Multiresolution Hierar chies on Unstructured Triangle MeshesLeif Kobbelt

�JensVorsatz Hans-PeterSeidel

Max-Planck-Institutfur Informatik

Abstract

Theuseof polygonalmeshesfor therepresentationof highly com-plex geometricobjectshasbecomethe de factostandardin mostcomputergraphicsapplications.Especiallytrianglemeshesarepre-ferreddueto theiralgorithmicsimplicity, numericalrobustness,andefficient display. The possibility to decomposea given trianglemeshinto a hierarchyof differently detailedapproximationsen-ablessophisticatedmodelingoperationslikethemodificationof theglobalshapeunderpreservationof thedetail features.

So far, multiresolutionhierarchieshave beenproposedmainlyfor mesheswith subdivision connectivity. This typeof connectiv-ity resultsfrom iteratively applyinga uniform split operatorto aninitially givencoarsebasemesh.In this paperwe demonstratehowa similar hierarchicalstructurecanbederivedfor arbitrarymesheswith norestrictionsontheconnectivity. Sincesmooth(subdivision)basisfunctionsareno longeravailablein this generalizedcontext,weuseconstrainedenergy minimizationto associatesmoothgeom-etry with coarse levels of detail. As the energy minimizationre-quiresoneto solve aglobalsparsesystem,we investigatetheeffectof variousparametersandboundaryconditionsin orderto optimizetheperformanceof iterative solvingalgorithms.

Another crucial ingredientfor an effective multiresolutionde-compositionof unstructuredmeshesis the flexible representationof detail information.We discussseveralapproaches.

1 Intr oduction

Subdivision techniquesprovide very efficient and flexible algo-rithms for the generationof free form surfacegeometry[2, 5, 6,18, 25, 39]. Startingwith an arbitrarycontrol mesh

�0 we can

apply thesubdivision rulesto computefiner andfiner meshes�

mwith control verticespm

i becomingmoreandmoredenseuntil thedesiredapproximationtolerancerequiredfor a givenapplicationisreached.Theresult is a smoothsurfacehaving thesametopologyastheinitial controlmesh.

The distinct subdivision levels�

m give rise to powerful mul-tiresolutionsemanticssincewe canconsidera subdivision schemeasthelow passreconstructionoperatorin thefilter bankalgorithmfor awavelet-typedecompositionof thegeometricshape.Thesub-division basisfunctionswhich areassociatedwith thecontrol ver-ticesgeneralizetheconceptof dyadicscalingfunctionsto polyhe-dral parameterdomains[26, 31, 40].�

ComputerGraphicsGroup, Max-Planck-Institutfur Informatik, ImStadtwald,66123Saarbrucken,Germany, kobbelt@mpi-sb.mpg.de

However, subdivision techniquesare genuinelybasedon thecoarse-to-finegenerationof hierarchicalgeometryrepresentations:a coarsebasemeshwith only few facesis iteratively refinedbyintroducinganexponentiallyincreasingnumberof degreesof free-dom for capturingfiner andfiner detail information. As a conse-quence,the control meshesmust have so-calledsubdivisioncon-nectivity which meansthat sub-regions of the refinedmesh

�m

whichcorrespondto onesinglefaceof theoriginalbasemesh�

0,have theconnectivity of regulargrids(cf. Fig. 1).

It turnsout thatthisrestrictionis notsuitablefor severalstandardapplicationscenarios.In practiceoneis oftengivensomeexistinggeometricmodelwhich is to bemodifiedby makinglocalor globaladjustments.Sincesuch triangularmeshesusually do not comewith the ratherspecialsubdivision connectivity, we cannotapplysubdivision techniqueswithoutpreprocessing.

Thispreprocessinghastoperformaglobalremeshingof thedata.Althoughseveralflexibleandrobustalgorithmshavebeenproposedfor this problem[7, 24, 22] therearestill difficultieswith automat-ically finding a suitablelayout for thebasemesh.Semi-automaticapproacheslike [23, 24] with constraintssetby the useronly par-tially solve this problem.Moreover, theremeshingis alwaysa re-samplingprocessandhenceevenanoptimal remeshingalgorithmcannotrecover theoriginal shapeexactly. High frequency artifactsdueto aliaserrorsareratherlikely to appear.

The rigidity of subdivision connectivity meshesemergesfromthe fact that the classificationof the detail coefficientsinto prede-finedrefinementlevelsisdonetopologically. Theactualsizeor geo-metricfrequencyassociatedwith a detailcoefficient hencestronglydependson thesizeof thecorrespondingbasetrianglein theunre-finedcontrolmesh.As it is usuallynotpossibleto haveall trianglesin thebasemeshof unit size,detailfeaturesonthesamerefinementlevel andtheir correspondingsupportcanvary by oneor moreor-dersof magnitude.Avoiding thisproblemby usingadaptive refine-mentstrategiesis notappropriatein someapplications.

Anotherproblemwhich is inherentto the multiresolutionrep-resentationof freeform geometrybasedon subdivision surfacesisthefixedsupportof themodifications.If controlverticesareusedashandlesto modify thesurfacegeometryon a certainlevel of detailthentheregion of themeshwhich actuallychanges,is determinedby thesupportof theassociatedbasisfunction. We couldsimulatemoreflexibility in thedefinition of thesupportby moving severalcontrolverticesfrom somefinerlevel simultaneouslybut thiswoulddiminishtheadvantagesof amultiresolutionrepresentation.

Moreover, thecoarsescalecontrolverticesin a subdivision rep-resentationarealignedto thecoarsescalegrid. Thismeansthatwelosespatialresolutionif we modify a surfaceon a low frequencyband.Consequently, wecanapplymodificationsof theglobalshapeonly at a very limited numberof locations. In fact, asevery con-trol vertex c in a subdivision connectivity meshis introducedon acertainrefinementlevel l � c � the supportof the modificationwhenmoving c is boundedby thesizeof thebasisfunctionsonthatlevel.

