Ellipsoid decomposition of 3D-models

StephanBischoff Leif Kobbelt

ComputerGraphicsGroupRWTHAachen,Germany

Abstract

In this paperwe presenta simpletechniqueto approximatethe volumeenclosedby a given trianglemeshwith a setofoverlappingellipsoids. This type of geometryrepresenta-tion allows usto approximatelyreconstruct3D-shapesfroma very small amountof informationbeingtransmitted.Thetwo centralquestionsthatwe addressare:how canwe com-puteoptimalfitting ellipsoidsthatlie in theinteriorof agiventriangle meshand how do we select the most significant(leastredundant)subsetfrom a hugenumberof candidateellipsoids.Ourmajormotivationfor computingellipsoidde-compositionsis therobusttransmissionof geometricobjectswherethereceivercanreconstructthe3D-shapeevenif partof thedatagetslostduringtransmission.

1 Intr oduction

Today the mostcommonrepresentationfor 3D objectsarepolygonmeshes.They areflexible enoughto approximatearbitraryshapesand the complexity of their processingin-creaseslinearly with the numberof polygonsbut is inde-pendentfrom thestructuralor topologicalcomplexity of theobject.Thereis ahugebodyof literatureaboutvarioustech-niquesfor the generation,modification,storage,transmis-sion,anddisplayof polygonalmodels[6, 21, 22, 20, 23].

Although polygons,i.e. piecewise linear surfaceshaveapproximationpropertiesthataresufficient for mostgraph-ics applicationsit turnsout thatrealisticmodelsstill requireup to severalmillions of triangles.This is why hierarchicalmodelsandmultiresolutionrepresentationshave beenintro-ducedinto computergraphics[9, 11, 20]. The basicobser-vation hereis that a coarseapproximationof a surfacecanalreadybeobtainedwith a surprisinglysmallnumberof tri-

angles[10, 18, 19]. If weusemoretriangles,theapproxima-tion getsbetterandbetterbut theefficiency, i.e., thequalitygainperpolygonis quickly decreasing.Progressive meshesexploit this effect by storingvery coarsebasemeshesplusa sequenceof refinementoperations[7, 12, 15]. Any prefixthetransmittedsequencethissequenceallowsthereceivertoreconstructan approximationof the original meshand therepresentationis progressive in the sensethat the most im-portantshapefeaturesshow up very early in the sequencewhile thebig numberof lessimportantdetailsfollow laterinthesequence.

Onedrawbackof polygonmeshes— even in their hier-archicalor progressive representation— arethestrict topo-logical consistency requirements.If we loseonepercentofthemeshdatathenit mightbecomeimpossibleto reconstructany partof thesurfacefrom theremaining99 percent.Thisis mostlydueto thenecessarycrossreferencesbetweenfacesandverticesandcanbesolvedonly by introducingahighde-greeof redundancy into therepresentation.Thisobservationmotivatesthe investigationof robust transmissionschemeswhich allow thereceiver to reconstructmostof thegeomet-ric informationevenif acertainpercentageof thedatais lostduringtransmission.

The key to robust transmissionis to divide the geom-etry into independentchunksof information (e.g. singlepoints) which allow the receiver to reconstructthe mani-fold neighborhoodrelation without relying on indexing orexplicit inter-vertex referenceinformation.In orderto avoidcomplex algorithmsfor generalsurfacereconstructionfrompoint cloudson thereceiver’s side,earlierapproachesto ro-bust transmissionassumedthat at least the global surfacetopologycanbetransmittedlosslessly[5]. Oncethis coarseshapeinformation is known, the receiver can insert addi-tional points into the meshbasedon spatialproximity (cf.Fig. 1 andFig. 2).

In this paperwe wantto extendthis earlierapproachsuchthatthecoarseshapeinformationcanbetransmittedrobustlyas well. The generalapproachis to approximatethe vol-umeenclosedby thegivenpolygonmeshwith a setof over-lapping ellipsoids. Theseellipsoidsrepresentindependentpiecesof geometricinformationandeachellipsoiddefinesitsown partof thegeometry. Redundancy is introducedby thefact that the ellipsoidsoverlapeachother. Hencethe over-all shapeandespeciallythesurfacetopologywill notchange

Figure1: In orderto avoid that therobusttransmissionbecomesascomplicatedasthegeneralsurfacereconstructionproblemfrom point clouds,thesender(who knows theoriginal mesh)providesadditionalinformationto thereceiver which simplifiesthe reconstruction(hintedreconstruction). In this casethe senderprovidesa coarsebasemeshto convey the global surfacetopology. All othermeshverticesaresentwithout any connectivity information. Thereceiver canapproximatelyreconstructtheorignalshapeby insertingtheverticesinto themeshbasedon spatialproximity.

