Computing Volumes of Solids Enclosed by Recursive Subdivision Surfaces

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1 EUROGRAPHI 97 / D. Fellner and L. zrmay-kalos (Guest Edtors) olume 6, (997), Number 3 omputng olumes of olds Enlosed by Reursve ubdvson urfaes Jörg Peters and Ahmad Nasr Abstrat The volume of a sold enlosed by a reursve subdvson surfae an be approxmated based on the losed-form representaton of regular parts of the subdvson surfae and a tght estmate of the loal onvex hull near extraordnary ponts. The approah presented s effent,.e. non-exponental, and robust n that t yelds rapdly ontratng error boundng boxes. An extenson to measurng hgher-order moments s skethed.. Introduton Due to oneptual smplty and dsplay by averagng, subdvson surfaes are a popular representaton for modelng, graphs and multresoluton (see e.g. 2, 5, 7, 7, 6, 8 and referenes theren). One of the noted drawbaks of subdvson surfaes for quanttatve applatons s ther lak of a fnte losed-form representaton. Already the bas queston, what s the volume enlosed by a subdvson surfae, s not treated n the lterature and does not seem to have a smple answer. Thus realst anmaton as opposed to artoons and geometr desgn usng ths onvenent representaton must rely on user ntuton rather than omputer gudane to generate objets that hold a gven volume or, n the ase of omputng the enter of mass, do not topple over. Ths gap n the subdvson arsenal s flled by the present paper. We develop an effent and robust algorthm for measurng volumes of subdvson surfaes and sketh how hgher-order moments an be treated n the same framework. For omparson, onsder frst the dret approah of splttng the volume enlosed by the subdvson surfae nto tetrahedra or sles as s the ommon approah when measurng polyhedra 0,, 9 ). The number of pees to be measured for exat alulaton s proportonal to the number of faets. ne eah polyhedron faet gves rse to a onstant number of new faets wth eah subdvson step, the resultng amount of work s exponental n the number of subdvson steps and hene unpratal for real tme applatons. A more sophstated approah s to ompute the volume of the upported by NF Natona Young Investgator grant R upported by an AUB grant ntal polyhedron and then trak the hange n volume due to the refnement, nterpretng the refnement as a sequene of uts appled to the sold enlosed by the polyhedron 3. However, ths trakng s not easly mplemented sne uts, n general, are nether smple nor planar and may add or remove materal dependng on loal onvexty. The approah s also not effent sne the amount of work s stll proportonal to the exponental nrease of the number of faets under subdvson. Yet there s an effent and robust approah to measurng the volume of most subdvson surfaes. Effent means that the amount of work s onstant at eah subdvson step and robust means that a two-sded error bound exsts and ontrats by a onstant multple less than, typally 2 3, wth eah refnement step. The two key observatons leadng to the result are () that peewse polynomal surfaes allow effent omputaton of the exat volume and hgher-order moments 6 and (2) that most popular subdvson shemes, e.g. 4, 2, are modfatons of box-splne subdvson rules, and hene generate a lmt surfae that s peewse polynomal exept at some solated ponts, alled extraordnary ponts. That s, whle subdvson surfaes lak a global losed-form representaton, nreasng submeshes of the subdvson polyhedron have suh a representaton. Ths sad, the overall strategy s lear namely to ompute the asymptot volume ontrbutons exatly for the regular regons of the subdvson surfae away from the extraordnary ponts, and to estmate and bound the volume ontrbutons of the neghborhood of eah extraordnary pont. For larty we lst our assumptons on the subdvson sheme. The Eurographs Assoaton 997. Publshed by Blakwell Publshers, 08 owley Road, Oxford OX4 JF, UK and 350 Man treet, Malden, MA 0248, UA.