For example,if wemoveacontrolvertex c0 which topologicallycorrespondsto a vertex in the basemeshthenwe canchoosethebasisfunctioncontrollingtheedit from any refinementlevel. How-ever, moving a directly adjacentvertex cm on refinementlevel mcanonly affect thefinestscalesincecm doesnot have a represen-tationon any coarserlevel. Hence,a coarsescalemodificationcanbecenteredat c0 but not at cm which lies only ε away. This is notintuitive for thedesignerto whomtheactualsurfacerepresentation

Figure1: Subdivision connectivity meshesresultfrom iteratively applyinga uniform split operationto the facesof an initial controlmesh.Only a fixednumberof isolatedextraordinaryverticeswith valence�� 6 remainin themesh.

Figure2: In a multiresolutionmodelingenvironment,the supportof themodificationandits characteristicshapeshouldadaptto thegivengeometry(here: thebust’s hair). Thelow-frequency modifi-cationaffectsexactly theregion definedby thedesigner. Thehighfrequency detail is preservedin a naturalway.

shouldbeopaque.With all thesedifficultiesenumerated,weunderstandthatcoarse-

to-finehierarchiesemergingfromsubdivisiontechniquesmightcer-tainly be the bestway to effectively representsmoothfree formgeometryin applicationslike surfacereconstruction,scattereddatainterpolation,or ab initio designwherethefacelayoutfor thebasemeshis definedby thedesigner. However, it doesnot appearto betheoptimalsolutionfor flexibly modifyingexistingmodelslike theonesobtainedfrom capturingreal objectgeometryby laserscan-ning devices.

Our goal is to enabletruefreeform multiresolutioneditswherethesupportandthecharacteristicsof amodificationcanadaptto thesurfacegeometry(cf. Fig. 2). In [21] we generalizedthe conceptof multiresolutiondecompositionandmodelingto mesheswith ar-bitrary topologyandconnectivity. The key observation is that wecanno longerstick to thenotionof surfacegeometrybeingrepre-sentedby thesuperpositionof smoothscalarvaluedbasisfunctionsover a nestedsequenceof grids.Thereasonfor this is thatwe can-

not make any assumptionson the actualdistribution of the meshverticesa priori. Hence,imposingany kind of vectorspacestruc-turewould requireusto constructexplicitly acustomtailoredbasisfunctionfor eachvertex.

Leaving theclassicalset-up,it turnsout thatfor merepolygonalmeshes(not control meshes),coarsenessand smoothnessare nolonger synonyms. While in the subdivision framework the basisfunctionson the coarsescalesarealsosmootherin the sensethatthey have lesscurvature,we find that for plain polygonalmeshestheeffect of shifting a controlvertex on a coarsescalestill causesa sharpfeature.To speakaboutsmoothpolygonalmesheswe needmoredegreesof freedomsincesmoothmeshesaretypically ratherfine tesselations.

Figure3: For plain trianglemesheswe have to distinguishcoarseandsmoothapproximations(upperandlower row). If meshesareconsideredas control mesheswith respectto scalarvaluedbasisfunctionsthentheconnectionbetweentheupperanthe lower rowis providedby evaluatingtheweightedsuperpositionof thecontrolvertices’influence.

Wehave to solve two centralproblemsin orderto developeffec-tive multiresolutionalgorithmsfor arbitrarymeshes.First we haveto constructa topological hierarchyof differentresolutionswith thefinestresolutionbeingtheoriginal mesh.This hierarchymustnotrely on any assumptionsabouttheconnectivity of thegivenmesh.

Besidesthetopologicallevelsof detailwe needa geometrichi-erarchy, i.e., we needa propercharacterizationof smoothcoarse-scalegeometry. In the subdivision basedmultiresolutionsetting,

we have the associatedscalingfunctionswhich fill in the smoothgeometry� betweenthecoarsescalecontrolvertices.In thegeneral-ized settingwe have to find analternative definitionsincea prioridefinedscalingfunctionsareno longeravailable. A possibleso-lution to this problemis to usediscreteenergy minimizationtech-niquesto obtainsmoothlow-detail approximationsto the originalmodel.

While thebasicprinciplesof this approachhave beenpresentedin [21], we discussmoretechnicaldetailsin this paper. After ex-plaining the generationof coarse-to-finehierarchiesand fine-to-coarsehierarchies,we comparedifferentwaysto representthede-tail informationbetweenthe resolutionlevels. The crucial issuesareherehow to definethe local frameswith respectto which thedetail is encodedandhow to chosethenumberof hierarchylevels.In the context of discreteenergy minimizationwe investigatetheeffect of variousparametersin the multi-level solving algorithm,namelythe numberof hierarchylevels and the numberof Gauß-Seideliterationsoneachlevel. Wedemonstratethatimposinginter-polationconstraintsat thecentersof thetriangularfacesacceleratestheglobalconvergenceof theiterativesolvercomparedto imposingtheconstraintsat thevertices.

2 Multiresolution representations

Most schemesfor themultiresolutionrepresentationandmodifica-tion of trianglemeshesemerge from generalizingharmonicanal-ysis techniqueslike the wavelet transform[1, 26, 31, 34]. Sincethefundamentalsarederivedin thescalar-valuedfunctionalsettingIRd � IR, difficultiesemerge from the fact thatmanifoldsin spacearein generalnot topologicallyequivalentto simply connectedre-gionsin IRd.