Figure2: If in a robusttransmissionschemeverticesarere-ceived without any connectivity information, they aresim-ply insertedinto thenearesttriangle.In orderto preservethemeshquality at all stages,every vertex insertionis followedby edge-flippingoperationsin orderimprovethevalencebal-anceor theaspectratioof thetriangles.

if only a few ellipsoidsaremissing. The redundancy doesnot significantly increasethe storagerequirementssinceanellipsoidcanbedefinedwith only 9 scalarvalues.Thiscorre-spondsto thestoragerequirementsof onevertex in ashared-vertex trianglemeshrepresentation1.

Thepaperis organizedasfollows: in thenext sectionwedescribethemainstepsof ourrobusttransmissionprocedure.Thenwe explain the detailsof our optimal ellipsoid fittingtechniquein Section3. Finally we discussresultsandfutureapplicationscenariosin Section4.

1Every vertex requires3 coordinatevalues,eachtriangle3 vertex in-dices.Sincetherearetwice asmany trianglesasvertices,we have 9 valuespervertex.

2 Robust geometr y transmission

For robusttransmissionwe have to guaranteethatwhenevera part of the geometryinformationgetslost, the remainingportionof thedatacanbeusedto reconstructat leasta goodapproximationof the object,i.e., the approximationqualityslowly degradeswith an increasingpercentageof lost data.Our basicidea is to decomposethe geometricshapeinto asetof ellipsoidsrepresentingthe coarseshapeof the objectplusa setof independentsamplepointswhich representthefinedetail.

Theellipsoidsaregeneratedsuchthatthey completelyfillout theinteriorof theobject,thesetof samplepointsis sim-ply thesetof originalmeshvertices.

Therobusttransmissionprocedureworksasfollows: Firsttheellipsoidsaretransmitted.Sincethey aregivenimplicitlyby asymmetric4 � 4 matrixQ, normalizedsuchthatxTQx �0 for exteriorandxTQx � 0 for interiorpointsx, it is easytocollecttheirshapeinformationin aspatialdistancefield. Forthis we definea sufficiently fine spatialgrid of scalarvalueswhich is initializedto � 1 everywhere.Whenanew ellipsoidQ is received, we computeits boundingbox and evaluatethe implicit functionxT Qx at all thegrid-pointswithin thatbox. Thegrid valuesareupdatedif thenew functionvalueissmallerthanthecurrentgrid value.

When all ellipsoids are received, we extract the zero-surfacefrom the spatialscalarfield by applyingthe march-ing cubesalgorithm[3, 4]. Obviously themethodstill workseven if someof the ellipsoidsare dropped. The extractedsurfacewill not changeits topologyaslong asthe remain-ing ellipsoidsarestill overlapping.Moreover, if themutualoverlapis strongenough,thelossof asmallportionof theel-lipsoidswill not affect thereconstructedshapesignificantly.

Oncewehavegeneratedaninitial mesh,westartinsertingthesamplepointswhich aretheoriginal meshvertices.The

detailsof this procedurearedescribedin [5]. All we needis agood� initial surfaceapproximationandthenmeshrecon-structionbasedon vertex insertionandedgeflipping worksrobustly. To increasethequality of thefinal result,we haveto eventuallyeliminatethosemeshverticeswhichhavebeengeneratedby themarchingcubesalgorithmsincethey arenotpartof theoriginalmeshgeometry.

3 Ellipsoid decomposition

Theproblemof coveringthevolumein theinteriorof agivenpolygonmeshis not well-definedsince,obviously, therearemany different possibleellipsoid decompositions.Our al-gorithmis designedto find onecandidateamongthis multi-tudeof decompositionsthatis locally optimalwith respecttoshape,orientationanddistribution of theellipsoids.A localoptimumin thiscontext meansthatwecheckafinite numberof configurationsandsearchfor thebestoneby somegreedyoptimizationscheme.

