subdivision schemes for thin plate splinesjwarren/papers/thinplate.pdfweimer and warren /...

EUROGRAPHICS’98 / N. FerreiraandM. Göbel(GuestEditors)

Volume17, (1998), Number3

Subdivision Schemes for Thin Plate Splines

HenrikWeimer JoeWarren

Departmentof ComputerScience,RiceUniversityP.O.Box 1892,Houston,Texas77005-1892, USA

{henrik,jwarren}@cs.rice.edu

AbstractThinplatesplinesarea well knownentityof geometricdesign.They aredefinedastheminimizerof a variationalproblemwhosedifferentialoperatorsapproximatea simplenotionof bendingenergy. Therefore, thin platesplinesapproximatesurfaceswith minimalbendingenergyandthey arewidelyconsideredasthestandard“fair” surfacemodel.Such surfacesaredesiredfor manymodelinganddesignapplications.Traditionally, thewaytoconstructsuchsurfacesis tosolvetheassociatedvariationalproblemusingfiniteelementsor by usinganalyticsolutionsbasedon radial basisfunctions.Thispaperpresentsa novelapproach for definingandcomputingthin platesplinesusingsubdivisionmethods.Wepresenttwo methodsfor theconstructionof thinplatesplinesbasedon subdivision:A globally supportedsubdivisionschemewhich exactlyminimizestheenergyfunctionalas well as a family of strictly local subdivisionschemeswhich only utilize a small, finite numberofdistinct subdivisionrules and approximatelysolvethe variational problem.A tradeoff betweenthe accuracyoftheapproximationandthelocality of thesubdivisionschemeis usedto pick a particular memberof this familyofsubdivisionschemes.Later, weshowapplicationsof theseapproximatingsubdivisionschemesto scattereddata interpolationand thedesignof fair surfaces.In particular wesuggestan efficientmethodologyfor findingcontrol pointsfor the localsubdivisionschemethat will leadto an interpolatinglimit surfaceanddemonstratehowtheschemescanbeusedfor theeffectiveandefficientdesignof fair surfaces.

Keywords: surfacemodeling,variationaldesign,subdivision,thin platesplines,optimization,scattereddatainter-polation,fair surfaces.

1. Introduction and Motivation

The classicalproblemof geometricdesignis thedefinitionandmanipulationof curvedshapes.For almostall modelingproblemstherepresentedshapeshave to bevisuallyappeal-ing or mechanicallystable.This in turn implies thatcertainconditionson the derivativesof the surfaceof the modeledobject aresatisfied.One very traditional approachto con-structsuchsurfacesis variationaldesignwherethe surfacemodelhasto minimize a certainquadraticfunctionalbuiltfrom a setof differentialoperators.The quality of the sur-faceis thenmeasuredin onescalarnumberby integratingthis functionalover thesurface.

Thin plate splinesare sucha surfacemodel whereonewantsthebendingenergy of thesurfaceto beminimal. Theminimality of the bendingenergy implies that the surface

tangentsvary as little as possiblewhich explains the termvariationaldesign.

Thefirst sectionsof this paperpresentamethodto modelthin platesplinesonboundedrectilineargridsthroughasub-divisionprocess.While theresultingsubdivisionschemeex-actly reproducesthis particularfamily of splinesit is glob-ally supported.Thesecondhalf of this paperfocuseson thederivation of a family of local subdivision schemeswhichbehave almost like thin plate splines.Theseschemesuseonly asmallnumberof distinctsubdivisionrulesthatcanbeprecomputedentirely. Theresultingsubdivisionsurfacescanbemanipulatedextremelyefficiently. Localitywill bethepa-rameterfor choosinga particularmemberof this family ofsubdivisionschemesandtheexactnessof theapproximationof thin platesplineswill dependon thesupport.

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TheEurographics AssociationandBlackwellPublishers1998.Publishedby BlackwellPublishers,108 Cowley Road,Oxford OX4 1JF, UK and350 Main Street,Malden,MA02148,USA.

WeimerandWarren/ SubdivisionSchemesfor ThinPlateSplines

1.1. Thin Plate Splines

Thin platesplineswere introducedto geometricdesignbyDuchon6 in 1976astheminimizerof thefunctional��

f ��IR2 f 2

uu 2 f 2uv f 2

vv � dudv (1)

with respectto someadditional interpolationconstraints.Thus,a thin plate splinecanbe definedas the function Fsuchthat � ��

F � is minimal (2)�F�T0 �� c0 (3)

for a given setof valuesc0 at a given setof knotsT0.��

f �approximatelymeasuresthe bendingenergy of a thin, in-finitely extendingelasticplatewhich is fixedin theinterpo-lationpointsc0.

TheEuler-Lagrangeequationimpliesthatsolutionsto thevariationalproblem(2) will satisfythebiharmonicequation

Fuuuu 2Fuuvv Fvvvv � 0 (4)

everywherein thedomainexceptat the locationof originalknotsT0 10.