Thephilosophybehindmultiresolutionmodelingon surfacesishenceto mimic the algorithmicstructureof the relatedfunctionaltransformsandpreserve someof the importantpropertieslike lo-cality, smoothness,stability or polynomial precisionwhich haverelatedmeaningin both settings[8, 13, 40]. Accordingly, thenestedsequenceof spacesunderlyingthe decompositioninto dis-joint frequency bandsis thoughtof beinggeneratedbottom-upfroma coarsebasemeshup to finer andfiner resolutions.This impliesthat subdivision connectivity is mandatoryon higherlevels of de-tail, i.e., themeshhasto consistof large regular regionswith iso-latedextra-ordinaryvertices. Additionally, we have to make surethat the topologicaldistancebetweenthe singularitiesis the samefor every pair of neighboringsingularitiesandthis topologicaldis-tancehasto be a power of 2. Obviously, sophisticatedmodelingoperationslike booleanoperationsnecessarilyrequirea completerestructuringof theresultingmeshto re-establishsubdivision con-nectivity.

Thesespecialtopologicalrequirementspreventsuchtechniquesfrom beingapplicableto arbitraryinputmeshes.To obtainaproperhierarchy, global remeshingand resamplingis necessarywhichgivesriseto alias-errorsandrequiresinvolvedcomputations[7, 24].

Luckily, therestrictedconnectivity is notnecessaryto definedif-ferent levels of resolutionor approximationfor a triangle mesh.In the literatureon meshdecimationwe find many examplesforhierarchiesbuilt on arbitrarymeshes[12, 17, 20, 27, 29, 32, 36].The key is alwaysto build the hierarchytop-down by eliminatingverticesfrom thecurrentmesh(incrementalreduction, cf. Fig. 4).Runningameshdecimationalgorithm,wecanstop,e.g.,everytimea certainpercentageof the verticesis removed. The intermediatemeshescanbeusedasa level-of-detailrepresentation[17, 26].

In bothcases,i.e.,thecoarse-to-fineor thefine-to-coarsegenera-tion of nested(vertex-) grids,themultiresolutionconceptis rigidlyattachedto topologicalentities.This makessenseif hierarchiesaremerelyusedto adjustthecomplexity of therepresentation.Wewillexploit thesequenceof nestedgridsemergingfrom this topologicalhierarchyto generalizethe conceptof multi-grid solversfor largesparsesystems.

In thecontext of multiresolutionmodeling, however, wewantthe

hierarchynot necessarilyto ratemeshesaccordingto their coarse-nessbut ratheraccordingto theirsmoothness. For thisweneedage-ometrichierarchyaccompanying thetopologicalone.To completeour basicequipmentfor themultiresolutionset-upon unstructuredmesheswe henceneed(besidesthestaticlevelsof detail)to definethedecompositionandreconstructionoperationswhichseparatethehigh-frequency detailfrom thelow-frequency shapeandeventuallyrecombinethe two to recover the original mesh. Here,the recon-structionoperatorhasto generatethesmoothlow-frequency shapeif thedetail informationis suppressedduringreconstruction.Thisis wherediscretefairing techniquescomein. Further, we have toencodethedetailinformationrelativeto thelow-frequency shapeinorderto guaranteeintuitive detailpreservationaftera globalmodi-fication(local frames).

2.1 Coarse-to-fine Hierarchies

For subdivision basedmultiresolution representationthe recon-structionoperatoris given by the underlyingsubdivision scheme.We transform a given mesh

�m to the next refinementlevel�

m� 1� S�

m by applying the stationarysubdivision operatorSandmovetheobtainedcontrolverticesby addingtheassociatedde-tail vectors:

�m� 1

� � m� 1 �� m. Thesupportof thesubdivision

maskimplies that eachcontrol vertex pmi in

�m hasinfluenceon

several control verticesin�

m� 1. Consequently, the modificationof pm

i ’s position eventually causesa smoothbump on the result-ing surface. The actualshapeof this bump can be computedbyapplyingthe subdivision operatorS without detail reconstruction,i.e. m : � 0. Obviously, thesupportof thebump dependson therefinementlevel mon which themodificationis applied.

The decompositionoperatorhasto be an inverseof the subdi-vision operator, i.e., given a fine mesh

�m� 1 we have to find a

mesh�

m suchthat�

m� 1 � S�

m. In this casethedetail vectors m : � � m� 1 � S�

m becomeas small as possible[40]. Due tothe uniform split which is part of the subdivision operatorS, it isobvious that this techniqueappliesonly if

�m� 1 hassubdivision

connectivity.

2.2 Fine-to-coar se Hierarchies

If we build thehierarchyby usinganincrementalmeshdecimationscheme,thedecompositionoperatorD appliesto arbitrarymeshes.Givena finemesh

�m� 1 wefind

�m� D

�m� 1, e.g.,by applying

a numberof edgecollapseoperations.However, it is not clearhowto definethedetailcoefficientssinceinversemeshdecimation(pro-gressivemeshes) alwaysreconstructstheoriginalmeshandthereisno canonicalway to generatesmoothlow-frequency geometrybysuppressingthedetail informationduringreconstruction.

To solve thisproblemwe split eachstepof theprogressive meshrefinementinto atopologicaloperation(vertex insertion)andageo-metricoperationwhich placesthere-insertedverticesat their orig-inal position. In analogyto the plain subdivision without detailreconstruction,we have to figure out a heuristicwhich placesthenew verticessuchthatthey lie onasmoothsurface(insteadof theiroriginalposition).Thedifferencevectorbetweenthispredictedpo-sitionandtheoriginal locationof thevertex canthenbeusedastheassociateddetailvector.

Sincewe operateon unstructuredmeshes,we cannotusefixed(stationary)rulesfor theplacementof the re-insertedvertices. In-steadwe usediscreteenergy minimization which meansthat there-insertedverticesareplacedsuchthat someglobal bendingen-ergy is minimized. In Section3 we review a simpletechniqueforthe effective generationof mesheswith minimum bendingenergywithout specificrequirementson theconnectivity.