3.1 Ellipsoid fitting

Fitting ellipsoidsto a givenvolumeis not easy—evenmuchsimpler problemslike e.g. finding the smallestenclosingspherearemathematicallyinvolved [17]. In a first attemptto fit ellipsoidsto the volumeboundedby a polygonmesh,onemightpick arandomseedpoint in theinterior, determinethreeorthogonalprincipal axesandthengrow the ellipsoidalongtheseaxesuntil it touchesthe mesh. However, thereareseveralcritical drawbacksin this straightforwardproce-dure.

First, randomdistribution of seedsamplesin the interiordoesnot take theglobalshapeof the3D objectinto account.For exampleif wehavea largeconvex shapewith somelongand thin featurepeekingout (cf. Fig 3) then the randomdistribution will pick seedpointswith a very high probabil-ity in the convex region of the object. Seedsamplesin thethin featurea very unlikely sincethe relative volumeis toosmall. Neverthelessthis small featurecarriesa significantvisualandgeometricalinformationandhenceit is moreim-portantthanan accurateapproximationof the convex part.Consequentlyit makesmoresenseto placetherandomseedpointson the surfaceof the polygonmeshandto grow theellipsoidsinto theinterior.

Thesecondproblemis that it is almostimpossibleto de-terminea goodchoicefor theprincipaldirectionsat a pointin the interior. For cylindric shapes,e.g.,we would like toalign oneprincipaldirectionto thecylinderaxis.This,how-ever is difficult to tell for a point lying in the interior of themesh. We could usea variantof the medialaxis transformandlocaldistancefield informationto optimizetheprincipalaxisorientationbut estimatingthemedialaxisof a polygonmeshis computationallycomplex andproneto numericalin-stabilities.

Figure3: A large convex objectwith a small feature. Vol-umetric randomsampleshit the featurewith a very smallprobabilitywhile randomsurfacesamplesaremorelikely tohit.

Again,startingtheellipsoidgrowth on theboundarysim-plifiesthesituation:Wecanstartgrowinganellipsoidfrom arandompoint p on theboundaryuntil it touchesanothersur-facepoint q. Thengrowth is continuedinto thedirection(s)orthogonalto the axis pq until a third point r is found. Fi-nally we grow in theorthogonaldirectionto theplanepqr.

3.2 Ellipsoid growth strategy

Theellipsoidgrowth startingfrom a surfacepoint p is nowdescribedin moredetail. We begin with theestimationof anormalvectornp at p. This is easyno matterif p lies in theinterior of a triangularface,on anedge,or coincideswith avertex of the mesh.We definea sphereQ1

�r � with radiusr

whosecenteris cr � p � rnp suchthat it touchesthe pointp. Thenwe increasethevalueof r until it touchesa secondsurfacepoint q.

To determinetheoptimal r we first computefor eachver-tex x theradiusr

�x � of thespheredefinedby theinterpolation

conditionsx, p andthenormalnp

r�x � � ��

x p��� 2

2�x p � Tnp

Thenwe set

ropt � min � r � x � � � x p � Tnp � 0 �After wefoundthelargestinscribedsphereQ1

�r � with the

two contactpointsp andq we continuegrowing in orthogo-naldirectionto theaxispq. For this we re-writethequadric

n�q �

nqp p p

Figure4: Left: Ellipsoid growth is impossibleif trianglesinsteadof verticesare testetfor intersection. Middle: Ellipsoidnormaln

�q � andmeshnormalnq do nothave to agree.Right: A smalloffsetenablesasignificantellipsoidgrowth.

Q1�r � asa 4 � 4 matrix:����� 1 0 0 cr � x�

0 1 0 cr � y�0 0 1 cr � z� cr � x�� cr � y�� cr � z� cr � x� 2 � cr � y� 2 � cr � z� 2 r2

������suchthat � x;1� T Q1 � x;1� � 0 implicitly definesthesphere.

Basedon the normalvectorsnp � c p andnq : � c qwedefineanotherquadricQ2 which is theproductof thetwotangentplanesat p andq respectively.