2 Peters and Nasr / omputng ubdvson olumes Doo-abn Extraordnary Pont enter of regular submesh atmull-lark Fgure : Damonds mark extraordnary ponts on the subdvson polyhedra. The enters of regular submeshes n the less refned polyhedra on the left are marked by rles. A regular submesh of the Doo-abn sheme (top) onssts of four quadrlaterals surroundng a vertex. A regular submesh of the atmull-lark sheme (bottom) onssts of eght quadrlaterals surroundng a quadrlateral.. The subdvson sheme generates regular submeshes that refne to regular submeshes so that the number of extraordnary ponts remans onstant. 2. Regular submeshes have a polynomal parametrzaton. 3. The subdvson algorthm has the loal onvex hull property. That s, new mesh ponts are onvex ombnatons of old ones nearby, e.g. when the subdvson mask has only non-negatve entres. We proeed by showng, n eton 2, how the volume s omputed by ntegratng over the surfae, and ompute the ontrbuton of regular subsurfaes. eton 3 defnes the approxmaton aps over the remanng volume and ther ntegrals, and eton 4 gves a onrete example n terms of the Doo-abn subdvson. eton 5 outlnes the omputaton of mass propertes. 2. olume ontrbutons from regular submeshes At eah step, a subdvson algorthm reates a new mesh of ponts from an old mesh. A desrable property of any subdvson algorthm s that t generates nreasng submeshes all of whose ponts have the same valene and whose faets all have the same number of edges. A submesh of ponts and faets wth ths standard valene and number of edges s alled regular. Popular subdvson shemes derve ther appeal from the fat that the lmt surfae s expltly known for regular submeshes. For example, the lmt surfae of the Doo-abn subdvson sheme 4 appled to a regular nne-pont submesh s a bquadrat tensor-produt splne surfae. The lmt surfae of the atmull-lark subdvson sheme 2 appled to a regular sxteen-pont submesh s a bub tensor-produt splne surfae. And the lmt surfae of Loop s subdvson s a lnear ombnaton of shfts of a 3-dreton box splne. Wth respet to these subsurfaes we wll apply Gauss dvergene theorem. Gven a parametrzaton x u x y z u v of a surfae wth normal x u v y u v z u v N n n where n u v U x u x v and gven a map f : R 3 R 3, the dvergene theorem (see e.g ) states that f d f N d.e. that the ntegral of the dvergene f x f over the volume equals the ntegral of the normal omponent f N f N over the surfae of. By hange of varable d U n dudv.e. the area element n s the nverse of the normalzaton fator of the normal dreton and hene f d f n n d U f n dudv Now f both f and x are polynomal then the ntegrand s polynomal; and f U s a smple doman, say a trangle or square, then the ntegral an be determned effently, expltly and exatly by averagng the Bernsten-Bézer oeffents 3. If the parametrzaton x of onssts of pathes x and U s the unon of the path domans U then U f n dudv U f n du dv hoosng f 0 0 z we need only ompute one omponent of n, n 3 to determne the volume : d x y u v x y v u 0 0 z d 0 0 z n n 2 n 3 d U z n 3 dudv The Eurographs Assoaton 997

3 Peters and Nasr / omputng ubdvson olumes Fgure 2: Unon of surfae layers at a 5-valent extraordnary pont. The algorthm Intally the algorthm omputes the volume ontrbuton 0 for all, possbly zero, surfae pees orrespondng to regular submeshes: for eah path (f. Fgure 3) z u v n 3 u v s omputed and ntegrated. Eah subsequent subdvson adds exatly one, geometrally ever smaller, layer of surfae pees defned by regular submeshes around eah extraordnary pont as shown n Fgure 2 and Fgure 3. ne the ontrbutons to the volume by these regular layers are omputed exatly and for the lmt surfae, the ontrbuton of submeshes obtaned by subdvdng a regular submesh need not be reomputed! Thus the work at eah subdvson step s onstant, proportonal only to the number of extraordnary ponts; n the mth step t onssts of omputng the surfae ntegral m of the mth layer added at the extraordnary pont. Wth W m the surfae ntegral of the ap whose onstruton s explaned n eton 3, the approxmaton to the volume at the mth step s m m 0 eton 4 gves a onrete example. W m 3. olume ontrbutons of the neghborhood of an extraordnary pont Havng dealt wth the regular parts of the mesh n the prevous seton, the goal of ths seton s to ap off the holes remanng n the regular surfae around the extraordnary ponts and ompute the ntegrals of the appng surfaes. To dretly apply the dvergene theorem, eah ap should jon the regular surfae ontnuously and wthout gap or overlap. Extraordnary ponts are typally ether meshponts wth a valene dfferent from the regular meshponts, or entrods of Fgure 3: On top, dots, some labelled jk, ndate the nodes of a submesh that determnes the subdvson matrx at an extraordnary pont n the Doo-abn sheme. The grey quadrlaterals eah represent a bquadrat path. Below rles ndate the nodes of the refned submesh. Agan the bquadrat pathes are delneated. mesh faets that have a dfferent than the standard number of edges. For example, f. Fgure, the Doo-abn subdvson sheme has a standard valene of four for both vertes and faets,.e. every mesh pont s surrounded by four quadrlaterals. The error boundng box By assumpton, the volume s bounded by the onvex hull of the submesh n the neghborhood of the extraordnary pont. Fgure 3 shematally shows suh a submesh for the Doo- abn sheme. The estmate an be mproved by knot nserton,.e. onverson to Bernsten-Bézer form at the boundary between the regular surfae pees and the neghborhood of the extraordnary pont. Fgure 4 shows the ndes of the The Eurographs Assoaton 997