Thin platesplineshave global support,i.e. at any pointin theresultingsurfaceall of thedatapointsc0 have a non-zeroinfluence.Furthermorethey havelinearprecisionwhichstatesthat if the given setof interpolationpointsc0 lies ina planethenthe resultingsplinewill reproducethis plane.Moreover they areinvariantundertranslation,rotationanduniformscaling.

Varioustechniquesfor solving this particularvariationalproblemhavebeenused.Traditionally, finite elementmeth-ods1 19 canbeappliedto approximatethesolutionin aniter-ativesolutionprocess.Alternatively logarithmicradialbasisfunctionscanbeusedto expressthesolutionanalytically18.

1.2. Subdivision

Subdivision is a way of representingcurved shapesas thelimit of iteratedlinear transformationswhich map coarsesetsof control points to finer ones.A subdivision schemeis describedby asequenceof subdivisionmatricesSk whichmapcontrol points pk of level k to control points pk � 1 atlevel k 1 by

pk� 1 � Skpk � (5)

Theactualsubdivisionobject(curve,surface,volumeandsoon) is now definedasthelimit of this process,

limk � ∞

pk � (6)

A subdivision schemeis calledlocal if the computationofa new control point at level k 1 only involvesold controlpointsof level k which lie in the topologicalneighborhoodof thenew point.

A very classicalexampleof sucha localsubdivisionpro-cessis theLane-Riesenfeldalgorithmfor repeatedknot in-sertionfor B-splines15 which is shown in figure1 for auni-form cubicspline.Depictedarethecontrolpointsfor level 0,1 and2 from topto bottom.ThesubdivisionmatrixS0 whichmapsthefivecontrolpointsof level 0 into ninecontrolpointsof level 1 hastheform

S0 � 18

��8 0 0 0 04 4 0 0 01 6 1 0 00 4 4 0 00 1 6 1 00 0 4 4 00 0 1 6 10 0 0 4 40 0 0 0 8

��(7)

and the subdivision matricesSk for k � 0 have a similarstructure.In particular, mostof the columnsof any Sk areshifts of the centercolumnof S0. Specialcolumnsappearonly at bothendsof theknot vector. For uniform univariatecubic B-splinesthesubdivision matricescontainonly threedifferentkindsof columnsandthreedifferentkindsof rowswhichareentirelypredetermined.

Figure 1: Approximatingsubdivisionfor uniform cubic B-splines:Theoriginal control polygonis shownon the top.Themiddleandbottomdepictthecontrol polygonafter oneand two roundsof subdivision.Thefinal spline is depictedasa dashedcurve.

Note that in figure 1 the limit curve doesnot interpolatetheoriginal controlpointsp0 andthat thesequenceof con-trol polygonsconvergesto thelimit curveveryquickly. Sub-divisionschemeswith thepropertythatthelimit objectdoesnot interpolatethecontrolpointsarecalledapproximating.

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TheEurographicsAssociationandBlackwellPublishers1998.


The benefitsof subdivision lie in thesimplicity andgen-erality of the approach.Implementationof a subdivisionschemeis very simplebecauseonly polyhedralmodelsareneeded.Furthermorethe subdivision schemeimplicitly de-finesseveral level of detail representationsof the limit ob-ject, dependingon thenumberof stepsthesubdivisionpro-cesshasbeenapplied.Subdivisionschemesareavery pow-erful way of representinga functionbecauseno explicit ba-sis for the solution spaceis needed.In particular it is notnecessaryto know whetherthesolutionspaceis polynomial,rational,logarithmic,hyperbolicandsoon.

Highly desirableare subdivision schemesthat are localandfor whichtheactualsubdivisionrulescanbeconstructedeasily. In this casethe resultingsurfacescan be computedand manipulatedextremely efficiently andthe relationbe-tween the original control points and the limit surfaceisfairly intuitive. Themostsuccessfulsubdivisionschemesinpracticalusetodayarelocalandrely onprecomputedsubdi-vision rules,e.g.15 7 16 9 23 24.

This paperwill makeuseof themethodologyintroducedby Warrenin 22 andwewill show how it canbeusedto con-structschemesfor oneof themostinterestingvariationalde-signproblems– thin platesplines.Usingthis approach,thesolutionto this particularvariationalproblemcanbe recastin the framework of subdivision andall the nicepropertiesandalgorithmsfor subdivisionareapplicable.

Unfortunatelyfor theexactsubdivisionschemetheactualsubdivision rulesareneitherlocal nor cheapto compute.Inparticular, the subdivision rules have to be recomputedatevery level in thescheme.

Many applicationsin geometricdesigndo not requireex-act minimizationof the thin platesplineenergy functional(1). Oneratherdesiressmoothsurfaceswhicharefair, well-behaved and can be computedefficiently. Minimizing thebendingenergy functionalonly guidesthesurfacein thatdi-rection.Thereforethe main point of this paperwill be thederivation of a family of subdivision schemesfor surfaceswhichmimic thebehavior of thin platesplines.Thesesubdi-visionschemesarelocalandrely only on asmallnumberofprecomputedsubdivision rules.They producesmoothlimitfunctionswhich almostsolve the original variationalprob-lemandcanbeevaluatedextremelyefficiently.