Q2 :�x p � T np � � q x � T nq � 0

Again, we re-write Q2 asa 4 � 4 matrix Q2 � �N � NT ��� 2

whereN is givenas�����! np " x# nq " x# np " x# nq " y# np " x# nq " z# np " x#%$ nTq q & np " y# nq " x# np " y# nq " y# np " y# nq " z# np " y#%$ nTq q & np " z# nq " x# np " z# nq " y# np " z# nq " z# np " z#%$ nTq q &

nq " x#%$ nTp p & nq " y#'$ nT

p p & nq " z#'$ nTp p & $ nT

p p &($ nTq q &

)+***,SincebothquadricsQ1 andQ2 aretouchingthetwo points

p andq with the correspondingnormalvectorsnp andnq,thesameis truefor all linearcombinations[1, 2]

Q1 � αQ2 (1)

Hencewe canusethecoefficient α to continuegrowing ourellipsoid. With increasingvalueα the ellipsoid will stretchwithin thewedgedefinedby thetwo tangentplanesat p andq (cf. Fig 5). As longasα staysbelow somethresholdα0 theresultingquadricwill still beof theellipsoidtype(andnot aparaboloidor hyperboloid). We cancheckthe type of thequadricduring the algorithmby computingthe eigenvaluesof the upperleft 3 � 3 sub-matrix: If all threeeigenvaluesarepositive, thequadricis of ellipsoidtype. To computetheoptimalα weobservethatanarbitrarypointx lieswithin theellipsoiddefinedby Q1 � αQ2 if andonly if

xT � Q1 � αQ2 � x � xTQ1x � αxTQ2x � 0

If α is positive,xTQ2x � 0 is a necessaryconditionfor x tolie within Q1 � αQ2 andhencewe find

α �- xTQ1xxTQ2x � : α

�x � (2)

p

q

r

sQ1

q

r

s

Q1 � αQ2

q

r

s

Q1 � αQ2 � βQ3

q

r

s

Figure5: Ellipsoid growth strategy: Startingout at a singlevertex p (upperleft, on top of pyramid)theellipsoidgrowthalgorithmsearchesin normaldirectionfor thebiggestemptysphereQ1 until it touchesthe vertex q (upperright). Nextthesphereis extendedto anellipsoidby equation(1) until ittouchesa third point r (lower left). Thethreecontactpointsp . q . r determinean ellipseE on the ellipsoid (lower right,black line) which servesasprofile curve of thecylinder Q3.Finally theellipsoidis elongatedalongthecylinderby equa-tion (3) until a fourthpoint s is touched(lower right).

Theoptimalα is easilycomputedas

αopt � min � α � x � � xTQ2x � 0 �By growing α we eventuallyfind a third touchingpoint

r. In cylindrical partsof themeshmodelthethreepointsp,q, andr have a high probabilityof lying in a planewhich isapproximatelyperpendicularto thecylinder axis(cf. Fig 5).Hencewecanfind acigar-shapedfitting ellipsoidbygrowinginto the directionorthogonalto the planespannedby p, q,andr.

Figure6: Upper left: Original horsemodelconsistingof about49000vertices. Upper right: Horsemodel reducedto 200verticesby GarlandandHeckbert’serror-quadricdecimationalgorithm[10]. Lowerrow, left to right: Ellipsoidapproximationsconsistingof 100,200and400ellipsoids(3600,7200and14400bytes)respectively. Notethemuchimproveddetailcomparingthe200ellipsoidmodelto thememory-wiseequivalent200verticesreducedmodel(7200bytes).

Thesimplestsolutionwould beto defineanotherquadricQ3 andto grow thecurrentellipsoidby findingthemaximumβ for which

Q1 � αQ2 � βQ3 (3)

liescompletelyin theinteriorvolumeof themeshandis stillof the ellipsoid type. However, as it turnsout, we cannotusethe sametechniqueasabove sincenow we have threetouchingpoints and tangentplanesand theseinterpolationconstraintsuniquelydefineasinglequadricsuchthatthereisno moredegreeof freedom.

