A priori computation of the number of surface subdivision levels

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1 ror comutaton of the number of surface subdvson levels Srne Lanquetn arc eveu LEI CS 55 F des Scences et Technques nversté de Bourgogne B 77 7 IJ Cedex France {slanquet mneveu}@u-bourgogne.fr bstract Subdvson surfaces are a owerful model wdely used n geometrc modellng. Controllng the accuracy of the aroxmaton of the lmt surface often nvolves the comutaton of the dstance between the control mesh the lmt surface. evertheless the a ror level or deth of subdvson based on a dstance crteron has not yet been exressed. The goal of ths aer s thus to comute ths level. Then the surface can be subdvded wth a gven accuracy wthout any dstance comutaton between the subdvson surface the lmt surface. Keywords: Subdvson surface dstance levels of subdvson. ITCTI Snce the 9 s subdvson surfaces have been used n varous felds such as multresoluton grahcs geometrc modellng or character anmaton for nstance: Ger s Game 997 Toy Story 999 onsters Inc.. Ther success s due to the fact that they combne the advantages of both olygonal mesh non unform ratonal B-slne BS. Indeed le olygonal modellng subdvson surfaces can be aled to arbtrary toology meshes. oreover le BS they nvolve a small set of control vertces. Subdvson surfaces are defned by an ntal control mesh a set of refnement rules. The alcaton of refnement rules generates a sequence of ncreasngly fne control meshes. These control meshes are often referred to as olygonal meshes or olyhedrons. The sequence of control meshes converges to a smooth surface called the lmt surface. There are two sorts of subdvson schemes: schemes whch rely on nterolaton e.g. Butterfly scheme [5] those whch mae use of aroxmaton e.g. Catmull-Clar [] oo-sabn [] Loo [] schemes. In ths aer only aroxmatng schemes are consdered. Ther control meshes converge to the lmt surface at each ste of refnement Fgure. The maxmum dstance at the ntal level s always larger than the dstance d at the next level so on. Fgure. stance n aroxmaton schemes. d Each subdvson ste rovdes a more accurate aroxmaton of the lmt surface. Subdvson s often stoed after a re-determned number of stes 6- generally loos correct when the surface seems smooth enough when the largest trangle dmenson or the average trangle sze s less than a gven threshold []. In most cases these crtera are suffcent but n certan cases such as comuter aded geometrc desgn the accuracy of obect modellng has to be recsely determned. sng the dstance between a vertex of the control mesh whch s an aroxmaton of the lmt surface the lmt surface offers two ways of controlng the subdvson level. In the frst oton ths dstance combned wth local roertes of subdvson allows us to subdvde the surface only where ths dstance s greater than a gven threshold savng a lot of trangles: ths s termed adatatve subdvson. In the second we can comute the number of a ror subdvsons requred to aroxmate the lmt surface wthn a gven accuracy. Ths allows us to redefne storage amount save dstance comutaton when subdvdng whch may be useful for real tme alcatons vdeo games vrtual realty... Ths also gves a ror nowledge of levels of detals whch s useful n some vew deendent mmersve alcatons for nstance. oreover when accuracy s not a maor constrant one may wsh to now how the levels of subdvson ncrease when accuracy ncreases. small beneft n recson may nvolve a huge memory ncrease one may wsh to stre a good trade off between accuracy levels of subdvson. Snce the ntroducton of subdvson surfaces by Catmull-Clar [] oo-sabn [] n 97 Loo [] n 97 varous subdvson schemes have been roosed analyss of the lmt surface has been carred out. Stam [ 5] derves an analytcal exresson for a set of egenbass functons whch evaluate the surface. Zorn s wor [6] extends these results by consderng the subdvson rules for ecewse smooth surfaces wth boundares deendng on arameters. arn et al. [] studed the dstance between a Bézer curve segment ts control olygon. The wor of Lutterort eters [9] develos a framewor for effcently comutng enclosures for multvarate olynomals. evertheless the a ror level or deth of subdvson based on a dstance crteron has not yet been exressed. The goal of ths aer s thus to comute ths level. Then the surface can be

2 subdvded wth a gven accuracy wthout any dstance comutaton between the subdvson surface the lmt surface. To our nowledge only Cheng [7] has roosed an exresson for the Catmull-Clar scheme. ur aer also deals wth the oo- Sabn Loo schemes. In addton ths method can be extended to other subdvson methods. The aer s organzed as follows. Secton s a bref revew of the man subdvson surface concets that we need n our aer. In secton we exlan how to comute the a ror number of levels of subdvson based on a gven dstance to the lmt surface. Ths comutaton s erformed on the tradtonal Catmull- Clar Loo oo-sabn schemes. esults are also shown n ths secton. Fnally drectons for future wor are roosed n the concluson.. SBIVISI SFCES In ths secton tradtonal subdvson schemes are brefly descrbed an uer bound of the