Thedistinguishingparameterfor themembersin thisfam-ily of subdivision schemeswill be the supportof the sub-division matrix. The larger the supportthe moreexact theapproximationof the original thin platesplinewill be, butalsothelessefficient canthesurfacesbemanipulated.Thuschoosingaparticularmemberof thisfamily involvesatrade-off betweenaccuracy andefficiency of thesolution.

1.3. Related Work

Variationaldesign,thin platesplinesandsubdivision haveall beenstudiedvery intensively over thepastdecades.Weseeourwork mainly influencedby thefollowing sources:

As mentionedearlier, Duchon 6 introducedthin platesplinesto geometricdesign.Meinguet18 suggestedtheirusefor scattereddata interpolation.Finally Franke8 extendedthin plate splinesby addingtensionparametersto the en-ergy functional.Moreconventionallyandin thegeneralcasevariationalproblemsaresolvedby finite elementapproaches1 19.

Modelingthroughsubdivisionbecameahot topic in geo-metricdesignafterthetwo landmarkpapersby CatmullandClark 2 andDooandSabin5 in thelate1970s.Theschemesintroducedby Dyn, Levin andGregory 7, its subsequent im-provementfor irregular meshesby Zorin et al. 23 24 and inparticularLoop’s subdivisionscheme16 9 foundmany prac-tical applications.A comprehensivetextbookonsubdivisionwill becomeavailablesoon21.

Smoothdiscreteinterpolationbasedon theminimizationof aroughnesscriterionwaspresentedby Mallet 17. Morere-centlytheideaof variationalsubdivisionhasbeendiscussedby Kobbelt and Schröderfor interpolatingcurves 12 14 aswell as for the fairing of interpolatingtriangular meshesusing the thin plate spline energy functional 13. A gen-eral methodologywhich canbe usedto solve elliptic vari-ationalproblemsusingapproximatingsubdivision schemeshasbeenpresentedin 22.

The work presentedhereis distinguishedfrom previouswork on variationalsubdivisionfor surfacesby theuseof anapproximatingsubdivision basis.Theuseof approximatingbasisfunctionsleads,justasfor B-splinesin theonedimen-sionalcase,to basisfunctionswhicharehighly local. In theendthis enablesus to find truly local subdivision schemeswhich almostsolve the variationalproblemwhile utilizingonly asmallnumberof precomputedrules.Thus,theselocalschemesaremuchmoreefficient in practicaluse

2. Variational Subdivision

The ideaof variationalsubdivision is to representthesolu-tion to avariationalproblemthatis describedin termsof itsdifferentialoperatorsthroughasubdivisionprocess.Warren22 presentedaparadigmfor deriving approximatingsubdivi-sionschemeswhichexactlysolvevariationalproblems.Thisframework will serve asthebasisfor our derivationsin thispaperandit will beaugmentedby amethodologyfor deriv-ing purely local subdivision schemeswhich almostsatisfythevariationaldescriptionof thin platesplines.Theselocalsubdivisionschemesutilize only asmallnumberof differentsubdivisionruleswhichcanbeentirelyprecomputed.

The goal is to representthe solution of the variationalthin plateproblem(1) asthe limit of a subdivision process.

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Later, thebehavior of this subdivisionschemefor thin platesplinesis approximatedby a small setof local subdivisionruleswhichcanbeprecomputedonce.Theremainderof thissectionwill briefly outline the methodpresentedin 22. Foramoredetailedandtheoreticallyjustifiedexpositionseetheoriginal reference.

2.1. Energy and Subdivision

As pointedout earlier in the introductionthin platesplinesaredefinedastheminimizersof theenergy functional(1)��

f �� IR2 f 2

uu 2 f 2uv f 2

vv � dudv �

difference mask location

corner masks

boundary masks

interior mask

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64 5 45

52 3 23

41 2 12

41 2 12

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-6

10

1

2

-6

2

1

-4

5

-6

11

-6

2

-6

2

1

1

1

Figure 2: Structure of the energy matrix E for thin platesplines.Thetop rowdepictsa rectilineargrid with theknotsnumbered.Theremainingrowsdepicttherulesusedto com-putea discreteenergy valuefor thedifferentkindsof knotsin thegrid.

Solvingthis problemin thediscreteframework of subdi-visionrequiresrecastingof this integralenergy measure

�to

adiscretesetting.Givenasetof controlpointspk for level k

of thesubdivisionprocess,acontinuousversionof this sub-division representationcanbe definedby choosinga setofbasisfunctionsBk andwriting thelevel k solutionas

Fk�u � v�� Bk

�u � v� pk � (8)

Thentheenergy of thecontrolpointspk canbedefinedastheenergy of Fk with respectto the original energy functional(1), i.e.as

��Fk � , whichyieldsthequadraticform��

Fk �� pTk Ekpk � (9)

Ek is asquarematrix,calledtheenergy matrix.Givenanar-bitrary setof controlpoints pk for level k theenergy of thecontrol grid cannow be assessedusingthis energy matrix.The entriesin Ek areinner productsof appropriatederiva-tivesof finiteelementbasisfunctionscenteredovertheknotsof thelevel k grid.