2.3 Detail encoding

In orderto guaranteeintuitive detail preservation undermodifica-tion of theglobalshape,we cannotsimply storethedetail vectors

Figure4: For multiresolutionrepresentationsbasedon subdivision techniques,the hierarchiesarebuilt from coarseto fine by applyingauniformsubdivisionoperator(toprow, left to right) while incrementalmeshdecimationgenerateshierarchiesfrom fineto coarseby iterativelyremoving vertices(bottomrow, left to right).

with respectto a globalcoordinatesystembut have to definethemwith respectto local frameswhicharealignedto thelow-frequencygeometry[10, 11]. Usually, theassociatedlocal framefor eachver-tex hasits origin at thelocationpredictedby thereconstructionop-eratorwith suppresseddetail.This is in analogyto decompositionsbasedona globalparameterizationof thesurfaces.

However, in many casesthis canleadto ratherlong detail vec-torswith asignificantcomponentwithin thelocal tangentplane(cf.Fig. 5). Sincewe prefershortdetail vectorsfor stability reasons,it makes senseto use a different origin for the local frame. Infact, the optimal choiceis to find that point on the low-frequencysurfacewhosenormalvectorpointsdirectly to theoriginal vertex.In this case,the detail is not given by a threedimensionalvector��� x ��� y��� z� T but ratherby a basepoint p � p � u � v� on the low-frequency geometryplus a scalarvalueh for the displacementinnormaldirection.If a localparameterizationof thesurfaceis avail-able thenthe basepoint p canbe specifiedby a two-dimensionalparametervalue � u � v� .

Figure5: The shortestdetail vectorsareobtainedby representingthedetail coefficientswith respectto thenearestlocal frame(left)insteadof attachingthe detail vectorsto the topologically corre-spondingoriginal vertices.

Thegeneralsettingfor detailcomputationis thatwe have giventwo meshes

�m� 1 and

� m� 1 where

�m� 1 is the original data

while�

m� 1 is reconstructedfrom the low-frequency approxima-tion

�m with suppresseddetail, i.e. for coarse-to-finehierachies,

themesh�

m� 1 is generatedby applyinga stationarysubdivisionschemeand for fine-to-coarsehierarchies

� m� 1 is optimal with

respectto someglobal bendingenergy functional. Encodingthegeometricdifferencebetweenbothmeshesrequiresusto associateeachvertex p of

�m� 1 with a correspondingbasepoint q on the

continuous(piecewise linear) surface�

m� 1 suchthat the differ-encevectorbetweentheoriginalpointandthebasepoint is parallelto thenormalvectorat thebasepoint. Any point q on

� m� 1 can

bespecifiedby atriangleindex i andbarycentriccoordinateswithinthereferredtriangle.

To actuallycomputethedetail coefficients,we have to defineanormalfield on themesh

� m� 1. Themostsimpleway to do this

is to usethenormalvectorsof thetriangularfacesfor thedefinitionof a piecewise constantnormalfield. However, sincetheorthogo-nalprismsspannedby a trianglemeshdo not completelycover thevicinity of themesh,we have to acceptnegative barycentriccoor-dinatesfor thebasepointsif anoriginalvertex liescloseto anedgeof�

m� 1 or if�

m� 1 is not smoothenough(cf. Fig 6). This leadsto non-intuitivedetailreconstructionif thelow-frequency geometryis modified(cf. Fig. 7).

A techniqueused in [21] is basedon the constructionof alocal quadraticinterpolantto the low-frequency geometry. Thebasepoint is found by Newton-iteration. Although this techniquereducesthe numberof pathologicalconfigurationswith negativebarycentriccoordinatesfor thebasepoint,we still observe artifactsin the reconstructedhigh-frequency surfacewhich are causedbythefactthattheresultingglobalnormalfield of thecombinedlocalpatchesis not continuous.

Wethereforeproposeadifferentapproachwhichadaptsthebasic

Figure6: Thepositionof a vertex in theoriginal mesh(high-frequency geometry)is givenby a basepoint on the low-frequency geometryplusa displacementin normaldirection.Therearemany waysto definea normalfield on a trianglemesh.With piecewiseconstantnormals(left) wedonotcover thewholespaceandhencewesometimeshaveto usevirtual basepointswith negativebarycentriccoordinates.Theuseof local quadraticpatchesandtheir normalfields(center)somewhat improvesthesituationbut problemsstill occursincetheoverall normalfield is not globally continuous.Suchdifficulties arecompletelyavoidedif we generatea Phong-typenormalfield by blendingestimatedvertex normals(right).

Figure7: We modifiedthe original surface(left) by usinga two-bandmultiresolutiondecomposition.Sincein this particularexperimentthelow-frequency geometrywaschosennot sufficiently smooth,many detailvectorshave basepointswith negative barycentriccoordinateswhenwe usea piecewise constantnormalfield. Consequently, no properdetail reconstructionis possibleafter the modification(center).Representingthedetailvectorswith respectto thePhongnormalfield on thelow-frequency meshleadsto theexpectedresult(right).

ideaof Phong-shading[9] wherenormalvectorsareestimatedattheverticesof atrianglemeshandacontinuousnormalfield for theinteriorof thetriangularfacesis computedby linearlyblendingthenormalvectorsat thecorners.