dstance between the control mesh the lmt surface s gven. We denote bold character the control mesh at the subdvson level.. Tradtonal subdvson schemes.. Catmull-Clar scheme e n f n e v v e e f e e f e stance to the lmt surface. Gven the revous notatons the mage v of the vertex valence n on the lmt surface can be wrtten [6]: + + nv e f v = n n+ 5.. Loo scheme Fgure. Left an ntal face. ght the new faces. v wth The Loo scheme generalzes quadratc trangular B-slnes the lmt surface obtaned s a quartc Box-slne. Ths scheme s based on slttng faces: each face of the control mesh at refnement level s subdvded nto four new trangular faces at level +. Ths frst ste s llustrated n Fgure. Consder a face: new vertces are nserted n the mddle of each edge they are named odd vertces those of the ntal face are named even vertces. - / a b / / / Fgure. Catmull-Clar subdvson around the vertex v wth valence n. Ths scheme can be aled on an arbtrary mesh ; t generates tensor roduct bcubc B-slne surfaces. egular vertces for ths + scheme have valence. Each vertex of can be assocated wth a face an edge or a vertex of These vertces are called resectvely face vertces edge vertces or vertex vertces are denoted f ev. Let v be a vertex of the ntal mesh wth valence n Fgure shows the subdvson rocess around v. The suerscrt reresents the refnement level. The face vertex s the centrod of the corresondng face. The edge vertex s comuted as follows: + + v + e + f + f + e = where subscrts are modulo n the valence of the vertex The vertex vertex s then comuted thus: n v = v + e + f n n n + + v. / / / / c d / Fgure. Loo mass where reresents old vertces the new oston of an even vertex left: a b an odd vertex rght: c d resectvely. In the second ste all vertces are dslaced by comutng a weghted average of the vertex ts neghbourng vertces. These averages can be substtuted by alyng dfferent mass accordng to vertex roertes: even/odd nteror/boundary Fgure. The a sub-fgure reresents the nteror even vertex mas where n denotes the vertex valence s chosen to be: = 6 π f n = 5 + cos f n > n n The b sub-fgure reresents the crease boundary even vertex mas. The c sub-fgure llustrates the nteror odd vertex mas. The d sub-fgure shows the crease boundary odd vertex mas.

3 stance to the lmt surface. Let be an even vertex wth valence n. Its mage on the lmt surface s obtaned usng the even neghbourng vertces. Thus the mage n can be wrtten as follows []: of the vertex wth valence n n. = + 5 n + n + =..... SBIVISI ETH Knowng the dstance from the control mesh vertces to the lmt surface allows us to determne the maxmum dstance between the control mesh the lmt surface. Indeed subdvson surface roertes are such that the control mesh vertces are the most dstant onts from the lmt surface. If a vertex s nserted between two vertces of the control mesh the dstance between ths vertex the lmt surface wll nevtably be smaller Fgure 7. Ths roerty s vald for any surface: convex or concave oen or closed Havng the mage of the vertex as a functon of ts neghbourhood allows us to comute the dstance between n. n.. : d = +.. n. +. n. + =.. oo-sabn scheme 5 α = + wth π + cos α = = Fgure 5. oo-sabn mas for nteror vertces. a. c. B B d. B b. B Fgure 6. Toology of oo-sabn subdvson surfaces. Ths scheme generates tensor roduct bquadratc B-slne surfaces. It can be aled on arbtrary meshes. Fgure 5 shows the mas to comute nteror vertces for ths scheme. When the new vertces are comuted faces are created as shown n Fgure 6. Consder a control mesh at an arbtrary level of subdvson. Sngle lnes reresent nteror edges double lnes reresent boundary edges Fgure 6.a. Interor vertces are denoted by crcles boundary vertces are denoted by crosses Fgure 6.b. For each face the new nteror vertces are connected to create face faces. ld faces are now reresented by dashes Fgure 6.c. For each edge new vertces generated from the ends of an edge are connected to roduce edge faces Fgure 6.d.For each vertex new vertces generated from ths vertex are oned to roduce vertex faces Fgure 6.e. e. B Fgure 7. stance from a mdont to the lmt surface. Cheng [] generalzes hs results on unform cubc B-slne curves to Catmull-Clar surfaces. He demonstrates the followng results for each segment at subdvson level : Theorem. Let be C t s the th cubc B-slne curve segment at subdvson level L t s the corresondng segment of the control olygon wrtten n barycentrc form = max wth the vertces of the + control olygon we have : L t C t Corollary. Wth as n Theorem for the control olygon to be close enough to the lmt curve wthn ε we need to erform at least levels of recursve subdvson wth log 6ε. For surfaces the result s smlar []. S u v be a subatch of a Catmull-Clar Theorem. Let surface after levels of recursve subdvson L uv the blnear arametrc reresentaton of the central face of the control S u v we have: mesh corresondng to L uv S uv 7 = max where l l + l l l l+ l wth the control vertces of the control mesh l Corollary. Wth as n Theorem the control mesh aroxmates the lmt surface wth the accuracy ε f the number of recursve subdvson s at least wth log ε.