This methodfor computingthe energy of the thin platesplineFk with control points pk is theoreticallysoundandgeneraland is applicablein the caseof irregular grids aswell. Becauseits implementationis fairly involved we willbeusingthewell knowndifferencemasksfor thebiharmonicequation(4) insteadwhich can be found in any text bookon finite differences,e.g.19. Thesedifferencemasksarede-picted in figure 2. The readerfamiliar with finite elementmethodsmayrecognizetheseenergy masksasthedifferenceschemesfor thebiharmonicequation.

2.2. Variational Subdivision for Thin Plate Splines

The idea of approximatingvariationalsubdivision as pre-sentedin 22 is to computethecontrolpointsfor thefinergridin suchawaythattheenergy of thecontrolgrid is preservedat old control pointsandenergy zerois achievedat all newcontrolpoints.

To formalize this constraintwe makeuseof an upsam-pling matrix Uk which carriescoefficients associatedwithcontrolpointsat level k over to level k 1 by insertingcoef-ficientzerofor all controlpointswhicharenew at level k 1.Figure3 illustratestheactionof theupsamplingmatrix:De-pictedarethreesuccessive levels of subdivision of a 3 � 3grid. Theenergy valueassociatedwith theoriginal,leftmostcontrol pointsarepaintedasblack dots.The energy valuesinsertedthroughupsamplingareshown as light gray dots.Note that thesubdividedgridswill maintaintheold energyvalues(blackdots)at thelocationof original, level 0 controlpoints,while adiscreteenergy of zero(graydots)is insertedfor all new knots. Intuitively the limit of this processwillproducecoefficientsthatsatisfythebiharmonicequation(4)everywhereexceptat theoriginal knots.

This idealeadsto a constrainton the subdivision matrixfor level k whichcanbeexpressedas

Ek � 1Sk � UkEk (10)

whereEk denotestheenergy matrixdefinedin (9), Sk is thesubdivisionmatrix for level k andUk representsupsampling

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U U

Figure 3: Theactionof theupsamplingmatrixU. Zero energyvaluesaredepictedby light graydots,non-zeroenergyis shownasblack dots.In thelimit this processwill producediscreteenergyzero everywhereexceptat theoriginal knots.

from level k to level k 1 by insertingzerosinbetweenoldcoefficients.

Equation(10) statesthat subdividing the control pointsand computingthe energy of the resultingfiner grid (lefthandside)shouldbe the sameascomputingthe energy atthecoarselevel andupsamplingthesevaluesto thefinergridby insertingzerosinbetween(right handside).

If the subdivision matricesSk satisfyequation(10) thenthelimit surfaceof thesubdivisionprocessis theminimizerof the original variationalproblem.For a completeformalproofsee22.

As theenergy matricesEk arerankdeficient,thesubdivi-sion matricesSk arenot well definedby (10) alone.Addi-tional constraintshave to beaddedto chooseoneparticularsubdivision basisfor the solutionspace.For example,onereasonablepropertythe subdivision schemeshouldhave isthepreservationof momentswhichstatesthattheinnerprod-ucts betweenlower degreepolynomialsand the finite ele-mentbasisfunctionusedfor theconstructionof E shouldre-mainconstantundersubdivision.If M is amatrix represent-ing theexpressionM �� uiv jBk

�u � v � dudu for somei � j � 0

thenthisconstraintcanbeexpressedas

Mk � 1Sk � Mk � (11)

This additionalconstrainton Sk will lead to a uniqueandwell-behavedsubdivisionscheme.For thin platesplinestheconstantand linear moments

�i � j ��

0 � 0� , � i � j �� 1 � 0� ,�

i � j � � �0 � 1� haveto beconsidered.

Solving equation (10) yields subdivision matrices Skwhich will producethe exact minimizer of the variationalproblemastheir limit surface.Figure4 depictsthreediffer-entstencilsof aparticularS. Shown arerows from theinte-rior of thegrid roundedto five digits.Theexactstencilsareglobally supportedandhave beentruncatedto a reasonablesize.Shown arethe rulesto computea new coefficient fora vertex, an edgeanda facein the interior of thegrid. Thelocationof thenew controlpoint is depictedby abox.

Inspectionof the resultingsubdivision matricessuggeststhefollowing statements:�

The subdivision rulesdo not dependsignificantlyon thelocationin thegrid. Rulescloserto theboundaryaredif-ferent from the onesin the interior, but the subdivision

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vertex mask

edge mask

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Figure 4: Exact subdivisionmasksfor thin plate splinesroundedto fivedigitsandtruncated.

rulesfor neighboringverticesin theinteriorof thegrid arealmostidentical,soarethosefor twoneighboringedgesorfaces.

�Therulesdonotdependsignificantlyonthelevel k of thesubdivision.Rulesfor vertices,edgesandfacesat level karealmostidenticalto therespectiverulesfor level j !� k.

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7. faceadjacentto boundary

Figure 5: Structureof thesubdivisionmatrix shownfor a 5 � 5 grid. Grid linesof thecoarsegrid aresolid, grid linesof thefinegrid aregray.�

The rulesarealmostlocal, i.e. for a new control coeffi-cient theold coefficientsin its neighborhoodhave a verylarge influencewhile control coefficients further awayhaveverysmallweights.