4 Peters and Nasr / omputng ubdvson olumes Bernsten-Bézer oeffents b 02, b 2, b 22, b 2, b 20 along the boundary for the Doo-abn subdvson. ne, dependng on the valene of the extraordnary pont, the exat onvex hull may be expensve to ompute, we settle for an effently omputable boundng box that enloses the mnmal and maxmal values of eah omponent of the ontrol ponts n the neghborhood of the extraordnary pont. To get an estmate of the ontraton of ths error box t s natural and standard to look at the subdvson matrx whh maps the submesh around the extraordnary pont to a refned submesh wth the same number of mesh ponts. For the Doo- abn sheme the matrx s of sze p p where p q 3 3 at a q-valent extraordnary pont, e.g. p 45 for the example n Fgure 3. For a smooth-surfang sheme the leadng egenvalue of ths matrx s and the two subdomnant, next largest egenvalues determne the frst-order behavor of the subdvson surfae and hene the asymptot shrnkage fator of the submesh surroundng the extraordnary pont 4. Denotng the seond-largest egenvalue by λ, the boundng box volume n three dmensons onverges asymptotally on the order of λ 3 wth eah refnement step. In pratal examples the asymptot rate s mmedately reahed (.f. Tables and 2). Fnally we defne the aps. To get a good estmate of the volume the ap pathes must le wthn the boundng box. We hoose them as a rng, orrespondng to the whte quadrlaterals n Fgure 4, of 0 -onneted pathes, of the same degree as the regular k subdvson surfae, and extendng the regular surfae parametrally k to over the neghborhood of the extraordnary pont. Ths avods gaps and determnes k layers of Bernsten-Bézer oeffents adjaent to the regular surfae. ewed as the result of knot nserton, the Bernsten-Bézer oeffents together wth the oeffent or rng of oeffents losest to the extraordnary pont form a onvex hull of the lmt surfae. The remanng nnermost rng(s) of oeffents may be determned by symmetry and knowledge of the lmtng surfae. For example, for the Doo-abn algorthm, the extenson determnes all but the Bernsten-Bézer oeffent wth ndex 00 (.f. Fgure 4). A natural hoe for ths oeffent s the entrod of the n-sded mesh ell whh s nterpolated by the lmt surfae. For the atmull-lark algorthm, the 2 extenson pns down all but one Bernsten-Bézer oeffent whh we may hoose to be the extraordnary pont of the subdvson polyhedron. 4. Example: The volume of Doo-abn subdvson surfaes The Doo-abn algorthm s a generalzaton of the subdvson sheme for bquadrat tensor produt B-splnes. For eah n-gon of the orgnal mesh, a new, smaller n-gon s reated and onneted wth ts neghbors as shown n Fgure 5. The masks for generatng a new n-gon from an old one are spefed n Fgure 6 for the regular ase n 4 (left), and the Fgure 4: Indes of the Bernsten-Bézer oeffents of a bquadrat extenson. Fgure 5: Mesh refnement by the Doo-abn algorthm. general ase (rght). In 4 Doo and abn suggest α j 3 2os 2π j n 4n 4 f j 0 0 else A regular Doo-abn submesh onssts of nne ponts j arranged as four quadrlateral faets surroundng a entral pont (.f. Fgure 3): Fgure 6: Masks for the Doo-abn algorthm. The Eurographs Assoaton 997

Peters and Nasr / omputng ubdvson olumes Fgure 7: olume error boxes of the frst three Doo-abn subdvsons of a ube-shaped mesh. subdvson error box volume 5.27343e-02 2.04675e-02 3.0476e-03 4.645e-04 5.

The ponts defne a bquadrat path wth Bernsten-Bézer oeffents b jk where b 02 4 02 2 0 b 02 2 2 2 b 22 4 22 2 b 0 2 0 b b 2 2 2 b 00 4 0 00 0 b 0 0 2 b 20 4 20 2 The Eurographs Assoaton 997 2 0 For ths