4 . Loo case otatons used for the vertces subdvson level are descrbed Fgure. + of the control mesh at Fgure. otatons used for the vertces of the control mesh. Theorem. Consder the dstance from the lmt surface S u v the central trangle Fgure at subdvson level : where L uv bold trangle n 5 n S u v L u v < ε n = max ' ' n + n + ' ' adacent to n s the valence of the vertex. roof. The Loo scheme s an aroxmatng scheme; the maxmum dstance between the control mesh the lmt surface s also obtaned for the control vertces. S u v L u v max max n n + + ' ' n ' ' adacent to For the next level of subdvson the same formula can be wrtten: + + S u v L u v We denote: max + + n n ' ' n + + ' ' adacent to n = max ' ' n + n + ' ' adacent to s bound u by a functon of + elacng the gves: at level + by ther exressons at level n = ' ' n + n ' ' adacent to 5 n n n + + ue to the exresson of : When n = when ' ' n ' ' adacent to = then 5 n = > π n n n 5 n + = + cos > then: We obtan the recurrence relaton: 5 n Corollary. Wth n as n Theorem the control mesh aroxmates the lmt surface wth the accuracy ε f the number of recursve subdvson s at least wth 5 n > ln ln. ε roof. The Theorem gves: 5 n S u v L u v < ε The last art of the nequalty can be wrtten: 5 5 n ε ln ln n < >. ε In artcular for a regular mesh wth n = 6 we have: > log ε. Table gves the subdvson deth accordng to the gven accuracy on a regular surface valence s 6 for every vertex the torus. In fgures to 6 the dar res. lght faces reresent the faces whose dstance to the lmt surface s greater res. less than ε. Excet for Fgure all the surfaces are subdvded usng the adatve subdvson for readablty we generate a smaller number of faces whch allows us to more easly dstngush dar lght faces n fgures. any crtera may regulate adatatve subdvson curvature [7] normal cone [] dhedral angle [] for nstance. Here we used a geometrc crteron: the dstance between the control mesh the lmt surface. We subdvde the surface only where ths dstance s greater than a gven threshold.

5 a. b. c. d. the maxmum valence s 6 because all vertces nserted durng the subdvson have valence 6. For nstance ntal valences of the box are or 5 so n = 5. For ε =.5 the subdvson deth found for n = 5 s. But s when the maxmum valence n s consdered to be 6 due to subdvsons. Fgure shows successve subdvsons of the box. t level some vertces are stll further than ε from the lmt surface. Indeed subdvson deth s. Fgure 9. fferent cases of subdvson n adatatve subdvson. Let us frst defne the terms used to exlan how faces are subdvded n ths adatatve subdvson: a vertex whch s not dslaced s called statc a vertex whch s dslaced s called moble. Faces are classfed nto categores accordng to the number of moble vertces. oble vertces are dected by crcles n Fgure 9 to. When all vertces are statc the face s not subdvded Fgure 5.a.. Fgure 9.b. llustrates the case where only one vertex s moble; only two among the three new vertces are then moble n order to avod cracs. When there are two moble vertces face subdvson s almost normal excet for the fact that one of the old vertces s statc Fgure 9.c.. Fnally when all vertces are moble subdvson s carred out n a normal way Fgure 9.d.. Ths adatatve subdvson scheme avods cracs but t does not allow a correct comutaton of the neghbourhood for all vertces. Indeed faces are generated but corresondences between edges are not udated. evertheless neghbourhood s not necessary because the dstance from a statc vertex to the lmt surface does not change. We smly need to save t from one level to another. In the other cases the neghbourhood s correct; the dstance can thus be comuted. ccuracy ε Subdvson deth 5 6 Table. Subdvson deth necessary for accuracy ε for the torus. Fgure shows the number of subdvsons necessary to obtan an accuracy ε of.5 for the torus model whch has regular valences everywhere. Fgure llustrates successve levels of subdvson to have an accuracy ε of.. We can easly verfy that subdvson level gves a correct result whereas level