Thesubdivisionschemepresentedin thissectionis notlo-calbut it producestheexactthin platesplineasthelimit sur-face.Thesubdivisionmatrixhasto berecomputedfor everylevel in thesubdivisionprocessandit is globallysupported.

3. A Local Subdivision Scheme for Thin Plate Splines

Theobservationsthat thesubdivision ruleshave fastdecay,look very similar at different levels, and are mostly inde-pendentof the actuallocationin the meshsuggestthat theschemecouldbeapproximatedto ahighdegreeof accuracyby a local subdivision schemewhich utilizes only a smallnumberof distinctsubdivisionrules.Thissectionfocusesonthe definitionof a family of suchsubdivision schemes.Anactualmemberof thisfamily ispickedby specifyingthesup-port of thesubdivision basisfunctionsandwe demonstrateherehow to computethe schemewith thesamesupportasbicubicB-splines,i.e.5 � 5 in a rectilineargrid.

Theselocal subdivision schemesno longerexactly solveequation(10).Thus,they no longerproducetheexactmini-mizerof thevariationalproblem(1).For many geometricde-signapplicationsthisexactenergy minimizationis notcriti-cal.Oneis ratherinterestedin having anefficientalgorithmfor constructingfair andsmoothsurfaces.We consequentlybelieve that thereare many real world applicationswhichcanmakeuseof suchsubdivisionschemeswhichalmostre-producethin platesplines.

To allow for stability andlocalsupportin thesubdivisionprocess,the exact equalityconstraintin (10) hasto be re-laxed.Theoriginalequation(10) reducesto

Ek � 1Sk - UkEk (12)

whereSk denotesthelocalsubdivisionmatrix.Oneveryrea-sonablemethodof solvingfor theentriesof Sk is to setupthesystem(12)symbolicallyandthento minimizetheL∞-normof theexpression ..

Ek� 1Sk / UkEk

..∞ (13)

whichwill leadto alinearprogrammingproblem.Minimiza-tion of theL∞-normof thisproblemis preferredovere.g.theL2-normbecauseconvergenceof asubdivisionschemeisde-terminedin theL∞-norm.

Equation(13) definesawholefamily of local subdivisionschemeswhich approximatethin plate splines.The actualsubdivision schemeis determinedby theparticularsupportand structureof the subdivision matrix Sk and the level kof subdivision.In thefollowing sectionit will becomeclearwhy the actualsolution is invariantunderthe level k. Thelocality and structureof the subdivision matrix remainasthe only parametersfor picking an actualmemberof thisfamily of subdivision schemes.We will show how to findtheschemewith supportequivalentto bicubicB-splinesona rectilineargrid.

3.1. Building the Local Subdivision Matrix from aSmall Set of Rules

Given a particulargrid of level k a local subdivision ma-trix Sk canbe constructedwhich usesonly a small number

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of distinct subdivision rules.The remainderof this sectionwill focus on the processof building this symbolic repre-sentationof thesubdivisionmatrixandsolvingtheresultinglinearprogram.

Therowsof theSk arepickeddependingonthelocationofthenew controlpoint in thegrid.For thesubdivisionschemewith thesupportof bicubicB-splineswe suggestusingdif-ferentrulesin thesubdivisionschemeasshown in figure5.Eachentryin thetablein figure5 correspondsto a differenttype of row in the subdivision matrix. All rows of a giventype are equivalent.Symmetriesand simple shifts can beusedto computethe actualstencil for a particularcontrolpoint.

The secondpoint of concernis the actualsupportof therows of the symbolicsubdivision matrix. We elect to clas-sify the locality of the subdivision schemesby the supportof the columnsof the subdivision matrix. A columnof thesubdivision matrix describeshow an old controlcoefficientis split over new coefficients.In general,a columnsupportof

�2n 1�0� � 2n 1� canbe chosenfor any n � 0. Only

odd supportsmakesensebecausethe rulesshouldbe sym-metricandthecolumnsarecenteredover theold coefficient.A columnsupportof

�2n 1�1� � 2n 1� implies that the

subdivision stencil to computea new face will have sup-port 2

�n / 1�1� 2

�n / 1� . The stencil for a vertex hassup-

port�2 2 n2 3 1�1� � 2 2 n2 3 1� anda control coefficient for

an edgehassupport�2 2 n2 3 1�0� 2

�n / 1� . Along thegrid

boundariesthe columnsof the subdivision matrix certainlyonly contributeto theinterior of thegrid.

If the subdivision basisfunctionsshouldhave the samesupportas bicubic B-splinesthencolumn i of S will havea 5 � 5 supportcenteredover control point i. This impliesthattherows of Shave thestencilsshown in figure6, e.g.anew control point for an interior faceis supportedover thecornerverticesof the face,a control point centeredover ainteriorvertex of thecoarsegrid is supportedoverthecoeffi-cientscenteredovertheknot itself andtheeightneighboringknotsandsoon.Theactualstructureof theparticularsubdi-vision rulesis alsodepictedin figure6. Two controlpointswill receivethesameweightdueto symmetrywhenthecor-respondingknotshavethesameiconsin thefigure.