5 Peters and Nasr / omputng ubdvson olumes Fgure 7: olume error boxes of the frst three Doo-abn subdvsons of a ube-shaped mesh. subdvson error box volume e e e e e e e e-08 Table : olume of eah error box when subdvdng the unt ube. The volume measured n the eghth subdvson s The ponts defne a bquadrat path wth Bernsten-Bézer oeffents b jk where b b b b b b b b b The Eurographs Assoaton For ths path, z n 3 s a salar-valued bqunt polynomal. Denotng ts Bernsten-Bézer oeffents by p j ts ntegral s j 0 p j. To ap the neghborhood of the extraordnary pont for the gven subdvson step, the oeffents jk generate the Bernsten-Bézer oeffents of a rng of bquadrat pathes defned by extenson and nterpolaton of the entrod: b 02 b 2 b 22 b b 00 b 2 b n n 00 b b The subdomnant egenvalue of the subdvson matrx s λ 2 mplyng an asymptot reduton of the error box volume by 2 3. Indeed, the error bounds are onfrmed by Table and Table 2 wth a graphal nterpretaton shown n 00 Fgure 4 and Fgure 8. A tghter error bound ould be obtaned by observng that 00 b mples that the lmt surfae les n the onvex hull of the Bernsten-Bézer oeffents of the ap. 5. Extenson to hgher-order Moments The proedure outlned for the zeroth order moment, the volume, s easly extended to hgher-order moments (.f. 6 ). For example, to ompute the x-omponent of the enter of mass we an hoose f 0 0 xz. As a vsual ad a mass error-box whose sze s the sum of the ndvdual error boxes may be plaed at the enter of mass of the approxmate surfae. Referenes. BERNARDINI, F. Integraton of polynomals over n- dmensonal polyhedra. omputer Aded Desgn 23() (99), ATMULL, E., AND LARK, J. Reursvely generated B-splne surfaes on arbtrary topologal meshes. omputer Aded Des. 0 (978), DE BOOR,. B-form bass. In Geometr Modelng: Algorthms and New Trends, G. Farn, Ed. IAM, Phladelpha, 987, pp

Peters and Nasr / omputng ubdvson olumes Fgure 8: olume error boxes for a stenl shape. val- subdvson ene 2 3 3 0.052734 0.00467 0.00047 4 0.25797 0.034572 0.004076 5 0.4377 0.033833 0.00327 6 0.

6 Peters and Nasr / omputng ubdvson olumes Fgure 8: olume error boxes for a stenl shape. val- subdvson ene Table 2: olume of error boxes for extraordnary ponts of valene 3,4,5,6 when subdvdng the stenl shape. 4. DOO, D., AND ABIN, M. Behavour of reursve subdvson surfaes near extraordnary ponts. omputer Aded Des. 0, 6 (Nov 978), DYN, N., LEIN, D., AND GREGORY, J. A. A butterfly subdvson sheme for surfae nterpolaton wth tenson ontrol. AM Trans. on Graphs 9 (990), GONZALEZ-OHOA,., MAMMON,., AND PE- TER, J. omputng moments of peewse polynomal surfaes. xxx xx (99x). submtted. 7. HOPPE, H., DEROE, T., DUHAMP, T., HALTEAD, M., JIN, H., MDONALD, J., HWEITZER, J., AND TUETZLE, W. Peewse smooth surfae reonstruton. omputer Graphs Proeedngs of ggraph 94 (994), KOBBELT, L. Interpolatory subdvson on open quadrlateral nets wth arbtrary topology. vol. 5, Basl Blakwell Ltd. Eurographs 96 onferene ssue. 9. LEE, Y. T., AND REQUIHA, A. A. G. Algorthms for omputng the volume and other ntegral propertes of solds. II a famly of algorthms based on representaton onverson and ellular approxmaton. ommun. AM (UA) 25 (ept. 982), LIEN,., AND KAJIYA, J. A symbol method for alulatng the ntegral propertes of arbtrary nononvex polyhedra. IEEE omputer Graphs and Applatons 4(9) (Ot 984), LOOP,. mooth subdvson for surfaes based on trangles. Thess, Unv. of Utah (987). 2. NARI, A. Polyhedral subdvson methods for freeform surfaes. AM Transatons on Graphs 6 (987), NARI, A. An algorthm to ompute volume of solds defned by reursve subdvson surfaes. Teh. rep., Ameran Unversty of Berut, 996. TR REIF, U. A unfed approah to subdvson algorthms near extraordnary vertes. AGD 2 (995), RUDIN, W. Real and omplex analyss. eres n Hgher Mathemats. MGraw-Hll, HRODER, WELDEN, AND ZORIN. Interpolatng subdvson for meshes wth arbtrary topology. omputer Graphs Proeedngs of ggraph 96 (996), TOLLNITZ, E. J., DEROE, T. D., AND ALEIN, D. H. Wavelets for omputer graphs: theory and applatons. Morgan Kaufmann, an Franso, alf, 996. The Eurographs Assoaton 997

Interpolation of the Irregular Curve Network of Ship Hull Form Using Subdivision Surfaces

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