leaves non accurate vertces dar areas. Fgure. Box dmensons:. Valence of nserted vertces must be taen nto account. In some cases usng the mean valence 6 s not suffcent. For nstance the surface n Fgure has a mnmum valence of a maxmum valence of. Wth an accuracy of.5 usng mean valence gves a subdvson deth of whereas the maxmum valence gves whch s the true result. Fgure. Snnng to dmensons:. ean valence 6 s not enough. n error remans at level near the ole vertex of valence. Table gves the subdvson deth as a functon of a gven accuracy n the dolhn model wth arbtrary valences. Fgure shows the number of subdvsons necessary to obtan an accuracy ε of.5. ccuracy ε Subdvson deth 5 n Table. Subdvson deth necessary as a functon of the accuracy ε n the dolhn model. Fgure. Subdvson of the torus n = 6 wth an accuracy ε of.5. Torus dmensons: Fgure. Subdvson of the torus n = 6 wth an accuracy ε of.. We have to ay attenton to the role of valence n. n denotes the maxmum valence of the mesh ncludng the subdvson stes. When valences are strctly less than 6 on the ntal control mesh Fgure. olhn dmensons: 9 9 n wth an accuracy of.5. Table gves the subdvson deth as a functon of a gven accuracy n the bunny model wth arbtrary valences. For an accuracy ε wth a dfference of. the n subdvson deth ncreases by from to 5 whch reresents an ncreased cost memory comutaton even wth adatatve subdvson. The same remar ales to table accuracy ε

6 asses from.55 to.5 whch leads to an ncrease n the subdvson level from to 5 ccuracy ε Subdvson deth 5 Table. Subdvson deth necessary as a functon of accuracy ε n the bunny model. Fgure 5 shows the number of subdvsons necessary to obtan an accuracy ε of.5 n the bunny model. d = + because the mage of verfes = + + [] so L t C t +. Fgure 5. The bunny dmensons: 6 n wth an accuracy of... oo-sabn case.. Quadratc B-slne curves The even odd vertces of the next level of subdvson are comuted as follows: + = = + + s for cubc B-Slne curves Theorem can be demonstrated: Theorem. Let be C t the quadratc B-slne th curve segment at subdvson level L t the corresondng segment of the control olygon wrtten n barycentrc form = max wth the vertces of the + control olygon. We have: L t C t roof. Let L t be the barycentrc form of the segments [ ] [ ] C t the corresondng curve segment of the quadratc B-slne curve. L t C t d because the control mesh s the furthest from the lmt curve at the control vertces n aroxmatng schemes. oreover 9 Fgure 6. nform quadratc B-slne curve ts ntal control olygon. Fgure 7. nform quadratc B-slne curve ts ntal control olygon control vertces of the next subdvson level. When the curve s subdvded ths result ales to both segments of curve yeldng: t t L C + t t L C +. sng formulas for even odd vertces we obtan: + = + + = + thus : wth = max the control vertces of + the control mesh. Ths gves the followng recurrence relaton: consequently: L t C t

7 Corollary. Wth as n Theorem for the control olygon to be close enough to the lmt curve wthn ε we need to erform at least levels of recursve subdvson wth log ε. roof. Let ε be a gven accuracy ε.e. log ε... B quadratc B-slne surfaces Theorem 5. Let S subdvsons L L t C t ε f u v be a atch of oo-sabn surface after uv the corresondng control mesh we then have: L uv S uv = max + + where l + l + l l l+ l+ l when are the control vertces of the control mesh l roof : Fgure. nform bquadratc B-slne surface ts ntal control olyhedron. L u v S u v d = S u v because the oo-sabn scheme s aroxmatng. oreover S u v B u B u B u B v = = = = = + = = = B u + B u B v + + because B u = wth = = max + +. l l l Fgure 9. nform bquadratc B-slne surface: ts ntal control olyhedron the control vertces at the next level. When the surface s subdvded ths result can be used agan on the four sub-atches of surface: L u v S u v max + + L { } l l l u v u v S max + + L { } l l l u v u v S max { + l l l + } L u v u v S max + + { } l l l oreover + = + l l l l l l + = + + = + + = + l l l l l l so where = max + +. l + l + l l l+ l+ l.

