coefficients.

Figure 11: ProgressiveTransmissionof thebrain model.Fromleft to right: 0%,1%,3%,and10%of thedetail coefficients.

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