Supposewe aregiven a triangle ��� a � b � c � with the associatednormalvectorsNa, Nb, andNc. For eachinterior point

q � αa � βb � γc

with α � β � γ � 1 we find theassociatednormalvectorNq by

Nq� αNa � βNb � γNc �

Whencomputingthedetailcoefficientsfor agivenpointp wehaveto find thebasepoint q suchthat�

p � q ��� Nq

hasall threecoordinatesvanishing. By plugging in the definitionof q andNq andeliminating γ � 1 � α � β we obtaina bivariatequadraticfunction

F : � u � v� � IR3

andwe have to find theparametervalue � α � β � suchthatF � α � β � �� 0 � 0 � 0� T . Thiscanbeaccomplishedby performingseveralstepsofNewton-iteration. Notice that F canbe interpretedasa quadraticsurfacepatchin IR3 which passesthroughthe origin. The Taylor-coefficientsof F canexplicitly begivenby

F � W � WWFu

� U � UW � W � 2WWFv

� V � VW � W � 2WWFuu

� UU � UW � WWFuv

� UV � UW � VW � 2WWFvv

� VV � VW � WW

whereU � p � NaV � p � NbW � p � NcUU � Na � aVV � Nb � bWW � Nc � cUV � � Nb � a � � � Na � b �UW � � Nc � a � � � Na � c �VW � � Nc � b � � � Nb � c �

In caseoneof thebarycentriccoordinatesof theresultingpointq isnegative,wecontinuethesearchfor abasepoint in thecorrespond-ing neighboringtriangle. SincethePhongnormalfield is globallycontinuouswe always find a basepoint with positive barycentriccoordinates.Fig. 6 depictsthe situationschematicallyandFig. 7shows an exampleedit wherethe piecewise constantnormalfieldcausesmeshartifactswhich do not occurif thePhongnormalfieldis used.

2.4 Hierarchy levels

For coarse-to-finehierarchiesthelevelsof detailaredeterminedbytheuniform refinementoperator. Startingwith thebasemesh

�0,

the mth refinementlevel is reachedafter applying the refinementoperatorm times. For fine-to-coarsehierarchiesthereis no suchcanonicalchoicefor the levels of resolution. Hencewe have tofigureoutsomeheuristicsto definesuchlevels.

In [21] a simpletwo-banddecompositionhasbeenproposedforthemodeling,i.e. thehighfrequency geometryis givenby theorig-inal meshandthelow-frequency geometryis thesolutionof someconstrainedoptimizationproblem.This simpledecompositionper-formswell if theoriginal geometrycanbeprojectedonto the low-frequency geometrywithoutself-intersections.Fig.8 schematically

Figure8: Whenthedifferencebetweentwo geometriclevelsof de-tail is too big, thehigh-frequency geometrycannotbeprojecteddi-rectly onto the low-frequency geometrywithout self-intersections.In orderto guaranteecorrectdetailreconstruction,we have to gen-erateintermediatelevels suchthat the mappingbetweentwo suc-cessive levelsis one-to-one.

shows a configurationwherethis requirementis not satisfiedandconsequentlythedetail featuredoesnotdeformintuitively with thechangeof theglobalshape.

This effect canbe avoidedby introducingseveral intermediatelevelsof detail,i.e.,by usinga truemulti-banddecomposition.Thenumberof hierarchylevelshasto bechosensuchthat the � i � 1� stlevel canbeprojectedonto level i without self-intersection.Detailinformationhasto becomputedfor every intermediatelevel.

The intermediatelevels can be generatedby the following al-gorithm. We startwith the original meshandapply an incremen-tal meshdecimationalgorithmwhich performsa sequenceof edgecollapseoperations.Whena certainmeshcomplexity is reached,we performthe reversesequenceof vertex split operationswhichreconstructstheoriginal meshconnectivity. Thepositionof there-insertedverticesis foundby solvinga globalbendingenergy mini-mizationproblem(discretefairing). Themeshthatresultsfrom thisprocedureis asmoothedversionof theoriginalmeshwherethede-greeby whichdetail informationhasbeenremoveddependson thetargetcomplexity of thedecimationalgorithm(cf. Fig 10)

Supposetheoriginal meshhasnm vertices,wherem is thenum-berof intermediatelevelsthatwewanttogenerate.Wecancomputethe meshes

�m � ����� � � 0 with fewer detail by applyingthe above

procedurewherethe decimationalgorithm stopsat a target reso-lution of nm � ����� � n0 remainingverticesrespectively. The resultingmeshesyield amulti-banddecompositionof theorignaldata.Whena modelingoperationchangestheshapeof

�0 we first reconstruct

thenext level�

1 by addingthestoreddetailvectorsandthenpro-ceedby successively reconstructing

��i � 1 from

� i .

The remainingquestionis how to determinethe numbersni .A simple way to do this is to build a geometricsequencewithni � 1 � ni

� const. This mimics the exponentialcomplexity growthof the coarse-to-finehierarchies.Anotherapproachis to stopthedecimationevery time a certainaverageedgelength l i in the re-mainingmeshis reached.

A morecomplicatedheuristictries to equalizethe sizesof thedifferencesbetweenlevels, i.e., thesizesof thedetail vectors.Wefirst computea multi-banddecompositionwith, say, 100 levels ofdetailwherewe choose i

�ni� const. For every pair of successive

levelswecancomputetheaveragelengthof thedetailvectors(dis-placementvalues). From this information we can easily chooseappropriatevaluesn j

� ni j suchthat the geometricdifferenceisdistributedevenly amongthedetaillevels.

In practiceit turnedout thataboutfive intermediatelevelsis usu-ally enoughto guaranteecorrectdetail reconstruction.Fig. 9 com-parestheresultsof a modelingoperationbasedon a two-bandanda multi-banddecomposition.