We thereforeuse anotherheuristic to define the thirdquadricQ3: Let the planethat containsthe threetouchingpointsp, q, andr begivenin parametericform by

x�u . v� � u

�q p �/� v

�r p �/� p 0 (4)

The intersectionof this planewith the quadricQ̄ : � Q1 �αQ2 is anellipseE in the

�u . v� -parameterspace.E is given

by a3 � 3 matrix

E �21 q p r p p0 0 1 3 T

Q̄ 1 q p r p p0 0 1 3 0

For thisellipsewecanfind the�u . v� -coordinatesof thecen-

terby solving

E

�� uv1

�� � 0

andwe find the�u . v� -coordinatesfor thetwo principalaxes

andcorrespondingradii by computingthe eigenvectorsandeigenvaluesof the upperleft 2 � 2-submatrixof E. Finallywe transformthe center, axes,andradii from the

�u . v� pa-

rameterspaceinto world coordinatesby using(4).The above constructiongivesus a planarellipseE in 3-

spacearoundthecentera0 with two principalaxesa1 anda2

andthecorrespondingradii r1 andr2. WedefineQ3 to betheelliptic cylinder that containsE andhasa3 � a1

� a2 asitsmainaxis. This cylinder containsall pointsx for which thefollowing equationholds:� � x a0 � T a1 � 2

r21

� � � x a0 � T a2 � 2r22

� 1 0We caneasily find a 4 � 4 matrix representationof Q3 byfactoringout the above equation.Oncewe have this repre-sentation,wecanproceedtheellipsoidgrowing processwith

respectto theβ parameterin (3) analogouslyto thedetermi-nation4 of theα parameter.

As we will demonstratein Section4, this ellipsoidgrow-ing procedureinducesagoodalignmentof theprincipalaxesto themainaxesof theenclosedvolume.

Remark When implementingthe above ellipsoid fittingprocedurethereare somedifficulties which arisefrom thefact that polygonmeshesdo not have a continuousnormalfield andhencea sphereor ellipsoid cantouchthe meshata surfacepoint q while the surfacenormaln

�q � andellip-

soidnormalnq do not necessarilyagree.This canleadto alocking situationwherethe ellipsoid cannotgrow anymorealthoughsomelocal ε modificationwouldenablesubstantialgrowth (Figure4).

To take this into accountwe modifedtheaboveconstruc-tion slightly in our implementation. Insteadof using theexact touchingpointsp, q, andr we usethe pointsp 5 , q 5 ,andr 5 which areobtainedby shifting theoriginal pointsbysomesmall offset ε in normaldirection into the interior ofthe mesh. This providessomeadditionalflexibility for thedeterminationof thecoefficientsα andβ.

3.3 Ellipsoid selection

We apply the above constructionto generatean extremeover-representationof the interior volumeof a given poly-gonmesh.Sincethecomputationof thegrowingellipsoidsisquitefastwecan,e.g.,computeafitting ellipsoidfor amod-eratelycomplex polygon meshby placing a seedpoint onevery vertex. Even thoughtheseellipsoidswill be stronglyoverlappingwe do not increasethe total memoryrequire-mentsbecause,aswediscussedin theintroduction,acollec-tion of n ellipsoidsrequiresthesameamountof storageasatrianglemeshwith n vertices.

To reducetheredundancy of the ellipsoiddecompositionweselectthemostsignificantonesfrom thesetof candidatesgeneratedin theinitializationphase.For this weusea rathersimplegreedyoptimization. We startwith the largestellip-soid,i.e.,with theellipsoidthathasthelargestvolume.Thenin every stepwe includethatellipsoidwhich addsthe mostto the total volume of the overlappingellipsoids. If thereare several ellipsoidscontributing approximatelythe sameadditionalvolume, we selectthe one that hasthe smallestminimumradius.

The only computationallyexpensive taskin the ellipsoidselectionprocessis the computationof the volume contri-bution of the individual ellipsoids. In every selectionstepwe have thesetof earlierincludedellipsoidsE1 .�060�0�. Ek andfor every new candidatewe have to estimatethe differencevolume

V � Enew 7 � E1 8 0�060 8 Ek �90Wesolvedthisproblembycomputinganapproximationovera spatialgrid of binary voxels. Whenever an ellipsoid is

addedto the selectionset,we rasterizeit andflag all vox-els the fall inside it ascovered. Thenwhenwe testa newcandidate,wealsorasterizetheellipsoidandcountthenum-berof uncoveredvoxelsto obtainanestimateof thevolumeincreaseV. This greedycandidateselectionschemeis eas-ily implementedby a heapdatastructure.All ellipsoidsaresortedin theheapaccordingto theirvolumecontribution. Onpoppinganellipsoidfrom thetopof theheapwere-calculatethevolumecontributionof all theintersectingellipsoidsandrebuild the heap. This techniquecan further be improvedby just flaggingthe volumecontribution of intersectingel-lipsoidsas invalid. Eachtime an elementis moved to thetop of the heap,we checkwhetherits volumecontributionis still valid. If not, we re-calculatethe volumeandrebuildthe heap. Using this “lazy evaluation” techniquekeepsre-dundantrasterizationstepsat a minimum und substantiallyspeedsup theselectionalgorithm.