Given the symbolicrepresentationof Sa linear programfor (13) canbe constructedby choosinga sufficiently largek, constructingSk andthencomputingthenormin (13).

Observe thattheL∞-normof a matrixneitherdependsonthe order of termsin a row of the matrix nor on the par-ticular orderof the rows. Therefore,the termsin the rowsof matrix (13) aswell astherows themselvescanbesortedinto a canonicallexicographicorder without changingtheL∞-normof theexpression.Usingsimplesetoperationsre-dundanciescanbeeliminatedby discardingduplicatetermsandrows.Thenormof thematrixexpression(13)is invariantunderthis reduction.

4 5 6 7

8 9 : ;

: < =

Figure 6: Thesupportand structure of the different kindsof rowsin thesubdivisionmatrix: Theinterior of thegrid isshownasa gray shadedregionandthe locationof thenewcontrol point is depictedby an emptybox. Indicesrefer tofigure 5. Insideonerule all coefficientswith the sameiconwill havethe samevalue,e.g. in rule 1 the top right andbottomleft coefficientsare forcedto beequalwhile thetworemainingentriesareunconstrained.

Closerobservation revealsthat in fact expression(13) isindependentof thelevel k of thesubdivision.BecauseSusesonly afixed,smallnumberof distinctrules,two subdivisionmatricesSk andSl havesimilarrowswhichareshiftsof eachother. Furthermoretheenergy matricesEk andEl havesim-ilar rows. Multiplication of E andS thusalsoleadsonly toa finite numberof different,similar rows. The redundancyeliminationdescribedabovedeterminestheunique,minimalsetof distinctrows in this processindependentof k. Onceacertainthresholdfor k hasbeenpassedall thedifferentkindsof non-zerocombinationsof rows of S and E show up in(13) andthe redundancy eliminationreducestheserows toa minimal setof expressionswhich entirely determinetheL∞-norm.

For the subdivision schemewith the supportof bicubicB-splinesandthe structurepreviously describedin figures5 and6 the systemreachesits maximalsizeafter 5 roundsof subdivision.Theredundancy eliminationproducesin thiscaseasetof 21 distinctrows in 83 distincttermsfor expres-sion(13).

A small linearprogramfor this reducedsetof constraintscanbeconstructed.After addingthreeadditionallinearcon-

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straintsto assertconstantandlinear precisionof thesubdi-visionscheme,thesystemcanbesolvedonceusingany rea-sonablelinearprogrammingpackage,e.g.CPLEX4. There-sultof this linearoptimizationarecoefficientsfor thediffer-entkindsof subdivisionrulesin S. Thesecanbeusedto con-structsubdivisionmatricesfor arbitrarysizerectilineargridsat arbitrarylevelsof subdivision.Thedifferentrulesresult-ing from solving this processfor subdivision matriceswiththe5 � 5 columnsupportof bicubicB-splinesareshown infigure7.

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Figure 7: Therulesof the local subdivisionschemefor theapproximationof thin platesplines.Indicesreferto figure5.

Thesubdivisionprocessdoesnotproducetheexactmini-mizerto thevariationalproblembut it is highly localandcanbe entirelyprecomputed– thusit canbe appliedextremelyefficiently.

4. Properties of the Subdivision Schemes

The exact solutionto the thin platesplinevariationalprob-lem hasC∞-continuityeverywhereexceptat thelocationofthe original knots– thereit is just C1. The global subdivi-sion schemepresentedin section2.2 solvesequation(10).Thus,it exactly reproducestheminimizerof thevariationalproblem(2) 22. Therefore,theexactsubdivisionschemepro-duceslimit surfaceswhich areC∞-continuouseverywhereexceptat the locationof original knots– therethey arejustC1.

Analyzing the smoothnessof the local schemesprovesslightly moreinvolved.The eigen-structureof the schemes

can be analyzed.The eigen-spectrumof the particularlo-cal schemepresentedin theprevioussectioncontainslead-ing eigen-values1 � 0 � 5 � 0 � 5 � 0 � 37�TSUSTS with all the remainingvalueshaving modulussmallerthan0 � 37. Thus the eigen-spectrumis consistentwith the subdivision schemebeingC1-continuous3 20. Furthermorethe schemeappearsvisu-ally smoothin all ourexamples.

Note that any degreeof continuity can be enforcedbyaddingappropriatedifferenceconstraintsfor thesubdivisionschemeto thelinearprogramandatthesametimeincreasingthecolumn-supportof S.

5. Applications

Thissectionpresentsapplicationsof theapproximatingsub-divisionprocess.Wewill show how theschemecanbeusedto interpolateagivensetof valuesandapplyit to definefairsurfaces.Note that to apply the subdivision schemewe donot have to re-runtheoptimizationprocessin (13). Instead,thecoefficientsin figure7 areused.