3 Constrained discrete fairing

In the previous sectionwe explainedhow to generatetopologicalhierarchiesfor mesheswith arbitraryconnectivity by incrementalmeshdecimation. An associatedgeometrichierarchycan be ob-tainedby re-insertingthe removed verticesandmoving themto anew positionsuchthata globalbendingenergy functionalis mini-mized.Theideais to computea meshwhich is assmoothaspossi-blewhile still containingacontrollableamountof geometricdetail.Fig. 10shows anexample.

FromCAGD it is well-known thatconstrainedenergy minimiza-tion is a very powerful techniqueto generatehigh quality surfaces[3, 14, 28, 30, 37]. For efficiency, one usually definesa sim-ple quadraticenergy functional � � f � andsearchesamongthe setof functionssatisfyingprescribedinterpolationconstraintsfor thatfunction f whichminimizes� .

Transferringthe continuousconceptof energy minimization tothe discretesettingof trianglemeshoptimizationleadsto the dis-cretefairing approach[19, 38]. Local polynomialinterpolantsareusedto estimatederivative informationat eachvertex by divideddifferenceoperators.Hence,the differentialequationcharacteriz-ing the functionswith minimumenergy is discretizedinto a linearsystemfor thevertex positions.

Sincethissystemis globalandsparse,weapplyiterativesolvingalgorithmslike theGauß-Seidel-scheme.For suchalgorithmsoneiterationstepmerelyconsistsin the applicationof a simple localaveragingoperator. This makesdiscretefairing aneasyaccessibletechniquefor meshoptimization.

For the most popularfairing functional, the thin-plate energy,thisapproachleadsto a simpleupdate-rule[21]

p ! p � 1ν " 2 � p � (1)

which has to be applied to all verticesof the mesh. Here, theumbrella-operator" is adiscretizationof theLaplace-operator[35]

" � p � � 1n

n # 1

∑j $ 0

p j � p

with p j being the directly adjacentneighbor vertices of p (cf.Fig. 11). The umbrella-operatorcan be appliedrecursively lead-ing to

" 2 � p � � 1n

n # 1

∑j $ 0 " � p j � � " � p �

asadiscretizationof thesquaredLaplacian.Thecoefficientν in (1)is givenby

ν � 1 � 1n ∑

j

1n j

wheren andn j arethe valencesof the centervertex p andits j thneighborp j respectively.

...

P

P

PP

P

2

1

0

n−1

Figure 11: To computethe discreteLaplacian,we needthe 1-neighborhoodof a vertex p ( � umbrella-operator).

In thecontext of discreteenergy minimization,the iterative ap-plicationof theupdate-rule(1) implementsaGauß-Seidelsolverfortheunderlyinglinearsystem.Froma moreabstractpoint of view,the rule canalsobe consideredasa mererelaxationoperatorthateffectively filters outhigh frequency noisefrom themesh[35].

Figure 9: Non-projectabledetail featuresare not reconstructedcorrectly. The original geometry(left) is modified by using a two-banddecompositionin thecenteranda multi-banddecompositionwith five intermediatelevelson theright.

Figure10: Four versionsof theStanfordbunny. Thesmootherversionsaregeneratedby applyingmeshdecimationdown to a certaintargetcomplexity andthenre-insertingtheverticesunderminimizationof somediscretefairnessfunctional.Thedegreeby which geometricdetailis removeddependson thecoarsenessof thebasemesh.Noticethatall shown mesheshave exactly thesameconnectivity.

3.1 Multi-le vel smoothing

A well-known negative result from numerical analysis is thatstraightforward iterative solvers like the Gauß-Seidelschemearenot appropriatefor largesparseproblems[33]. More sophisticatedsolversexploit knowledgeaboutthestructure of theproblem.Theimportantclassof multi-grid solversachieve linear runningtimesin thenumberof degreesof freedomby solvingthesameproblemon grids with differentstepsizesandcombiningthe approximatesolutions[16].

For difference( � discretedifferential)equationsof elliptic typetheGauß-Seideliterationmatriceshaveaspecialeigenstructurethatcauseshigh frequenciesin the error to be attenuatedvery quicklywhile for lower frequenciesno practically useful rate of conver-gencecanbe observed. Multi-level schemeshencesolve a givenproblemonaverycoarsescalefirst. Thissolutionis usedto predictinitial valuesfor a solutionof thesameproblemon thenext refine-ment level. If thesepredictedvalueshave only small deviationsfrom the true solution in low-frequency sub-spaces,then Gauß-Seidelperformswell in reducingtheremaininghigh-frequency er-ror. Thealternatingrefinementandsmoothingleadsto highly effi-cientvariationalsubdivisionschemes[19] whichgeneratefair high-resolutionmesheswith a rateof several thousandtrianglespersec-ond(linearcomplexity!).

We canapplythesameprincipleto hierarchicalmeshstructureswhicharegeneratedfrom fine-to-coarse.Insteadof iteratively solv-ing the discretizedoptimizationproblem on the finest level, wesolve it on coarserintermediatelevelsfirst andthenusethecoarsesolutionsto estimatebetterstartingvaluesfor theiterativesolveronthefiner levels.

A completeV-cycle multi-grid solver recursively appliesopera-torsΦi

� ΨPΦi # 1 RΨ wherethefirst (right) Ψ is a generic(pre-)smoothingoperator— a Gauß-Seidelschemein our case. R is arestrictionoperatorto go onelevel coarser. This is wherethemeshdecimationcomesin. On thecoarserlevel, thesameschemeis ap-plied recursively, Φi # 1, until on the coarsestlevel the numberofdegreesof freedomis smallenoughto solve thesystemdirectly (orany otherstoppingcriterion is met). On theway back-up,thepro-longationoperatorP insertsthepreviously removed verticesto go

onelevel finer again. P canbeconsideredasa non-regular subdi-vision operatorwhich hasto predictthepositionsof theverticesinthenext level’s solution.There-subdividedmeshis anapproxima-tive solutionwith mostly high frequency error. (Post-)smoothingby somemoreiterationsΨ removesthe noiseandyields the finalsolution.