8 Ths gves the followng recurrence relaton. then L uv S uv Corollary 5. The control mesh aroxmates the lmt surface wth an accuracy of ε f the number of recursve subdvsons verfes: log ε. roof. Let ε be a gven accuracy ε.e. log ε. L u v S u v ε f Ths result s demonstrated n the artcular case where valence s regular for oo-sabn.. CCLSI We studed the subdvson deth necessary to obtan a gven accuracy ε n the tradtonal schemes namely Loo oo-sabn Catmull-Clar schemes. The subdvson can then be erformed wthout any dstance comutaton snce we are sure of reachng the desred accuracy. oreover t allows us to redefne memory requrements the ntal mesh the level of subdvson are enough to comute the number of faces n the fnal mesh. The subdvson can also be adatatve. In ths case we do not save comutaton tme as the dstance crteron has to be evaluated at each subdvson. But the user may modfy hs accuracy crteron f the subdvson level s too hgh. In Loo subdvson the demonstraton s carred out n a general case whatever the valences of the surface vertces. The subdvson deth for oo-sabn subdvson s only treated for vertces of regular valence. But ths s not a restrcton because after one ste of subdvson the valence of any vertex of the oo-sabn surface s four. ne of our future roects s to generalze the demonstraton for Catmull-Clar surfaces for arbtrary valences. oreover snce we now the subdvson level we have to verfy whether a n -adc subdvson may be used to drectly comute the fnal surface wth 5. EFEECES n =. [] mresh. G. Farn. azdan datve subdvson schemes for trangular meshes n Herarchcal Geometrc ethods n Scentfc Vsualzaton H.H. G. Farn B. Hamann edtors Edtor [] Catmull E. J. Clar ecursvely generated B-slne surfaces on arbtrary toologcal meshes. Comuter ded esgn : [] Cheng F. J. ong. Subdvson eth Comutaton datve esh efnement for Catmull-Clar Subdvson Surfaces. n SI Conference on Geometrc esgn Comutng Seattle ov. -.. [] oo.. Sabn Behavour of recursve subdvson surfaces near extraordnary onts. Comuter ded esgn : [5] yn.. Levn J. Gregory butterfly subdvson scheme for surface nterolaton wth tenson control. C Transactons on Grahcs 99. 9: [6] Halstead.. Kaas T. eose Effcent Far Interolaton usng Catmull-Clar surfaces. Comuter Grahcs SIGGH roceedngs 99 99:. 5-. [7] Km.S. Intersectng Surfaces of Secal Tyes. Shae odelng Internatonal :. -. [] Loo C. Smooth Subdvson Surfaces Based on Trangles. aster's thess nversty of tah nversty of tah 97. [9] Lutterort. J. eters tmzed refnable enclosures of multvarate olynomal eces. Comuter ded Geometrc esgn. : [] ueller H.. Jaesche datve Subdvson Curves Surfaces. roceedngs of Comuter Grahcs Internatonal 9 99:. -5. [] üller K. S. Havemann Subdvson Surface Tesselaton on the Fly usng a Versatle esh data Structure. Eurograhcs'. 9: [] arn. J. eters. Lutterort Shar quanttatve bounds on the dstance between a olynomal ece ts Bézer control olygon. Comuter ded Geometrc esgn : [] 'Bren... anocha Calculatng Intersecton Curve roxmatons for Subdvson Surfaces.. htt:// [] Stam J. Evaluaton of Loo Subdvson Surfaces n Subdvson for odelng nmaton SIGGH 999 Course ote S.C. ote Edtor [5] Stam J. Exact Evaluaton f Catmull-Clar Subdvson Surfaces t rbtrary arameter Values n Subdvson for odelng nmaton SIGGH 999 Course ote [6] Zorn.. Krstansson Evaluaton of ecewse smooth subdvson surfaces. The Vsual Comuter. : [7] Zorn.. Schröder W. Sweldens Interactve multresoluton mesh edtng. SIGGH'9 roceedngs 99: bout the authors Srne Lanquetn s a h student n the eartment of Comuter Scence LEI at Burgundy nversty France. Her contact emal s srne.lanquetn@u-bourgogne.fr. arc eveu s a rofessor n the eartment of Comuter Scence LEI at the Burgundy nversty France. Hs contact emal s marc.neveu@u-bourgogne.fr.