4 Results and future work

In this sectionwe demonstratethe effectivenessof our ap-proachby applyingit to someof thestandardpolygonmeshdatasetsin thegraphicscommmunity.

4.1 Ellipsoid decomposition

Figure6 shows the resultsof applyingour algorithmto thewell-known horsemodel. Theoriginal meshcontainsabout49000vertices.In orderto avoid ellipsoidgrowth beingre-strictedby the immediateneighboringvertices(1-ring), wesmoothedthe meshin a preprocessingstep. Thenwe ran-domly selected2500seedverticesandappliedour ellipsoidgrowth algorithm(ε � 0 0 1) (seeremarkin section3.2). Thecomputationof the ellipsoidstook only a few seconds.Fi-nally we reordedthe ellipsoidsaccordingto their volumecontributionsusinga100 � 100 � 100voxelgrid asdescribedin theprevioussection.Thislaststeptookabouthalf anhour,but notethattheseareoff-line preprocessingtimings.

4.2 Robust transmission

Figure7 depictsa typical reconstructionsequence.First theserver transmitsthebasegeometryandtopologyby sendinga numberof ellipsoids.Theclient mayalreadyrendertheseellipsoidsasa first approximationof the final model. Thentheclientextractsa trianglemeshfrom theellipsoidapprox-imation,by e.g. a marchingcubesalgorithm[3] or by somekind of shrink-wrappingapproach[8]. Becauseof the pre-smoothingandthe offset ε, the resultingmeshwill slightlyshrink comparedto the original mesh. Hencewe have toadaptthereconstructionschemein thefollowing way: Eachprogressively receivedvertex is first projectedonto thecur-rentmeshandtheninsertedinto theclosesttrianglevia a 1-to-3 split operation.A local edgeflipping heuristicrestoresa generalizedDelauny-criterion to preserve the tesselation

Figure7: The sequenceabove depictsa typical reconstructionsequence:From the ellipsoid modelof the bunny (upperleft,300ellipsoids)the reconstructionalgorithmextractsa marchingcubesmesh(uppermiddle,10000vertices).Thefirst chunkof receivedverticesis projectedontothemeshandtheninsertedinto themesh(upperright, 5000verticesinserted).After thatthepointsof themarchingcubesmeshareremovedby adecimationalgorithm.Theremainingverticesareshiftedbackto theiroriginal positions(lower left, 5000vertices).Finally the remainingverticesareinsertedinto themesh(lower middle,10000verticesinserted,lower right 30000verticesinserted).

quality of the mesh. A detaileddescriptionof this schemeandits efficient implementationusingspacepartitioningcanbefoundin [5].

Sincetheverticesthatwerecreatedby themarchingcubessteparenotpartof theoriginalgeometryandthey haveto beremovedassoonasa sufficient numberof otherverticesarereceivedto maintainthecoarsemeshgeometry. Theremovalis doneby a simplemeshdecimationscheme,theremainingverticesarethenshiftedbackto theiroriginalpositions.Nowthe restof the verticescanbe insertedasbefore. Sincenointer-vertex/facereferencesarenecessary, this schemeis ro-bustin thesensethata certainpercentageof ellipsoid/vertexlossis toleratedby thereconstructionalgorithm.

4.3 Future work

We plan a more detailedinvestigationof variousselectioncriteria to determinethe optimal ellipsoid decomposition.Theselectionhasto guaranteesufficientoverlapbetweentheellipsoidsbut on theotherhandit hasto leadto a significantredundancy reduction.

A possiblefutureapplicationcouldbetheautomaticshaperecognition.Deriving somestatisticalmeasuresfrom thesetof ellipsoids,liketheaverageandstandarddeviationof ellip-soidradii andtheir ratiocouldbeanidentifyerthatis invari-ant undervariousshapemodifications.This will be a topicof futureresearch.

5 Ackno wledg ementThis work was supportedby the DeutscheForschungsge-meinschaftundergrantKO2064/1-1,“DistributedProcess-ing andDeliveryof Digital Documents”.

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