In our particularsettingof thin-plateoptimizationon fine-to-coarsehierarchies,theΨ-operatoris simply theupdate-rule(1) andthe restrictionoperatoris a sequenceof edge-collapseor vertexremoval stepswhich areperformedby the meshdecimationalgo-rithm. Theprolongationoperatorre-insertsthevertices.Sincetheprolongationoperatorcanbedesignedto insertthenew verticesto alocally optimalposition,i.e., thecenterof gravity of its directneig-borssuchthat " � p � � 0, thereis no needto actuallyperformanypre-smoothing.In fact,thewholemulti-level smoothingalgorithmreducesto meshdecimationdown to a certainresolutionandthenalternatingthe re-insertingand Gauß-Seidelsmoothing. Anotherconsequenceis thatmoresophisticatedW-cycleschedulesareveryunlikely to improve theconvergenceof thealgorithm.

Theareseveralalgorithmicparametersin thisgenericmulti-levelscheme.First,we have to choosethenumberof Gauß-Seidelstepswhich areperformedon every level. As this is themosttime con-sumingstepof thealgorithmandsinceour goal is to run theopti-mizationin real-timewith a prescribednumberof framespersec-ond, we cannotallow the iteration to proceeduntil the residuumdropsbelow somegiventhreshold.Weratherperformafixednum-ber of iterationson eachlevel. By adjustingthat numberwe di-rectly tradethe quality of the resultingmeshfor the speedof thealgorithm.

Anotheralgorithmicparameteris thenumberof hierarchylevels.The two extremepositionsare either to re-insertall verticesandthenperformGauß-Seidelon the finest level only or to apply (1)after the insertionof every single vertex. From a practicalpointof view, the upperboundfor the granulatityof hierarchylevels isreachedif theverticeswhichareinsertedwhengoingfrom level

�i

to�

i � 1, are independentfrom eachother, i.e., their topologicaldistanceis larger than somethreshold. This is becausethe localupdateoperation(1) propagatesgeometricchangesveryslowly. Analternative to combininga sequenceof independentvertex splits

Figure12: Thisdiagramshows thelogarithmof theapproximationerror (vertical axis) vs. the computationtime (horizonticalaxis).Theknotson eachpolygonmark themeasurementsfor a differentnumberof Gauß-Seideliterations(1 � �%��� � 20). The differentpoly-gonsconnectthemeasurementsfor thesamenumberof hierarchylevels(from bottomto top: 27� 14� 9 � 7 � 6 levels). Themonotony ofthecurvesshows thatfor a fixedamountof computationtime (ver-tical line) or a prescribedapproximationerror (horizontalline) themulti-level smoothingschedulewith the highernumberof levelsalwaysoutperformstheothers.

(or edgecollapses)is proposedin [15] wherethe local smoothingoperatoris appliedonly in thevicinity of thenewly insertedvertex.

Since the eigenstructureof the Gauß-Seideliteration matrixandhencetheconvergencebehavior of thegeneralizedmulti-levelschemestronglydependson the actualconnectivity of the mesh,we cannotderive generalestimatesfor theconvergencerates.Nev-erthelesswe can analyzethe typical behavior of the multi-levelsmoothingon fine-to-coarsehierarchiesby numericalexperiments.We madesomeexperimentswherewe performedthe multi-levelsmoothingwith a varying numberof hierarchylevels and Gauß-Seideliterationsperlevel. Theresultsareshown in Fig. 12.

Obviously the approximationerror decreaseswith increasingnumberof Gauß-Seidelstepsandwith increasingnumberof lev-els but also the computationalcostsbecomehigher. Whenusingthemulti-level smoothingin practicalapplicationswetypically pre-scribethemaximumtimeor themaximumapproximationerror, i.e.,wewanttofind thebestapproximationwithin agivenperiodof timeor we want to find a solutionwith a prescribedapproximationer-ror as fastaspossible. In Fig. 12 theseconstraintscorrespondtoverticalor horizonticallinesrespectively.

As a generalrule of thumbit turnedout thatmoreGauß-Seideliterationsperlevel only marginally improve thefinal result.This isdueto thebadconvergenceon eachindividual level. Betterresultscanbe achieved if morehierarchylevels areusedbut with feweriterationsperlevel.

Noticethatthenumberof topological hierarchylevelsasoneal-gorithmicparameterin themulti-level smoothingschemehasnoth-ing to do with thenumberof geometrichierarchylevels in thege-ometricmulti-banddecomposition(topologicalvs. geometrichier-archy). Oneis usedto make thedetail reconstructionmorerobustwhile theotheris usedto acceleratetheglobaloptimizationproce-dure.

3.2 Boundar y constraints

In orderto enableintuitive modelingfunctionality we have to im-plementa simpleandeffective interactionmetaphor. As theshapeof the meshis controlledby discretecurvatureminimization, themostsimpleway to influencetheresultis by imposingappropriateboundaryconstraints.Theseconstraintsdeterminethesupportandtheshapeof themodification.

In [21] weproposedasimplemetaphorwherethedesignerstartsby markinganarbitraryregion on themesh

�m. In fact,shepicks

a sequenceof surfacepoints(not necessarilyvertices)on thetrian-gle meshandthesepointsareconnectedeitherby geodesicsor byprojectedlines.Thestripof triangles& whichareintersectedby thegeodesic(projected)boundarypolygonseparatesaninterior region�('

andanexterior region�

i ) � �('+* &,� . Theinterior region�('

isto beaffectedby thefollowing modification.