5.1. Control Points for Interpolating SubdivisionSurfaces

Thepresentedsubdivisionschemesareapproximating.Thismeansthat the subdivision surfacedoesnot interpolatetheverticesof the original control net. Like in the caseof B-splinecurves,thesequenceof controlnetsdescribescloserandcloserapproximationsof thelimit surface.Hence,in or-der to interpolatea particularsetof points,adequateinitialcontrolcoefficientsfor thesubdivisionhavetobedeterminedfirst.

Formally thisproblemcanbedescribedasfollows:Givena setof interpolationconditionsc0 centeredover the knotsof theoriginalcontrolgrid, find asetof initial controlpointsp0 centeredover thesameknotssuchthat

I0p0 � c0 � (14)

HereI0 denotestheinterpolationmatrix for level 0. Thein-terpolationmatrix Ik containsthe valuesof the basisfunc-tionswhichareinducedby thesubdivisionschemesampledat theknotsof level k. The

�i � j � -th entryof I0 containsthe

valueof the basisfunction centeredover knot i sampledatknot j.

The interpolationmatrix Ik is characterizedby the equa-tion

UTk Ik � 1Sk � Ik � (15)

As mentionedearlierIk containsthevaluesof thebasisfunc-tionsof levelk sampledattheknots.Equation(15)statesthatcomputingthevaluesof thebasisfunctionsat level k (righthandside) is the sameas subdividing the basisfunctions,thencomputingtheir valuesat thefinergrid andthendown-samplingbackto thecoarsegrid usingthe transposeof theupsamplingmatrix (left handside).

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Thesubdivisionschemepresentedin section3 is localandusesonly a small numberof precomputedrules.Thus,therows of the interpolationmatricesIk arealsodrawn from asmall set of commonrows. In fact, the supportof the in-terpolationmatrix Ik is the sameas the supportof UT

k Sk.A symbolicrepresentationsof Ik canbe constructedeasilyusingthe representationfor Sk. Togetherwith anadditionalconstraintthattherowsof Ik haveto sumto onetheinterpo-lation matricesarecompletelydeterminedby equation(15).A linearprogramfor V UT

k Ik � 1Sk / Ik V ∞ � 0 canbeusedtosolve for theentriesin I .

Given a setof interpolationconstraintsc0 centeredoverthe knotsof level 0 a suitablesetof control points p0 canthenbe determinedvery efficiently usingthe inverseof theinterpolationmatrix I0:

p0 � I W 10 c0 � (16)

The following sectionextends this paradigmto the casewherethesetof interpolationconstraintsc0 doesnot coverthefull setof knotsof level 0.

5.2. Scattered Data Interpolation

For this examplethe scattereddataproblemis, given a setof functionalsamplesX � ui � vi � wi �ZY i � 1 �USTSUS[� n\ find asurfacewhich interpolatesthesepoints.

Given sucha setof samplesan interpolatingsubdivisionsurfacebasedon the local subdivisionschemefrom section3 canbe constructedby picking a sufficiently denseinitialgrid of knotsandthenwarpingthesamples

�ui � vi � wi � to the

closestknotwith respectto u andv asshown in figure8. Thevaluewi is usedasthedesiredinterpolationvalueassociatedwith theknot.

Figure 8: Thesamples(solid dots)are warpedto knotsofa grid basedon u and v. This leadsonly to interpolationconditionsfor a subsetof the knots.Theremainingcontrolpoints, depictedby gray diamonds,can be determinedbyconstrainingthediscreteenergyat theknotto bezero.

As the set of samplesusually doesnot cover all of theknots,only a subsetof themhave associatedinterpolationconditions.Thus,it is notpossibleto find thecompletecon-trol net by simply applyingthe inverseof the interpolation

matrixasdescribedin theprevioussection.Instead,thecon-trol points have to be computedsuch that interpolationisachievedat thoseknotswhich have anassociatedinterpola-tion constraintandenergy zerois achievedatall theremain-ing knots.This constructionwill producea control coeffi-cientassociatedwith everyknotandthesubdivisionschemecanbeapplied.We planto publisha paperfocusingon thedetailsof thismethodat a latertime.

An exampleis shownin figure9.Theoriginalsurfacewasgivenby

f�u � v�� / 0 � 4eW^]_] u W 1 2 � ] v � 1 2 ` 0 � 6eW^]_] u � 1 a 2 2 � ] v� 1 a 2 2 ` eW^]_] u W 1 2 � ] v W 1 2 ` �

5.3. Design of Fair Surfaces

A very commonproblemin geometricdesignis the mod-eling of fair surfaces.In this framework a fair andsmoothhasto beconstructedwhichmatchesupwith existingcurvesalongpatchboundaries.This sectionwill demonstratehowthis problemcanbesolvedefficiently usingour local subdi-visionscheme.

The naive methodfor applyinga functional subdivisionschemeto a three-dimensionalproblemis to applytherulesseparatelyto the three parameterfunctions.The resultingsurfaceis smoothbut the variationalpropertiesof the ob-ject cannot beguaranteed.A morereasonablesolutionis tolocally reparametrizethemeshby projectingthelocalneigh-borhoodinto anestimatefor thetangentplaneandapplyingtheschemein thereparametrizedspace.