A secondpolygon(not necessarilyclosed)is markedwithin thefirst one to define the handle. The semanticsof this arbitrarilyshapedhandleis quitesimilar to thehandlemetaphorin [37]: whenthe designermovesor scalesthe virtual tool, the samegeometrictransformationis appliedto the rigid handleand the surroundingmesh

�('follows accordingto a constrainedenergy minimization

principle.Thefreedomto definetheboundarystrip & andthehandlegeom-

etry allows thedesignerto build ”customtailored” basisfunctionsfor theintendedmodification.Particularly interestingis thedefini-tion of a closedhandlepolygonwhich allows to control the char-acteristicsof a bell-shapeddent: For the sameregion

�(', a tiny

ring-shapedhandlein themiddlecausesarathersharppeakwhile abiggerring causesawiderbubble(cf. Fig 13). Noticethatthemeshverticesin the interior of thehandlepolygonalsomove accordingto theenergy minimization.

Figure13: Controllingthecharacteristicsof themodificationby thesizeof a closedhandlepolygon.

Sincewe areworking on trianglemeshes,theenergy minimiza-tion on

� 'is done by discretefairing techniques. To enable

realtimeediting we use the multi-level smoothingapproach(cf.Fig. 14). While Fig. 15 depictsthe generalmodelingset-upforageometrictwo-banddecomposition,moreintermediatelevelscanbe usedfor the detail reconstructionif the original geometrycan-notbeprojectedontotheoptimizedmeshwithoutself-intersections.The boundarytriangles& provide the correctC1 boundarycondi-tions for minimizing the thin plateenergy functional. The handleimposesadditionalinterpolatoryconstraintsonthelocationonly —derivativesshouldnot beaffect by thehandle.

In [21] we proposedto imposethe handleinterpolationcon-straintsto theoptimizationproblemby simply freezingeveryothervertex of thehandlepolygon. On onehandthis is a simpleway toimplementinterpolationconstraints,onetheotherhandit preventsany influenceon thetangentplane.

Anotherway to imposeinterpolationconstraintsis to prescribethemfor centersof triangles.Suchconstraintscaneasilybeembed-dedinto theiterativeenergy minimizationby allowing Gauß-Seidelupdatesfor all verticesandre-enforcingtheconstraintsafter eachiteration. This meansthat we shift the constrainedtrianglessuchthat their centerscoincidewith the interpolationpointsaftereverysmoothingcycle. By shifting the triangleswithout rotationwe al-low the tangentat the interpolationpoint to be controlledby theoptimizationprocess(andhencewedonot imposeaC1 constraint).Fig 16 demonstratesthat the convergencebehavior is muchbetterfor this kind of interpolationconstraintcomparedto freezingver-tices.

Figure14: During thereal-timemodeling,themulti-level smoothingalwaysstartson thecoarseslevel down to which�('

is reduced(left).We alternatevertex re-insertionandGauß-Seidelsmoothing(centerleft) until themeshwith minimumthin plateenergy with respectto thecurrentinterpolationconstraintsis found(centerright). To thissmoothmesh,weaddthedetailcoefficientsto reconstructthemodifiedsurface(right).

Figure15: A flexible metaphorfor multiresolutionedits. On the left, theoriginal meshis shown. The black line definesthe region of themeshwhich is subjectto themodification.Thewhite line definesthehandlegeometrywhichcanbemovedby thedesigner. Bothboundariescanhave anarbitraryshapeandhencethey can,e.g.,bealignedto geometricfeaturesin themesh.TheboundaryandthehandleimposeC1

andC0 boundaryconditionsto themeshandthesmoothversionof theoriginal meshis foundby applyingdiscretefairing while observingtheseboundaryconstraints.Thecenterleft shows the resultof the curvatureminimization(theboundaryandthe handleareinterpolated).The geometricdifferencebetweenthe two left meshesis storedasdetail informationwith respectto loacalframes.Now the designercanmove thehandlepolygonandthis changestheboundaryconstraintsfor thecurvatureminimization. Hencethediscretefairing generatesamodifiedsmoothmesh(centerright). Adding thepreviously storeddetail informationyieldsthefinal resulton theright. Sincewe canapplyfastmulti-level smoothingwhensolvingtheoptimizationproblem,themodifiedmeshcanbeupdatedwith several framespersecondduringthemodelingoperation.Noticethatall four mesheshave thesameconnectivity.

4 Conc lusions and future work

We explainedhow to addressvarioustechnicalproblemswhenus-ing the fine-to-coarsemultiresolutionmeshrepresentationwhichhasbeenproposedin [21]. We presenteda new way to encodethegeometricdetailinformationby usingacontinuousnormalfieldon the low-frequency geometry. This makesthedetail reconstruc-tion morerobust thanotherlocal framebasedtechniques.We alsoshowed how the use of several intermediatelevels of detail en-ablesthehandlingof geometricconfigurationswhichcannotbepro-cessedcorrectlywith a plain two-banddecomposition.We furtherinvestigatedthe influenceof variousalgorithmic parametersontothe overall performanceof multi-level smoothingschemeswhenappliedto a fine-to-coarsehierarchyon arbitrarymeshes.

In ourcurrentimplementationof themultiresolutionmeshmod-eling technique,the supportingmeshwhich is controlledby con-strainedoptimizationduringtheinteractive modelinghasthesameconnectivity astheoriginal mesh.In thefutureit might bepromis-ing to dropthis restriction.Wecouldimprove thestabilityandcon-vergencespeedof the multi-level schemeby usingregularly con-nectedmeshesinstead.Moreover, this couldprovide thepossibil-ity to use”better” local parameterizationsfor thediscreteLaplaceoperatorwith reasonablecomputationaleffort [4, 15]. However,imposingtheboundaryconditionsinto theoptimizationwould be-comemoreinvolved.

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