An exampleapplicationfor thedesignof fair surfacesin3D is shown in figure 10. A polygonalmodelof the spaceshuttlehasbeenmodified to replacethe part representingthe noseby a densemeshgeneratedthrough the subdivi-sionschemedescribedin section3. The fair surfaceshownhereinterpolatestheverticesthatwerepresentin theorigi-nalmodel.Thecontrolpointsfor thesubdivisionweredeter-minedusingtheinverseof theinterpolationmatrix.

6. Conclusion and Future Work

Wepresentedtwo subdivisionschemesfor thin platesplineson a rectilineargrid. The first schemereproducesthe exactsolutionto thevariationalproblem.It hasglobalsupportandthesubdivisionmatrixhasto berecomputedatevery level ofthesubdivisionprocess.Thesecondschemeis localandusesonly asmallnumberof ruleswhichcanbeprecomputed,butit onlyapproximatesthin platesplines.Thequalityof theap-proximationdependsonthesupportof thesubdivisionmask.Arbitrarily smallresidualscanbeachievedby increasingthesupportof thebasisfunctions.

In the future,we plan to extendthis approachto handleirregular gridsof arbitrarytopologicaltype.We first planto

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work out a formulationof theexactsubdivisionschemeforthin platesplinesin this framework by generalizingtheen-ergy matrix to irregular grids. The ultimategoal will be tofind againa simplesetof subdivisionrulesthatcanbeusedto approximatetheexact solutionto a high degreeof accu-racy. Becausethelocalgeometryof thegrid canbearbitraryin thecaseof irregulargridsit maybenecessaryto parame-terizethesubdivisionrulesover thelocalgeometry.

Finally, we plan to generalizethe methodfor construct-ing interpolationmatricesto local, stationarysubdivisionschemes,suchasLoop’sscheme16.

Acknowledgments

Thisworkhasbeensupportedin partunderNationalScienceFoundationgrantCCR-9500572,TexasAdvancedTechnol-ogy Programgrant003604-010andby WesternGeophysi-cal.

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4. CPLEXOptimization,Inc, UsingtheCPLEXCallableLibrary (1995).

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6. J.Duchon,Splinesminimizingrotationinvariant semi-normsin Sobolev spaces, in W. SchemppandK. Zeller(eds.): ConstructiveTheory of Functions of severalvariables: proceedingsof a conferenceheld at Ober-wolfach,April 25-May1, 1976, LectureNotesin Math-ematics,571, pp.85–100,Springer-Verlag(1977).

7. N. Dyn, D. Levin andJ. A. Gregory, A ButterflySub-divisionSchemefor SurfaceInterpolationwith TensionControl, Transactionson Graphics,9, 2, pp. 160–169(1990).

8. R. Franke,Thin Plate Splineswith Tension, in R. E.Barnhill andW. Böhm(eds.),Surfacesin CADG1984,North-Holland,pp.87–95(1985),

9. H. Hoppe,T. DeRose,T. Duchamp,M. Halstead,H.Jin,J.McDonald,J.SchweitzerandW. Stuetzle,Piece-wiseSmoothSurfaceReconstruction,ComputerGraph-ics Proceedings,Annual ConferenceSeries,pp. 295–300(1994).

10. J. HoschekandD. Lasser, Grundlagender Geometri-schenDatenverarbeitung, 2nd edition, B. G. Teubner(1992).

11. L. Kobbelt, A variational approach to subdivision,ComputerAided GeometricDesign,13, pp. 743–761(1992).

12. L. Kobbelt,InterpolatorySubdivisiononOpenQuadri-lateral Netswith Arbitrary Topology, ProceedingsofEurographics1996, ComputerGraphicsForum, pp.409–420(1996).

13. L. Kobbelt,DiscreteFairing, Proceedingsof the Sev-enth IMA Conferenceon Mathematicsof Surfaces(1996).

14. L. KobbeltandP. Schröder, ConstructingVariationallyOptimal CurvesthroughSubdivision, California Insti-tute of Technology, Departmentof ComputerScience,TechnicalReportCS-TR-97-05(1997).

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16. C. Loop, Smoothsubdivisionsurfacesbasedon trian-gles, Master’s thesis,Departmentof Mathematics,Uni-versityof Utah(1987).

17. J.-L. Mallet, Discretesmoothinterpolationin geomet-ric modeling, ComputerAided Design,24, 4, pp.178–191(1992).

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Weimer and Warren, Figure 9: Scattereddatainterpolationusingthin platesplinesubdivision.Theoriginal surfaceis shownon thetop.Thirty randomsampleswereusedto constructthecontrol meshon thebottomleft which resultsin theinterpolatingsubdivisionsurfaceon thebottomright.

Weimer and Warren, Figure 10: Thenoseof theoriginal spaceshuttlemodel(left) replacedby a fair subdivisionsurfacewhich interpolatestheoriginal vertices(right). Thebottomleft figurealsoshowsthecontrol polygonproducedby theinverseof theinterpolationmatrix.

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subdivision schemes for thin plate splinesjwarren/papers/thinplate.pdfweimer and warren /...

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